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

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1 Lecture 9: Panel Data Model (Chapter 14, Wooldridge Textbook) 1

2 2 Panel Data Panel data is obtained by observing the same person, firm, county, etc over several periods. Unlike the pooled cross sections, the observations for the same cross section unit (panel, entity, cluster) in general are dependent. Thus cluster-robust statistics that account for correlation within panel should be used.

3 3 Organizing Panel Data It is important to have an ID variable that distinguishes one entity from others, such as patient ID, firm ID and county name. The observations for the same panel (over several periods) should be adjacent. This is called long form required by Stata command xtreg. The data for the minimum wage paper is wide form. Stata command reshape can be used to transform the wide form to the long form.

4 4 Panel Data and Causality Panel data can be used to control for time invariant unobserved heterogeneity, and therefore is widely used for causality research. By contrast, cross sectional data cannot control for time invariant unobserved heterogeneity, so may suffer bigger omitted variable bias than panel data. The idea is simple. We take various forms of difference, and the time invariant unobserved heterogeneity is removed. Effectively, the panel data use the same panel as both treatment group and control group, and by invoking the before and after comparison, remove the time invariant omitted variables. The limitation of panel data is that time varying omitted variables are still present. But overall, the omitted variable bias gets smaller than cross sectional data.

5 5 Unobserved Effect Panel Data Model Consider a two-period unobserved effect model The subscript i indexes panels, while t indexes periods. y it = β 0 + δ 0 d t + β 1 x it + a i + e it (1) a i is time constant unobserved heterogeneity. e it is the idiosyncratic error, or time-varying unobserved heterogeneity. a i + e it is the composite error term. d t is time dummy, so is panel constant and time varying; you can think of a i as panel dummy, so is time constant and panel varying.

6 6 Endogeneity The main reason to use panel data is to correct for the endogeneity caused by unobserved time constant effect, i.e., cov(x it,a i ) 0 (2) Given that nonzero covariance, the pooled OLS estimator applied to (1) is inconsistent.

7 7 First Difference (FD) Estimator I The repeated observations for the same panel make it possible to remove a i via differencing First write down the regression for period 2 and period 1 explicitly as y it=2 = β 0 + δ β 1 x it=2 + a i + e it=2 (3) y it=1 = β 0 + δ β 1 x it=1 + a i + e it=1 (4) Now it is clear that a i can be removed by subtracting the second equation from the first one.

8 First Difference (FD) Estimator II 8 So we compute the first time difference for each panel y i = y i,t=2 y i,t=1 (5) x i = x i,t=2 x i,t=1 (6) e i = e i,t=2 e i,t=1 (7) Finally, run the regression using the first-differened data, called first difference equation: y i = δ 0 + β 1 x i + e i (8) Notice that both a i and β 0 disappear. In general, differencing removes all time constant variables (such as gender). OLS applied to the FD regression (8) yields the so called first-difference estimator. The FD estimator is consistent and has causal interpretation if the regressor in (8) is exogenous, i.e., E( x i, e i ) = 0 (9)

9 9 Serial Correlation In general the error term in the difference regression (8), e i, is negatively serially correlated when e it is serially uncorrelated. For example, if data have three periods, then E( e i,t e i,t 1 ) = E[(e i,t=3 e i,t=2 )(e i,t=2 e i,t=1 )] = σ 2 e < 0 So cluster-robust statistics should be used.

10 10 Diminishing Variation Typically, the variation in the differenced independent variable is much smaller than the variation in the original independent variable. Thus imprecise estimate can be expected from FD estimator. Like the IV estimator, here we face the same tradeoff of efficiency versus unbiasedness.

11 11 STATA Command The STATA command to get the time differenced data is by panelid: gen dy = y[_n]-y[_n-1] by panelid: gen dx = x[_n]-x[_n-1] This will produce missing value for the first observation of each entity. The by panelid part is important. Without that part you will get overall difference, which is meaningless for our purpose.

