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Likelihood ratio test of model specification

`h = lratiotest(uLogL,rLogL,dof)`

`h = lratiotest(uLogL,rLogL,dof,alpha)`

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

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

returns
a logical value (`h`

= lratiotest(`uLogL`

,`rLogL`

,`dof`

)`h`

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

`lratiotest`

constructs the test statistic
using the loglikelihood objective function evaluated at the unrestricted
model parameter estimates (`uLogL`

) and the restricted
model parameter estimates (`rLogL`

). The test statistic
distribution has `dof`

degrees of freedom.

If

`uLogL`

or`rLogL`

is a vector, then the other must be a scalar or vector of equal length.`lratiotest(uLogL,rLogL,dof)`

treats each element of a vector input as a separate test, and returns a vector of rejection decisions.If

`uLogL`

or`rLogL`

is a row vector, then`lratiotest(uLogL,rLogL,dof)`

returns a row vector.

Estimate unrestricted and restricted univariate linear time series models, such as

`arima`

or`garch`

, or time series regression models (`regARIMA`

) using`estimate`

. Estimate unrestricted and restricted VAR models (`varm`

) using`estimate`

.The

`estimate`

functions return loglikelihood maxima, which you can use as inputs to`lratiotest`

.If you can easily compute both restricted and unrestricted parameter estimates, then use

`lratiotest`

. By comparison:`waldtest`

only requires unrestricted parameter estimates.`lmtest`

requires restricted parameter estimates.

`lratiotest`

performs multiple, independent tests when the unrestricted or restricted model loglikelihood maxima (`uLogL`

and`rLogL`

, respectively) is a vector.If

`rLogL`

is a vector and`uLogL`

is a scalar, then`lratiotest`

“tests down” against multiple restricted models.If

`uLogL`

is a vector and`rLogL`

is a scalar, then`lratiotest`

“tests up” against multiple unrestricted models.Otherwise,

`lratiotest`

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.

[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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