CHAPTER9.doc

CHAPTER 9TEACHING NOTESThe coverage of RESET in this chapter recognizes that it is a test for neglected nonlinearities, and it should not be expected to be more than that. (Formally, it can be shown that if an omitted variable has a conditional mean that is linear in the included explanatory variables, RESET has no ability to detect the omitted variable. Interested readers may consult my chapter in Companion to Theoretical Econometrics, 2001, edited by Badi Baltagi.) I just teach students the F statistic version of the test.The Davidson-MacKinnon test can be useful for detecting functional form misspecification, especially when one has in mind a specific alternative, nonnested model. It has the advantage of always being a one degree of freedom test.I think the proxy variable material is important, but the main points can be made with Examples 9.3 and 9.4. The first shows that controlling for IQ can substantially change the estimated return to education, and the omitted ability bias is in the expected direction. Interestingly, education and ability do not appear to have an interactive effect. Example 9.4 is a nice example of how controlling for a previous value of the dependent variable something that is often possible with survey and nonsurvey data can greatly affect a policy conclusion. Computer Exercise 9.3 is also a good illustration of this method.I rarely get to teach the measurement error material, although the attenuation bias result for classical errors-in-variables is worth mentioning.The result on exogenous sample selection is easy to discuss, with more details given in Chapter 17. The effects of outliers can be illustrated using the examples. I think the infant mortality example, Example 9.10, is useful for illustrating how a single influential observation can have a large effect on the OLS estimates.With the growing importance of least absolute deviations, it makes sense to at least discuss the merits of LAD, at least in more advanced courses. Computer Exercise 9.9 is a good example to show how mean and median effects can be very different, even though there may not be “outliers” in the usual sense.SOLUTIONS TO PROBLEMS9.1There is functional form misspecification if 0 or 0, where these are the population parameters on ceoten2 and comten2, respectively. Therefore, we test the joint significance of these variables using the R-squared form of the F test: F= (.375- .353)/(1- .375)(177 8)/2 2.97. With 2 and df, the 10% critical value is 2.30 awhile the 5% critical value is 3.00. Thus, the p-value is slightly above .05, which is reasonable evidence of functional form misspecification. (Of course, whether this has a practical impact on the estimated partial effects for various levels of the explanatory variables is a different matter.)9.2Instructors Note: Out of the 186 records in VOTE2.RAW, three have voteA88 less than 50, which means the incumbent running in 1990 cannot be the candidate who received voteA88 percent of the vote in 1988. You might want to reestimate the equation dropping these three observations.(i) The coefficient on voteA88 implies that if candidate A had one more percentage point of the vote in 1988, she/he is predicted to have only .067 more percentage points in 1990. Or, 10 more percentage points in 1988 implies .67 points, or less than one point, in 1990. The t statistic is only about 1.26, and so the variable is insignificant at the 10% level against the positive one-sided alternative. (The critical value is 1.282.) While this small effect initially seems surprising, it is much less so when we remember that candidate A in 1990 is always the incumbent. Therefore, what we are finding is that, conditional on being the incumbent, the percent of the vote received in 1988 does not have a strong effect on the percent of the vote in 1990.(ii) Naturally, the coefficients change, but not in important ways, especially once statistical significance is taken into account. For example, while the coefficient on log(expendA) goes from -.929 to -.839, the coefficient is not statistically or practically significant anyway (and its sign is not what we expect). The magnitudes of the coefficients in both equations are quite similar, and there are certainly no sign changes. This is not surprising given the insignificance of voteA88.9.3(i) Eligibility for the federally funded school lunch program is very tightly linked to being economically disadvantaged. Therefore, the percentage of students eligible for the lunch program is very similar to the percentage of students living in poverty.