CQLIM Procedure
Tests on Parameters
In general, you can write the tested hypothesis as
where 
is an 
vector-valued function of the parameters 
that is given by the r expressions that you specify in the TEST statement.
Let 
be the estimate of the covariance matrix of 
. Let be the unconstrained estimate of , and let 
be the constrained estimate of such that 
. Let
Using this notation, you compute the test statistics for the three types of tests as follows:
-
The Wald test statistic is defined as
The Wald test is not invariant to reparameterization of the model (Gregory and Veall 1985; Gallant 1987, p. 219). For more information about 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 likelihood ratio test statistic is
where
represents the constrained estimate of and 
is the concentrated log-likelihood value.
For each type 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 in the TEST statement. The p-values that are 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 of its small sample distribution than that of the other two tests. However, the Wald test has the lowest computational cost, because it does not require computation of the constrained estimate .
The following statements use the TEST statement to perform a likelihood ratio test:
proc cqlim;
model y = x1 x2 x3;
test x1 = 0, x2 * .5 + 2 * x3 = 0 /lr;
run;
For more information, see SAS/ETS User's Guide.