Week 2: Pooling Cross Section across Time (Wooldridge Chapter 13)

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1 Week 2: Pooling Cross Section across Time (Wooldridge Chapter 13) Tsun-Feng Chiang* *School of Economics, Henan University, Kaifeng, China March 3, / 30

2 Pooling Cross Sections across Time Pooled Cross-Section Data For each time period, samples are randomly drew within a region. Observations are independently, but not identically. Sample (Individual) Year y x1 x2 x3 x / 30

3 Panel Data (Longitudinal Data) For each sample, its observations are collected across time. Therefore, the observations are not independent. That is, some time-invariant sample characterisitcs could affect the observations over time. Balanced Panel: each sample (Individual) has the same time period. Sample (Individual) Year y x1 x2 x3 x /

4 Pooled Cross Sections Pooled Cross Sections Since each obervation is independent, we treat all samples like they are ordinary cross-sectional data, but need to control the effect of time on the dependent variable of interest. The regression model of pooled cross section: y i = β 0 + δ 2 d2 + δ 3 d3 + + δ T dt + β 1 x i1 + β 2 x i2 + + β k x ik + u i (eq. 1) where d2, d3,, dt, is a dummy variable of time when the sample i was drew. Given the Assumption 1 to 6, the model with pooled data can be estimated by OLS. 4 / 30

5 Pooled Cross Sections Example 13.1 Figure: Determinants of Women s Fertility 5 / 30

6 Pooled Cross Sections Structural Change across Time The previous estimations assume the effects of independent variables are time invariant. What if they are not? The possible solution lies on the Chow Test (Ch. 7). Method 1 Run one restricted model (eq. 1) and T unrestricted models for separate time period: y i = β 0 + β 1 x i1 + β 2 x i2 + + β k x ik + u i for t = 1, 2,, T Obtain SSRs from both the unrestricted and restricted model, then apply the F test (Chow statistic): F = [ SSRr SSRur (n T Tk) SSR ur ][ (T 1)k ] where SSR ur = SSR SSR T ; k: # of explanatory variables. 6 / 30

7 Pooled Cross Sections Example 13.2 Figure: Change in the Return to Education and Gender Gap 7 / 30

8 Pooled Cross Sections Chow Test (Example 13.2) Figure: SSR for the first period unrestricted model Figure: SSR for the second period unrestricted model 8 / 30

9 Pooled Cross Sections Chow Test (Example 13.2, continued) Figure: SSR for the restricted model F (5,1072) = [ ( ) ( ) ][ ( ) (2 1) 5 ] = 1.99 Do not reject the null hypothesis that no strucutual change exist. 9 / 30

10 Pooled Cross Sections Chow Test (Example 13.2, continued) My R code: >gender_return = read.csv(file.choose(), header = TRUE) Separate data collected from two different years: > subset_1978 <-subset(gender_return, y85<1) > subset_1985 <-subset(gender_return, y85>0) > gender_return_all <- lm(lwage y85+educ+exper+i(exper^2)+union+female,data = gender_return ) > anova(gender_return_all) > gender_return_1978 <- lm(lwage educ+exper+i(exper^2)+union+female,data = subset_1978 ) > anova(gender_return_1978) > gender_return_1985 <- lm(lwage educ+exper+i(exper^2)+union+female,data = subset_1985 ) > anova(gender_return_1985) 10 / 30

11 Pooled Cross Sections Method 2 (applicable to two-period models only) Add a time dummy in the model and Interact each explanatory variable with the time dummy: y i = β 0 +δ 0 d2+δ 1 d2 x i1 +β 1 x i1 +δ 2 d2 x i2 +β 2 x i2 + +δ k d2 x ik +β k x ik +u i Jointly test the linear hypothesis: H 0 : δ 0 = δ 1 = = δ k = 0. If rejected, then there is the structural change. 11 / 30

12 Pooled Cross Sections Policy Analysis Using the pooled cross section data, it is possible to do policy analysis. Suppose a policy (mostly a dummy) was employed at a time, then we can see whether it caused effect on the dependent variable hereafter. The change of the environment where the samples operate cause by some exogenous event is called a natural experiment (quasi-experiment). Let z be the scale of a policy, then with the two-period pooled cross-section data, the regression model is: y = β 0 + δ 0 d2 + β 1 z + δ 1 d2 z+ other control variables. (eq. 2) when z = 1, it means the sample is in the scale of a policy (treatment group in the experiment); otherwise z = 0 (control group in the experiment); d2 = 1 is the time after the policy was employed, otherwise, d2 = / 30

