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Question

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Answers

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 response
variable 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 does
this 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 lunch
variables in the model?

So Chevy Jebbie Shelves rule is 100%. Are 100 times 1 -1 of her case squared percent of the data values is within K standard deviations where K is greater than one. So our solution is the first time the Z score, which is x minus mu divided by sigma. Ah this is equal to 17.28 -14, divided by 0.4. This equals 8.2. So using Chevy chiefs inequality or sorry, the rule, we'll just plug that in 100 times one minus 1/8 10.2 squared and we get approximately mhm 98 51%. With the other weights are at least the 17 oz.

First one. The test for a unit root in series you rate unemployment rate using the usual dickey fuller test with a constant yeah. And the augmented dickey fuller with two legs of change of unemployment rate. I find that seven both times we are unable to reject the now hypothesis that unemployment rate series is a unit fruit. The legs are not significant. However, the significance of the legs matters. So the outcome of the unit root test, we will repeat what we have done in part one two series vacancy rate and report the result in part two. I guess similar result. So the rate is a unit root. Well part one and two. I use package the R. Package A. T. S. A. And the function is a D. F. Dot test. R. Three. We assuming that unemployment rate and vacation re rate are both integrated of level one. We test for co integration using the angle grandeur test with no legs. So the step the steps are as follow. We first regress, you read on the rate then we yet the residual and we run the key fuller has on the residual to see whether the residuals our unit root. I find that you're right and we rate Arco integrated at the 5% level. Yeah Heart Forest. I get the leads and lacks estimator of the change in vacancy rate and I did note that uh CB rates up minus one. This is for the lack and plus one is for the lead. This is a regression result. So the usual centered errors are in green and in round brackets, the robots that Iran's are in blue and in square brackets you can see that the main estimate on vacancy rate is highly significant. This one is not correct. So the centered errol the usual one for the estimate of the first lack of change in vacancy rate is 164 In all cases except for the estimate of the lead of C. V. Right. The robust standard Iran's are larger than the usual standard errors. This is usually the case it happens but rare that the robot standard errors are smaller than the usual standard errors. The r square of this regression is 0.77 So for the rate, because the robot standard error is larger than the usual standard error. So we will get a wider confidence interval if we use a robot standard error and for confidence interval you will run this function in our count in and you impose the name of the regression. It was spits all the 95% confidence intervals for all explanatory variables. The default version is the 95% interval. But because the standard barrel of this estimate is are very close, two versions are very close to each other so the confidence intervals should be roughly equal. Yeah. Last part. What you could say about real business of the claim that you rate and the rate are co integrated. Yeah. When I run the test and good grandeur, the results are not consistent across alternative types of process. In one case I can reject the notion that the residuals are united and for all the cases I cannot reject. So I conclude that the claim that you rate and be rate our co integrated is not robust.


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