Convert ARMA model to MA model
garchma
will be removed in a future release.
Use arma2ma
instead.
InfiniteMA = garchma(AR,MA,NumLags)
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 useful for calculating the standard errors of minimum mean square error forecasts of univariate ARMA models.
| 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 MA coefficients that |
| Vector of coefficients of the infinite-order MA representation
associated with the finite-order ARMA model specified by |
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 garchma
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. garchma
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 ψ weights
returned in InfiniteMA
to approximate y_{t} as
a pure MA process.
$${y}_{t}={\epsilon}_{t}+{\displaystyle \sum _{i=1}^{\infty}{\psi}_{i}}{\epsilon}_{t-i}$$
The jth element of the truncated infinite-order
moving-average output vector, ψ_{j} or InfiniteMA(j)
,
is consistently the coefficient of the jth lag
of the innovations process, ε_{t-j},
in this equation. See Box, Jenkins, and Reinsel [15], Section 5.2.2, pages 139-141.
Calculate a forecast horizon of 10 periods for the following ARMA(2,2) model:
$${y}_{t}=0.5{y}_{t-1}-0.8{y}_{t-2}+{\epsilon}_{t}-0.6{\epsilon}_{t-1}+0.08{\epsilon}_{t-2}$$
To obtain probability limits for these forecasts, use garchma
to
compute the first 9 (that is, 10 - 1
)
weights of the infinite order MA approximation.
PSI = garchma([0.5 -0.8], [-0.6 0.08], 9)'
Warning: GARCHMA will be removed in a future release. Use ARMA2MA instead. PSI = -0.1000 -0.7700 -0.3050 0.4635 0.4758 -0.1329 -0.4471 -0.1172 0.2991
From the model, AR = [0.5 -0.8]
and MA
= [-0.6 0.08]
.
Note:
Since the current-time-index coefficients of y_{t} and ε_{t} are |
[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.