Convert ARMA model to AR model
garchar
will be removed in a future release.
Use arma2ar
instead.
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 |
| Vector of coefficients of the infinite-order AR representation
associated with the finite-order ARMA model specified by the |
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)'
Warning: GARCHAR will be removed in a future release. Use ARMA2AR instead. 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.