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Use the data in MATHPNL for this exercise. You will do a fixed effects version of the first differencingdone in Computer Exercise 11 in Chapter $13 .$ The model of interest is$\begin{aligned} \operatorname{math} 4_{i t}=& \delta_{1} y 94_{t}+\ldots+\delta_{5} y 98_{t}+\gamma_{1} \log \left(r e x p p_{i t}\right)+\gamma_{2} \log \left(r e x p p_{i, t-1}\right) \\ &+\psi_{1} \log \left(e n r o l_{i t}\right)+\psi_{2} \operatorname{lunch}_{i t}+a_{i}+u_{i t} \end{aligned}$where the first available year (the base year) is 1993 because of the lagged spending variable.(i) Estimate the model by pooled OLS and report the usual standard errors. You should include anintercept along with the year dummies to allow $a_{i}$ to have a nonzero expected value. What arethe estimated effects of the spending variables? Obtain the OLS residuals, $\hat{v}_{i r}$(ii) Is the sign of the lunch_{it} \text { coefficient what you expected? Interpret the magnitude of the } coefficient. Would you say that the district poverty rate has a big effect on test pass rates?(iii) Compute a test for AR $(1)$ serial correlation using the regression $\hat{v}_{i t-1} .$ You should use the years 1994 through 1998 in the regression. Verify that there is strong positive serial correlation and discuss why.(iv) Now, estimate the equation by fixed effects. Is the lagged spending variable still significant?(v) Why do you think, in the fixed effects estimation, the enrollment and lunch program variablesare jointly insignificant?(vi) Define the total, or long-run, effect of spending as $\theta_{1}=\gamma_{1}+\gamma_{2} .$ Use the substitution $\gamma_{1}=\theta_{1}-\gamma_{2}$ to obtain a standard error for $\theta_{1} .$ IHint: Standard fixed effects estimation using $\log \left(r e x p p_{i t}\right)$ and $z_{i t}=\log \left(r e x p p_{i, t-1}\right)-\log \left(r e x p p_{i t}\right)$ as explanatory variables should do it.]

The coefficient of $\log \left($rexpp$_{i t}\right)$ is $\theta_{1} \cdot$ Its estimate is 6.591824 and its standard error is 2.637934

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Chapter 14

Advanced Panel Data Methods

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These are the estimates for Part one and Part four. These regressions have the same number of observations and equals 5 50 Same number of time periods. T equals six, but they differ by the are square. Actually, there are squares are pretty close. Our square for part one is 10.5, and for part four is going sixth. Part two is based on part one in part two. We examine the estimate on the lunch variable. This variable is the percent of students in the district eligible for free or reduced price lunches, which is determined by poverty status. Therefore, the lunch variable is effectively a poverty rate. The estimate for this variable is minus 0.41 and this variable is highly significant. We can tell that the district poverty rate has a large impact on the math past rate. A 1% point increase in the lunch variable reduces the past rate by about 10.0.41 percentage point. Part three. We will examine whether they're standard Iran's now the era in terms. Exhibit A R. One serial correlation. Oh, we will take the residuals from the Poland l s regression in part one. Let's call that you had. Yeah, and we regret this residual on its first leg. Yeah. The estimate on the first lack of the residual denoted row has an estimate of 0.5 oh four with a standard error of 0.17 we are able to reject the null hypothesis that row equals zero or the null hypothesis of no serial correlation. In other words, there exists a strong evidence of a R one First order auto correlation. The positive serial correlation is caused by many reasons. It could be the presence of a time constant and observed effect. R four is, as we have seen above and comparing part one and part four regression results. We see that the coefficient on the leg spending variable mhm okay, has become smaller. But the T statistic is still very high. Almost three. So there is still evidence of a leg spending effect after controlling for and observed district effects. Yeah, Yeah. What? Mhm Part five. Now we can notice the change in the estimate on the lunch variable. The lunch variable is now having a positive estimate and no longer significant. Oh, we could notice the same reduction in the magnitude of effect and the loss of significance for the variable lock of enrollment. We can explain that by thinking about the meaning of enrollment and lunch variables. This changed slowly over time, which means that their effects are largely captured by an observed effect. A sub i mhm because of the time demeaning. Also the fixed effects. Yes, the effects of these variables are hard to capture or they can't be estimate precisely. The story is different for the spending coefficients. There is a policy change during this period and because of the change in policy spending shift substantially in 1994 after the passage of a proposal in Michigan and the proposal change the way schools were funded. Heart Sixth. The estimated long run spending effect is yeah, Feta hat equals 6.59 with a standard ERA of 2.64

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