En línea Opciones Binarias gambita en español: Modelo De Media Móvil

Net. sourceforge. openforecast. models Clase MovingAverageModel

Un modelo de predicción de media móvil se basa en una serie temporal construida artificialmente en la que el valor para un período de tiempo dado se sustituye por la media de ese valor y los valores de cierto número de períodos de tiempo anteriores y posteriores. Como puede haber adivinado a partir de la descripción, este modelo es el más adecuado para datos de series de tiempo; Es decir, datos que cambian con el tiempo. Por ejemplo, muchos gráficos de acciones individuales en el mercado de valores muestran 20, 50, 100 o 200 días promedios móviles como una forma de mostrar tendencias.

Dado que el valor pronosticado para cualquier período dado es el promedio de los períodos anteriores, entonces el pronóstico aparecerá siempre "rezagado" detrás de los aumentos o disminuciones en los valores observados (dependientes). Por ejemplo, si una serie de datos tiene una tendencia alcista alcista, entonces un pronóstico de media móvil proporcionará generalmente una subestimación de los valores de la variable dependiente.

El método del promedio móvil tiene una ventaja sobre otros modelos de predicción en el sentido de que suaviza los picos y valles (o valles) en un conjunto de observaciones. Sin embargo, también tiene varias desventajas. En particular, este modelo no produce una ecuación real. Por lo tanto, no es tan útil como una herramienta de pronóstico a medio y largo plazo. Sólo se puede utilizar con fiabilidad para prever uno o dos períodos en el futuro.

El modelo de media móvil es un caso especial de la media móvil ponderada más general. En la media móvil simple, todos los pesos son iguales.

Desde: 0,3 Autor: Steven R. Gould

Campos heredados de la clase net. sourceforge. openforecast. models. AbstractForecastingModel

MovingAverageModel () Construye un nuevo modelo de pronóstico de media móvil.

MovingAverageModel (período int) Construye un nuevo modelo de pronóstico de media móvil, utilizando el período especificado.

GetForecastType () Devuelve un nombre de una o dos palabras de este tipo de modelo de pronóstico.

Init (DataSet dataSet) Se utiliza para inicializar el modelo de media móvil.

ToString () Esto debe anularse para proporcionar una descripción textual del modelo de pronóstico actual incluyendo, cuando sea posible, cualquier parámetro derivado utilizado.

Métodos heredados de la clase net. sourceforge. openforecast. models. WeightedMovingAverageModel

MovingAverageModel

Construye un nuevo modelo de pronóstico de media móvil. Para que un modelo válido sea construido, debe llamar a init y pasar en un conjunto de datos que contiene una serie de puntos de datos con la variable de tiempo inicializada para identificar la variable independiente.

MovingAverageModel

Construye un nuevo modelo de pronóstico de media móvil, usando el nombre dado como variable independiente.

Parámetros: independentVariable - el nombre de la variable independiente que se va a utilizar en este modelo.

MovingAverageModel

Construye un nuevo modelo de pronóstico de media móvil, utilizando el período especificado. Para que un modelo válido sea construido, debe llamar a init y pasar en un conjunto de datos que contiene una serie de puntos de datos con la variable de tiempo inicializada para identificar la variable independiente.

El valor del período se utiliza para determinar el número de observaciones que se utilizarán para calcular el promedio móvil. Por ejemplo, para un promedio móvil de 50 días donde los puntos de datos son observaciones diarias, entonces el período debe ser 50.

El período también se utiliza para determinar la cantidad de períodos futuros que se pueden pronosticar con eficacia. Con una media móvil de 50 días, entonces no podemos razonablemente - con ningún grado de exactitud - pronosticar más de 50 días más allá del último período para el cual los datos están disponibles. Esto puede ser más beneficioso que, digamos, un período de 10 días, donde sólo podríamos prever razonablemente 10 días más allá del último período.

Parámetros: período - el número de observaciones que se utilizarán para calcular la media móvil.

MovingAverageModel

Construye un nuevo modelo de pronóstico de media móvil, usando el nombre dado como variable independiente y el período especificado.

Parámetros: independentVariable - el nombre de la variable independiente que se va a utilizar en este modelo. Período - el número de observaciones que se utilizarán para calcular la media móvil.

en eso

Se utiliza para inicializar el modelo de media móvil. Este método debe ser llamado antes de cualquier otro método en la clase. Dado que el modelo de media móvil no deduce ninguna ecuación para la predicción, este método utiliza el DataSet de entrada para calcular los valores de pronóstico para todos los valores válidos de la variable de tiempo independiente.

Especificado por: init in interface PredeterminaciónModel Overrides: init en la clase AbstractTimeBasedModel Parámetros: dataSet - conjunto de datos de observaciones que se pueden utilizar para inicializar los parámetros de pronóstico del modelo de pronóstico.

getForecastType

Devuelve un nombre de una o dos palabras de este tipo de modelo de pronóstico. Mantenga esto corto. Una descripción más larga debe implementarse en el método toString.

Encadenar

Esto debería anularse para proporcionar una descripción textual del modelo de pronóstico actual incluyendo, cuando sea posible, cualquier parámetro derivado utilizado.

Especificado por: toString en la interfaz ForecastingModel Overrides: toString en clase WeightedMovingAverageModel Devuelve: una representación de cadena del modelo de pronóstico actual y sus parámetros.

Net. sourceforge. openforecast. models Clase WeightedMovingAverageModel

Un modelo de predicción de la media móvil ponderada se basa en una serie temporal construida artificialmente en la que el valor para un período de tiempo dado se sustituye por la media ponderada de ese valor y los valores de cierto número de períodos de tiempo precedentes. Como puede haber adivinado a partir de la descripción, este modelo es el más adecuado para datos de series de tiempo; Es decir, datos que cambian con el tiempo.

Dado que el valor pronosticado para cualquier período dado es un promedio ponderado de los períodos anteriores, entonces el pronóstico siempre aparecerá "rezagado" detrás de los aumentos o disminuciones en los valores observados (dependientes). Por ejemplo, si una serie de datos tiene una tendencia alcista hacia arriba, entonces un promedio ponderado del promedio móvil proporcionará generalmente una subestimación de los valores de la variable dependiente.

El modelo de media móvil ponderada, al igual que el modelo de media móvil, tiene una ventaja sobre otros modelos de predicción en el sentido de que suaviza los picos y valles (o valles) en un conjunto de observaciones. Sin embargo, al igual que el modelo de media móvil, también tiene varias desventajas. En particular, este modelo no produce una ecuación real. Por lo tanto, no es tan útil como una herramienta de pronóstico a medio y largo plazo. Sólo se puede utilizar de forma fiable para predecir unos cuantos períodos en el futuro.

Desde: 0.4 Autor: Steven R. Gould

Campos heredados de la clase net. sourceforge. openforecast. models. AbstractForecastingModel

WeightedMovingAverageModel () Construye un nuevo modelo de predicción del promedio móvil ponderado.

WeightedMovingAverageModel (pesos dobles []] Construye un nuevo modelo de predicción del promedio móvil ponderado, usando los pesos especificados.

Forecast (double timeValue) Devuelve el valor de pronóstico de la variable dependiente para el valor dado de la variable de tiempo independiente.

GetForecastType () Devuelve un nombre de una o dos palabras de este tipo de modelo de pronóstico.

GetNumberOfPeriods () Devuelve el número actual de períodos utilizados en este modelo.

GetNumberOfPredictors () Devuelve el número de predictores utilizados por el modelo subyacente.

SetWeights (pesos dobles []) Establece los pesos utilizados por este modelo de predicción del promedio móvil ponderado a los pesos dados.

ToString () Esto debe anularse para proporcionar una descripción textual del modelo de pronóstico actual incluyendo, cuando sea posible, cualquier parámetro derivado utilizado.

Métodos heredados de la clase net. sourceforge. openforecast. models. AbstractTimeBasedModel

WeightedMovingAverageModel

Construye un nuevo modelo de predicción del promedio móvil ponderado, usando los pesos especificados. Para que un modelo válido sea construido, debe llamar a init y pasar en un conjunto de datos que contiene una serie de puntos de datos con la variable de tiempo inicializada para identificar la variable independiente.

El tamaño de la matriz de pesos se utiliza para determinar el número de observaciones que se utilizarán para calcular la media móvil ponderada. Además, el período más reciente se dará el peso definido por el primer elemento de la matriz; Es decir, pesos [0].

El tamaño de la matriz de pesos también se utiliza para determinar la cantidad de períodos futuros que pueden ser pronosticados con eficacia. Con una media móvil ponderada de 50 días, no podemos razonablemente - con ningún grado de exactitud - pronosticar más de 50 días más allá del último período para el cual los datos están disponibles. Incluso los pronósticos cercanos al final de este rango probablemente no serán confiables.

Nota sobre los pesos

En general, los pesos que se pasan a este constructor deben sumar hasta 1,0. Sin embargo, como conveniencia, si la suma de los pesos no suma 1.0, esta implementación escala todos los pesos proporcionalmente de modo que sumen a 1.0.

Parámetros: pesos - una serie de pesos a asignar a las observaciones históricas al calcular la media móvil ponderada.

WeightedMovingAverageModel

Construye un nuevo modelo de predicción del promedio móvil ponderado, utilizando la variable nombrada como variable independiente y los pesos especificados.

Parámetros: independentVariable - el nombre de la variable independiente que se va a utilizar en este modelo. Pesos - una serie de pesos para asignar a las observaciones históricas al calcular el promedio móvil ponderado.

WeightedMovingAverageModel

Construye un nuevo modelo de predicción del promedio móvil ponderado. Este constructor está destinado a ser utilizado sólo por subclases (por lo tanto, está protegido). Cualquier subclase que utilice este constructor debe invocar posteriormente el método (protected) setWeights para inicializar los pesos que usará este modelo.

WeightedMovingAverageModel

Construye un nuevo modelo de predicción del promedio móvil ponderado usando la variable independiente dada.

Parámetros: independentVariable - el nombre de la variable independiente que se va a utilizar en este modelo.

pesajes

Establece los pesos utilizados por este modelo de predicción de promedio móvil ponderado a los pesos dados. Este método está destinado a ser utilizado sólo por subclases (por lo tanto está protegido), y sólo en conjunción con el (protegido) constructor de un argumento.

Cualquier subclase que utiliza el constructor de un argumento debe llamar a setWeights antes de invocar el método AbstractTimeBasedModel. init (net. sourceforge. openforecast. DataSet) para inicializar el modelo.

Nota sobre los pesos

En general, los pesos que se pasan a este método deben sumar hasta 1,0. Sin embargo, como conveniencia, si la suma de los pesos no suma 1.0, esta implementación escala todos los pesos proporcionalmente de modo que sumen a 1.0.

Parámetros: pesos - una serie de pesos a asignar a las observaciones históricas al calcular la media móvil ponderada.

pronóstico

Devuelve el valor de pronóstico de la variable dependiente para el valor dado de la variable de tiempo independiente. Las subclases deben implementar este método de manera consistente con el modelo de predicción que implementan. Las subclases pueden hacer uso de los métodos getForecastValue y getObservedValue para obtener pronósticos y observaciones "anteriores", respectivamente.

Especificado por: forecast en la clase AbstractTimeBasedModel Parámetros: timeValue - el valor de la variable de tiempo para la que se requiere un valor de pronóstico. Devuelve: el valor de pronóstico de la variable dependiente para el tiempo dado. Tiros: IllegalArgumentException - si no hay datos históricos suficientes - observaciones pasadas a init - para generar una previsión para el valor de tiempo dado.

getNumberOfPredictors

Devuelve el número de predictores utilizados por el modelo subyacente.

Devuelve: el número de predictores utilizados por el modelo subyacente.

getNumberOfPeriods

Devuelve el número actual de períodos utilizados en este modelo.

Especificado por: getNumberOfPeriods en la clase AbstractTimeBasedModel Devuelve: el número actual de períodos utilizados en este modelo.

getForecastType

Devuelve un nombre de una o dos palabras de este tipo de modelo de pronóstico. Mantenga esto corto. Una descripción más larga debe implementarse en el método toString.

Encadenar

Esto debería anularse para proporcionar una descripción textual del modelo de pronóstico actual incluyendo, cuando sea posible, cualquier parámetro derivado utilizado.

Especificado por: toString en la interfaz ForecastingModel Overrides: toString en la clase AbstractTimeBasedModel Devuelve: una representación de cadena del modelo de pronóstico actual y sus parámetros.

Promedios móviles ponderados: Lo básico

Con los años, los técnicos han encontrado dos problemas con la media móvil simple. El primer problema radica en el marco temporal del promedio móvil (MA). La mayoría de los analistas técnicos creen que la acción del precio, el precio de las acciones de apertura o cierre, no es suficiente de lo que depender para predecir correctamente las señales de compra o venta de la acción de cruce de la MA. Para resolver este problema, los analistas asignan ahora más peso a los datos de precios más recientes utilizando el promedio móvil con suavidad exponencial (EMA). (Obtenga más información sobre cómo explorar la media móvil ponderada exponencialmente.)

Un ejemplo Por ejemplo, usando un MA de 10 días, un analista tomaría el precio de cierre del décimo día y multiplicaría este número por 10, el noveno día por nueve, el octavo día por ocho y así sucesivamente al primero del MA . Una vez que se ha determinado el total, el analista dividirá el número por la adición de los multiplicadores. Si agrega los multiplicadores del ejemplo de MA de 10 días, el número es 55. Este indicador se conoce como el promedio móvil ponderado linealmente. (Para la lectura relacionada, compruebe hacia fuera Los promedios móviles simples hacen que las tendencias se destacan.)

Muchos técnicos son creyentes firmes en la media móvil exponencialmente suavizada (EMA). Este indicador se ha explicado de muchas maneras diferentes que confunde tanto a los estudiantes como a los inversores. Tal vez la mejor explicación viene de John J. Murphy "Análisis Técnico de los Mercados Financieros", (publicado por el New York Institute of Finance, 1999):

El promedio movible exponencialmente suavizado aborda ambos problemas asociados con el promedio móvil simple. En primer lugar, el promedio suavizado exponencial asigna un mayor peso a los datos más recientes. Por lo tanto, es una media móvil ponderada. Pero si bien asigna menor importancia a los datos de precios pasados, incluye en su cálculo todos los datos en la vida útil del instrumento. Además, el usuario puede ajustar la ponderación para dar mayor o menor peso al precio del día más reciente, que se añade a un porcentaje del valor del día anterior. La suma de ambos valores porcentuales se suma a 100.

Por ejemplo, el precio del último día se podría asignar un peso de 10% (.10), que se agrega al peso de los días anteriores del 90% (.90). Esto da el último día 10% de la ponderación total. Esto sería el equivalente a un promedio de 20 días, al dar al precio de los últimos días un valor menor del 5% (0,05).

