(a) Briefly discuss the advantage of using panel data in this situation rather than pure crosssections

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Donald Sullivan
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1 Answer Key Fixed Effect and First Difference Models 1. See discussion in class.. David Neumark and William Wascher published a study in 199 of the effect of minimum wages on teenage employment using a U.S. state panel. The paper used annual observations for the years and included all 50 states plus the District of Columbia. The estimated equation is of the following type: Eit = β0 + β1 (Mit /Wit ) + γdi + + γnd51i + δbt + + δtb13t + uit where E is the employment to population ratio of teenagers, M is the nominal minimum wage, and W is average wage in the state. In addition, other explanatory variables, such as the primeage male unemployment rate, and the teenage population share were included. D-D51 are the state dummies, B-B13 are the time dummies. (a) Briefly discuss the advantage of using panel data in this situation rather than pure crosssections or time series. There are likely to be omitted variables in the above regression. On way to deal with some of these is to introduce state and time dummy variables to capture the fixed component of the error term. State dummy variables will capture the influence of omitted variables that are state specific and do not vary over time, while the time dummies will capture any country-wide variables that are common to all states at a point in time. (Furthermore, there are also more observations when using panel data, resulting in more variation.) (b) Estimating the model by OLS but including only time dummy variables results in the following output Eit = β (Mit /Wit ) (SHY it ) 1.53 uram it ; (.08) (.8) (.13) R =.0 where SHY is the proportion of teenagers in the population, and uram is the prime-age male unemployment rate. Coefficients for the time fixed effects are not reported. Numbers in parentheses are homoskedasticity-only standard errors. Comment on the above results. Are the coefficients statistically significant? Since these are level regressions, how would you calculate elasticities? There is a negative relationship between minimum wages and the employment to population ratio. Increases in the share of teenagers in the population result in higher employment to population ratio, and increases in the prime-age male unemployment rate lower the employment to population ratio. 0 percent of employment to population of teenagers variation is explained by the above regression. The relative minimum wage and the prime-age male unemployment rate are significant using a 1% significance level, while the proportion of teenagers in the population is not.

2 The important point is to note that elasticities vary with levels. One possibility is to report elasticities at the sample means. (c) Adding state fixed effects changes the above equation as follows: Eit = β (Mit /Wit ) 0.19 (SHY it ) 0.54 uram it ; R = 0.69 (0.10) (0.) (0.11) Compare the two results. Why would the inclusion of state dummy variables change the coefficients in this way? The parameter of interest here is the coefficient on the relative minimum wage. While it was highly significant in the previous regression, it has now changed signs and is statistically insignificant. The explanatory power of the equation has increased substantially. The size of the other two coefficients has also decreased. The results suggest that omitted variables, which are now captured by state dummy variables, were correlated with the regressors and caused omitted variable bias. (d) The significance of each coefficient decreased, yet R increased. How is that possible? What does this result tell you about testing the hypothesis that all of the state fixed effects can be restricted to have the same coefficient? How would you test for such a hypothesis? The influence of the state dummy variables is large. These are bound to be statistically significant and the hypothesis to restrict these coefficients is zero is bound to fail. Since these are linear hypothesis that are supposed to hold simultaneously, an F-test is appropriate here. C13.6 (i) You may use STATA to directly tests restrictions such as H 0 : 1 = after estimating the unrestricted model in (13.). See hmk_ch_13.log. But we can also simply rewrite the equation to test this using any regression software. Write the differenced equation as log(crime) = clrprc -1 + clrprc - + u. Following the hint, we define 1 = 1, and then write 1 = 1 +. Plugging this into the differenced equation and rearranging gives log(crime) = clrprc -1 + ( clrprc -1 + clrprc - ) + u. Estimating this equation by OLS (again, see hmk_ch_13.log)gives ˆ 1 =.0091, se( ˆ 1 ) = The t statistic for H 0 : 1 = is.0091/ , which is not statistically significant. (ii) With 1 = the equation becomes (without the i subscript)

3 log(crime) = ( clrprc -1 + clrprc - ) + u = [( clrprc -1 + clrprc - )/] + u, where 1 = 1. But ( clrprc -1 + clrprc - )/ = avgclr. (iii) The estimated equation is (See hmk_ch_13.log) log( crime) = avgclr (.063) (.0051) n = 53, R =.175, R =.159. Since we did not reject the hypothesis in part (i), we would be justified in using the simpler model with avgclr. Based on adjusted R-squared, we have a slightly worse fit with the restriction imposed. But this is a minor consideration. Ideally, we could get more data to determine whether the fairly different unconstrained estimates of 1 and in equation (13.) reveal true differences in 1 and.. use "D:\Courses\grad econometrics\homework\crime3.dta". ***You can test this restriction using the test command:. reg clcrime cclrprc1 cclrprc Source SS df MS Number of obs = F(, 50) = 5.99 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = clcrime Coef. Std. Err. t P> t [95% Conf. Interval] cclrprc cclrprc _cons test cclrprc1= cclrprc ( 1) cclrprc1 - cclrprc = 0 F( 1, 50) = 1.15 Prob > F = ***Or you can make a transformation as suggested in the notes and use a t-tes > t:. gen changesum = cclrprc1+ cclrprc (53 missing values generated). reg clcrime cclrprc1 changesum Source SS df MS Number of obs = F(, 50) = 5.99 Model Prob > F = Residual R-squared =