13 Pooled Cross Sections Suppose there is a representative sample ( x) who is: (i) not covered by the policy (z = 0), then in the first period (d2 = 0) before the policy is employed, then the average effect of the policy and time is ȳ z=0,d2=0 = ˆβ 0. (ii) not covered by the policy (z = 0), then in the second period (d2 = 1) after the policy is employed, the average effect of the policy and time is ȳ z=0,d2=1 = ˆβ 0 + ˆδ 0. (iii) covered by the policy (z = 1), then in the first period (d2 = 0) before the policy is employed, the average effect of the policy and time is ȳ z=1,d2=0 = ˆβ 0 + ˆβ 1. (iv) covered by the policy (z = 1), then in the second period (d2 = 1) after the policy, the average effect of the policy and time is ȳ z=1,d2=1 = ˆβ 0 + ˆδ 0 + ˆβ 1 + ˆδ 1. the average treatment effect is: ˆδ 1 = (ȳ z=1,d2=1 ȳ z=1,d2=0 ) (ȳ z=0,d2=1 ȳ z=0,d2=0 ) = ( y treatment y control ). 13 / 30

14 Pooled Cross Sections Example 13.3 Figure: Effect of a Garbage Incinerator s Location on Housing Prices (Column 3, Table 13.2) 14 / 30

15 Two-Period Panel Data Analysis Two-Period Panel Data Analysis Omitted Variables Motivation: there are some important control variables, but not observable or not available in the data sets. The fixed effect the capture the effect of unobservable characteristics of samples. Let i denote a unit of sample, such as a firm, an individual, a county, and t denote time. A fixed effects model can be written as: y it = β 0 + δ 0 d2 t + β 1 x it1 + β 2 x it2 + + β k x itk + a i + u it, t = 1, 2 (eq. 3) where a i is called fixed effect, a i s unobservable charactersitic (so it is called unobserved heterogeneity). u it is called the idiosyncratic error or time-varying error. 15 / 30

16 Two-Period Panel Data Analysis If both u it and a i are uncorrelated with all variables x it s, let v it = u it + a i, which is also uncorrelated with x it s. (eq. 3) can be written as: y it = β 0 + δ 0 d2 t + β 1 x it1 + β 2 x it2 + + β k x itk + v it v it is called the composite error. Since Assumption 1-4 hold, the ˆβ estimated using pooled OLS is unbiased (Assumption 5 is not guaranteed, so the test of homoskedasticity might follow). However, it is rare that a i are uncorrelated with all variables x it s. When a i affects any x it, the pooled OLS estimates are biased (called heterogeneity bias). Remedy: take a i away. 16 / 30

17 Two-Period Panel Data Analysis In the second period: y i2 = (β 0 + δ 0 ) + β 1 x i21 + β 2 x i β k x i2k + a i + u i2, (d2 t = 1) In the first period: y i1 = β 0 + β 1 x i11 + β 2 x i β k x i1k + a i + u i1, (d2 t = 0) Difference the two equations to obtain the following first-differenced equation: (y i2 y i1 ) = δ 0 + β 1 (x i21 x i11 ) + β 2 (x i22 x i12 ) + + β k (x i2k x i1k ) + (u i2 u i1 ) y i = δ 0 + β 1 x i,1 + β 2 x i,2 + + β k x i,k + u i (eq. 4) where (x i21 x i11 ) = x i,1, (x i22 x i12 ) = x i,2 and so on. 17 / 30

18 Two-Period Panel Data Analysis For (eq. 4), if the strict exogeneity holds, that is u i is uncorrelated with all of x i,1 and x i,2 and x i,k, the OLS method is applicable to estimate δ and β. ˆδ and ˆβ are unbiased. However, (eq. 4) is not always working when Strict exogeneity fails Using the OLS leads to biased estimators. Adding more time-varying variables can solve this problem in some extent. Variation in x itj is small across time Little variation in x ij can lead to large standard error for β j. Using longer differences over time is a way to increase variation. If x ij = 0, it is impossible to estimate (eq. 4) using the OLS since the assumption of no mulitcollinearity is violated. These kinds of time-invariant should be taken away from the model, because it is a part of a i. 18 / 30

19 Two-Period Panel Data Analysis Example 13.5 Figure: Two-Period Panel Data 19 / 30

20 Two-Period Panel Data Analysis Example 13.5 (continued) can t use pooled cross-section data because individual fixed characteristics are correlated with other independent variables. Figure: Sleeping versus Working 20 / 30