Figura 1: Promedio Movido Suavizado Exponencialmente

El gráfico anterior muestra el índice Nasdaq Composite desde la primera semana de agosto de 2000 hasta el 1 de junio de 2001. Como puede ver claramente, la EMA, que en este caso está utilizando los datos de precios de cierre durante un período de nueve días, Vender señales el 8 de septiembre (marcado por una flecha negra hacia abajo). Este fue el día en que el índice se rompió por debajo del nivel de los 4.000. La segunda flecha negra muestra otra pierna abajo que los técnicos esperaban. El Nasdaq no pudo generar suficiente volumen e interés de los inversores minoristas para romper la marca de 3.000. Luego se zambulló de nuevo hasta el fondo en 1619.58 el 4 de abril. La tendencia alcista del 12 de abril está marcada por una flecha. Aquí el índice cerró en 1,961.46, y los técnicos comenzaron a ver a los gestores de fondos institucionales comenzando a recoger algunos negocios como Cisco, Microsoft y algunos de los temas relacionados con la energía. (Lea nuestros artículos relacionados: Moving Average Envelopes: Refinación de una herramienta de comercio popular y rebote promedio móvil).

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Una relación de deuda y rentabilidad utilizada para determinar la facilidad con que una empresa puede pagar intereses sobre la deuda pendiente.

Una cuenta que se puede encontrar en la parte de activos del balance de una empresa. La buena voluntad a menudo puede surgir cuando una empresa.

Un fondo de índice es un tipo de fondo mutuo con una cartera construida para igualar o rastrear los componentes de un índice de mercado, tales.

Un contrato de derivados mediante el cual dos partes intercambian instrumentos financieros. Estos instrumentos pueden ser casi cualquier cosa.

Aprenda lo que es EBITDA, vea un video corto para aprender más y con lecturas le enseñamos cómo calcularlo usando MS.

Documentación

Modelo de media móvil autorregresiva

ARMA (p, q) Modelo

Para algunas series de tiempo observadas, se necesita un modelo de AR o MA de muy alto orden para modelar bien el proceso subyacente. En este caso, un modelo combinado de media móvil autorregresiva (ARMA) puede ser a veces una opción más parcimoniosa.

Un modelo ARMA expresa la media condicional de y t como una función de ambas observaciones pasadas, y t & # x2212; 1. & # x2026 ;. Y t & # x2212; pag. Y las innovaciones pasadas, & # x03B5; T & # x2212; 1. & # x2026 ;. & # X03B5; T & # x2212; Q. El número de observaciones pasadas de las que y t depende, p. Es el grado AR. El número de innovaciones pasadas de las que depende t, q. Es el grado de MA. En general, estos modelos son denotados por ARMA (p, q).

La forma del modelo ARMA (p, q) en Econometrics Toolbox & # x2122; es

Y t = c + & lt; x03D5; 1 y t & # x2212; 1 + & # x2026; + & # X03D5; P y t & # x2212; P + & lt; x03B5; T + + x03B8; 1 & # x03B5; T & # x2212; 1 + & # x2026; + & # X03B8; Q & # x03B5; T & # x2212; Q.

Donde & # x03B5; T es un proceso de innovación no correlacionado con cero medio.

En la notación polinómica del operador de lag, L i y t = y t & # x2212; yo. Definir el grado p AR polinomio operador lag & # x03D5; (L) = (1 & lt; L & lt; x & lt; x & lt; Defina el grado q MA polinomio operador lag & # x03B8; (L) = (1 + & lt; x03B8; 1 L + + # x03B8; q L q). Puede escribir el modelo ARMA (p, q) como

& # X03D5; (L) yt = c + & lt; x03B8; (L) & # x03B5; T

Los signos de los coeficientes en el polinomio de operador de retraso AR, & # x03D5; (L). Son opuestos al lado derecho de la Ecuación 5-10. Cuando se especifican e interpretan los coeficientes AR en Econometrics Toolbox, utilice el formulario de la ecuación 5-10.

Estacionariedad e Invertibilidad del Modelo ARMA

Considere el modelo ARMA (p, q) en la notación del operador de lag,

& # X03D5; (L) yt = c + & lt; x03B8; (L) & # x03B5; T

A partir de esta expresión, se puede ver que

Es la media incondicional del proceso, y & # x03C8; (L) es un polinomio racional de operador de retraso de grado infinito, (1 + + x03C8; 1 L + & # x03C8; 2 L2 + & # x2026;).

Nota: La propiedad Constant de un objeto modelo arima corresponde a c. Y no la media incondicional & # 956; .

Por la descomposición de Wold [1]. La ecuación 5-12 corresponde a un proceso estocástico estacionario siempre que los coeficientes & # x03C8; Soy absolutamente summable. Este es el caso cuando el polinomio AR, & # x03D5; (L). es estable . Lo que significa que todas sus raíces están fuera del círculo unitario. Adicionalmente, el proceso es causal siempre que el polinomio MA sea invertible. Lo que significa que todas sus raíces están fuera del círculo unitario.

Econometrics Toolbox refuerza la estabilidad y la invertibilidad de los procesos ARMA. Cuando se especifica un modelo ARMA utilizando arima. Se obtiene un error si se introducen coeficientes que no corresponden a un polinomio AR estable oa un polinomio MA inversible. De forma similar, la estimación impone restricciones de estacionariedad e invertibilidad durante la estimación.

Referencias

[1] Wold, H. Un estudio en el análisis de series de tiempo estacionarias. Uppsala, Suecia: Almqvist & amp; Wiksell, 1938.

Selecciona tu pais

Hay una serie de enfoques para modelar series de tiempo. Describimos algunos de los enfoques más comunes a continuación.

Decomposiciones estacionales, residuales y de tendencia

Un enfoque consiste en descomponer las series temporales en un componente de tendencia, estacional y residual.

El triple alisado exponencial es un ejemplo de este enfoque. Otro ejemplo, denominado loess estacional, se basa en mínimos cuadrados ponderados localmente y es discutido por Cleveland (1993). No discutimos el loess estacional en este manual.

Métodos Basados en Frecuencia

Otro enfoque, comúnmente utilizado en aplicaciones científicas y de ingeniería, es analizar las series en el dominio de la frecuencia. Un ejemplo de este enfoque en el modelado de un conjunto de datos de tipo sinusoidal se muestra en el estudio de caso de desviación de haz. La gráfica espectral es la principal herramienta para el análisis de frecuencia de series temporales.

Modelos Autoregresivos (AR)

Un enfoque común para el modelado de series de tiempo univariadas es el modelo autorregresivo (AR): $$ X_t = \ delta + \ phi_1 X_ + \ phi_2 X_ + \ cdots + \ phi_p X_ + A_t \, $$ donde \ (X_t \) es La serie de tiempo, \ (A_t \) es ruido blanco, y $$ \ delta = \ left (1 - \ sum_ ^ p \ phi_i \ derecha) \ mu \. $$ con \ (\ mu \) denotando la media del proceso.

Un modelo autorregresivo es simplemente una regresión lineal del valor actual de la serie contra uno o más valores previos de la serie. El valor de \ (p \) se denomina el orden del modelo AR.

Los modelos AR pueden ser analizados con uno de varios métodos, incluyendo técnicas lineales lineales por mínimos cuadrados. También tienen una interpretación directa.

Modelos de media móvil (MA)

Otro enfoque común para el modelado de modelos de series temporales univariadas es el modelo del promedio móvil (MA): $$ X_t = \ mu + A_t - \ theta_1 A_ - \ theta_2 A_ - \ cdots - \ theta_q A_ \, $$ donde \ ) Es la serie de tiempo, \ (\ mu \) es la media de la serie, \ (A_ \) son términos de ruido blanco, y \ (\ theta_1, \, \ ldots, \, \ theta_q \) son los parámetros de el modelo. El valor de \ (q \) se denomina el orden del modelo MA.

Es decir, un modelo de media móvil es conceptualmente una regresión lineal del valor actual de la serie contra el ruido blanco o choques aleatorios de uno o más valores anteriores de la serie. Se supone que los choques aleatorios en cada punto provienen de la misma distribución, normalmente una distribución normal, con ubicación a cero y escala constante. La distinción en este modelo es que estos choques aleatorios se propagan a los valores futuros de las series temporales. El ajuste de las estimaciones de MA es más complicado que con los modelos de AR porque los términos de error no son observables. Esto significa que los procedimientos de ajuste no lineales iterativos deben ser usados en lugar de mínimos cuadrados lineales. Los modelos MA también tienen una interpretación menos obvia que los modelos AR.

A veces, el ACF y PACF sugieren que un modelo de MA sería una mejor elección de modelo y, a veces, ambos AR y MA términos se deben utilizar en el mismo modelo (véase la Sección 6.4.4.5).

Tenga en cuenta, sin embargo, que los términos de error después de que el modelo sea apropiado deberían ser independientes y seguir las suposiciones estándar para un proceso univariante.

Box y Jenkins popularizaron un enfoque que combina el promedio móvil y los enfoques autorregresivos en el libro "Análisis de series temporales: pronóstico y control" (Box, Jenkins y Reinsel, 1994).

Aunque tanto los enfoques de media móvil como autoregresivos ya eran conocidos (y fueron investigados originalmente por Yule), la contribución de Box y Jenkins fue desarrollar una metodología sistemática para identificar y estimar modelos que podrían incorporar ambos enfoques. Esto hace que los modelos de Box-Jenkins sean una clase poderosa de modelos. Las siguientes secciones discutirán estos modelos en detalle.

Meb Faber Investigación

Modelo de tiempo

preguntas frecuentes

Trato de ser tan abierto y honesto acerca de los beneficios, así como los inconvenientes de cada estrategia y enfoque que la investigación.

De la mayor importancia es encontrar un programa de gestión de activos y el proceso que es adecuado para usted.

El modelo de tiempo se publicó sólo como un ejemplo simple. Hay mejoras considerables que se pueden hacer al modelo y no ejecutar fondos de clientes con los parámetros exactos en el libro blanco o libro.

A continuación se presentan las preguntas más frecuentes que recibo por correo electrónico. Si tiene más preguntas, envíeme un correo electrónico a [correo electrónico & # 160; protegido] con línea de asunto Preguntas frecuentes:

1. Cómo se actualiza este modelo? Qué quiere decir "precio mensual"?

El modelo, tal como se publica, sólo se actualiza una vez al mes el último día del mes. La acción del mercado en el ínterin es ignorada. El modelo publicado sólo pretendía ser ampliamente representativo del rendimiento que se podría esperar de un sistema tan simple.

2. Ha examinado una versión all-in en la que invierte el 100% de los activos en cualquier clase de activos en una señal de compra?

Sí, pero esto elimina los beneficios de la diversificación y expone la cartera a grandes riesgos cuando sólo unas pocas clases de activos están en una señal de compra. Además, introduce costos innecesarios de transacción. Las devoluciones son más altas pero con un aumento innecesario en el riesgo.

3. Ha examinado una versión larga y corta en la que corta la clase de activos en lugar de pasar a efectivo?

Sí. Los resultados están en el apéndice del libro.

4. Reequilibra las clases de activos mensualmente?

Sí. Aunque mostramos en el libro que es importante reequilibrar en algún momento, la frecuencia no es tan importante. Recomendamos un reequilibrio anual en cuentas exentas de impuestos, y un reequilibrio basado en los flujos de efectivo en cuentas imponibles.

5. Has probado varias medias móviles?

Sí. Existe una amplia estabilidad de los parámetros de 3 meses a más de 12 meses. Idem para EMAs.

6. Me gusta la estrategia y quiero implementarla, debería esperar hasta el siguiente reequilibrio?

Por lo general invertimos inmediatamente en el punto de reequilibrio. Si bien esto puede tener un efecto significativo en los resultados a corto plazo, debe ser un lavado en el largo plazo. Los inversores preocupados por el corto plazo pueden escalonar sus compras a lo largo de varios meses o trimestres.

7. Dónde puedo seguir la estrategia?

8. Qué pasa con el uso de datos diarios o semanales? No sólo actualizando mensualmente exponer a un inversor a los movimientos dramáticos del precio en el interino?

Hemos visto datos de confirmación para varios plazos, algunos superiores, otros inferiores. Su pregunta es válida, pero también considere lo contrario. Qué sucede con un sistema que se actualiza diariamente cuando un mercado baja rápidamente, luego se invierte y vuelve a subir? El inversor habría sido whipshawed y perdido capital.

9. Cuál es la mejor manera para que un individuo implemente el modelo apalancado?

Esto es complicado. Idealmente, pueden utilizar apalancamiento a una tasa de margen razonable. Interactive Brokers es siempre justo aquí. El uso de ETFs apalancados es una idea horrible. Para los inversores familiarizados con el producto, los futuros son una buena opción. También se puede usar un sistema de rotación cross-in todo-en-mercado.

10. Alguna vez ha pensado en combinar los sistemas de sincronización y rotación?

11. Por qué está tomando crédito por utilizar el modelo de 200 días de media móvil?

12. Para el sistema de rotación que ha escrito acerca de dónde adquirió el intérprete más alto durante los últimos 3, 6 y 12 meses, simplemente está utilizando la media del rendimiento de 3, 6 y 12 meses para calcular el rendimiento máximo ?

13. Se ha optimizado el crossover sma de 10 meses para todas las clases de activos (cinco), o es posible que diferentes marcos de tiempo funcionen mejor para diferentes clases de activos?

Diferentes plazos seguramente funcionarán mejor (en el pasado), pero hay una estabilidad de parámetros amplia a través de diferentes longitudes promedio móvil.

14. Alguna vez ha intentado agregar oro a su modelo (o cualquier otra clase de activo)?

Sí, usamos más de 50 clases de activos en Cambria & # 8211; El papel está destinado a ser instructivo. 15. Por qué escogió el SMA de 10 meses?

Sólo para ser representativos de la estrategia, y también corresponde más cercano a la media móvil de 200 días. Elegimos mensualmente ya que los datos diarios no se remontan hasta ahora para muchas de las clases de activos.

16. De dónde sacó sus datos históricos?

Datos Financieros Globales.

17. Qué software utilizó para realizar los backtests históricos?

18. A veces se menciona el uso de BND o AGG en lugar de IEF. Porqué es eso?

Mencionamos en el libro que la sincronización de los bonos de volatilidad más baja no hace mucha diferencia (mayores bonos de vol como empresas, emergentes y basura funcionan bien sin embargo). Mencionamos que un inversor podría comprar y mantener un índice de bonos como AGG o BND en lugar de tiempo IEF.

19. Estoy tratando de replicar sus resultados con la base de datos X (Yahoo, Google, etc) y mis resultados no coinciden. Lo que da?

Los índices revelados en el documento y en el libro se obtienen de Global Financial Data. No puedo comprobar todas las fuentes de datos para ver cómo calculan sus números, pero asegúrese de que los números son rendimiento total, incluidos los dividendos y los ingresos. Para Yahoo Finance uno necesita usar los números ajustados & # 8211; Y asegúrese de ajustar cada mes (o registrar las nuevas vueltas para ese mes), un proceso tedioso.

Meb Faber es cofundador y Director de Inversiones de Cambria Investment Management, y autor de cinco libros.

