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Lagrange multiplier test of model specification

`h = lmtest(score,ParamCov,dof)`

`h = lmtest(score,ParamCov,dof,alpha)`

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

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

returns
a logical value (`h`

= lmtest(`score`

,`ParamCov`

,`dof`

)`h`

) with the rejection decision
from conducting a Lagrange multiplier test of
model specification at the 5% significance level. `lmtest`

constructs
the test statistic using the score function (`score`

),
the estimated parameter covariance (`ParamCov`

),
and the degrees of freedom (`dof`

).

returns the rejection decision of the Lagrange multiplier test conducted at
significance level `h`

= lmtest(`score`

,`ParamCov`

,`dof`

,`alpha`

)`alpha`

.

If

`score`

and`ParamCov`

are length*k*cell arrays, then all other arguments must be length*k*vectors or scalars.`lmtest`

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

`score`

is a row cell array, then`lmtest`

returns a row vector.

`lmtest`

requires the unrestricted model score and parameter covariance estimator evaluated at parameter estimates for the restricted model. For example, to compare competing, nested`arima`

models:Analytically compute the score and parameter covariance estimator based on the innovation distribution.

Use

`estimate`

to estimate the restricted model parameters.Evaluate the score and covariance estimator at the restricted model estimates.

Pass the evaluated score, restricted covariance estimate, and the number of restrictions (i.e., the degrees of freedom) into

`lmtest`

.

If you find estimating parameters in the unrestricted model difficult, then use

`lmtest`

. By comparison:`waldtest`

only requires unrestricted parameter estimates.`lratiotest`

requires both unrestricted and restricted parameter estimates.

`lmtest`

performs multiple, independent tests when inputs are cell arrays.If the gradients and covariance estimates are the same for all tests, but the restricted parameter estimates vary, then

`lmtest`

“tests down” against multiple restricted models.If the gradients and covariance estimates vary, but the restricted parameter estimates do not, then

`lmtest`

“tests up” against multiple unrestricted models.Otherwise,

`lmtest`

compares model specifications pair-wise.

`alpha`

is nominal in that it specifies a rejection probability in the asymptotic distribution. The actual rejection probability can differ from the nominal significance. Lagrange multiplier tests tend to under-reject for small values of`alpha`

, and over-reject for large values of`alpha`

.Lagrange multiplier tests typically yield lower rejection errors than likelihood ratio and Wald tests.

[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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