# garchar

Convert ARMA model to AR model

`garchar` will be removed in a future release. Use `arma2ar` instead.

## Syntax

`InfiniteAR = garchar(AR,MA,NumLags)`

## Description

`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.

## Input Arguments

 `AR` 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. `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 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`.

## Output Arguments

 `InfiniteAR` 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, {yt} is the return series of interest and {εt} the innovations noise process.

${y}_{t}=\sum _{i=1}^{R}{\varphi }_{i}{y}_{t-1}+{\epsilon }_{t}\sum _{j=1}^{M}{\theta }_{j}{\epsilon }_{j-1}$

If you write this model equation as

${y}_{t}={\varphi }_{1}{y}_{t-1}+...+{\varphi }_{R}{y}_{t-R}+{\epsilon }_{t}+{\theta }_{1}{\epsilon }_{t-1}+...+{\theta }_{M}{\epsilon }_{t-M}$

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 yt-j and εt-j, respectively. `garchar` assumes that the current-time-index coefficients of yt and εt are `1` and are not part of `AR` and `MA`.

In theory, you can use the π weights returned in `InfiniteAR` to approximateyt as a pure AR process.

${y}_{t}=\sum _{i=1}^{\infty }{\pi }_{i}{y}_{t-i}+{\epsilon }_{t}$

In this equation, the jth element of the truncated infinite-order autoregressive output vector,πj or `InfiniteAR(j)`, is consistently the coefficient of the jth lag of the observed return series, yt-j. See Box, Jenkins, and Reinsel [15], Section 4.2.3, pages 106-109.

## Examples

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 yt and εt 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 ```

## References

[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.