Question

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?

Name: Problem 11, Chapter 13: Introductory Econometrics
Uploaded: 2021-02-15T21:23:53-08:00
Duration: 11 min 26 s
Channel: Heather Duong

   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?

Introductory Econometrics

Jeffrey M.… 6th Edition

Chapter 13, Problem 11

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Solved by Expert Heather Duong

02/15/2021

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Therefore, a 10% increase in $rexpp_{it}$ leads to a $\frac{\beta_{1}}{10}$ percentage point increase in $math4_{it}$. Show more…

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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 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?

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Key Concepts

Panel Data Models with Unobserved Effects

This concept involves models where data is observed over multiple time periods for each individual (or district, in this context) and there exist unobserved individual-specific factors that do not vary over time. These models account for these time?invariant differences by including fixed effects (or random effects) in order to control for potential omitted variable bias.

Logarithmic Transformation and Coefficient Interpretation

Taking the logarithm of an explanatory variable allows for interpreting the estimated coefficient as an elasticity or a percentage change effect. For instance, if the explanatory variable is log-transformed, a small change in the log variable approximates a percentage change in the original variable, and the coefficient can be divided (or scaled) appropriately to reflect the change in the dependent variable in percentage points.

First Differencing

First differencing is a technique used in panel data analysis to remove unobserved time-invariant effects by subtracting the previous time period's observations from the current period's. This method helps to eliminate bias arising from omitted variables that do not change over time, ensuring that the estimation focuses on within-individual changes rather than between-individual differences.

Lagged Explanatory Variables in Panel Data

Including lagged values of an explanatory variable in a panel data model is a way to capture dynamic relationships, allowing the model to account for delayed effects of a variable. When using first differencing, adding lagged variables requires careful interpretation since the differencing process may alter the timing and dynamics captured in the model.

Robust Standard Errors

Robust standard errors are adjustments made to the standard errors of estimated coefficients to account for potential heteroskedasticity—situations where the variability of the error term is not constant across observations. Such adjustments ensure that inferences (like hypothesis tests) remain valid even when these issues are present.

Serial Correlation in Panel Data and AR(1) Testing

Serial correlation, particularly first-order autoregressive (AR(1)) correlation, occurs when error terms in one period are correlated with those in the previous period. In first-differenced models, the differencing process itself may induce negative serial correlation, so tests for AR(1) are used to confirm and adjust for any such patterns, thereby ensuring valid inference.

Joint Significance Testing of Additional Variables

This concept involves testing whether a set of additional explanatory variables (or controls) jointly contribute to the explanatory power of the model. In a panel data setting, such a test can determine if the inclusion of additional variables, like measures of enrollment or poverty, significantly improves the model, helping to assess model specification and the potential need for these controls.

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