Modelo de media móvil

En el análisis de series de tiempo. El modelo de media móvil (MA) es un enfoque común para el modelado de series temporales univariadas. La notación MA (q) se refiere al modelo de media móvil de orden q:

Donde μ es la media de la serie, el θ 1. Θ q son los parámetros del modelo y el ε t. Ε t -1. Ε t - q son términos de error de ruido blanco. El valor de q se llama el orden del modelo MA. Esto se puede escribir de forma equivalente en términos del operador de retroceso B como

Por lo tanto, un modelo de media móvil es conceptualmente una regresión lineal del valor actual de la serie frente a los actuales y anteriores (no observados) términos de error de ruido blanco o choques aleatorios. Se supone que los choques aleatorios en cada punto son mutuamente independientes y provienen de la misma distribución, normalmente una distribución normal. Con ubicación a cero y escala constante.

Contenido

Interpretación

Decidir la conveniencia del modelo de MA

A veces, la función de autocorrelación (ACF) y la función de autocorrelación parcial (PACF) sugerirán que un modelo de MA sería una mejor elección de modelo y, a veces, tanto los términos AR como MA deben usarse en el mismo modelo (véase Box-Jenkins # ).

Colocación del modelo

El ajuste de las estimaciones de MA es más complicado que con modelos autorregresivos (modelos AR) porque los términos de error retrasados no son observables. Esto significa que los procedimientos de ajuste no lineales iterativos deben ser usados en lugar de mínimos cuadrados lineales.

Elegir el orden q

La función de autocorrelación de un proceso de MA (q) se convierte en cero a un retardo q + 1 y mayor, por lo que determinamos el retraso máximo apropiado para la estimación examinando la función de autocorrelación de la muestra para ver dónde se vuelve insignificantemente diferente de cero para todos los retrasos más allá de a Cierto retraso, que se designa como el retardo máximo q.

Ver también

Otras lecturas

Enders, Walter (2004). "Modelos estacionarios de series temporales". Applied Econometric Time Series (Segunda edición). Nueva York: Wiley. Pp. & Lt; 160; 48-107. ISBN & # 160; 0-471-45173-8. & # 160;

enlaces externos

Modelos ARMA (p, q) para el análisis de series temporales - Parte 3

Por Michael Halls-Moore el 7 de septiembre de 2015

Este es el tercer y último post de la mini-serie sobre modelos de media móvil autoregresiva (ARMA) para el análisis de series de tiempo. Hemos introducido modelos autorregresivos y modelos de media móvil en los dos artículos anteriores. Ahora es el momento de combinarlos para producir un modelo más sofisticado.

En última instancia, esto nos llevará a los modelos ARIMA y GARCH que nos permitirán predecir los rendimientos de los activos y predecir la volatilidad. Estos modelos constituirán la base para las señales comerciales y las técnicas de gestión de riesgos.

Si ha leído la Parte 1 y la Parte 2, habrá visto que tendemos a seguir un patrón para nuestro análisis de un modelo de series temporales. Lo repetiré brevemente aquí:

Justificación - Por qué nos interesa este modelo en particular?

Definición - Definición matemática para reducir la ambigüedad.

Correlograma - Trazado de un correlograma muestral para visualizar el comportamiento de un modelo.

Simulación y ajuste - Ajuste del modelo a simulaciones, para asegurar que hemos entendido el modelo correctamente.

Datos financieros reales - Aplicar el modelo a los precios reales de los activos históricos.

Prediction - Forecast subsequent values to build trading signals or filters.

In order to follow this article it is advisable to take a look at the prior articles on time series analysis. They can all be found here .

Bayesian Information Criterion

In Part 1 of this article series we looked at the Akaike Information Criterion (AIC) as a means of helping us choose between separate "best" time series models.

A closely related tool is the Bayesian Information Criterion (BIC). Essentially it has similar behaviour to the AIC in that it penalises models for having too many parameters. This may lead to overfitting. The difference between the BIC and AIC is that the BIC is more stringent with its penalisation of additional parameters.

Bayesian Information Criterion

If we take the likelihood function for a statistical model, which has $k$ parameters, and $L$ maximises the likelihood. then the Bayesian Information Criterion is given by:

Where $n$ is the number of data points in the time series.

We will be using the AIC and BIC below when choosing appropriate ARMA(p, q) models.

Ljung-Box Test

In Part 1 of this article series Rajan mentioned in the Disqus comments that the Ljung-Box test was more appropriate than using the Akaike Information Criterion of the Bayesian Information Criterion in deciding whether an ARMA model was a good fit to a time series.

The Ljung-Box test is a classical hypothesis test that is designed to test whether a set of autocorrelations of a fitted time series model differ significantly from zero. The test does not test each individual lag for randomness, but rather tests the randomness over a group of lags.

Ljung-Box Test

We define the null hypothesis $ $ as: The time series data at each lag are i. i.d.. that is, the correlations between the population series values are zero.

We define the alternate hypothesis $ $ as: The time series data are not i. i.d. and possess serial correlation.

We calculate the following test statistic. $Q$:

Where $n$ is the length of the time series sample, $\hat _k$ is the sample autocorrelation at lag $k$ and $h$ is the number of lags under the test.

The decision rule as to whether to reject the null hypothesis $ $ is to check whether $Q > \chi^2_ $, for a chi-squared distribution with $h$ degrees of freedom at the $100(1-\alpha)$th percentile.

While the details of the test may seem slightly complex, we can in fact use R to calculate the test for us, simplifying the procedure somewhat.

Autogressive Moving Average (ARMA) Models of order p, q

Now that we've discussed the BIC and the Ljung-Box test, we're ready to discuss our first mixed model, namely the Autoregressive Moving Average of order p, q, or ARMA(p, q).

Rationale

To date we have considered autoregressive processes and moving average processes.

The former model considers its own past behaviour as inputs for the model and as such attempts to capture market participant effects, such as momentum and mean-reversion in stock trading.

The latter model is used to characterise "shock" information to a series, such as a surprise earnings announcement or unexpected event (such as the BP Deepwater Horizon oil spill ).

Hence, an ARMA model attempts to capture both of these aspects when modelling financial time series.

Note that an ARMA model does not take into account volatility clustering, a key empirical phenomena of many financial time series. It is not a conditionally heteroscedastic model. For that we will need to wait for the ARCH and GARCH models.

Definición

The ARMA(p, q) model is a linear combination of two linear models and thus is itself still linear:

Autoregressive Moving Average Model of order p, q

A time series model, $\ $, is an autoregressive moving average model of order $p, q$ . ARMA(p, q), if:

\begin x_t = \alpha_1 x_ + \alpha_2 x_ + \ldots + w_t + \beta_1 w_ + \beta_2 w_ \ldots + \beta_q w_ \end

Where $\ $ is white noise with $E(w_t) = 0$ and variance $\sigma^2$.

If we consider the Backward Shift Operator . $ $ (see a previous article ) then we can rewrite the above as a function $\theta$ and $\phi$ of $ $:

We can straightforwardly see that by setting $p \neq 0$ and $q=0$ we recover the AR(p) model. Similarly if we set $p = 0$ and $q \neq 0$ we recover the MA(q) model.

One of the key features of the ARMA model is that it is parsimonious and redundant in its parameters. That is, an ARMA model will often require fewer parameters than an AR(p) or MA(q) model alone. In addition if we rewrite the equation in terms of the BSO, then the $\theta$ and $\phi$ polynomials can sometimes share a common factor, thus leading to a simpler model.

Simulations and Correlograms

As with the autoregressive and moving average models we will now simulate various ARMA series and then attempt to fit ARMA models to these realisations. We carry this out because we want to ensure that we understand the fitting procedure, including how to calculate confidence intervals for the models, as well as ensure that the procedure does actually recover reasonable estimates for the original ARMA parameters.

In Part 1 and Part 2 we manually constructed the AR and MA series by drawing $N$ samples from a normal distribution and then crafting the specific time series model using lags of these samples.

However, there is a more straightforward way to simulate AR, MA, ARMA and even ARIMA data, simply by using the arima. sim method in R.

Let's start with the simplest possible non-trivial ARMA model, namely the ARMA(1,1) model. That is, an autoregressive model of order one combined with a moving average model of order one. Such a model has only two coefficients, $\alpha$ and $\beta$, which represent the first lags of the time series itself and the "shock" white noise terms. Such a model is given by:

We need to specify the coefficients prior to simulation. Let's take $\alpha = 0.5$ and $\beta = -0.5$:

The output is as follows:

Let's also plot the correlogram:

We can see that there is no significant autocorrelation, which is to be expected from an ARMA(1,1) model.

Finally, let's try and determine the coefficients and their standard errors using the arima function:

We can calculate the confidence intervals for each parameter using the standard errors:

The confidence intervals do contain the true parameter values for both cases, however we should note that the 95% confidence intervals are very wide (a consequence of the reasonably large standard errors).

Let's now try an ARMA(2,2) model. That is, an AR(2) model combined with a MA(2) model. We need to specify four parameters for this model: $\alpha_1$, $\alpha_2$, $\beta_1$ and $\beta_2$. Let's take $\alpha_1 = 0.5$, $\alpha_2=-0.25$ $\beta_1=0.5$ and $\beta_2=-0.3$:

The output of our ARMA(2,2) model is as follows:

And the corresponding autocorelation:

We can now try fitting an ARMA(2,2) model to the data:

We can also calculate the confidence intervals for each parameter:

Notice that the confidence intervals for the coefficients for the moving average component ($\beta_1$ and $\beta_2$) do not actually contain the original parameter value. This outlines the danger of attempting to fit models to data, even when we know the true parameter values!

However, for trading purposes we just need to have a predictive power that exceeds chance and produces enough profit above transaction costs, in order to be profitable in the long run.

Now that we've seen some examples of simulated ARMA models we need mechanism for choosing the values of $p$ and $q$ when fitting to the models to real financial data.

Choosing the Best ARMA(p, q) Model

In order to determine which order $p, q$ of the ARMA model is appropriate for a series, we need to use the AIC (or BIC) across a subset of values for $p, q$, and then apply the Ljung-Box test to determine if a good fit has been achieved, for particular values of $p, q$ .

To show this method we are going to firstly simulate a particular ARMA(p, q) process. We will then loop over all pairwise values of $p \in \ $ and $q \in \ $ and calculate the AIC. We will select the model with the lowest AIC and then run a Ljung-Box test on the residuals to determine if we have achieved a good fit.

Let's begin by simulating an ARMA(3,2) series:

We will now create an object final to store the best model fit and lowest AIC value. We loop over the various $p, q$ combinations and use the current object to store the fit of an ARMA(i, j) model, for the looping variables $i$ and $j$.

If the current AIC is less than any previously calculated AIC we set the final AIC to this current value and select that order. Upon termination of the loop we have the order of the ARMA model stored in final. order and the ARIMA(p, d,q) fit itself (with the "Integrated" $d$ component set to 0) stored as final. arma :

Let's output the AIC, order and ARIMA coefficients:

We can see that the original order of the simulated ARMA model was recovered, namely with $p=3$ and $q=2$. We can plot the corelogram of the residuals of the model to see if they look like a realisation of discrete white noise (DWN):

The corelogram does indeed look like a realisation of DWN. Finally, we perform the Ljung-Box test for 20 lags to confirm this:

Notice that the p-value is greater than 0.05, which states that the residuals are independent at the 95% level and thus an ARMA(3,2) model provides a good model fit.

Clearly this should be the case since we've simulated the data ourselves! However, this is precisely the procedure we will use when we come to fit ARMA(p, q) models to the S&P500 index in the following section.

Financial Data

Now that we've outlined the procedure for choosing the optimal time series model for a simulated series, it is rather straightforward to apply it to financial data. For this example we are going to once again choose the S&P500 US Equity Index.

Let's download the daily closing prices using quantmod and then create the log returns stream:

Let's perform the same fitting procedure as for the simulated ARMA(3,2) series above on the log returns series of the S&P500 using the AIC:

The best fitting model has order ARMA(3,3):

Let's plot the residuals of the fitted model to the S&P500 log daily returns stream:

Notice that there are some significant peaks, especially at higher lags. This is indicative of a poor fit. Let's perform a Ljung-Box test to see if we have statistical evidence for this:

As we suspected, the p-value is less that 0.05 and as such we cannot say that the residuals are a realisation of discrete white noise. Hence there is additional autocorrelation in the residuals that is not explained by the fitted ARMA(3,3) model.

Próximos pasos

As we've discussed all along in this article series we have seen evidence of conditional heteroscedasticity (volatility clustering) in the S&P500 series, especially in the periods around 2007-2008. When we use a GARCH model later in the article series we will see how to eliminate these autocorrelations.

In practice, ARMA models are never generally good fits for log equities returns. We need to take into account the conditional heteroscedasticity and use a combination of ARIMA and GARCH. The next article will consider ARIMA and show how the "Integrated" component differs from the ARMA model we have been considering in this article.

Michael Halls-Moore

Mike is the founder of QuantStart and has been involved in the quantitative finance industry for the last five years, primarily as a quant developer and later as a quant trader consulting for hedge funds.

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Ways to estimate volatility

Some Advanced Methods for Volatility estimation

When we calculate volatility using the customary methods we don't take into account the order of observations. Additionally, all observations have equal weights in the formulas. But the most recent data about asset's return movements is more important for volatility forecasting than more dated data. That is why, the recently recorded statistical data should be given more weight for forecasting purposes than older data. One of the models that operate off of this assumption is the exponentially weighted moving average.

Promedio móvil simple (SMA)

The Moving Average is an average of a set of variables such as stock prices over time. The term "moving" steams from the fact that as each new price is added, the oldest price is subsequently deleted. The n-day Simple Moving Average takes the sum of the last n days prices. The SMA model is probably the most widely used volatility model in Value at Risk studies.

The disadvantage of the SMA is that it is inherently a memory-less function. A major drop or rise in the price is forgotten and does not manifest itself quantitatively in the simple moving average. As you can see in the following table, on day 9 there is a big step in the simple moving average, while price has been constant at $170. This distortion is caused by the low price on day 4 - dropped from the SMA on day 9.

Exponentially Weighted Moving Average (EWMA)

This section discusses the J. P. Morgan RiskMetrics© approach to estimating and forecasting volatility that uses an exponentially weighted moving average model (EWMA).

The EWMA model allows one to calculate a value for a given time on the basis of the previous day's value. The EWMA model has an advantage in comparison with SMA, because the EWMA has a memory. The EWMA remembers a fraction of its past by a factor A, that makes the EWMA a good indicator of the history of the price movement if a wise choice of the term is made. Using the exponential moving average of historical observations allows one to capture the dynamic features of volatility. The model uses the latest observations with the highest weights in the volatility estimate.

Initial value of volatility is taken as standard deviation of recent N returns (r 1 . r 2 . r N ) of N+1 days.

where is mean of N returns.

The exponentially weighted moving average model depends on

which is often referred to as the decay factor.

