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Convert ARMA model to AR model

`ar = arma2ar(ar0,ma0)`

`ar = arma2ar(ar0,ma0,numLags)`

returns
the coefficients of the truncated, infinite-order AR model approximation
to an ARMA model having AR and MA coefficients specified by `ar`

= arma2ar(`ar0`

,`ma0`

)`ar0`

and `ma0`

,
respectively.

`arma2ar:`

Accepts:

Vectors or cell vectors of matrices in difference-equation notation.

`LagOp`

lag operator polynomials corresponding to the AR and MA polynomials in lag operator notation.

Accommodates time series models that are univariate or multivariate (i.e.,

`numVars`

variables compose the model), stationary or integrated, structural or in reduced form, and invertible.Assumes that the model constant

*c*is 0.

To accommodate structural ARMA models, specify the input arguments

`ar0`

and`ma0`

as`LagOp`

lag operator polynomials.To access the cell vector of the lag operator polynomial coefficients of the output argument

`ar`

, enter`toCellArray(ar)`

.To convert the model coefficients of the output argument from lag operator notation to the model coefficients in difference-equation notation, enter

arDEN = toCellArray(reflect(ar));

`arDEN`

is a cell vector containing at most`numLags`

+ 1 coefficients corresponding to the lag terms in`ar.Lags`

of the AR model equivalent of the input ARMA model in difference-equation notation. The first element is the coefficient of*y*, the second element is the coefficient of_{t}*y*_{t–1}, and so on.

The software computes the infinite-lag polynomial of the resulting AR model according to this equation in lag operator notation:

where $$\Phi (L)={\displaystyle \sum _{j=0}^{p}{\Phi}_{j}}{L}^{j}$$ and $$\Theta (L)={\displaystyle \sum _{k=0}^{q}{\Theta}_{k}}{L}^{k}.$$$${\Theta}^{-1}(L)\Phi (L){y}_{t}={\epsilon}_{t},$$

`arma2ar`

approximates the AR model coefficients whether`ar0`

and`ma0`

compose a stable polynomial (a polynomial that is stationary or invertible). To check for stability, use`isStable`

.`isStable`

requires a`LagOp`

lag operator polynomial as input. For example, if`ar0`

is a vector, enter the following code to check`ar0`

for stationarity.ar0LagOp = LagOp([1 -ar0]); isStable(ar0LagOp)

A

`0`

indicates that the polynomial is not stable.You can similarly check whether the AR approximation to the ARMA model (

`ar`

) is stationary.

[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.

[2] Hamilton, J. D. *Time Series Analysis*.
Princeton, NJ: Princeton University Press, 1994.

[3] Lutkepohl, H. *New Introduction to Multiple
Time Series Analysis.* Springer-Verlag, 2007.

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