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Convert ARMA model to AR model
InfiniteAR = garchar(AR,MA,NumLags)
InfiniteAR = garchar(AR,MA,NumLags) computes the coefficients of an infinite-order AR model, using the coefficients of the equivalent univariate, stationary, invertible, finite-order ARMA(R,M) model as input. garchar truncates the infinite-order AR coefficients to accommodate a user-specified number of lagged AR coefficients.
R-element vector of autoregressive coefficients associated with the lagged observations of a univariate return series modeled as a finite-order, stationary, invertible ARMA(R,M) model. | |
M-element vector of moving-average coefficients associated with the lagged innovations of a finite-order, stationary, invertible univariate ARMA(R,M) model. | |
(optional) Number of lagged AR coefficients that garchar includes in the approximation of the infinite-order AR representation. NumLags is an integer scalar and determines the length of the infinite-order AR output vector. If NumLags = [] or is unspecified, the default is 10. |
Vector of coefficients of the infinite-order AR representation associated with the finite-order ARMA model specified by the AR and MA input vectors. InfiniteAR is a vector of length NumLags. The jth element of InfiniteAR is the coefficient of the jth lag of the input series in an infinite-order AR representation. Box, Jenkins, and Reinsel refer to the infinite-order AR coefficients as "π weights." |
In the following ARMA(R,M) model, {y_{t}} is the return series of interest and {ε_{t}} the innovations noise process.
$${y}_{t}={\displaystyle \sum _{i=1}^{R}{\varphi}_{i}}{y}_{t-1}+{\epsilon}_{t}{\displaystyle \sum _{j=1}^{M}{\theta}_{j}}{\epsilon}_{j-1}$$
If you write this model equation as
$${y}_{t}={\varphi}_{1}{y}_{t-1}+\mathrm{...}+{\varphi}_{R}{y}_{t-R}+{\epsilon}_{t}+{\theta}_{1}{\epsilon}_{t-1}+\mathrm{...}+{\theta}_{M}{\epsilon}_{t-M}$$
you can specify the garchar input coefficient vectors, AR and MA, 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 ε_{t-j}, respectively. garchar assumes that the current-time-index coefficients of y_{t} and ε_{t} are 1 and are not part of AR and MA.
In theory, you can use the π weights returned in InfiniteAR to approximatey_{t} as a pure AR process.
$${y}_{t}={\displaystyle \sum _{i=1}^{\infty}{\pi}_{i}}{y}_{t-i}+{\epsilon}_{t}$$
In this equation, the jth element of the truncated infinite-order autoregressive output vector,π_{j} or InfiniteAR(j), is consistently the coefficient of the jth lag of the observed return series, y_{t-j}. See Box, Jenkins, and Reinsel [15], Section 4.2.3, pages 106-109.
For the following ARMA(2,2) model, use garchar to obtain the first 20 weights of the infinite-order AR approximation.
$${y}_{t}=0.5{y}_{t-1}-0.8{y}_{t-2}+{\epsilon}_{t}-0.6{\epsilon}_{t-1}+0.08{\epsilon}_{t-2}$$
From this model,
AR = [0.5 -0.8]; MA = [-0.6 0.08]; lagLength = 20;
Since the current-time-index coefficients of y_{t} and ε_{t} are 1, the example omits them from AR and MA. This saves time and effort when you specify parameters using dot notation on a garch model.
PI = garchar(AR,MA,lagLength)'
PI = -0.1000 -0.7800 -0.4600 -0.2136 -0.0914 -0.0377 -0.0153 -0.0062 -0.0025 -0.0010 -0.0004 -0.0002 -0.0001 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000
[1] Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.