Firstly, this parameter defines a relative weight

that is applied to the last return.

This weight also defines the effective amount of data used in estimating volatility. The more the value, the less last observation affects the current dispersion estimation. Secondly, defines the rate of dispersion return to the previous level. The greater the value, the faster dispersion will come back to the previous level after strong return change. The optimal value for current daily dispersion (volatility) is =0.94. For such a value, the evaluation of dispersion can be done on the basis of 50 observations, and return of first day (r 1 ) will be considered with the relative weight of (1-0.94)*0.94^49=0.0029. Even for 30 observations the error will be insignificant.

The formula of the EWMA model can be rearranged to the following form: Thus, the older returns have the lower weights, which are close to zero.

Note, in the standard formula we take all returns with the same weight 1/(N-1)

The chart below shows 30 day historical volatility calculated by the EWMA method and the ordinary historical volatility calculated as a standard deviation of stock returns. As you can see EWMA Volatility almost agrees with ordinary historical volatility, but advantage of using EWMA is that this model requires only the last day's data and no additional recalculations.

The high low historical volatility also can be calculated by the EWMA method. In this case return r t must be calculated as the natural logarithm of the ratio of a stock's high price from the day n to stock's low price from the day t.

And the initial volatility value

is taken as Parkinson's number for the recent N days.

Time Series Definition of ARIMA Models

ARIMA (auto-regressive integrated moving average ) models establish a powerful class of models which can be applied to many real time series. ARIMA models are based on three parts: (1) an autoregressive part, (2) a contribution from a moving average, and (3) a part involving the first derivative of the time series:

The auto-regressive part (AR) of the model has its origin in the theory that individual values of time series can be described by linear models based on preceding observations. For instance: x(t) = 3 x(t-1) - 4 x(t-2). The general formula for describing AR[p]-models (auto-regressive models) is:

The consideration leading to moving average models (MA models) is that time series values can be expressed as being dependent on the preceding estimation errors. Past estimation or forecasting errors are taken into account when estimating the next time series value. The difference between the estimation x(t) and the actually observed value x(t) is denoted e (t). For instance: x(t) = 3 e (t-1) - 4 e (t-2).

The general description of MA[q]-models is:

When combining both AR and MA models, ARMA models are obtained. In general, forecasting with an ARMA [p, q]-model is described using the following equation:

After additional differentiation of the time series, and integrating it after application of the model, one speaks of ARIMA models. They are used when trend filtering is required. The parameter d of the ARIMA[p, d,q]-model determines the number of differentiation steps.

First, the time series is derived d times until it is stationary. Then, a suitable ARMA[p, q] model is fitted to the resulting series. Finally, the estimated forecasts have to be integrated d times.

Many more variants of ARIMA models have been introduced to treat specific cases. Here, the whole group of such models is subsumed under the term ARIMA models. Since their characteristics are determined by the three parameters p, d, and q, they are also referred to as ARIMA[p, d,q]-models. The parameter p denotes the order of the auto-regressive part, the parameter q the order of the moving average part, and d the number of differentiation steps.

Last Update: 2004-Jul-03

Forecasting with time series analysis

What is forecasting?

Forecasting is a method that is used extensively in time series analysis to predict a response variable, such as monthly profits, stock performance, or unemployment figures, for a specified period of time. Forecasts are based on patterns in existing data. For example, a warehouse manager can model how much product to order for the next 3 months based on the previous 12 months of orders.

You can use a variety of time series methods, such as trend analysis, decomposition, or single exponential smoothing, to model patterns in the data and extrapolate those patterns to the future. Choose an analysis method by whether the patterns are static (constant over time) or dynamic (change over time), the nature of the trend and seasonal components, and how far ahead you want to forecast. Before producing forecasts, fit several candidate models to the data to determine which model is the most stable and accurate.

Forecasts for a moving average analysis

The fitted value at time t is the uncentered moving average at time t -1. The forecasts are the fitted values at the forecast origin. If you forecast 10 time units ahead, the forecasted value for each time will be the fitted value at the origin. Data up to the origin are used for calculating the moving averages.

You can use the linear moving averages method by calculating consecutive moving averages. The linear moving averages method is often used when there is a trend in the data. First, calculate and store the moving average of the original series. Then, calculate and store the moving average of the previously stored column to obtain a second moving average.

In naive forecasting, the forecast for time t is the data value at time t -1. Using moving average procedure with a moving average of length one gives naive forecasting.

Forecasts for a single exponential smoothing analysis

The fitted value at time t is the smoothed value at time t-1. The forecasts are the fitted value at the forecast origin. If you forecast 10 time units ahead, the forecasted value for each time will be the fitted value at the origin. Data up to the origin are used for the smoothing.

In naive forecasting, the forecast for time t is the data value at time t-1. Perform single exponential smoothing with a weight of one to do naive forecasting.

Forecasts for a double exponential smoothing analysis

Double exponential smoothing uses the level and trend components to generate forecasts. The forecast for m periods ahead from a point at time t is

L t + mT t . where L t is the level and T t is the trend at time t.

Data up to the forecast origin time will be used for the smoothing.

Forecasts for Winters' method

Winters' method uses the level, trend, and seasonal components to generate forecasts. The forecast for m periods ahead from a point at time t is:

where L t is the level and T t is the trend at time t, multiplied by (or added to for an additive model) the seasonal component for the same period from the previous year.

Winters' Method uses data up to the forecast origin time to generate the forecasts.

Autoregressive moving average model

Visión de conjunto

In statistics. autoregressive moving average ( ARMA ) models . sometimes called Box-Jenkins models after the iterative Box-Jenkins methodology usually used to estimate them, are typically applied to time series data.

Autoregressive model

The notation AR( p ) refers to the autoregressive model of order p . The AR( p ) model is written

X t = c + ∑ i = 1 p φ i X t − i + ε T =c+\sum _ ^ \varphi _ X_ +\varepsilon _ .\,>

where φ 1. …. φ p ,\ldots ,\varphi _ > are the parameters of the model, c is a constant and ε t > is an error term (see below). The constant term is omitted by many authors for simplicity.

An autoregressive model is essentially an infinite impulse response filter with some additional interpretation placed on it.

Example: An AR(1)-process

An AR(1)-process is given by:

X t = c + φ X t − 1 + ε T =c+\varphi X_ +\varepsilon _ ,\,>

where ε t > is a white noise process with zero mean and variance σ 2 >. (Note: The subscript on φ 1 > has been dropped.) The process is covariance-stationary if | φ | < 1. If φ = 1 then X t > exhibits a unit root and can also be considered as a random walk. which is not covariance-stationary. Otherwise, the calculation of the expectation of X t > is straightforward. Assuming covariance-stationarity we get

where μ is the mean. For c = 0, then the mean = 0 and the variance is found to be:

It can be seen that the autocovariance function decays with a decay time of τ = − 1 / ln ⁡ ( φ ) [to see this, write B n = K ϕ | n | =K\phi ^ > where K is independent of n . Then note that ϕ | n | = e | n | ln ⁡ ϕ =e^ > and match this to the exponential decay law e − n / τ > ] The spectral density function is the Fourier transform of the autocovariance function. In discrete terms this will be the discrete-time Fourier transform:

This expression contains aliasing due to the discrete nature of the X j >. which is manifested as the cosine term in the denominator. If we assume that the sampling time ( Δ t = 1 ) is much smaller than the decay time ( τ ), then we can use a continuum approximation to B n >.

which yields a Lorentzian profile for the spectral density:

where γ = 1 / τ is the angular frequency associated with the decay time τ .

An alternative expression for X t > can be derived by first substituting c + φ X t − 2 + ε t − 1 +\varepsilon _ > for X t − 1 > in the defining equation. Continuing this process N times yields

X t = c ∑ k = 0 N − 1 φ k + φ N X t − N + ∑ k = 0 N − 1 φ k ε t − K =c\sum _ ^ \varphi ^ +\varphi ^ X_ +\sum _ ^ \varphi ^ \varepsilon _ .>

For N approaching infinity, φ N > will approach zero and:

It is seen that X t > is white noise convolved with the φ k > kernel plus the constant mean. By the central limit theorem. the X t > will be normally distributed as will any sample of X t > which is much longer than the decay time of the autocorrelation function.

Calculation of the AR parameters

The AR( p ) model is given by the equation

X t = ∑ i = 1 p φ i X t − i + ε T =\sum _ ^ \varphi _ X_ +\varepsilon _ .\,>

It is based on parameters φ i > where i = 1. p . Those parameters may be calculated using least squares regression or the Yule-Walker equations .

where m = 0. p . yielding p + 1 equations. γ m > is the autocorrelation function of X, σ ε > is the standard deviation of the input noise process, and δ m is the Kronecker delta function.

Because the last part of the equation is non-zero only if m = 0, the equation is usually solved by representing it as a matrix for m > 0, thus getting equation

[ γ 1 γ 2 γ 3 ⋮ ] = [ γ 0 γ − 1 γ − 2 … γ 1 γ 0 γ − 1 … γ 2 γ 1 γ 0 … … … … … ] [ φ 1 φ 2 φ 3 ⋮ ] \gamma _ \\\gamma _ \\\gamma _ \\\vdots \\\end >= \gamma _ &\gamma _ &\gamma _ &\dots \\\gamma _ &\gamma _ &\gamma _ &\dots \\\gamma _ &\gamma _ &\gamma _ &\dots \\\dots &\dots &\dots &\dots \\\end > \varphi _ \\\varphi _ \\\varphi _ \\\vdots \\\end >>

Derivation

The equation defining the AR process is

X t = ∑ i = 1 p φ i X t − i + ε T =\sum _ ^ \varphi _ \,X_ +\varepsilon _ .\,>

Multiplying both sides by X t − m and taking expected value yields

E [ X t X t − m ] = E [ ∑ i = 1 p φ i X t − i X t − m ] + E [ ε t X t − m ]. X_ ]=E\left[\sum _ ^ \varphi _ \,X_ X_ \right]+E[\varepsilon _ X_ ].>

Now, E[ X t X t − m ] = γ m by definition of the autocorrelation function. The values of the noise function are independent of each other, and X t − m is independent of ε t where m is greater than zero. For m > 0, E[ε t X t − m ] = 0. For m = 0,

E [ ε t X t ] = E [ ε t ( ∑ i = 1 p φ i X t − i + ε t ) ] = ∑ i = 1 p φ i E [ ε t X t − i ] + E [ ε t 2 ] = 0 + σ ε 2. X_ ]=E\left[\varepsilon _ \left(\sum _ ^ \varphi _ \,X_ +\varepsilon _ \right)\right]=\sum _ ^ \varphi _ \,E[\varepsilon _ \,X_ ]+E[\varepsilon _ ^ ]=0+\sigma _ ^ ,>

Now we have, for m ≥ 0,

γ m = E [ ∑ i = 1 p φ i X t − i X t − m ] + σ ε 2 δ metro. =E\left[\sum _ ^ \varphi _ \,X_ X_ \right]+\sigma _ ^ \delta _ .>

E [ ∑ i = 1 p φ i X t − i X t − m ] = ∑ i = 1 p φ i E [ X t X t − m + i ] = ∑ i = 1 p φ i γ m − yo. ^ \varphi _ \,X_ X_ \right]=\sum _ ^ \varphi _ \,E[X_ X_ ]=\sum _ ^ \varphi _ \,\gamma _ ,>

which yields the Yule-Walker equations:

Moving average model

The notation MA( q ) refers to the moving average model of order q .

X t = ε t + ∑ i = 1 q θ i ε t − i =\varepsilon _ +\sum _ ^ \theta _ \varepsilon _ \,>

Autoregressive moving average model

The notation ARMA( p . q ) refers to the model with p autoregressive terms and q moving average terms. This model contains the AR( p ) and MA( q ) models,

X t = ε t + ∑ i = 1 p φ i X t − i + ∑ i = 1 q θ i ε t − yo. =\varepsilon _ +\sum _ ^ \varphi _ X_ +\sum _ ^ \theta _ \varepsilon _ .\,>

Note about the error terms

Specification in terms of lag operator

In some texts the models will be specified in terms of the lag operator L . In these terms then the AR( p ) model is given by

ε t = ( 1 − ∑ i = 1 p φ i L i ) X t = φ X t =\left(1-\sum _ ^ \varphi _ L^ \right)X_ =\varphi X_ \,>

where φ represents the polynomial

The MA( q ) model is given by

X t = ( 1 + ∑ i = 1 q θ i L i ) ε t = θ ε t =\left(1+\sum _ ^ \theta _ L^ \right)\varepsilon _ =\theta \varepsilon _ \,>

where θ represents the polynomial

Finally, the combined ARMA( p . q ) model is given by

( 1 − ∑ i = 1 p φ i L i ) X t = ( 1 + ∑ i = 1 q θ i L i ) ε t ^ \varphi _ L^ \right)X_ =\left(1+\sum _ ^ \theta _ L^ \right)\varepsilon _ \,>

or more concisely,

Fitting models

ARMA models in general can, after choosing p and q, be fitted by least squares regression to find the values of the parameters which minimize the error term. It is generally considered good practice to find the smallest values of p and q which provide an acceptable fit to the data. For a pure AR model the Yule-Walker equations may be used to provide a fit.

Aplicaciones

ARMA is appropriate when a system is a function of a series of unobserved shocks (the MA part) as well as its own behavior. For example, stock prices may be shocked by fundamental information as well as exhibiting technical trending and mean-reversion effects due to market participants.

Generalizations

Autoregressive moving average models can be generalized in other ways. See also autoregressive conditional heteroskedasticity (ARCH) models and autoregressive integrated moving average (ARIMA) models. If multiple time series are to be fitted then a vector ARIMA (or VARIMA) model may be fitted. If the time-series in question exhibits long memory then fractional ARIMA (FARIMA, sometimes called ARFIMA) modelling is appropriate. If the data is thought to contain seasonal effects, it may be modeled by a SARIMA (seasonal ARIMA) or a periodic ARMA model.

Autoregressive moving average model with exogenous inputs model (ARMAX model)

The notation ARMAX( p . q . b ) refers to the model with p autoregressive terms, q moving average terms and b eXogenous inputs terms. This model contains the AR( p ) and MA( q ) models and a linear combination of the last b terms of a known and external time series d t >. It is given by:

X t = ε t + ∑ i = 1 p φ i X t − i + ∑ i = 1 q θ i ε t − i + ∑ i = 1 b η i d t − yo. =\varepsilon _ +\sum _ ^ \varphi _ X_ +\sum _ ^ \theta _ \varepsilon _ +\sum _ ^ \eta _ d_ .\,>

Ver también

Referencias

George Box. Gwilym M. Jenkins. and Gregory C. Reinsel. Time Series Analysis: Forecasting and Control . third edition. Prentice-Hall, 1994.

Mills, Terence C. Time Series Techniques for Economists. Cambridge University Press, 1990.

Percival, Donald B. and Andrew T. Walden. Spectral Analysis for Physical Applications. Cambridge University Press, 1993.