12 FD Estimator can be used to control for time-constant unobserved heterogeneity. FD estimator cannot be used when the regressor of interest is time-constant. FD estimator is imprecise when the regressor changes little over time. 12

13 13 Fixed Effect (FE) Estimator I For concreteness let t = (1,2,3) in the following causal model y it = β 0 + δ 1 d1 t + δ 2 d2 t + β 1 x it + a i + e it (10) Note that there are two time-dummies in (10) because there are three periods. 1, period 1; d1 t = 0, period 2 3. d2 t = 1, period 2; 0, period 1 3. (11) (12) so period 3 is the base period.

14 14 Fixed Effect (FE) Estimator II Averaging (10) across i leads to the so called between regression where the time averages are ȳ i = β 0 + δ 1 d1 t + δ 2 d2 t + β 1 x i + a i + ē i (13) ȳ i = 1 3 x i = 1 3 ē i = 1 3 The average of a i is itself since it is time-invariant. 3 y it (14) t=1 3 x it (15) t=1 3 e it (16) t=1

15 15 Time Average Note that the time average is different from the overall average ȳ i ȳ 1 3n n 3 i=1 t=1 y it (17) The STATA command to get time average is by panelid: egen ybar = mean(y)

16 16 Between Regression OLS estimator applied to the between regression is inconsistent since cov( x i a i ) = 1 n t cov(x it a i ) 0, see (2). However later we will use between equation to get estimate for the panel dummy â i.

17 17 Fixed Effect (FE) Estimator III Subtracting the between regression (13) from (10) leads to the so called within regression y demean it = δ 1 d1 demean t + δ 2 d2 demean t + β 1 x demean it + e demean it (18) where y demean it = y it ȳ i (19) x demean it = x it x i (20) e demean it = e it ē i (21) Note a i is removed. Finally OLS applied to the within regression (18) is the FE estimator.

18 18 Dummy Variable Regression The FE estimator can be alternatively obtained from a dummy variable regression y it = β 0 + δ 1 d1 t + δ 2 d2 t + n 1 a j c j + β 1 x it + u it (22) j=1 where c j is the dummy variable for the j-th cross unit: 1, cross unit j; c j = 0, other cross units. (23) and a j is its coefficient.the STATA command is areg y x, absorb(panelid)

19 19 Frisch-Waugh Theorem Essentially the dummy variable regression treats a i in (10) as entity-specific intercept terms. The Frisch-Waugh theorem indicates that ˆβ 1 in the dummy variable regression (22) is numerically equivalent to the FE estimator obtained from the within regression (18).

20 20 STATA Command The STATA command xtreg y x, fe vce(robust) produces the FE estimator along with cluster-robust standard error. If you include a time constant independent variable such as gender, it will be dropped.

21 21 Fixed Effect You can estimate the fixed effect â i by replacing coefficient with its FE estimates in the between regression â i = ȳ i ˆβ 0 ˆδ 1 d1 t δ ˆ 2 d2 t ˆβ 1 x i (24) STATA also reports the F test for the joint significance of fixed effects: H 0 : â 1 = â 2 =... = â n 1 = 0 (25)

22 22 Remarks FE estimator cannot be used when the regressor of interest x it is time-invariant (such as gender). Cluster Robust Standard Error should be used in the within regression (18) since e demean it serially correlated. is FE and FD estimators are inconsistent when E(x it e it ) 0 in the main model (10). In that case, IV variable is needed to completely resolve the endogeneity issue.

23 23 Three Questions Q: Why do we need time dummy? A: The time dummy d1 t and d2 t in (10) can control for time varying but panel constant unobserved effect. Example is national trend. It affects every panel and evolves over time. Q : Why do we need panel dummy? The panel dummy c j in (22) can control for panel varying but time constant unobserved effect. Example is ability. It varies across persons but remains unchanged over time. Q: What if there are time-varying omitted variables? A: IV is still needed if there is time-varying omitted variable. Nevertheless finding IV for the within regression is easier than finding IV for the original regression (Why?)

24 Panel data model is useful when the omitted variable is time-invariant. Panel data model cannot be used when the key regressor is time-invariant. IV Estimator applied to the Within Regression should be considered when the omitted variable is time-varying. 24

Empirical Application of Panel Data Regression

Empirical Application of Panel Data Regression 1. We use Fatality data, and we are interested in whether rising beer tax rate can help lower traffic death. So the dependent variable is traffic death, while