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