External links

Asset Allocation Models Based on Moving Averages Are Dumb

Moving average crossover models are dumb

I f monthly MA(3) > monthly MA(10) buy at the next open Exit at the next open if MA(3) < monthly MA(10)

The equity curve of this system is shown below:

Below are some key performance statistics of this system:

It may be seen that the timing models generated about 110 basis points of annual excess return as compared to buy and hold but at a much lower drawdown.

Each random run is repeated 20,000 times and the CAR is calculated. Then the cumulative frequency distribution of CAR is plotted as shown below:

Moving average crossover models are based on wishful thinking

Below are some performance details:

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8 Responses to Asset Allocation Models Based on Moving Averages Are Dumb

Fred Dobbs says:

I don't know how much you can rely on testing something like the SPY system you present that only showed 8 trades even after you do a simulation. The EEM slice of time you present is only 5.5 years, which is a very short period. One can always find short periods like that, but they may not represent long run expectations.

Have the seen this paper by Zakamulin? http://papers. ssrn. com/sol3/papers. cfm? abstract_id=2585056 He tests a number of different MA methods on the S&P Composite index on over 158 years of data. Yes, I know costs are not included, but the results are very strong and low-cost index funds are available now. His conclusion is: "Whereas over very long-term horizons the market timing strategy is almost sure to outperform the market on a risk-adjusted basis, over more realistic medium-term horizons the market timing strategy is equally likely to outperform as to underperform. Yet we find that the average outperformance is greater than the average underperformance." He also has another paper on robustness testing of MAs.

"I don't know how much you can rely on testing something like the SPY system you present that only showed 8 trades even after you do a simulation."

I did not present the system. This system is a part of some well-known allocation methods (ex. Faber).

"The EEM slice of time you present is only 5.5 years, which is a very short period."

Do you think that 5.5 years of devastating losses is a short period of time? This is a recent market unlike studies that go back when there were no cars, computers, even electricity or telephones and people moved around on horses. I wonder why an sane person would pay attention to these studies that only reflect data-mining bias and wishful thinking.

"Have the seen this paper by Zakamulin?"

I have learnt over the years to rely on my own work. There are many issues with backtests, many assumptions and data-mining bias. I started backtesting systems in the mid 1980s unlike some authors who only discovered backtesting in the last few years. Backtesting is more of an art than a science.

MAs are a dangerous indicator for market timing. Performance deteriorates fast during sideways and fast markets. Relying on MAs is indistinguishable from gambling with money. Outperformance is due to luck as a general rule.

"over more realistic medium-term horizons the market timing strategy is equally likely to outperform as to underperform"

This is wishful thinking, it is not science. But before that he concludes:

"Third, we did find support for the claim that one can beat the market by timing it. Yet the chances for beating the market depend on the length of the investment horizon"

Fred, it boils down to this: very long backtests fool naive market researchers due to stock market structural bias. Rules play no role. See for example:

Michael, I agree with you that moving average cross over systems are largely random. A trend following system based on this will do very poorly on equity indices in the past three years even though the underline equity indices themself are doing very well. A very long back test only ensure that you are more likely to run into a period where this strategy does extremly well so as to lift the overall metrics to a good level. It does not say anything about goinng forward it would work or not. It's unknow. The best I think we can do is to apply this kind of systems equally on a variety of uncorrelated assets. For example, while trend following does not work on equity indices in the past three years, it seems to work very well on currencies.

Again great article!

"The best I think we can do is to apply this kind of systems equally on a variety of uncorrelated assets."

I think this is the key but one problem is that correlation varies and instruments may get correlated during certain period. CTAs have struggled in recent years although their trend-following methods still carry a positive skew from the 1990s. Take a look at performance here in the last 5 years.

Note that 2011, 2012 and 2013 marked the first two and three consecutive losing streaks for CTAs. Remember that CTAs mostly use MAs and other similar longer-term indicators.

thx for your analysis. i read it with a lot of interest, because some time ago i was advicing friends of mine to an asset allocation system with ma. they are just getting started with their jobs and so they wanted to know what to do with their money. as i did read some ma and ma with asset allocation studies i thought this would be a good way to go (implemented with ETFs, yes i know ETFs are no holy grail). i want to express that expection of an investment is one important point, like the possible risk (i told them that if you cant take a 50% drawdown you should not start investing in stocks, with or without ma's) in deciding what to do for my friends, but also there is an effort factor. if you have a job and not much time/interest in investing your possibilities are llimited. you can simply go buy and hold, buy fonds (i do think that the probability that a fond beats buy and hold is pretty low) or do some very simply strategie (like ma's), which hopefully gives you a realistic chance for outperformance versus buy and hold after several years. so my question: do you think my advice for my friends using such a ma asset allocation is a good/bad advice, do you know of possible alternative strategies or do you have other ideas to this topic ?

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Forecasting Computer Usage

Julie M. Hays University of St. Thomas

Journal of Statistics Education Volume 11, Number 1 (2003), www. amstat. org/publications/jse/v11n1/datasets. hays. html

Copyright y copia; 2003 by Julie M. Hays, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the author and advance notification of the editor.

Key Words: Causal forecasting; Model-building; Seasonal Variation; Simple linear regression; Time-series forecasting; Transformations.

Abstracto

The dataset bestbuy. dat. txt contains actual monthly data on computer usage (Millions of Instructions Per Second, MIPS) and total number of stores from August 1996 to July 2000. Additionally, information on the planned number of stores through December 2001 is available. This dataset can be used to compare time-series forecasting with trend and seasonality components and causal forecasting based on simple linear regression. The simple linear regression model exhibits unequal error variances, suggesting a transformation of the dependent variable.

1. Introduction

One of the most prevalent uses of regression analysis in actual business settings is for forecasting. For a summary of some forecasting methods see Armstrong (2001) or Arsham (2002). The bestbuy. dat. txt dataset can be used to demonstrate and discuss both time-series and causal forecasting. Time constraints and the interests and needs of the students determine whether I supply the analyses or have the students perform the analyses.

I have used this dataset throughout the semester in an MBA Decision Analyses class. This class is a core requirement for all evening MBA students and covers a range of decision analysis and statistical topics, including regression analysis and forecasting. Most students are required to take an introductory business statistics course prior to this course, so they have had some exposure to statistical topics, but few students have any academic experience with forecasting.

Best Buy Co. Inc. (NYSE:BBY), headquartered in Eden Prairie, Minnesota, is the largest volume specialty retailer of consumer electronics, personal computers, entertainment software and appliances. In August of each year, Best Buy purchases mainframe MIPS (Millions of Instructions Per Second, a measure of computing resources) in anticipation of the coming holiday season. Computing resources are needed to track and analyze retail information needed for billing, inventory, and sales. For planning and budgeting purposes they also wish to forecast the number of MIPS needed the following year. Best Buy Corporation actually used this dataset to predict computer usage in order to budget for and purchase an appropriate amount of computing power. However, prior to 2001, Best Buy did not do any statistical analysis of this data. Best Buy only looked at the numbers (they did not even graph the data) and then guessed at the amount of MIPS needed in the coming year.

Students are asked to forecast the MIPS needed for December 2000 and December 2001 using the bestbuy. dat. txt dataset. This dataset was obtained from the Best Buy Corporation and contains monthly data on computer usage (MIPS) and total number of stores from August 1996 to July 2000. Additionally, information on the planned number of stores through December 2001 is available.

Students can easily understand the seasonality that retail operations experience. Best Buy Corporation has experienced significant growth over the past few years and most students understand that as a firm grows, their need for computing power also increases. Therefore, this dataset can be used to demonstrate time-series forecasting with both a trend and seasonality.

This dataset can also be used to demonstrate causal forecasting based on simple linear regression of computer usage and number of stores. The simple linear regression model exhibits unequal error variances, suggesting a transformation of the dependent variable.

Finally, a comparison between the time-series model and causal model can be made and discussed with the students.

2. Time Series Forecasting

Before I allow the students to begin any numerical analyses, I have the students plot computer usage versus time. I have the students “forecast” the number of MIPS needed for December 2000 and December 2001 using only the plot of computer usage (MIPS) versus time, Figure 1. The plot clearly shows a trend in MIPS usage with time. Typically, students “eyeball” the graph and predict MIPS usage of 500 for December 2000 and 600 for December 2001.

Figure 1. MIPS vs Time.

Students who actually fit a line to the data forecast MIPS usage of 527 for December 2000 and 624 for December 2001 (Figure 2 ).

Figure 2. MIPS vs Time.

I introduce simple moving average, weighted moving average and exponential smoothing forecasting techniques to the students before they attempt to use these forecasting models to predict future MIPS usage. I also discuss the evaluation of forecasting models using MAD and CFE (explained below). The interested reader can find more detailed discussions of these topics in Stevenson (2002) or at Sparling (2002).

Moving Average An n - period moving average is the average value over the previous n time periods. As you move forward in time, the oldest time period is dropped from the analysis.

Weighted Moving Average An n - period weighted moving average allows you to place more weight on more recent time periods by weighting those time periods more heavily.

Exponential Smoothing The forecast for the next period using exponential smoothing is a smoothing constant, ( 0 1), times the demand in the current period plus (1- smoothing constant) times the forecast for the current period. where F t+1 is the forecast for the next time period, F t is the forecast for the current time period, D t is the demand in the current time period, and 0 1 is the smoothing constant. To initiate the forecast, assume F 1 = D 1 . Higher values of a place more weight on the more current time periods.

Because this model is less intuitive, I usually expand this equation to help the students understand that demand from time periods prior to the current period is included in this model. and where D t-1 is the demand in the previous time period, D t-2 is the demand in the time period before the previous time period, and F t-1 is the forecast in the previous time period, and F t-2 is the forecast in the time period before the previous time period.

Because the data storage requirements are considerably less than for the moving average model, this type of model was used extensively in the past. Now, although data storage is not usually an issue, it is typical of real-world business applications because of its historical usage.

Mean Absolute Deviation (MAD) The evaluation of forecasting models is based on the desire to produce forecasts that are unbiased and accurate. The Mean Absolute Deviation (MAD) is one common measure of forecast accuracy.

Cumulative sum of Forecast Errors (CFE) The Cumulative sum of Forecast Errors (CFE) is a common measure of forecast bias.

“Better” models would have lower MAD and CFE close to zero.

After explaining these techniques, I have the students work through the following simple example in class. I give the students the demand profile (Table 1 ) and have them calculate forecasts using a 3-period moving average and exponential smoothing with a smoothing constant of 0.2. I also have them calculate the MAD and CFE for both models. We discuss using the MAD and CFE to determine the “best” model.

I also point out to the students that I have arbitrarily chosen the number of periods for the moving average model and the smoothing constant for the exponential smoothed model. I discuss using MAD and CFE to determine the “best” choice for these variables.

Table 1. In-class forecasting example.

All numbers rounded to the nearest hundredth

Once the students are familiar with these techniques, I have them estimate MIPS for December 2000 and 2001 using a 3-period moving average and exponential smoothing with a smoothing constant of 0.2 (Figure 3 ). This can be done using Excel, Minitab or any statistics package. The forecast for the 3-period moving average is 463 MIPS and for the exponential smoothed is 450 MIPS.

Figure 3. Actual and forecast MIPS.

The students can easily see that there is a “problem” with their forecasts. Although I have told the students that exponential smoothing and moving average forecasting models are only appropriate for stationary data, they don’t really understand this until they try to use the technique. This exercise helps the students understand that moving average and exponential smoothing are really only averaging techniques and helps them comprehend the need to account for trends in forecasting. I demonstrate adjusting for trends by using double exponential smoothing. Double exponential smoothing is a modification of simple exponential smoothing that effectively handles linear trends. Good explanations of this technique can be found in Wilson and Keating (2002) or at Group6 (2002).

Double Exponential Smoothing

where F t+1 is the forecast for the next time period,

A t is the exponentially smoothed level component in the current period where F t is the forecast for the current time period, D t is the demand in the current time period, and 0 1 is the smoothing constant and T t is the exponentially smoothed trend component in the current period. where 0 1 is the smoothing constant for the trend, T t-1 is the trend in the previous period, and C t is the current trend

The forecast for n periods into the future is

After I explain this model, I have the students go back and re-estimate their forecast using this model (Figure 4 ). Minitab has these functions built in and will compute the optimal smoothing parameters, and , based on minimizing the sum of squared errors, but any statistics package could be used. Minitab will also compute Mean Absolute Prediction Error (MAPE), Mean Absolute Deviation (MAD), Mean Square Error (MSE) and provides 95% confidence prediction intervals (see Figure 4 ).

The forecasts obtained are essentially the same as the forecasts obtained from fitting a line to the data, MIPS usage of 527 for December 2000 and 624 for December 2001.

Figure 4. Optimal double exponential smoothed.

I ask the students if they are happy with their forecast now or if there is anything else they need to do. I supply a plot of errors versus time for the double exponential smoothed model with the December errors highlighted (Figure 5 ). Most students are aware that retail firms have their highest sales during the Christmas season (December). Therefore, students typically mention seasonality and we discuss the possible ways that we could account for seasonality.

Figure 5. Double exponential smoothed model errors.

The students usually mention both an additive and multiplicative adjustment for seasonality using all the past data or only some of the past data. Simple explanations of these two techniques can be found in Hanke and Reitsch (1998) or at Nau (2002). In other words, we could compare the forecast for December 1999 to the actual for December 1999 and for the additive model we would add this difference to our forecast for December 2000. Or, for the multiplicative model, we would multiply the forecast for December 2000 by the actual December 1999/forecast December 1999. They carry this further and discuss using the data from 1998, 1997, and 1996 to produce an average adjustment. I lead the discussion towards the smoothing techniques we have been discussing and how we could use these types of techniques to come up with seasonal adjustments for our forecasts. I explain that Winter developed just such a technique of triple exponential smoothing. Winter’s technique basically adds (or multiplies) a smoothed seasonal adjustment to the model, similar to the addition of a smoothed adjustment for a trend in the double exponential smoothed model. The interested reader can find the calculation formulas and explanations of triple exponential smoothing (or Winter’s method) in Minitab (1998b) or Prins (2002a).

I use Minitab to demonstrate Winter’s model (Figure 6 ) because the calculations for this method are fairly complex and most students only need to have a general understanding of this type of technique. Using Winter’s model the forecast for December 2000 is 521 MIPS and the forecast for December 2001 is 606 MIPS.

I also use this opportunity to mention ARIMA models and direct interested students to resources such as Minitab (1998a) for more information about ARIMA models.

Figure 6. Winter's Method.

3. Causal Forecasting

I supply the students with a plot of computer usage (MIPS) vs. number of stores (Figure 7 ) and again have them “forecast” computer usage for December 2000 and December 2001. Best Buy believes that they will have 394 stores in December of 2000 and 445 stores in December of 2001.

Figure 7. MIPS vs number of stores.

Again most students “eyeball” the graph and use graphical linear extrapolation to arrive at their forecast. They predict usage of 600 MIPS for December 2000 and 800 MIPS for December 2001.

