GARCH Toolbox

garchma

Convert finite-order ARMA models to infinite-order moving average (MA) models

Syntax

• InfiniteMA = garchma(AR, MA, NumLags)
  

Arguments

AR	R-element vector of auto-regressive coefficients associated with the lagged observations of a univariate return series modeled as a finite order, stationary, invertible ARMA(R,M) model.
MA	M-element vector of moving-average coefficients associated with the lagged innovations of a finite-order, stationary, invertible univariate ARMA(R,M) model.
NumLags	(optional) Number of lagged MA coefficients that garchma includes in the approximation of the infinite-order MA representation. NumLags is an integer scalar and determines the length of the infinite-order MA output vector. If NumLags = [] or is not specified, the default is 10.

Description

InfiniteMA = garchma(AR, MA, NumLags) computes the coefficients of an infinite-order MA model, using the coefficients of the equivalent univariate, stationary, invertible finite-order ARMA(R,M) model as input. garchma truncates the infinite-order MA coefficients to accommodate the number of lagged MA coefficients you specify in NumLags.

This function is particularly useful for calculating the standard errors of minimum mean square error forecasts of univariate ARMA models.

InfiniteMA

Vector of coefficients of the infinite-order MA representation associated with the finite-order ARMA model specified by AR and MA. InfiniteMA is a vector of length NumLags. The jth element of InfiniteMA is the coefficient of the jth lag of the innovations noise sequence in an infinite-order MA representation. Note that Box, Jenkins, and Reinsel refer to the infinite-order MA coefficients as the " weights."

In the following ARMA(R,M) model, {y_t} is the return series of interest and {_t} the innovations noise process.

If you write this model equation as

you can specify the garchar input coefficient vectors, AR and MA, exactly as you read them from the model. In general, the jth elements of AR and MA are the coefficients of the jth lag of the return series and innovations processes y_{t - j} and _{t - j}, respectively. garchma assumes that the current-time-index coefficients of y_t and _t are 1 and are not part of AR and MA.

In theory, you can use the weights returned in InfiniteMA to approximate y_t as a pure MA process.

Consistently, the jth element of the truncated infinite-order moving-average output vector, _j or InfiniteMA(j), is the coefficient of the jth lag of the innovations process, _{t - j}, in this equation. See Box, Jenkins, and Reinsel [7], Section 5.2.2, pages 139-141.

Given the above discussion, the AR and MA vectors differ from the corresponding AR and MA polynomials formally presented in time series references such as Box, Jenkins, and Reinsel. The conversion from GARCH Toolbox vectors to the corresponding GARCH Toolbox polynomials is as follows:

• AR polynomial tested for stationarity = [1 ; -AR]
• MA polynomial tested for invertibility = [1 ; MA]

Example

Suppose you want a forecast horizon of 10 periods for the following ARMA(2,2) model.

To obtain probability limits for these forecasts, use garchma to compute the first 9 (i.e., 10 - 1) weights of the infinite order MA approximation.

From the model, AR = [0.5 -0.8] and MA = [-0.6 0.08].

Since the current-time-index coefficients of y_t and _t are 1, the example omits them from AR and MA. This saves time and effort when you specify parameters via the garchset and garchget user interfaces.

• PSI = garchma([0.5 -0.8], [-0.6 0.08], 9);
  PSI'
  
  ans =
  
     -0.1000
     -0.7700
     -0.3050
      0.4635
      0.4758
     -0.1329
     -0.4471
     -0.1172
      0.2991
  

See Also
garchar, garchpred

Reference

[1]  Box, G.E.P., G.M. Jenkins, G.C. Reinsel, Time Series Analysis: Forecasting and Control, third edition, Prentice Hall, 1994.

garchllfn

garchplot