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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.

| |

| |

(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.

If you write this model equation as

you can specify the `garchar` input coefficient
vectors, `AR` and `MA,` as you read
them from the model. In general, the *j*th elements
of `AR` and `MA` are the coefficients
of the *j*th 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 approximate*y*_{t} as
a pure AR process.

In this equation, the *j*th element of the
truncated infinite-order autoregressive output vector,*π*_{j} or `InfiniteAR(j)`,
is consistently the coefficient of the *j*th 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.

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

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.

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