Specification tests

Summary

Setup

A random sample \(\left( y_i \right)\) with density/distribution function \(f\left( y_i;θ \right)\) or a random sample \(\left( y_i,x_i \right)\) with conditional density/distribution function \(f\left( y_i|x_i;θ \right)\) where \(y_i\) is a scalar, \(x_i\) is \(k×1\) , and \(θ\) is a \(p×1\) vector of unknown parameters.
Individual log-likelihood functions, \(l_i\left( θ \right)=f\left( y_i;θ \right)\) or \(l_i\left( θ \right)=f\left( y_i|x_i;θ \right)\)
A log-likelihood function \(l\left( θ \right)=∑l_i\left( θ \right)\)
Individual score vectors \(s_i\left( θ \right)=∂l_i/∂θ\) and the score vector \(s\left( θ \right)=∑s_i\left( θ \right)\)
A maximum likelihood estimator \(\hat{θ}\)
Information matrix \(I\left( θ \right)=-E\left( \frac{∂^2l_i\left( θ \right)}{∂θ∂θ'} \right)=-E\left( \frac{∂s_i\left( θ \right)}{∂θ'} \right)\)
\(V=I{\left( θ \right)}^{-1}\) is the asymptotic variance matrix of \({\hat{θ}}_{ML}\)

Hypothesis

\[H_0:Rθ=q\]

where \(R\) is a given \(J×p\) matrix and \(q\) is a given \(J×1\) vector. \(J≤p\) and \(R\) has full rank \(J\) .
We define \(\tilde{θ}\) as the constrained MLE, the argument that maximizes the likelihood function under the null hypothesis,

\[\tilde{θ}=\arg max_{θ} l\left( θ \right) s.t. Rθ=q\]

Wald test

\[ξ_W=n{\left( R\hat{θ}-q \right)}'{\left( R\hat{V}R' \right)}^{-1}\left( R\hat{θ}-q \right) \sim χ_J^2\]

where \(\hat{V}\) is a consistent estimate of \(V\) . We reject \(H_0\) if \(ξ_W>χ_{J,α}^2\) .

Likelihood ratio (LR) test

\[ξ_{LR}=2\left( l\left( \hat{θ} \right)-l\left( \tilde{θ} \right) \right) \sim χ_J^2\]

Lagrange multiplier (LM) test

\[ξ_{LM}=n^{-1}s{\left( \tilde{θ} \right)}' {\tilde{I}}^{-1}s\left( \tilde{θ} \right) \sim χ_J^2\]

where \(\tilde{I}\) is any estimate of the information matrix \(I\left( θ \right)\) evaluated at \(\tilde{θ}\) . Typically,

\[\tilde{I}= \frac{1}{n}\sum_{i=1}^{n}{ s_i\left( \tilde{θ} \right)s_i{\left( \tilde{θ} \right)}' }\]