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The file MATHPNL contains panel data on school districts in Michigan for the years 1992 through$1998 .$ It is the district-level analogue of the school-level data used by Papke $(2005) .$ The responsevariable of interest in this question is $m a t h 4,$ the percentage of fourth graders in a district receiving a passing score on a standardized math test. The key explanatory variable is rexpp, which is real expenditures per pupil in the district. The amounts are in 1997 dollars. The spending variable will appear in logarithmic form.(i) Consider the static unobserved effects model$\begin{aligned} m a t h 4_{i t}=& \delta_{1} y 93_{t}+\ldots+\delta_{6} y 98_{I}+\beta_{1} \log \left(r e x p p_{i t}\right) \\ &+\beta_{2} \log \left(e n r o l_{i t}\right)+\beta_{3} \operatorname{lunch}_{i t}+a_{i}+u_{i t} \end{aligned}$where enrol_{it} \text { is total district enrollment and lunch. } is the percentage of students in the district eligible for the school lunch program. (So lunch_iis a pretty good measure of the district-wide poverty rate.) Argue that $\beta_{1} / 10$ is the percentage point change in math $4_{i t}$ when real per-student spending increases by roughly 10$\%$(ii) Use first differencing to estimate the model in part(i). The simplest approach is to allow an intercept in the first-differenced equation and to include dummy variables for the years 1994 through 1998 . Interpret the coefficient on the spending variable.(iii) Now, add one lag of the spending variable to the model and reestimate using first differencing.Note that you lose another year of data, so you are only using changes starting in $1994 .$ Discuss.the coefficients and significance on the current and lagged spending variables.(iv) Obtain heteroskedasticity-robust standard errors for the first-differenced regression in part (iii).How do these standard errors compare with those from part (iii) for the spending variables?(v) Now, obtain standard errors robust to both heteroskedasticity and serial correlation. What doesthis do to the significance of the lagged spending variable?(vi) Verify that the differenced errors $r_{i t}=\Delta u_{i t}$ have negative serial correlation by carrying out a test of $\mathrm{AR}(1)$ serial correlation.(vii) Based on a fully robust joint test, does it appear necessary to include the enrollment and lunchvariables in the model?

(i) (ii) (iii) see video (iv) SE of $\delta log(rexpp)$ increases but the variable still significant (v) (vi) (vii) see video

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Chapter 13

Pooling Cross Sections Across Time: Simple Panel Data Methods

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part one. We will take changes of both sides of the equation holding the other variables fixed the change in math four equals beta, one times the change in lock of riel spending per student which is beta one over 100 times 100 times change in lock of riel spending per student. Basically, we just divide and multiply 100 to the equation and that is roughly equal beta one over 100 times the percentage change in riel spending per student. So if the percentage change in real spending per student is 10, then the change in math four. Yeah, okay. Is beta one over 100 times 10. You just pluck 10 into Yeah, this side of the equation. And you wouldn't get beta one over 10. Mhm. This is the estimation result using Poland old l s in first differences. Except for the year dummies, we have five year Demis And besides the change in lock of riel spending per student, we have the change in lock enrollment and the change in lunch option. This is another dummy. We have 3300 observations and our square is 33000.208 The spending coefficient implies that a 10% increase in real spending mhm her student increases the math. Four pass rate by about 3.45 divided by 10. This is follows part one, and that equals 0.35% points. Part three. We add the lag spending change and drop another year. This is what we get. This is variable. We just add to the equation, and we drop ear 1994. The number of observations we used in this equation is 27 50 and the our square is 500.238 So how do the estimates on the rial spending per student change? Well, the contemporaneous spending variable. This one still have a negative coefficient, but it is not at all statistically significant. Meanwhile, the coefficient on the leg spending variable is highly significant and implies that a 10% increase in spending last year increases the math. Four pass rate by about 1.1 percentage points. Part four Hetero scad elasticity Robots standard. Errol for the coefficient of the lock change of real spending per student is 4.28 which reduces the significance of the change in lock of real spending per student even further. The hetero Scholastic City Robert Standard error of the estimate on lack real spending is about 4.38 That also lowers to t statistic and subsequently the significance of this variable. Still, this variable is highly significant at just over the 1% level. Part five asked you to calculate the fully robbers standard error, and that for the contemporary nous, spending is five point no. 4.4, 4.94 and that for the lack spending is five point on three. So for the court temporary spending again the significance level is further reduced. And for the past, spending the new standard their own gives a T statistic of 2.15 and that is associated with a P value of 0.0 32 So again, this variable is still significant at the 5% level. Part six. We wouldn't use four years of data for the test. We will do a pull old L s regression of our I t had on our I t minus one head using years 1995 1996 1997 1998 data The regression gives a row hat of minus 0.4 23 with a standard error of 0.19 So Roll Head is statistically significant and this is a strong evidence for serial correlation. Part seven. We will test the joint significance of the change in lock of enrollment and the change of lunch. We will use a nef test that is corrected for hetero scholastic, city and serial correlation. In other words, the F test must be fully robust. This test this statistic has two and 549 degrees of freedom, with the value of 0.93 and a P value of 0.4. So we are unable to reject the null hypothesis. The two variables are jointly insignificant, and so we could drop them from their regression.

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