MLE #8. Econ 674. Purdue University. Justin L. Tobias (Purdue) MLE #8 1 / 20

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1 MLE #8 Econ 674 Purdue University Justin L. Tobias (Purdue) MLE #8 1 / 20

2 We begin our lecture today by illustrating how the Wald, Score and Likelihood ratio tests are implemented within the context of the Box-Cox regression model: and assume σ 2 is known. We will also write this equation as: y i (λ) = µ + ɛ i. Note that: 1 2 Justin L. Tobias (Purdue) MLE #8 2 / 20

3 Complete on your own: Derive the density of a representative scalar observation y within the Box-Cox model. First, note that the Jacobian of the transformation in this case is not unity, and specifically, Thus, Justin L. Tobias (Purdue) MLE #8 3 / 20

4 Complete on your own: Derive the log-likelihood for the Box-Cox sampling model. By the assumed independence across observations, we have: Thus, (dropping irrelevant constants from the expression) Justin L. Tobias (Purdue) MLE #8 4 / 20

5 Complete on your own: Characterize, as best you can, the restricted and unrestricted MLE s for this problem. Note: For the restricted model, we impose the null H 0 : λ = c. For the unrestricted case, This implies that Justin L. Tobias (Purdue) MLE #8 5 / 20

6 Likewise, differentiating with respect to λ gives: where Justin L. Tobias (Purdue) MLE #8 6 / 20

7 To obtain the unrestricted MLE, we can then concentrate the likelihood function and find the ˆλ that solves: Once this is done (which is just a scalar optimization problem), we can then calculate: Justin L. Tobias (Purdue) MLE #8 7 / 20

8 As for the restricted model, note that giving so that no optimization is required for the restricted model. Note the computational advantage of the Score test here! Justin L. Tobias (Purdue) MLE #8 8 / 20

9 Complete On Your Own: Describe how to implement the likelihood ratio test for this problem. Note (again, dropping irrelevant constants, as they cancel in the difference): Likewise, Simply form and compare it to critical values from the χ 2 1 table. Justin L. Tobias (Purdue) MLE #8 9 / 20

10 Complete On Your Own: Describe how to implement the Wald test for this problem. To implement this test, we first need to estimate the information matrix from the unrestricted MLE. There are alternate ways to do this. One way is to calculate: Note that these are derivatives from a single observation s contribution to the likelihood function (hence the y i notation) rather than the derivative of the full log-likelihood. Justin L. Tobias (Purdue) MLE #8 10 / 20

11 With this in hand, we can then calculate ˆV, the (2, 2) element of Î 1 (ˆµ, ˆλ) and calculate in order to implement the test. Justin L. Tobias (Purdue) MLE #8 11 / 20

12 Complete On Your Own: Describe how to implement the Score test for this problem. First, in an identical manner to that described within the Wald test, we estimate the information matrix, but this time we evaluate the derivatives at ˆµ r and λ = c. This gives We then calculate the restricted score vector, Note that the first element of the restricted score vector must be zero since this is how ˆµ r is obtained. Justin L. Tobias (Purdue) MLE #8 12 / 20

13 Complete On Your Own: The Score test is completed by calculating which simply picks off the appropriate part of the (inverse) information matrix when calculating the distance from L λ (ˆµ r, c) from zero. In this (and most) cases, the three test statistics will not be identical and can potentially result in different conclusions. Justin L. Tobias (Purdue) MLE #8 13 / 20

14 In a problem set, you are asked to generate data from a Box-Cox regression model: yi λ 1 = µ + ɛ i, ɛ N (0, σ 2 I n ), y > 0 λ with µ = 2, σ 2 = 1 and λ =.4. To do this, first generate ɛ from its Normal distribution, and then solve for y as a function of ɛ and the parameters of the data generation process. Justin L. Tobias (Purdue) MLE #8 14 / 20

15 Generating 1,000 observations in this way, we obtain: ˆµ = 2.039, ˆλ = Furthermore, we obtain the following test statistics for the hypothesis λ =.4 Wald = 1.66, Score = 1.85, LR = so we would fail to reject at, say, the 5 percent level, since the associated critical value is Justin L. Tobias (Purdue) MLE #8 15 / 20

16 We repeat this exercise again (this time with a new data set) and obtain: ˆµ = 2.01, ˆλ =.394. Furthermore, we obtain the following test statistics for the hypothesis λ =.2 Wald = 190.0, Score = 246.9, LR = so we would clearly reject at, say, the 5 percent level, since the associated critical value is Justin L. Tobias (Purdue) MLE #8 16 / 20

17 Similarly, we can repeat this process over and over again to examine properties of our test statistic. Specifically, we generated 10, 000 data sets in this fashion, each time calculating the Wald, Score and LR test. For each such iteration, we counted whether or not the hypothesis was rejected at the 10 % level (the corresponding critical value is 2.71). Here are the percentages of times that the null was rejected for each test: Wald =.097, Score =.105, LR =.101 consistent with our expectations. Justin L. Tobias (Purdue) MLE #8 17 / 20

18 Similarly, we can repeat this process over and over again to examine properties of our test statistic. Specifically, we generated 10, 000 data sets in this fashion, each time calculating the Wald, Score and LR test. For each such iteration, we counted whether or not the hypothesis λ =.2 was rejected at the 10 % level (the corresponding critical value is 2.71). Here are the percentages of times that the null was rejected for each test: Wald = 1, Score = 1, LR = 1. consistent with our expectations. Justin L. Tobias (Purdue) MLE #8 18 / 20

19 Invariance Invariance Another property enjoyed by MLE s is invariance. (Loosely speaking, the MLE of a function of the parameters is the function of the MLE of those parameters). Suppose the model is parameterized in terms of θ yet we are interested in γ = g(θ) for some continuous, one-to-one function g. Then the MLE of γ is g of the MLE of θ, i.e., Justin L. Tobias (Purdue) MLE #8 19 / 20

20 Invariance Invariance Complete Proof on your Own: Proof. Justin L. Tobias (Purdue) MLE #8 20 / 20

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