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The MODEL Procedure |

In general, the hypothesis tested can be written as

where is a vector valued function of
the parameters given by the *r* expressions
specified on the TEST statement.

Let be the estimate of the covariance matrix of .Let be the unconstrained estimate of and be the constrained estimate of such that .Let

The Wald test statistic is defined as

The Wald test is not invariant to reparameterization of the model (Gregory 1985, Gallant 1987, p. 219). For more information on the theoretical properties of the Wald test see Phillips and Park 1988.

The Lagrange multiplier test statistic is

where is the vector of Lagrange multipliers from the computation of the restricted estimate .

The Lagrange multiplier test statistic is equivalent to Rao's efficient score test statistic:

where *L* is the log likelihood function for the estimation method used.
For OLS and SUR the Lagrange multiplier test statistic is computed as:

The likelihood ratio test statistic is

where represents the constrained estimate of
and *L* is the concentrated log likelihood value.

For OLS and SUR, the likelihood ratio test statistic is computed as:

where *df* is the difference in degrees of freedom for the full and restricted
models, and *nparms* is the number of parameters in the full system.

The Likelihood ratio test is not appropriate for models with nonstationary serially correlated errors (Gallant 1987, p. 139). The likelihood ratio test should not be used for dynamic systems, for systems with lagged dependent variables, or with the FIML estimation method unless certain conditions are met (see Gallant 1987, p. 479).

For each kind of test, under the null hypothesis the test statistic is
asymptotically distributed as a random variable with
*r* degrees of freedom, where *r* is the number of expressions on
the TEST statement.
The *p*-values reported for the tests are computed from the
distribution
and are only asymptotically valid.

Monte Carlo simulations suggest that the asymptotic distribution of the Wald test is a poorer approximation to its small sample distribution than the other two tests. However, the Wald test has the least computational cost, since it does not require computation of the constrained estimate .

The following is an example of using the TEST statement to perform a likelihood ratio test for a compound hypothesis.

test a*exp(-k) = 1-k, d = 0 ,/ lr;

It is important to keep in mind that although
individual *t* tests for each parameter are printed
by default into the parameter estimates table,
they are only asymptotically valid for nonlinear models.
You should be cautious in drawing any inferences from these *t*
tests for small samples.

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