I have the students perform a simple linear regression of MIPS on number of stores and produce the residual plot (Figures 8 and 9 ). I use this opportunity to emphasize the usefulness of the residual plot in evaluating the model. I highlight the “megaphone” shape of the residual plot (the residuals are increasing as the number of stores increases) and explain that this implies that a transformation of the dependent variable is indicated.

Figure 8. MIPS vs number of stores.

Figure 9. MIPS vs number of stores residual plot.

Although I used the Box-Cox procedure (Box and Cox 1964 ) to determine the appropriate transformation, this technique is beyond the scope of this class. Therefore, I just tell the students that the appropriate transformation is square root of MIPS and mention that there are mathematical techniques that can be used to determine the appropriate transformation. I direct interested students to Neter, Kutner, Nachtsheim, and Wasserman (1996) or Prins (2002b) for descriptions of this technique.

I have the students re-estimate the regression equation and produce the residual plot for this regression (Figures 10 and 11 ). Although the R 2 is slightly lower, the residuals are now more evenly distributed.

Figure 10. Square root MIPS vs number of stores.

Figure 11. Square root MIPS vs number of stores residual plot.

I also have the students predict computer usage for December 2000 and December 2001 using the fitted equation. If students have difficulty predicting MIPS, because of the square root transformation of MIPS, I explain the calculations in class. The new predictions are 664 MIPS for December 2000 and 977 MIPS for December 2001.

Again, an adjustment for seasonality could be made. Although, any of the seasonality adjustments discussed in the previous section could be used here, I usually have the students use an average multiplicative adjustment. This could be done by calculating actual/predicted for all months, averaging these seasonal factors for each particular month and multiplying the resulting seasonal factor by the predicted value. If this is done, the new predictions for December 2000 and December 2001 are 700 MIPS and 1029 MIPS.

4. Comparison of Methods

After the students have used the various methods to predict MIPS usage, I have them discuss which method they have most confidence in and why they believe that that model is the best. Several important points can be made here.

First, I emphasize that forecasting is a very imperfect science and no technique can perfectly predict the future. The “best” technique will balance the accuracy needed with the complexity (or cost) of the model.

Second, I emphasize the value of “plotting the data.” One of the best (and easiest) methods to evaluate various models is a visual examination of the data and forecasts that would be produced by the method under consideration.

Third, I emphasize the need to account for trends and seasonality if those are present in the data. Moving averages and exponential smoothing are appropriate forecasting methods only if the data are stationary. If there are trends and/or seasonalities present, more sophisticated methods should be used.

Finally, we discuss the difficulty inherent in finding a causal predictor for most values we wish to predict in business environments.

5. Conclusion

The dataset bestbuy. dat. txt can be used to demonstrate both time series and causal forecasting. Analysis of the dataset leads to a discussion and comparison of the positives and negatives of various forecasting methods.

6. Obtaining the Data

The file bestbuy. dat. txt contains the raw data. The file bestbuy. txt is a documentation file containing a brief description of the dataset.

Appendix - Key to the variables in bestbuy. dat. txt

Trend-following is one of the oldest investment methods

Labeled as technical analysis, trend-following went largely un-researched by academics

Research of cross-sectional momentum exploded after Narasimhan Jegadeesh and Sheridan Titman published their seminal 1992 study, but time-series momentum remained largely ignored until after 2008

Price-based trend-following techniques, like moving average systems, remained separate from return-based time-series momentum techniques.

New research shows that moving average systems and time-series momentum are mathematically-linked techniques

In 1838, James Grant published The Great Metropolis, Volume 2. Within, he spoke of David Ricardo, an English political economist that was active in the London markets in the late 1700s and early 1800s. Ricardo amassed a large fortune trading both bonds and stocks. According to Grant, Ricardo’s success was attributed to three golden rules:

“As I have mentioned the name of Mr. Ricardo, I may observe that he amassed his immense fortune by a scrupulous attention to what he called his own three golden rules, the observance of which he used to press on his private friends. These were, “Never refuse an option* when you can get it,”—”Cut short your losses,”—”Let your profits run on.” By cutting short one’s losses, Mr. Ricardo meant that when a member had made a purchase of stock, and prices were falling, he ought to resell immediately. And by letting one’s profits run on he meant, that when a member possessed stock, and prices were raising, he ought not to sell until prices had reached their highest, and were beginning again to fall. These are, indeed, golden rules, and may be applied with advantage to innumerable other transactions than those connected with the Stock Exchange.”

“Cut short your losses” and “let your profits run on” became the core tenets of trend-following.

Other prominent early trend-followers include:

Charles H. Dow, founder and first editor of the Wall Street Journal as well as co-founder of Dow Jones and Company

Jesse Livermore, who is quoted by Edwin Lefèvre as having said, "[t]he big money was not in the individual fluctuations but in the main movements. sizing up the entire market and its trend."

Richard Wyckoff, whose method involved entering long positions only when the market was trending up and shorting when the market was trending down.

There was even an early academic study of trend-following performed by Alfred Cowles III and Herbert Jones in 1933. In the study, titled Some A Posteriori Probabilities in Stock Market Action . they focus on counting the number of sequences – times when positive returns were followed by positive returns, or negative returns were followed by negative returns – to reversals – times when positive returns are followed by negative returns, and vice versa.

Cowles and Jones evaluated the ratio of these sequences and reversals in stock prices over periods ranging 20 minutes to 3 years. Their results:

It was found that, for every series with intervals between observations of from 20 minutes up to and including 3 years, the sequences out-numbered the reversals. For example, in the case of the monthly series from 1835 to 1935, a total of 1200 observations, there were 748 sequences and 450 reversals. That is, the probability appeared to be .625 that, if the market had risen in a given month, it would rise in the succeeding month, or, if it had fallen, that it would continue to decline for another month. The standard deviation for such a long series constructed by random penny tossing would be 17.3; therefore the deviation of 149 from the expected value of 599 is in excess of eight times the standard deviation. The probability of obtaining such a result in a penny-tossing series is infinitesimal.

Despite promising empirical and theoretical results for trend-following, the next academic studies would not come until nearly a century later.

In 1934, Benjamin Graham and David Dodd published Security Analysis . Later, in 1949, they published The Intelligent Investor.

In these weighty tomes, they outline their methods for successful investing. Graham and Dodd’s method focused on evaluating the financial state of the underlying business. Their objective was to identify a company’s intrinsic value and purchase stock when the market offered a substantial discount to that value.

For Graham and Dodd, anything else was mere speculation.

Graham and Dodd gave fundamental investors – and specifically value investors – their bible.

Anything, then, that was not fundamental investing was technical analysis. And since trend-following relied only on evaluating past prices, it was labeled technical analysis.

Unfortunately, academics largely dismissed technical analysis through the 1900s. This is likely due to the fact that it was difficult to study and test. Practitioners follow a large number of different techniques. Sometimes these different techniques can lead to contradictory predictions between technicians.

But in 1993, Narasimhan Jegadeesh and Sheridan Titman published Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency. In their paper, they outlined an investment strategy that purchased stocks that had outperformed their peers and sold stocks that had underperformed.

Jegadeesh and Titman called their approach relative strength – a term that had been long used by technicians. Now it is sometimes called cross-sectional momentum . relative momentum, or often just momentum .

This simple method outlined by Jegadeesh and Titman created statistically significant positive returns that could not be explained by common risk factors.

This paper ushered in an era of momentum research, with academics exploring how the technique fared across geographies, time-frames, and asset classes. The results were that momentum was surprisingly robust.

Despite the success of relative strength . interest in its close cousin trend-following was still nowhere to be found.

Until the financial crisis of 2008.

Technically, one of the most popular research papers about trend-following – Mebane Faber’s A Quantitative Approach to Tactical Asset Allocation – was published in 2006. But the majority of interest from academics occurred post-2008.

We attribute this interest to trend-following’s risk mitigation properties.

The studies typically fall into two camps.

In the first camp was the study of trend-following, which tended to follow simple mechanical systems, like moving averages. Faber (2006) fell into this camp, using a 10-month moving average cross-over.

There are several variations of these systems. For example, one might use the cross of price over the moving average as a signal. Another might use the cross of a shorter moving average over a longer. Finally, some may even use directional changes in the moving average as the signal.

Others tended to focus on what would become known as time-series momentum . In time-series momentum, the trading signal is generated when the total return over a given period crosses over the zero-line.

One of the most prominent studies for time-series momentum was Moskowitz, Ooi, and Pedersen (2011), which demonstrated the anomaly was significant in 58 liquid equity index, currency, commodity, and bond futures.

Trend-following moving average rules were still considered to be technical trading rules versus the quantitative approach of time-series momentum. Perhaps the biggest difference is that the trend-following camp tended to focus on techniques using prices while the momentum camp focused on returns.

However, research over the last half-decade actually shows that they are mathematically related strategies.

Bruder, Dao, Richard, and Roncalli’s 2011 Trend Filtering Methods for Momentum Strategies united moving-average cross-over strategies and time-series momentum by showing that cross-overs were really just an alternative weighting scheme for returns in time-series momentum. To quote,

The weighting of each return … forms a triangle, and the biggest weighting is given at the horizon of the smallest moving average. Therefore, depending on the horizon n 2 of the shortest moving average, the indicator can be focused toward the current trend (if n 2 is small) or toward past trends (if n 2 is as large as n 1 /2 for instance).

In Marshall, Nguyen and Visaltanachoti’s Time-Series Momentum versus Moving Average Trading Rules . published in 2012, time-series momentum is shown to be related to changes in direction of a moving average. In fact, time-series momentum signals will not occur until the moving average changes direction.

Therefore, moving average rules which rely on price crossing the moving average are likely to occur before a change in signal from time-series momentum.

Similar to Bruder, Dao, Richard, and Roncalli, Levine and Pedersen show that time-series momentum and moving average cross-overs are highly related in their 2015 paper Which Trend is Your Friend? . They also find that time-series momentum and moving-average cross-over strategies perform similarly across 58 liquid futures and forward contracts.

In their 2015 paper Uncovering Trend Rules, Beekhuizen and Hallerbach also link moving averages with returns, but further explore trend rules with skip periods and the popular MACD (moving average convergence divergence) rule. Using the implied link of moving averages and returns, they show that the MACD is as much trend following as it is mean-reversion.

These studies are important because they help validate the approach of price-based systems. Being mathematically linked, technical approaches like moving averages can now be linked to the same theoretical basis as the growing body of work in time-series momentum.

Market practitioners have long held that the trend is your friend and academic literature has finally begun to agree.

But perhaps, most importantly, we now know that it doesn’t matter whether you take the technical approach using moving averages or the quantitative approach of measuring returns. At the end of the day, they’re more or less the same thing.

Incoming search terms:

Autoregressive–moving-average model

In the statistical analysis of time series, autoregressive–moving-average (ARMA) models provide a parsimonious description of a (weakly) stationary stochastic process in terms of two po. (展开) lynomials, one for the auto-regression and the second for the moving average. The general ARMA model was described in the 1951 thesis of Peter Whittle, Hypothesis testing in time series analysis, and it was popularized in the 1971 book by George E. P. Box and Gwilym Jenkins. Given a time series of data Xt, the ARMA model is a tool for understanding and, perhaps, predicting future values in this series. The model consists of two parts, an autoregressive (AR) part and a moving average (MA) part. The model is usually then referred to as the ARMA(p, q) model where p is the order of the autoregressive part and q is the order of the moving average part (as defined below).

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Autoregressive–moving-average model - Wikipedia, the free.

Адаптировано для мобильных устройств · The notation AR(p) refers to the autoregressive model of order p. The AR(p) model is written. where are parameters, is a constant, and the random variable is white noise

autoregressive+moving+average (ARMA) model 的翻译查词.

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自动回归移动模型 - 小司 - 博客园

2009-10-16 · From Wikipedia In statistics and signal processing, autoregressive moving average (ARMA) models . sometimes called Box-Jenkins models after the iterative Box.

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Autoregressive Moving Average Model - MATLAB & Simulink

Адаптировано для мобильных устройств · ARMA(p, q) Model . For some observed time series, a very high-order AR or MA model is needed to model the underlying process well. In this case, a combined.

autoregressive integrated moving average model - Free.

autoregressive integrated moving average model - Free definition results from over 1700 online dictionaries. ARIMA( Autoregressive Integrated Moving Average ) 模型，差分自.

Articles > Investing > Importance of Moving Averages in FOREX Trading

Importance of Moving Averages in FOREX Trading

- Michael Duane Archer

The moving average (MA ) is another instrument used to study trends and generate market entry and exit signals. It is the arithmetic average of closing prices over a given period. The longer the period studied, the weaker the magnitude of the moving average curve. The number of closes in the given period is called the moving average index.

Market signals are generated by calculating the residual value:

When the residual crosses into the positive area, a buy signal is generated.

When the residual drops below zero, a sell signal is generated.

A significant refinement to this residual method (also called moving average convergence divergence, or MACD for short) is the use of two moving averages. When the MA with the shorter MA index (called the oscillating MA index) crosses above the MA with the longer MA index (called the basis MA index), a sell signal is generated.

Residual = Basis MA(X ) - Oscillating MA(X )

Again, we use the EUR/USD currency pair to illustrate the moving average method.

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403 errors usually mean that the server does not have permission to view the requested file or resource. These errors are often caused by IP Deny rules, File protections, or permission problems.

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File and Directory Ownership

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Symbolic Representation

The first character indicates the file type and is not related to permissions. The remaining nine characters are in three sets, each representing a class of permissions as three characters. The first set represents the user class. The second set represents the group class. The third set represents the others class.

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- rwx r-x r-x a regular file whose user class has full permissions and whose group and others classes have only the read and execute permissions.

c rw - rw - r-- a character special file whose user and group classes have the read and write permissions and whose others class has only the read permission.

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Numeric Representation

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These values never produce ambiguous combinations. each sum represents a specific set of permissions. More technically, this is an octal representation of a bit field вЂ“ each bit references a separate permission, and grouping 3 bits at a time in octal corresponds to grouping these permissions by user . group . and others .

Permission mode 0 7 5 5

Permission mode 0 6 4 4

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How to Edit. htaccess files in cPanel's File Manager

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How to Edit file permissions in cPanel's File Manager

Before you do anything, it is suggested that you backup your website so that you can revert back to a previous version if something goes wrong.

Open the File Manager

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In the Files section, click on the File Manager icon.

Check the box for Document Root for and select the domain name you wish to access from the drop-down menu.

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An Introduction to State and Local Public Finance Thomas A. Garrett and John C. Leatherman

PART 2 - SELECTED APPLICATIONS IN PUBLIC FINANCE

IV. Revenue Forecasting

Revenue forecasting involves the use of analytical techniques to project the amount of financial resources available in the future. In the public sector, revenues come from taxes, fees, license sales or intergovernmental transfers. Forecasting attempts to identify the relationship between the factors that drive revenues (tax rates, building permits issued, retail sales) and the revenues government collects (property taxes, user fees, sales taxes). The ability to accurately project future resources is critical to avoiding budgetary shortfalls or collecting excess taxes or fees. For the federal government, even small errors in projecting revenue can result in serious budget problems such as large surpluses or deficits. Thus, revenue forecasting is fundamental to both state and federal governments, as well as many larger municipalities. As local governments continue to shift reliance from the property tax to user fee-based revenues, forecasting will be increasingly important to smaller units of government and department administrators.