4 Adj R-squared = Total Root MSE = clcrime Coef. Std. Err. t P> t [95% Conf. Interval] cclrprc changesum _cons reg clcrime cavgclr Source SS df MS Number of obs = F( 1, 51) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = clcrime Coef. Std. Err. t P> t [95% Conf. Interval] cavgclr _cons C13.1. (i) The estimated equation using pooled OLS is mrdrte = d exec +.53 unem (4.43) (.14) (.63) (0.78) n = 10, R =.10, R = reg mrdrte d93 exec unem if year==90 year==93 Source SS df MS Number of obs = F( 3, 98) = 3.69 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = 10.3 mrdrte Coef. Std. Err. t P> t [95% Conf. Interval] d exec unem _cons Because the coefficient on exec is positive (but statistically insignificant), there is no evidence of a deterrent effect. In using pooled OLS, we are exploiting only the cross-sectional variation in the data. If states that have had high murder rates in the past have reacted by implementing capital punishment, we can see a positive relationship between murder rates and capital punishment even if there is a deterrent effect. (Yet again, we must distinguish between correlation and causality.)

5 (ii) If we difference away the unobserved state effects which can include historical factors that lead to higher murder rates and aggressive use of capital punishment the story is different. The FD estimates are mrdrte = exec.067 unem (.09) (.043) (.159) n = 51, R =.110, R = reg cmrdrte cexec cunem if year== F(, 48) =.96 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = cmrdrte Coef. Std. Err. t P> t [95% Conf. Interval] cexec cunem _cons Now we find a deterrent effect: one more execution in the prior three years is estimated to decrease the murder rate by about.10, or about one murder per million people (because mrdrte is measured as murders per 100,000 people). The t statistic on exec is about.4, and so the effect is statistically significant. [The estimated deterrent effect turns out not to be robust to small changes in the data used. See Computer Exercise C14.7.] Note how the unemployment effect has become statistically insignificant. (iii) The BP and White tests both test two restrictions in this case. The BP and White F statistics are both about.6. Both have p-values above.50, so there is no evidence of heteroskedasticity in the FD equation. (See below for construction by hand). reg cmrdrte cexec cunem if year== F(, 48) =.96 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = cmrdrte Coef. Std. Err. t P> t [95% Conf. Interval] cexec cunem _cons predict resid, resid. gen resid = resid^

6 . predict yhat (option xb assumed; fitted values). gen yhat = yhat^. reg resid cexec cunem if year== F(, 48) = 0.60 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = resid Coef. Std. Err. t P> t [95% Conf. Interval] cexec cunem _cons display 51* display 1-chi(,1.393) ***If you do the F version of LM statistic it is ***[(.043/)]/[(1-.043)/(51--1)] =.598 distributed F(,51--1) ***1-F(, 48,.598 ) = so about the same level of significance as the chi- ***See Wooldridge Chapter 8 for more details. reg resid yhat yhat if year== F(, 48) = 0.58 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = resid Coef. Std. Err. t P> t [95% Conf. Interval] yhat yhat _cons (iv) The heteroskedasticity-robust t statistic on exec is 6.11, which is a huge increase in magnitude. This is a bit puzzling for two reasons. First, the tests for heteroskedasticity find essentially no evidence for heteroskedasticity. Second, it is rare to find a heteroskedasticityrobust standard error that is so much smaller than the usual OLS standard error.. reg cmrdrte cexec cunem if year==93, robust Linear regression Number of obs = 51 F(, 48) = 18.9 Prob > F = R-squared = Root MSE = Robust

7 cmrdrte Coef. Std. Err. t P> t [95% Conf. Interval] cexec cunem _cons (v) I would tend to go with the usual OLS t statistic because it gives a more cautious conclusion and there is no evidence of heteroskedasticity that should affect the t statistics. The usual two-sided p-value is about.0. The heteroskedasticity-robust p-value is zero to many decimal places, and it is hard to believe we have that much confidence in finding an effect. This is a case where it is important to remember that the robust standard errors (and, therefore, the robust t statistics) are only justified in large samples. n = 51 may just not be a large enough sample size with this kind of data set to produce reliable heteroskedasticity-robust statistics.

Problem Set 5 ANSWERS

Economics 20 Problem Set 5 ANSWERS Prof. Patricia M. Anderson 1, 2 and 3 Suppose that Vermont has passed a law requiring employers to provide 6 months of paid maternity leave. You are concerned that women