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Wald test of model specification

`h = waldtest(r,R,EstCov)`

`h = waldtest(r,R,EstCov,alpha)`

```
[h,pValue]
= waldtest(___)
```

```
[h,pValue,stat,cValue]
= waldtest(___)
```

returns
a logical value (`h`

= waldtest(`r`

,`R`

,`EstCov`

)`h`

) with the rejection decision
from conducting a Wald
test of model specification.

`waldtest`

constructs the test statistic using
the restriction function and its Jacobian, and the value of the unrestricted
model covariance estimator, all evaluated at the unrestricted parameter
estimates (`r`

, `R`

, and `EstCov`

,
respectively).

If any input argument is a cell vector of length

> 1, then the other input arguments must be cell arrays of length*k*.*k*`waldtest`

(`r`

,`R`

,`EstCov`

) treats each cell as a separate, independent test, and returns a vector of rejection decisions.If any input argument is a row vector, then the software returns output arguments as row vectors.

Estimate unrestricted univariate linear time series models, such as

`arima`

or`garch`

, or time series regression models (`regARIMA`

) using`estimate`

. Estimate unrestricted multivariate linear time series models using`vgxvarx`

.`estimate`

and`vgxvarx`

return parameter estimates and their covariance estimates, which you can process and use as inputs to`waldtest`

.If you cannot easily compute restricted parameter estimates, then use

`waldtest`

. By comparison:`lratiotest`

requires both restricted and unrestricted parameter estimates.`lmtest`

requires restricted parameter estimates.

`waldtest`

performs multiple, independent tests when the restriction function vector, its Jacobian, and the unrestricted model parameter covariance matrix (`r`

,`R`

, and`EstCov`

, respectively) are equal-length cell vectors.If

`EstCov`

is the same for all tests, but`r`

varies, then`waldtest`

"tests down" against multiple restricted models.If

`EstCov`

varies among tests, but`r`

does not, then`waldtest`

"tests up" against multiple unrestricted models.Otherwise,

`waldtest`

compares model specifications pair-wise.

`alpha`

is nominal in that it specifies a rejection probability in the asymptotic distribution. The actual rejection probability is generally greater than the nominal significance.The Wald test rejection error is generally greater than the likelihood ratio and Lagrange multiplier test rejection errors.

[1] Davidson, R. and J. G. MacKinnon. *Econometric
Theory and Methods*. Oxford, UK: Oxford University Press,
2004.

[2] Godfrey, L. G. *Misspecification Tests in Econometrics*.
Cambridge, UK: Cambridge University Press, 1997.

[3] Greene, W. H. *Econometric Analysis*.
6th ed. Upper Saddle River, NJ: Pearson Prentice Hall, 2008.

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

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