# pyblp.ProblemResults.run_wald_test¶

ProblemResults.run_wald_test(restrictions, restrictions_jacobian)

Test the validity of model restrictions with the Wald test.

Following Newey and West (1987), the Wald statistic is

(1)$\text{Wald} = Nr(\hat{\theta})'[R(\hat{\theta})VR(\hat{\theta})']^{-1}r(\hat{\theta})$

where the restrictions are $$r(\theta) = 0$$ under the test’s null hypothesis, their Jacobian is $$R(\theta) = \frac{\partial r(\theta)}{\partial\theta}$$, and $$V$$ is the covariance matrix of $$\sqrt{N}(\hat{\theta} - \theta)$$ in (30).

If the restrictions are valid, the Wald statistic is asymptotically $$\chi^2$$ with degrees of freedom equal to the number of restrictions.

Parameters
• restrictions (array-like) – Column vector of the model restrictions evaluated at the estimated parameters, $$r(\hat{\theta})$$.

• restrictions_jacobian (array-like) – Estimated Jacobian of the restrictions with respect to all parameters, $$R(\hat{\theta})$$. This matrix should have as many rows as restrictions and as many columns as ProblemResults.parameter_covariances.

Returns

The Wald statistic.

Return type

float

Examples