mletestingslides - Hypothesis Testing in a Likelihood...

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Hypothesis Testing in a Likelihood Framework Eric Zivot November 18, 2009
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Wald Statistic The Wald statistic is based directly on the asymptotic normal distribution of ˆ θ mle : ˆ θ mle A N ³ θ, ˆ I ( ˆ θ mle | x ) 1 ´ An implication of the asymptotic normality result is that the usual t -ratio for testing H 0 : θ = θ 0 t = ˆ θ mle θ 0 d SE ( ˆ θ mle ) = ˆ θ mle θ 0 q ˆ I ( ˆ θ mle | x ) 1 = ³ ˆ θ mle θ 0 ´ q ˆ I ( ˆ θ mle | x ) A N (0 , 1)
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Using the continuous mapping theorem, it follows that Wald = ³ ˆ θ mle θ 0 ´ 2 ˆ I ( ˆ θ mle | x ) 1 = ³ ˆ θ mle θ 0 ´ 2 ˆ I ( ˆ θ mle | x ) A χ 2 (1) Note: if the curvature of ln L ( θ | x ) near θ = ˆ θ mle is big (high information) then the squared distance ³ ˆ θ mle θ 0 ´ 2 gets blown up when constructing the Wald statistic. If the curvature of ln L ( θ | x ) near θ = ˆ θ mle is low, then I ( ˆ θ mle | x ) is small and the squared distance ³ ˆ θ mle θ 0 ´ 2 gets attenuated when constucting the Wald statistic.
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