(ii) We can use our usual reasoning on omitting important variables from a regression equation. The variables log(expend) and lnchprg are negatively correlated: school districts with poorer children spend, on average, less on schools. Further, 0, the measurement error can be positive or negative, but, since tvhours 0, e0 must satisfy e0 -tvhours*. So e0 and tvhours* are likely to be correlated. As mentioned in part (i), because it is the dependent variable that is measured with error, what is important is that e0 is uncorrelated with the explanatory variables. But this is unlikely to be the case, because tvhours* depends directly on the explanatory variables. Or, we might argue directly that more highly educated parents tend to underreport how much television their children watch, which means e0 and the education variables are negatively correlated.9.5The sample selection in this case is arguably endogenous. Because prospective students may look at campus crime as one factor in deciding where to attend college, colleges with high crime rates have an incentive not to report crime statistics. If this is the case, then the chance of appearing in the sample is negatively related to u in the crime equation. (For a given school size, higher u means more crime, and therefore a smaller probability that the school reports its crime figures.)SOLUTIONS TO COMPUTER EXERCISESC9.1(i) To obtain the RESET F statistic, we estimate the model in Computer Exercise 7.5 and obtain the fitted values, say . To use the version of RESET in (9.3), we add ()2 and ()3 and obtain the F test for joint significance of these variables. With 2 and 203 df, the F statistic is about 1.33 and p-value .27, which means that there is not much concern about functional form misspecification.(ii) Interestingly, the heteroskedasticity-robust F-type statistic is about 2.24 with p-value .11, so there is stronger evidence of some functional form misspecification with the robust test. But it is probably not strong enough to worry about.C9.2Instructors Note: If educKWW is used along with KWW, the interaction term is significant. This is in contrast to when IQ is used as the proxy. You may want to pursue this as an additional part to the exercise.(i) We estimate the model from column (2) but with KWW in place of IQ. The coefficient on educ becomes about .058 (se .006), so this is similar to the estimate obtained with IQ, although slightly larger and more precisely estimated.(ii) When KWW and IQ are both used as proxies, the coefficient on educ becomes about .049 (se .007). Compared with the estimate when only KWW is used as a proxy, the return to education has fallen by almost a full percentage point.(iii) The t statistic on IQ is about 3.08 while that on KWW is about 2.07, so each is significant at the 5% level against a two-sided alternative. They are jointly very significant, with F2,925 8.59 and p-value .0002.C9.3(i) If the grants were awarded to firms based on firm or worker characteristics, grant could easily be correlated with such factors that affect productivity. In the simple regression model, these are contained in u.(ii) The simple regression estimates using the 1988 data are =.409+.057 grant(.241)(.406)n = 54, R2 = .0004.The coefficient on grant is actually positive, but not statistically different from zero.(iii) When we add log(scrap87) to the equation, we obtain=.021-.254 grant88+.831 log(scrap87)(.089)(.147)(.044)n = 54, R2 = .873,where the year subscripts are for clarity. The t statistic for H0: = 0 is -.254/.147 1.73. We use the 5% critical value for 40 df in Table G.2: 1.68. Because t= -1.73 -1.68, we reject H0 in favor of H1: 0 at the 5% level. (iv) The t statistic is (.831 1)/.044 -3.84, which is a strong rejection of H0.(v) With the heteroskedasticity-robust standard error, the t statistic for grant88 is -.254/.142 -1.79, so the coefficient is even more significantly less than zero when we use the heteroskedasticity-robust standard error. The t statistic for H0: = 1 is (.831 1)/.071 -2.38, which is notably smaller than before, but it is still pretty significant.C9.4(i) Adding DC to the regression in equation (9.37) gives =23.95-.567 log(pcinc)-2.74 log(physic)+.629 log(popul)+16.03 DC(12.42)(1.641)(1.19)(.191)(1.77) n = 51, R2 = .691, = .664.The coefficient on DC means that even if there was a state that had the same per capita income, per capita physicians, and population as Washington D.C., we predict that D.C. has an infant mortality rate that is about 16 deaths per 1000 live births higher. This is a very large “D.C. effect.” (ii) In the regression from part (i), the intercept and all slope coefficients, along with their standard errors, are identical to those in equation (9.38), which simply excludes D.C. Of course, equation (9.38) does not have DC in it, so we have nothing to compare with its coefficient and standard error. Therefore, for the purposes of obtaining the effects and statistical significance of the other explanatory variables, including a dummy variable for a single observation is identical to just dropping that observation when doing the estimation.The R-squareds and adjusted R-squareds from (9.38) and the regression in part (i) are not the same. They are much larger when DC is included as an explanatory variable because we are predicting the infant mortality rate perfectly for D.C. You might want to confirm that the residual for the observation corresponding to D.C. is identically zero.C9.5 With sales defined to be in billions of