In Example $8.7,$ we computed the OLS and a set of WLS estimates in a cigarette demand equation.
(i) Obtain the OLS estimates in equation $(8.35) .$
(ii) Obtain the $\hat{h}_{i}$ used in the WLS estimation of equation $(8.36)$ and reproduce equation $(8.36) .$ From this equation, obtain the unweighted residuals and fitted values; call these $\hat{u}_{i}$ and $\hat{u}_{i}$ respectively. (For example, in Stata, the unweighted residuals and fitted values are given by default.)
(iii) Let $\ddot{u}_{i}=\hat{u} / \sqrt{\hat{h}}_{i}$ and $\check{y}_{i}=\hat{y}_{i} \sqrt{\hat{h}}_{i}$ be the weighted quantities. Carry out the special case of the White test for heteroskedasticity by regressing $\breve{u}_{i}^{2}$ on $\breve{y}_{i}, \check{y}_{i}^{2},$ being sure to include an intercept, as always. Do you find heteroskedasticity in the weighted residuals?
(iv) What does the finding from part (iii) imply about the proposed form of heteroskedasticity used
in obtaining $(8.36) ?$
(v) Obtain valid standard errors for the WLS estimates that allow the variance function to be
misspecified.

Answer

(i) See eq. 8.35. (ii) Follow eq. 8.32 (iii) yes (iv) invalid hypothesis test results (v) see video

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Introductory Econometrics

Chapter 8

Heteroskedasticity

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Video Transcript

part one. The result is simply Equation 8.35 Sure. In the textbook, you could use regression command in Seita or L m in our to obtain this equation. Part two. I will follow their textbooks. Notation you will need to find the weight. Uh, h I had used in their waited list. Square estimation. So what do you do first is you get the s result and then you will replace Thea observed you with all l s residuals. The we can call there old l s residuals you had. Then you regress. You run the lock of you hat on all the x variables, you wouldn't get the fitted value. And let's call it G. I had. And then you can find the weight. So the weight h I had is e to the power of G I hat. All right, Part three, it. So for part three, you win. Regress. They're transformed residual on their transformed outcome. Variable. That is the basic of the white test works. The regression equation is this is the intercept. This is the first slope, and this is the second slope. The null hypothesis of the white test is that There is no hetero scared Atis ity or homo scared elasticity. And the equation for that is thes two parameters should equal zero. Okay, continue. You will need to find the are square of the regression. And the Oscar is about 0.27 and we will get We will need a parameter k of to. And that give is an f statistic of 11.15 11 15 with a p value pups very small. So we are able to reject the null hypothesis. Who, if the null hypothesis is home most congested city. So we conclude that there's evidence against that. This means the symbol means exist, right? Yeah. Part four in part four. Well, part four is based on part three, and the existence of hetero Scholastic city implies that what we did in, um, they're feasible. Uh, generalized list square we used in the example does not eliminate hetero Scott Estes City. Okay, Thank you. You speak when hetero Scholastic city happens, The usual standard errors T statistic and F statistic school reported with the weighted least square are not valid. Mhm. Okay. Parts five u. N. Estimate the equation with weighted least square and also with rubbers. Standard errors. Let me write down their estimation. Result real quick number of cigarettes with the intercept. Yes, on income should be a lot of income. Lack of cigarettes, Bryce, Education. Each age square and restaurants. We have 807 observations with the R square of 0.11 3. Okay, the intercept we got. Okay, So all the estimate, the beta estimates are exactly the same as Theo Equation 8.36 in their textbook. The only difference is the standard errors. You will see that we will get a much larger standard errors in, uh, when we use the option Robust. These are their beta estimates. Okay, I will put the standard heroines in bracket and in green ink. We're lacking space here. Okay, This part is pretty small. So the standard error of age is 0.115 of a square is 0.12 and a restaurant is 0.72 So you can see there are big differences in standard errands. Comparing this equation with equation 8.36 It indicates that ah proposed correction for hetero Statistics City did not fully solve the problem.

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