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Convert ARMA model to MA model

`ma = arma2ma(ar0,ma0)`

`ma = arma2ma(ar0,ma0,numLags)`

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

= arma2ma(`ar0`

,`ma0`

)`ar0`

and `ma0`

,
respectively.

`arma2ma:`

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.

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

$${y}_{t}={\Phi}^{-1}(L)\Theta (L){\epsilon}_{t}$$

where $$\Phi (L)={\displaystyle \sum _{j=0}^{p}{\Phi}_{j}}{L}^{j}$$ and $$\Theta (L)={\displaystyle \sum _{k=0}^{q}{\Theta}_{k}}{L}^{k}.$$

`arma2ma`

approximates the MA 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 MA approximation to the ARMA model (

`ma`

) is invertible.

[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.