dollars, we obtain the following estimated equation using all companies in the sample: =2.06+.317 sales-.0074 sales2+.053 profmarg(0.63)(.139)(.0037)(.044) n = 32, R2 = .191, = .104.When we drop the largest company (with sales of roughly $39.7 billion), we obtain =1.98+.361 sales-.0103 sales2+.055 profmarg(0.72)(.239)(.0131)(.046) n = 31, R2 = .191, = .101.When the largest company is left in the sample, the quadratic term is statistically significant, even though the coefficient on the quadratic is less in absolute value than when we drop the largest firm. What is happening is that by leaving in the large sales figure, we greatly increase the variation in both sales and sales2; as we know, this reduces the variances of the OLS estimators (see Section 3.4). The t statistic on sales2 in the first regression is about 2, which makes it almost significant at the 5% level against a two-sided alternative. If we look at Figure 9.1, it is not surprising that a quadratic is significant when the large firm is included in the regression: rdintens is relatively small for this firm even though its sales are very large compared with the other firms. Without the largest firm, a linear relationship between rdintens and sales seems to suffice.C9.6 (i) Only four of the 408 schools have b/s less than .01.(ii) We estimate the model in column (3) of Table 4.3, omitting schools with b/s 40. (Data are missing for some variables, so not all of the 1,989 observations are used in the regressions.)(ii) When observations with obrat 40 are excluded from the regression in part (iii) of Problem 7.16, we are left with 1,768 observations. The coefficient on white is about .129 (se .020). To three decimal places, these are the same estimates we got when using the entire sample (see Computer Exercise C7.8). Perhaps this is not very surprising since we only lost 203 out of 1,971 observations. However, regression results can be very sensitive when we drop over 10% of the observations, as we have here. (iii) The estimates from part (ii) show that does not seem very sensitive to the sample used, although we have tried only one way of reducing the sample.C9.8 (i) The mean of stotal is .047, its standard deviation is .854, the minimum value is 3.32, and the maximum value is 2.24.(ii) In the regression jc on stotal, the slope coefficient is .011 (se = .011). Therefore, while the estimated relationship is positive, the t statistic is only one: the correlation between jc and stotal is weak at best. In the regression univ on stotal, the slope coefficient is 1.170 (se = .029), for a t statistic of 38.5. Therefore, univ and stotal are positively correlated (with correlation = .435).(iii) When we add stotal to (4.17) and estimate the resulting equation by OLS, we get 1.495 + .0631 jc + .0686 univ + .00488 exper + .0494 stotal(.021)(.0068)(.0026)(.00016)(.0068)n = 6,758, R2 = .228For testing bjc = buniv, we can use the same trick as in Section 4.4 to get the standard error of the difference: replace univ with totcoll = jc + univ, and then the coefficient on jc is the difference in the estimated returns, along with its standard error. Let q1 = bjc - buniv. Then . Compared with what we found without stotal, the evidence is even weaker against H1: bjc buniv. The t statistic from equation (4.27) is about 1.48, while here we have obtained only -.80. (iv) When stotal2 is added to the equation, its coefficient is .0019 (t statistic = .40). Therefore, there is no reason to add the quadratic term.(v) The F statistic for testing exclusion of the interaction terms stotaljc and stotaluniv is about 1.96; with 2 and 6,756 df, this gives p-value = .141. So, even at the 10% level, the interaction terms are jointly insignificant. It is probably not worth complicating the basic model estimated in part (iii).(vi) I would just use the model from part (iii), where stotal appears only in level form. The other embellishments were not statistically significant at small enough significance levels to warrant the additional complications.C9.9 (i) The equation estimated by OLS is = 21.198 - .270 inc + .0102 inc2 - 1.940 age + .0346 age2 ( 9.992)(.075)(.0006)(.483)(.0055)+ 3.369 male + 9.713 e401k(1.486)(1.277)n = 9,275, R2 = .202The coefficient on e401k means that, holding other things in the equation fixed, the average level of net financial assets is about $9,713 higher for a family eligible for a 401(k) than for a family not eligible.(ii) The OLS regression of on inci, , agei, , malei, and e401ki gives .0374, which translates into F = 59.97. The associated p-value, with 6 and 9,268 df, is essentially zero. Consequently, there is strong evidence of heteroskedasticity, which means that u and the explanatory variables cannot be independent even though E(u|x1,x2,xk) = 0 is possible.(iii) The equation estimated by LAD is = 12.491 - .262 inc + .00709 inc2 - .723 age + .0111 age2 ( 1.382)(.010)(.00008)(.067)(.0008)+ 1.018 male + 3.737 e401k(.205)(.177)n = 9,275, Psuedo R2 = .109Now, the coefficient on e401k means that, at given income, age, and gen