Revenue forecasts can apply to aggregate total revenue or to single revenue sources such as sales tax revenues or property tax revenues. There is no single method for projecting revenues. Rather, different methods tend to work better depending on the type of revenue. Similarly, there is no standard time-frame over which to attempt a forecast. State government might look ahead to the next year’s budget, while managers of a city water system may be concerned about a twenty year time horizon. Finally, revenue forecasting is intimately tied to the public policy process and is thus subject to considerable scrutiny and even political pressure.

B. The Forecasting Process

Government fiscal policy is affected by the context in which it is formed. It deals with not only economic but also political concerns. It is essential to establish assumptions and procedures that concerned parties agree upon, as well as a mechanism for evaluating the validity of revenue forecasts. Thus, a disciplined process is needed. Guajardo and Miranda (2000) suggest a seven step process. The following steps are applied to each type of revenue to be forecast.

The first step involves selecting a time period over which revenue data is examined. The length of time depends on the availability and quality of data, the type of revenue to be forecasted, and the degree of accuracy sought.

In the second step, the data is examined to determine any patterns, rates of change, or trends that may be evident. Patterns may suggest that the rates of change are relatively stable or changing exponentially. Once the trend is identified, the forecaster needs to decide to what degree the revenue is predictable. This is done by examining the underlying characteristics of the revenue, such as the rate structures used to collect the revenue, changes in demand, or seasonal or cyclical variation.

Forecasters next need to understand the underlying assumptions associated with the revenue source. They need to consider to what degree the revenue is affected by economic conditions, changing citizen demand, and changes in government policies. These assumptions help determine which forecasting method is most appropriate.

The next step is to actually project revenue collections in future years. The method selected to perform the projection depends on the nature and type of revenue. Revenue sources with a high degree of uncertainty, such as new revenues and grants or asset sales, may employ a qualitative forecasting method, such as consensus or expert forecasting. Revenues that are generally predictable will typically be forecast using a quantitative method, such as a trend analysis or regression analysis.

After the projections have been made, the estimates need to be evaluated for their reliability and validity. To evaluate the validity of the estimates, the assumptions associated with the revenue source are re-examined. If the assumptions associated with existing economic, administrative, and political environment are sound, the projections are assumed valid. Reliability is assessed by conducting a sensitivity analysis. This involves varying key parameters used to create the estimates. If large changes in the estimates result, the projection is assumed to have a low degree of reliability.

In the sixth step, actual revenue collections are monitored and compared against the estimates. Monitoring serves both to assess the accuracy of the projections and to determine whether there is likely to be any budget shortfall or surplus.

Finally, as conditions affecting revenue generation change, the forecast will need updating. Fluctuations in collections may be caused by unexpected changes in economic conditions, policy and administrative adjustments, or in patterns of consumer demand.

C. Forecasting Methods

There are a wide range of forecasting techniques available (Frank, 1993; Makridakis and Wheelwright, 1987, 1989; Guajardo and Miranda, 2000). They range from relatively informal qualitative techniques to highly sophisticated quantitative techniques. In revenue forecasting, more sophisticated does not necessarily mean more accurate. In fact, an experienced finance officer can often "guess" what is likely to happen with a great deal of accuracy. In general, forecasters use a variety of techniques, recognizing that some perform better than others depending on the nature of the revenue source.

yo. Qualitative Forecasting Methods

Qualitative forecasting methods rely on judgements about future revenue collection. These techniques are often referred to as judgmental or nonextrapolative approaches. In addition to their relatively small dependence on numbers, these techniques frequently do not provide a rigorous specification of underlying assumptions.

a. Judgmental Forecasting

Among the most commonly used methods of forecasting is judgmental forecasting . This technique involves having an individual or small group of people make assessments of likely future conditions. While sounding ad hoc, the technique can produce very good estimates, especially when experienced persons are involved. The forecaster will utilize experience in conjunction with consideration of historical trends, current economic conditions, and other factors relevant to the revenue source.

Judgmental approaches tend to work best when background conditions are changing rapidly. When economic, political or administrative conditions are in flux, quantitative methods may not capture important information about factors that are likely to alter historical patterns.

A variation of the judgmental approach is consensus forecasting . Here, experts familiar with factors affecting a particular type of revenue meet to discuss near-term conditions in order to reach agreement about what is likely to happen to revenue collections. For example, municipal public administrators might meet with persons familiar with the local real estate market, economists monitoring local, state and national conditions, and representatives of local financial institutions to come up with a consensus forecast of future building permit applications. Consensus forecasting tends to work best when there is little historical information to draw upon that might be used with a quantitative forecasting method.

Judgmental forecasting approaches certainly have their place among forecasting methods. To some extent, a judgmental perspective needs to supplement any forecasting technique, even the most quantitatively rigorous methods. As might be suspected, however, judgmental approaches can be subject to bias and other sources of error. Guajardo and Miranda (2000) provide the following list of the major weaknesses of qualitative forecasting methods:

anchoring events – allowing recent events to influence perceptions about future events, e. g. the city hosting a recent major convention influencing perceptions about future room taxes

information availability – over-weighting the use of readily available information

false correlation – forecasters incorporating information about factors that are assumed to influence revenues, but do not

inconsistency in methods and judgements – forecasters using different strategies over time to make their judgements, making them less reliable

selective perceptions – ignoring important information that conflicts with the forecaster’s view about causal relationships

wishful thinking – giving undue weight to what forecasters and government officials would like to see happen

group think – when the dynamics of forming a consensus tends to lead individuals to reinforce each other’s views rather than maintaining independent judgements

political pressure – where forecasters adjust estimates to meet the imperatives of budgetary constraints or balanced budgets.

Ii. Quantitative Forecasting Methods

Quantitative methods relay on numerical data relevant to the revenue source. Quantitative methods also make explicit the assumptions and procedures used to generate forecasts. Finally, quantitative methods will also generally assign a margin of error to forecasts, providing a indication of the degree of uncertainty associated with the estimates.

There are two general types of quantitative forecasting methods. The first is a time series approach that consists of a large number of techniques that generally use past trends to project future revenues. The second general approach, while still incorporating time series data, constructs causal models that use the variables assumed to influence the level of a particular revenue.

In general, quantitative methods do a better job of predicting future revenues than do qualitative methods (Cirincione, et al. 1999; Makridakis and Wheelwright, 1989). Simpler quantitative methods also generally perform as well as more complex methods (Makridakis, et al. 1984). Finally, the time series approach typically outperforms the causal modeling approaches, at least in the near-term, given the uncertainty associated with capturing all the relevant economic factors that influence revenue generation (Frank, 1993).

a. Time Series Approaches

Time series approaches are the "bread and butter" of forecasting. They have been used extensively in the private sector and have been subject to substantial evaluation. Today, computer software exists that automatically applies the appropriate technique given the characteristics of the data entered. The underlying assumption of time series techniques is that patterns associated with past values in a data series can be used to project future values.

In using time series techniques, Frank (1993) identifies several essential concepts that need consideration prior to the selection of technique. The first is what constitutes a trend . This fundamentally questions how long a data series is required for the technique to be able to identify any underlying pattern in the data. There are no definitive guidelines as to the number of data points required in constructing a data set. Generally, the data should cover a period of at least several years and, depending on the technique used, should include a minimum of 24 observations and perhaps as many as 50 or more observations.

Cyclicality in time series refers to the extent to which the revenue source is influenced by general business cycles. Again, with local governments moving away from the relatively stable and predictable property tax to sales taxes and user fees, the need to take into consideration the effects of business cycles becomes relatively more important.

Similarly, seasonality is another cyclic phenomenon that needs consideration. This is typically the case when the observations are monthly or quarterly. The mathematical formulas employed can be adjusted to determine both the degree of seasonality that may exist as well as whether seasonality is increasing or decreasing over time.

Randomness is another factor that affects time series data. Randomness refers to unexpected events that may distort trends that otherwise exist over the long-term. Events such as natural disasters, political crisis, and the outbreak of war can result in temporary distortions in trends. Randomness can also result from natural variations around average or typical behavior. When the data series have a constant mean and variance over time, this is known a stationarity . Stationarity exists if the data series were divided into several parts and the independent averages of the means and variances of each part were about equal. If the average of each mean or variance were substantially different, nonstationarity would be suggested. When randomness tends to characterize a data series time series techniques do not perform very well, as performing econometric analyses on nonstationary data can often result in biased estimates.

segundo. Descriptions of Time Series Forecasting Models

There are a large number of time series approaches that are used in forecasting. Cirincione, et al. (1999) discuss a number of issues in their use and provide a nice summary of a variety of techniques in an appendix to their article. This presentation builds on the technical description found there.

1. Naive Forecasting

A naive forecasting model simply assumes the revenue available at time t is the same amount available at time t -1. This is also known as the random walk approach .

where F t is the forecast at time t . and A t -1 is the actual value at time t -1.

A variation of this involves averaging the two prior periods to generate the estimate. Yet another variation involves adjusting for any seasonality that may be present. Naive forecasting is often used when the data series is unpredictable. It is also used in expert forecasting as the starting point for estimates that are then adjusted mentally.

2. Moving Average Models

Moving average models are probably the most commonly used time series approach among local governments. As implied by the name, the future value to be forecast is based on the average of N previous periods. It is a moving average because the oldest data points are dropped off as new ones are added.

where F is the forecast at time t . A t-i is the actual value at time t-i . and N is the number of time periods averaged.

The length of time to include in the average depends on the degree of variation present in the data series. To the extent there appears a high degree of randomness in the data, a longer period is used. Similarly, to the extent cyclicality or seasonality is present in the data, longer time periods are required. An amount of trial and error will be needed to find the best fitting model, although new software can very rapidly identify the time period producing the minimum forecast error. While more complex time series techniques can perform better than the moving average, it does a reasonably good job and is often used as the benchmark against which other methods are compared.

3. Exponential Smoothing Models

The single exponential smoothing model is one of the common forecasting techniques used in the private sector. The model is a moving average of forecasts that have been corrected for the error observed in preceding forecasts. In the first smoothing model, there is assumed no trend or seasonal pattern.

where F t is the forecast at time t . A t-i is the actual value at time t - i . and N is the number of time periods averaged.

The parameter is the smoothing coefficient and has an estimated value between zero and one. It is referred to as an exponential smoothing model because the value of tends to affect past values exponentially. As approaches one, the forecast resembles a short-term moving average, while an closer to zero tends to resemble long-term moving averages. Regardless of the value of , however, exponential smoothing tends to give more recent values higher implicit weights. Again, is typically estimated using trial and error to secure the best fitting model, but software today can rapidly find the model that minimizes forecast error.

The single parameter smoothing model presented above can be adapted to take into account trends that may be present in the data. The form presented here is called the Holt Model. In addition to the smoothing parameter estimated in the exponential smoothing model, a parameter representing the trend is also estimated.

Following the exposition found in Cirincione, et al. (1999), the forecast at time t for k periods into the future equals the level of the series at t plus the product of k and the trend at time t . The level of series is estimated as a function of the actual value of the series at time t . the level of the series at a previous time, and the estimated trend at a previous time. The parameter is a smoothing coefficient. The trend at time t is estimated to be a function of the smoothed value of the change in level between the two time periods and the estimated trend for the previous time period. The values for the smoothing parameters, and , are between zero and one.

where F t+k is the forecast at time k periods in the future, A t is the actual value at time t . S t is the level of the series at time t . T t is the trend at time t, and and are smoothing parameters.

5. Damped Trend Exponential Smoothing

While the Holt Model takes into consideration the trend that may be inherent in the data series, it somewhat unrealistically assumes the trend continues in perpetuity. This means it can overshoot estimates several time periods in the future. A variation known as damped trend exponential smoothing has the effect of dampening the trend as time continues into subsequent periods. It includes a third parameter, , with a value between zero and one that specifies a rate of decay in the trend.

where is the forecast at time k periods in the future, A t is the actual value at time t . S t is the level of the series at time t . T t is the trend at time t . and , . are smoothing parameters.

6. Holt-Winter’s Linear Seasonal Smoothing

This model adapts Holt’s method to include a seasonal component in addition to a smoothing coefficient and a trend parameter. The first variant of the model is additive. This assumes the seasonality is constant over the series being forecast.

where F t+k is the forecast at time k periods in the future, A t is the actual value at time t . S t is the level of the series at time t . T t is the trend at time t . I t is the seasonal index at time t . s is the seasonal index counter, and , , and are smoothing parameters.

The multiplicative variant of this model assumes that the seasonality is changing over the length of the series.

Incorporating seasonality, of course, increases the data requirements – typically three to four years of monthly data. The model is also quite complex, estimating smoothing, trend and seasonal parameters simultaneously. Because of these difficulties, many communities use simpler methods such as single or double exponential smoothing methods.

7. Box-Jenkins ARIMA Models

ARIMA is an acronym for autoregressive integrated moving average. Autoregressive and moving average refer to two of the components of the model, while integrated refers to the process of translating the calculations into a metric that can be interpreted.

ARIMA modeling has three components (Frank, 1993). In the model identification stage . the forecaster must decide whether the time series is autoregressive, moving average, or both. This is usually done by visually inspecting diagrams of the data or employing various statistical techniques. In the second stage, model estimation and diagnostic checks . the forecaster verifies the original model identification is correct. This requires subjecting the model to a variety of diagnostics. If the model checks out, the forecaster then proceeds to the third stage, forecasting .

The principle advantage of using the ARIMA approach is that the method can generate confidence intervals around the forecasts. This actually serves as another check of the validity of the model. If it predicts a high degree of confidence about a dubious forecast, the modeler may have to respecify the form of the model.

In order to achieve best results using the Box-Jenkins ARIMA approach, three assumptions need be met. The first is the generally accepted threshold of 50 data points. This tends to be a significant obstacle for many local governments who may collect data only annually for some types of revenue.

The second assumption is that the data series is stationary, i. e. that the data series varies around a constant mean and variance. Running a regression on two non-stationary variables can result in spurious results. If the data is non-stationary, the data series needs differencing and/or the addition of a time trend. If the data is trend non-stationary only, then adding a linear time trend to the model will render the series stationary. Trend non-stationary data have a mean and variance that change over time by a constant amount. If the data is first-difference non-stationary, then first differencing of the data will render the series stationary. Differencing involves subtracting the observation in time t by the observation in time t-1 for all observations. Whether the data requires these types of treatment should become apparent at the identification stage, and is generally easily accomplished with econometric software programs.

