Hypothesis Testing Daniel Schmierer Econ 312 March 30, 2007
Basics Parameter of interest: θ Θ Structure of the test: H 0 : θ Θ 0 H 1 : θ Θ 1 for some sets Θ 0, Θ 1 Θ where Θ 0 Θ 1 = (often Θ 1 = Θ Θ 0 ).
Basics Type I error occurs when we reject a true null hypothesis. Type II error occurs when we accept a false null hypothesis. Size of a test (α): Pr(Type I error). Power of a test (1 β): 1 Pr(Type II error). A test is better if (all else equal) it has a higher power and/or lower size.
Consistency For a given critical (rejection) region C n at sample size n, the size of the test is and the power is α n = Pr(y C n θ Θ 0 ) π n (θ) = Pr(y C n θ) for θ Θ 1 Size generally doesn t depend on alternative hypothesis, but power does. A test is called consistent if lim n π n (θ) = 1 for all θ Θ 1 (if in the limit, the test always rejects false null hypotheses).
Standard case Maximum Likelihood Objective function is ln L(y; θ) = n i=1 ln f (y i; θ) with θ R k. Let the test be H 0 : θ = θ 0 H 1 : θ θ 0 Note that n(ˆθ θ 0 ) N(0, I 1 θ 0 ) and ( ) n 1 d N(0, I θ0 ) where θ0 n ln L(y;θ) θ d I θ0 = E [ 2 ] ln L θ θ So standardizing and squaring these quantities will give asymptotic χ 2 distributions.
Trinity Wald test statistic: n(ˆθ θ 0 ) Iˆθ n(ˆθ θ0 ) a χ 2 k Rao s ( Score (LM) test statistic: ) ( n 1 I 1 n 1 θ0 n ln L(y;θ) θ ˆθ n ln L(y;θ) θ ) θ0 a χ 2 k Likelihood Ratio test statistic: 2 ln [ ] L(y;θ0 ) a χ 2 L(y;ˆθ) k
Graphical Interpretation
Composite Hypotheses A test has a composite null hypothesis if the null hypothesis contains more than one possible value of θ. In particular, say θ = ( θ 1, θ 2 ), then a common test is (k 1) (l 1) H 0 : θ 1 = θ 0 1, θ 2 unrestricted H 1 : θ unrestricted Can go over an example of this for maximum likelihood.
ML Example Let true parameter vector be θ 0 ( θ 1, θ 2 ) Define estimated parameter vector (unconstrained) as ˆθ (ˆθ 1, ˆθ 2 ) Define estimated parameter vector (constrained, i.e. under the null hypothesis setting θ 1 = θ 0 1 ) as θ (θ 0 1, θ 2 ) Decompose the information matrix into the following blocks ( ) Iθ1 θ I θ0 1 I θ1 θ 2 I θ2 θ 1 I θ2 θ 2
ML Example From asymptotic normality results we know that n(ˆθ1 θ 0 1) d N(0, upper left block of I 1 θ 0 ) By partitioned inverse results, the upper left block of I 1 θ 0 is Also define I 11 θ 0 = I 11 1 θ 0 (k k) = I 1 θ 1 θ 1 (k k) I θ1 θ 2 I 1 θ 2 θ 2 I θ2 θ 1 (k l)(l l)(l k) ( I 11 1 θ 0 ) 1 = ( I 1 θ 1 θ 1 I θ1 θ 2 I 1 θ 2 θ 2 I θ2 θ 1 ) 1
ML Example Wald test statistic: W = n(ˆθ 1 θ 0 1) I 11 θ 0 n(ˆθ1 θ 0 1) Wald uses unrestricted estimate. It checks whether the null hypothesis and the relevant portion of the unrestricted estimate (which is the best choice of parameters under the alternative hypothesis) are very far apart. Intuitively, if the null hypothesis were true, then by the consistency of the ML estimator, the best choice under the alternative hypothesis should be getting close to the null hypothesis.
ML Example Score test statistic: ( ) 1 ln L ( ) LM = n 1 ln L Iθ 11 1 θ 0 n 1 θ 1 θ LM uses the restricted estimate. It checks whether the relevant portion of the score vector at the restricted estimate (which is the best choice of parameters under the null hypothesis) is close to zero. Intuitively, if the null hypothesis were true, then the gradient of the likelihood should be zero in the population at that parameter value and so restricting ourselves to that parameter value should produce a gradient close to zero. θ
ML Example Likelihood ratio test statistic: [ ] L(y; θ) LR = 2 ln L(y; ˆθ) Compares the highest value of the likelihood under the null hypothesis with the highest value of the likelihood under the alternative hypothesis. Intuitively, if the null hypothesis were true (and under our regularity conditions about the uniform convergence of the likelihood function), then the maximum of the likelihood under the null hypothesis and the maximum of the likelihood under the alternative hypothesis should be close.
ML Example All three are asymptotically distributed as χ 2 k. Show asymptotic equivalence using Taylor expansions see Asymptotic Theory Part IV notes.
Good references: Engle, R. (1984), Wald, Likelihood Ratio and Lagrange Multiplier Tests in Econometrics, Handbook of Econometrics, Vol. II, Ch. 13. Newey, W. and McFadden, D. (1994), Large Sample Estimation and Hypothesis Testing, Handbook of Econometrics, Vol. IV, Ch. 36. Both are on Google Scholar.