Econ 1123: Section 5. Review. Internal Validity. Panel Data. Clustered SE. STATA help for Problem Set 5. Econ 1123: Section 5.

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1 Outline 1 Elena Llaudet October 6, based on Common Mistakes on P. Set 4 lnftmpop = higdppc -.25 lackpf +.65 higdppc * lackpf 2 lnftmpop = β 0 + β 1 higdppc + β 2 lackpf + β 3 lackpf 2 + β 4 higdppc * lackpf + β 5 higdppc * lackpf 2 Question 2: Write out the estimated regression lines for the two groups Question 3c: What does testing Ho: higdppc * lackpf = higdppc * lackpf 2 =0meanineverydayterms? Question 3a: Is the difference between the two slopes statistically different from zero? Question 3d: What does testing Ho: lackpf 2 = higdppc * lackpf 2 =0meanineverydayterms?

2 (A) Reasons for Biased Coefficients Astatisticalanalysisisinternallyvalidifthestatistical inferences about causal effects are valid for the population being studied. Studies are internally valid (A) if the estimated regression coefficients are unbiased and consistent, and (B) if their standard errors yield confidence intervals with the desired confidence level. 1. Omitted Variable Bias: Four potential solutions: (1) if omitted var. is observed, include it in the regression; (2) if not observed but unchanged over time, use panel data; (3) instrumental variables regression; (4) randomized experiment. 2. Misspecification of the Functional Form: Ifthetrue population regression function is nonlinear but the estimated regression is linear, then this function form misspecification makes the estimator biased. [Type of omitted var. bias]. 3. Measurement Error: Errors-in-variablesbiasintheOLS estimator arises when an independent variable is measured imprecisely. Potential solutions: (1) get an accurate measure; (2) instrumental variables regression; (3) develop a mathematical model of the measurement error. (B) Reasons for Inconsistent Standard Errors 4. Sample Selection: Sample selection bias occurs when the availability of the data is influenced by a selection process that is related to the value of the dependent variable. 4. Endogeneity: Simultaneous causality bias arises in a regression of Y on X when, in addition to the causal link of interest from X to Y, there is a causal link from Y to X. Potential solutions: (1) instrumental variables regression, (2) randomized experiment. 1. Heteroskedasticity when we have assumed homoskedasticity 2. Correlation of the error term across observations: Observations are not independent. Serial correlation in the error term can arise in panel data and time series data. Potential solutions: (1) Use an alternative formula for SE.

3 Panel data: data on multiple units (states, schools, etc...) observed over multiple time periods. Using fixed effects to solve omitted variable (OV) bias: What do unit fixed effects control for? It controls for omitted variables that vary across the units but are constant over time. What do time fixed effects control for? It controls for omitted variables that vary over time but are constant across the units. How does one include time fixed effects into a regression? We first need to generate a dummy variable for each year, and then include them all in the regression. Using STATA: tabulate year, generate (yr) reg Y X yr*, r Using asterisk tells STATA to include all of the variables that begin with yr. Just be careful that you do not have other variables that also begin with yr. What type of fixed effects do we need to include to control for omitted variables that vary both across units and over time? Fixed effects can not help you in this case. STATA will automatically drop one of the dummies (to avoid perfect mutlicolinearity). Alternatively, one could use areg Y X.., absorb(year) Preferred Models: What happens if you instead include the variable year in the regression? You are including the variable as continuous, thus, it is not equivalent to running time fixed effects (which allows for each time period to have a different intercept in the regression and places no restrictions on them). How do we know that a model with time fixed effects is preferable than a model without them? We shall test the following joint hypothesis: H O :alltimefixedeffects = 0 H A : at least one of the time fixed effects = 0 Using STATA: tabulate year, generate (yr) reg Y X yr*, r testparm yr*

4 An Example: Suppose that we have students test score data for three schools in one city over a period of 10 years, and we want to measure the effect of per pupil expenditures on student performance. Initially, we could run the following model: MI: TestsScores it = β 0 + β 1 Per pupil expenditure it +u it However, there might be some characteristics of the schools that are both (a) determinant of test scores, and (b) correlated with per pupil expenditures. For example, one of the schools might have very problematic students. The performance of the students suffers as a result & the teachers might demand to be paid more to go to teach in this school. If we do not have a way to measure those characteristics, but can assume that they are constant over time, then we could control for them by running a model with schools fixed effects. If there are three schools in the data, how many dummies should we add to the regression? Two. MII: TestsScores it = β 0 + β 1 Per pupil expenditure it + School B it + School C it +u it What would be now the interpretation of β 0? This is the intercept for the students in school A. It is the average performance of students in school A when per pupil expenditures are equal to zero (no substantive interpretation). Now, suppose that we realize that during the 10 years under study there was an upward trend in the student performance of these three schools due to a higher level of education of the parents in all three schools, who also pressured the city into spending more money in the schools. To get rid of the omitted variable bias caused by this, we could add to our regression time fixed effects. Given that we have data for 10 years, how many dummies should we add to the regression? Nine. MIII: TestsScores it = β 0 + β 1 Per pupil expenditure it + School B it + School C it +T2 it +T3 it... + T10 it +u it Potential Problem with : Observations might not be independent (violation of one of the OLS assumptions, observations i.i.d.). This would lead to inconsistent standard errors and would be a threat to the internal validity of the analysis. Solution:. allow for the errors to be correlated within a cluster but assume that errors not in the same cluster are uncorrelated. What would be now the interpretation of β 0? This is the intercept for the students in school A at time 1.

5 How does one cluster standard errors in STATA? We first need to tell STATA that our data is a panel xtset schools year (Notice that the cross-sectional entity goes first and the time entity goes last) Then, we can run the regression: xtreg test scores x1 x2, fe vce(cluster schools) This regression has clustered standard errors at the school level and school fixed effects. If you want to include year fixed effects, then: xtreg test scores x1 x2 yr*, fe vce(cluster schools) In the problem set you will have a dataset of observations within states and across years. Something like this: 1: state A year 1 x : state A year 2 x n: state B year 1 x1 n where x1 is a dummy gen change = ( x1[ n] == 1) & ( x1[ n-1] ==0) will generate a dummy variable called change that will take values of 1 on the year where the state experienced a change of x1 from 0 to 1. In the problem set, to cluster SE at the state level, use the variable stateid (it s a numeric variable) not the variable state (it s a string variable). STATA does not accept string variables for clustering.

Lecture 9: Panel Data Model (Chapter 14, Wooldridge Textbook)

Lecture 9: Panel Data Model (Chapter 14, Wooldridge Textbook) 1 2 Panel Data Panel data is obtained by observing the same person, firm, county, etc over several periods. Unlike the pooled cross sections,