The third assumption of ARIMA models is that the series be homoscedastic . i. e. has a constant variance. If the amplitude of the variance around the mean is great even after differencing, the series is considered heteroscedastic. The remedy for this problem may be simple or complex and involves measures such as using the natural logarithm of the data, using square or cubed roots, or truncating the data series (cutting out certain values).

The first component of the ARIMA process is the autoregressive component. The autoregressive component predicts future values based on a linear combination of prior values. An autoregressive process of order p can be shown as:

where F t is the predicted value at time t . A t-p is the actual value at time t . and p ’s are the estimated parameters.

The moving average component provides forecasts based on prior forecasting errors. The moving average component of a model for a q - order process can be shown as:

where F t is the predicted value at time t . t-q is the forecast error at time t . and the p ‘s are the estimated parameters.

These two components together form autoregressive moving average (ARMA) models. ARMA models assume a stationary data series before first differencing or the inclusion of a time trend. If a series has been rendered trend or difference stationary, the above models form the Box-Jenkins ARIMA model (Box and Jenkins, 1976). The number of autoregressive and moving average lags in an ARIMA model is represented as ARIMA( p, d,q ), where d is the degree of difference, i. e. d =1 if the data is first differenced, d =2 if the data is second differenced, etc. If d =0, the ARIMA model is an ARMA (p, q ). Further derivations can also take into account seasonality by considering autoregressive or moving average trends that occur at certain points in time. In the case of seasonality the ARIMA model is expressed as ARIMA( p, d,q )( P, Q ) where P is the number of seasonal autoregressive lags and Q is the number of seasonal moving average lags. Seasonality is a consideration with relatively frequent data, such as weekly, monthly, or possibly quarterly data.

Causal forecasting models generally tend to be among the more complex techniques, having large data requirements and requiring a high degree of statistical skill. These approaches tend to work best for revenues that are heavily influenced by economic factors, such as business license fees, income taxes, and retail sales taxes. Thus, external data representing relevant economic performance indicators are used to predict the level of revenue expected. Some of the common economic information incorporated into these models include local population, income, and price information (Wong ,1995).

The complexity of causal models varies. The simplest type would be a simple linear regression model that might attempt to project revenue as a function of time, for example. Multiple regression models incorporate any number of relevant explanatory variables, including important policy variables as binary dummy variables. Binary variables take the value of one if a specific time period is represented and a value of zero otherwise. To illustrate, following Cirincione et al. (1999), four common regression models employing ordinary least squares can be show as:

where F t is the predicted value at time t . T t is the value of the time at time t . T t 2 is the squared value of time at time t . is the linear trend parameter associated with time, is the quadratic trend parameter associated with time squared, D s is a binary dummy variable for each of the s seasons, and is the parameter value associated with each season. The estimated values on the dummy variables reveal the average level of the dependent variable during the designated time period. Testing the equality of the dummy coefficients can reveal whether there are significant differences in the average level of the dependent variable across seasonal periods.

Econometric forecasts are structured similar to regression equations, but can include estimates of change across multiple dimensions. Thus, complex events and relationships can be modeled where the output from one equation is fed into another equation as they are solved simultaneously. The types of revenue for which econometric forecasts are most useful include corporate tax, personal income tax, real estate tax, sales tax, and user charges and fees, such as building and construction permits.

This brief overview of revenue forecasting belies the fact that forecasting is a major field of economics. The intricacies and variations can not be represented thoroughly in a brief section. Yet, for those concerned with public finance, the topic is one of growing importance, especially at the municipal level. While some of these techniques are likely beyond the capability of local government managers, improvements in computer software and assistance available from universities and outreach providers increases the plausibility of using these tools even in smaller units of government.

Referencias

[1] An, H. Z. and Huang, F. C. (1996). The geometrical ergodicity of nonlinear autoregressive models. Statist. Sinica 6 943–956.

[2] Brockwell, P. J. Liu, J. and Tweedie, R. L. (1992). On the existence of stationary threshold autoregressive moving-average processes. J. Time Ser. Anal. 13 95–107.

[3] Chan, K. S. Petruccelli, J. D. Tong, H. and Woolford, S. W. (1985). A multiple-threshold AR(1) model. J. Appl. Probab. 22 267–279.

[4] Chan, K. S. and Tong, H. (1985). On the use of the deterministic Lyapunov function for the ergodicity of stochastic difference equations. Adv. in Appl. Probab. 17 666–678.

[5] Chen, R. and Tsay, R. S. (1991). On the ergodicity of TAR(1) processes. Ann. Appl. Probab. 1 613–634.

[6] Cline, D. B.H. and Pu, H. m.H. (1999). Geometric ergodicity of nonlinear time series. Statist. Sinica 9 1103–1118.

[7] Cline, D. B.H. and Pu, H. M.H. (2004). Stability and the Lyapounov exponent of threshold AR-ARCH models. Ann. Appl. Probab. 14 1920–1949.

[8] Ling, S. (1999). On the probabilistic properties of a double threshold ARMA conditional heteroskedastic model. J. Appl. Probab. 36 688–705.

[9] Ling, S. Tong, H. and Li, D. (2007). Ergodicity and invertibility of threshold moving-average models. Bernoulli 13 161–168.

[10] Liu, J. and Susko, E. (1992). On strict stationarity and ergodicity of a nonlinear ARMA model. J. Appl. Probab. 29 363–373.

[11] Lu, Z. (1998). On the geometric ergodicity of a non-linear autoregressive model with an autoregressive conditional heteroscedastic term. Statist. Sinica 8 1205–1217.

Mathematical Reviews (MathSciNet): MR1666249

[12] Robinson, P. M. (1977). The estimation of a nonlinear moving average model. Stochastic Process. Appl. 5 81–90.

[13] Slutsky, E. (1927). The summation of random causes as the source of cyclic processes. Voprosy Koniunktury 3 34–64. English translation in Econometrika 5 105–146..

[14] Tong, H. (1978). On a thresold model. In Pattern Recognition and Signal Processing (C. H. Chen, ed.) 575–586. Amsterdam: Sijthoff and Noordhoff.

[15] Tong, H. (1990). Nonlinear Time Series . A Dynamical System Approach. Oxford Statistical Science Series 6 . New York: Oxford Univ. Press.

Mathematical Reviews (MathSciNet): MR1079320

[16] Tong, H. (2011). Threshold models in time series analysis – 30 years on. Stat. Interface 4 107–118.

Mathematical Reviews (MathSciNet): MR2812802 Zentralblatt MATH: 05983884

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definition - Autoregressive moving average model

Autoregressive–moving-average model

For other uses of ARMA, see Arma .

In statistics and signal processing. autoregressive–moving-average ( ARMA ) models . sometimes called Box–Jenkins models after the iterative Box–Jenkins methodology usually used to estimate them, are typically applied to autocorrelated time series data.

Contenido

Autoregressive model

The notation AR( p ) refers to the autoregressive model of order p . The AR( p ) model is written

An autoregressive model is essentially an all-pole infinite impulse response filter with some additional interpretation placed on it.

Some constraints are necessary on the values of the parameters of this model in order that the model remains stationary. For example, processes in the AR(1) model with | φ 1 | ≥ 1 are not stationary.

Moving-average model

The notation MA( q ) refers to the moving average model of order q :

Autoregressive–moving-average model

The notation ARMA( p . q ) refers to the model with p autoregressive terms and q moving-average terms. This model contains the AR( p ) and MA( q ) models,

Note about the error terms

Specification in terms of lag operator

In some texts the models will be specified in terms of the lag operator L . In these terms then the AR( p ) model is given by

The MA( q ) model is given by

where θ represents the polynomial

Finally, the combined ARMA( p . q ) model is given by

or more concisely,

Alternative notation

Some authors, including Box, Jenkins & Reinsel [ 1 ] use a different convention for the autoregression coefficients. This allows all the polynomials involving the lag operator to appear in a similar form throughout. Thus the ARMA model would be written as

Fitting models

ARMA models in general can, after choosing p and q, be fitted by least squares regression to find the values of the parameters which minimize the error term. It is generally considered good practice to find the smallest values of p and q which provide an acceptable fit to the data. For a pure AR model the Yule-Walker equations may be used to provide a fit.

Finding appropriate values of p and q in the ARMA( p , q ) model can be facilitated by plotting the partial autocorrelation functions for an estimate of p . and likewise using the autocorrelation functions for an estimate of q . Further information can be gleaned by considering the same functions for the residuals of a model fitted with an initial selection of p and q .

Brockwell and Davis [ 2 ] (p.273) recommend using AICc for finding p and q .

Implementations in statistics packages

In R. the tseries package includes an arma function. The function is documented in "Fit ARMA Models to Time Series". or use stats::arima

Mathematica has a complete library of time series functions including ARMA [ 3 ]

MATLAB includes a function ar to estimate AR models, see here for more details .

IMSL Numerical Libraries are libraries of numerical analysis functionality including ARMA and ARIMA procedures implemented in standard programming languages like C, Java, C#.NET, and Fortran.

gretl can also estimate ARMA models, see here where it's mentioned .

GNU Octave can estimate AR models using functions from the extra package octave-forge .

Stata includes the function arima which can estimate ARMA and ARIMA models. see here for more details

SuanShu is a Java library of numerical methods, including comprehensive statistics packages, in which univariate/multivariate ARMA, ARIMA, ARMAX, etc. models are implemented in an object-oriented approach. These implementations are documented in "SuanShu, a Java numerical and statistical library" .

SAS has a econometric package, ETS, that estimates ARIMA models see here for more details .

Aplicaciones

ARMA is appropriate when a system is a function of a series of unobserved shocks (the MA part) [ clarification needed ] as well as its own behavior. For example, stock prices may be shocked by fundamental information as well as exhibiting technical trending and mean-reversion effects due to market participants.

Generalizations

Autoregressive–moving-average models can be generalized in other ways. See also autoregressive conditional heteroskedasticity (ARCH) models and autoregressive integrated moving average (ARIMA) models. If multiple time series are to be fitted then a vector ARIMA (or VARIMA) model may be fitted. If the time-series in question exhibits long memory then fractional ARIMA (FARIMA, sometimes called ARFIMA) modelling may be appropriate: see Autoregressive fractionally integrated moving average. If the data is thought to contain seasonal effects, it may be modeled by a SARIMA (seasonal ARIMA) or a periodic ARMA model.

Another generalization is the multiscale autoregressive (MAR) model. A MAR model is indexed by the nodes of a tree, whereas a standard (discrete time) autoregressive model is indexed by integers.

Note that the ARMA model is a univariate model. Extensions for the multivariate case are the Vector Autoregression (VAR) and Vector Autoregression Moving-Average (VARMA).

Autoregressive–moving-average model with exogenous inputs model (ARMAX model)

The notation ARMAX( p . q . b ) refers to the model with p autoregressive terms, q moving average terms and b exogenous inputs terms. This model contains the AR( p ) and MA( q ) models and a linear combination of the last b terms of a known and external time series . It is given by:

Some nonlinear variants of models with exogenous variables have been defined: see for example Nonlinear autoregressive exogenous model .

Statistical packages implement the ARMAX model through the use of "exogenous" or "independent" variables. Care must be taken when interpreting the output of those packages, because the estimated parameters usually (for example, in R [ 4 ] and gretl ) refer to the regression:

where m t incorporates all exogenous (or independent) variables:

Ver también

This article includes a list of references. but its sources remain unclear because it has insufficient inline citations . Please help to improve this article by introducing more precise citations. (August 2010)

Referencias

^ George Box. Gwilym M. Jenkins. and Gregory C. Reinsel. Time Series Analysis: Forecasting and Control . third edition. Prentice-Hall, 1994.

^ Brockwell, P. J. and Davis, R. A. Time Series: Theory and Methods . 2nd ed. Springer, 2009.

^ Time series features in Mathematica

^ ARIMA Modelling of Time Series. R documentation

Mills, Terence C. Time Series Techniques for Economists. Cambridge University Press, 1990.

Percival, Donald B. and Andrew T. Walden. Spectral Analysis for Physical Applications. Cambridge University Press, 1993.

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer )

A Moving Average Bayesian Model for Spatial Surface and Coverage Prediction from Environmental Point-Source Data

This paper addresses the general problem of estimating at arbitrary locations the value of an unobserved quantity that varies over space, such as ozone concentration in air or nitrate concentrations in surface groundwater, on the basis of approximate measurements of the quantity and perhaps of associated covariates at specificied locations. A nonparametric Bayesian approach is proposed, in which a joint prior distribution for the unobserved spatially-varying quantity is constructed as a moving average of independent-increment random measures. A reversible jump Markov chain Monte Carlo computational approach is proposed for approximating the posterior distribution of the unobserved quantity at all spatial locations, as well as averages of the quantity over arbitrary regions and other summaries of interest. The moving average Bayesian approach is compared with more conventional nitrate concentrations in groundwater. The surfaces and coverages are intended for use as part of the Regional Vulnerability Assessment (ReVA) program in the mid-Atlantic region.

Wolpert, R. L. E. R. Smith, and M. O'connell. A Moving Average Bayesian Model for Spatial Surface and Coverage Prediction from Environmental Point-Source Data. Presented at US EPA 23rd Annual National Conference on Managing Environmental Quality Systems, Tampa, FL, April 13-16, 2004.

7.3.7 Exponentially Weighted Moving Average (EWMA)

7.3.7 Exponentially Weighted Moving Average

T o reconcile the assumptions of uniformly weighted moving average (UWMA) estimation with the realities of market heteroskedasticity, we might apply estimator [7.10 ] to only the most recent historical data t q . which should be most reflective of current market conditions. Doing so is self-defeating, as applying estimator [7.10 ] to a small amount of data will increase its standard error. Consequently, UWMA entails a quandary: applying it to a lot of data is bad, but so is applying it to a little data.

This motivated Zangari (1994 ) to propose a modification of UWMA called exponentially weighted moving average (EWMA) estimation.[1] This applies a nonuniform weighting to time series data, so that a lot of data can be used, but recent data is weighted more heavily. As the name suggests, weights are based upon the exponential function. Exponentially weighted moving average estimation replaces estimator [7.10 ] with

where decay factor λ is generally assigned a value between .95 and .99. Lower decay factors tend to weight recent data more heavily. Tenga en cuenta que

but the weights do not sum to 1 for finite . To remedy this, we may modify estimator [7.18] as

Exponentially weighted moving average estimation is widely used, but it is a modest improvement over UWMA. It does not attempt to model market conditional heteroskedasticity any more than UWMA does. Its weighting scheme replaces the quandary of how much data to use with a similar quandary as to how aggressive a decay factor λ to use.

Consider again Exhibit 7.6 and our example of the USD 10MM position is SGD. Let’s estimate 1|0 σ 1 using exponentially weighted moving average estimator [7.20]. If we use λ = .99, we obtain an estimate for 1|0 σ 1 of .0054. If we use λ = .95, we obtain an estimate of .0067. These correspond to position value-at-risk results of USD 89,000 and USD 110,000, respectively.

Exercises

Exhibit 7.7 indicates 30 days of data for 1-month CHF Libor.

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Stata 11 moving average

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