21 Two-Period Panel Data Analysis Policy Analysis Like the pooled cross-section data, the two-period panel data are convenient to analyze the effects of policies. Let z be a dummy variable of policy coverage for the sample (z it = 1 if the sample is covered at time t; z it = 0, otherwise) and d2 for the time dummy (d2 = 1 when the sample is in the second period; d2 = 0 otherwise). The the fixed-effect model is: y it = β 0 + δ 0 d2 + β 1 z it + a i + u it for t = 1, 2 Take the difference of the two equations, the model becomes y i = δ 0 + β 1 z i + u i (eq. 5) Use OLS to run the model if Assumption 1 to 4 hold. β 1 is the average treatment effect. 21 / 30

22 Two-Period Panel Data Analysis Why no interaction term? Suppose we had estimated (eq. 5), and obtained OLS coefficients ˆδ 0 and ˆβ 1. The average effect of the policy is: y = ˆδ 0 + ˆβ 1 z For a sample who is in the treatment group ( z = 1), the average effect of the policy is y treatment = ˆδ 0 + ˆβ 1. For a sample who is in the control group ( z = 0), the average effect of the policy is y control = ˆδ 0. Therefore, ˆβ 1 = y treatment y control which is equivalent to the definition of ˆδ 1 estimated in pooled cross-section model (eq. 2). 22 / 30

23 Two-Period Panel Data Analysis Example 13.7 Figure: Effect of Drunk Driving Laws on Traffic Fatalities 23 / 30

24 Multi-Period Panel Data Analysis Panel Data with More than Two Time Periods When there are T>2 periods, more time dummies should be added in (eq. 3), y it = δ 1 + δ 2 d2 t + δ 3 d3 t + + δ T dt t + β 1 x it1 + β 2 x it2 + + β k x itk + a i + u it, t = 1, 2,, T Assume strict endogeneity holds, or cov(u is, x itj ) = 0 for all t, s, j. This rules out the effect of current changes in the error on the future explanatory variables (see Chapter 15 if it is violated). To remove the fixed effect a i from all T equations, substract the model of the t-th period from the (t + 1)th period. There will be T 1 differenced equations, y it = δ 2 d2 t + δ 3 d3 t + + δ T dt t + β 1 x it1 + + β k x itk + u it, t = 2, 3,, T (eq. 6) 24 / 30

25 Multi-Period Panel Data Analysis (eq. 6) has no intercept, it is inconvenient to calculate R 2. Look closer, the differenced equations can be shown in the following form for each period, δ 2 + β 1 x it1 + + β k x itk + u it, for t = 2 δ 2 + δ 3 + β 1 x it1 + + β k x itk + u it, for t = 3 δ 3 + δ 4 + β 1 x it1 + + β k x itk + u it, for t = 4. δ T 1 + δ T + β 1 x it1 + + β k x itk + u it, for t = T There are actually an intercept and a time dummy in the equation for each t. (The equation for t = 2 is an exception, we treat its time dummy as an intercept.) Therefore, (eq. 6) can be re-written as: y it = α 0 + α 3 d3 t + + α T dt t + β 1 x it1 + + β k x itk + u it, t = 2, 3,, T (eq. 7) Using OLS for pooled cross-section data to estimate (eq. 7); the estimates of β j are identical in (eq. 6). 25 / 30

26 Multi-Period Panel Data Analysis A New Assupmtion of Uncorrelation When there are more than two-period data, the new assumption that u it is uncorrelated over time. That is, cov( u it, u is ) = 0, for all t s (This assumption is not guaranteed If we assume u it in the pre-differenced equation is uncorrelatd over time.) Only when u it follows the random walk will this assumption be satisfied. Definition: Autogression Model (AR Model) A regression model is called AR(P) y t = a 0 + a 1 y t 1 + a 2 y t a p y t p + e t where e t is the disturbance white noise which satisfies (1) E(e t ) = 0 for all t, (2) E(e 2 t ) = σ2 for all t, and (3) cov(e t, e s ) = 0 for all t s. Random Walk is an AR(1) with a 0 = 0 and a 1 = / 30

27 Multi-Period Panel Data Analysis Example 13.8 Panel data with more than two periods Figure: Data 27 / 30

28 Multi-Period Panel Data Analysis Example 13.8 (Continued) Figure: Missing Value in the Data Set I use the command na.omit( ) to get away the NA in the data. > ezone_nomissing <- na.omit(ezone) 28 / 30

29 Multi-Period Panel Data Analysis Example 13.8 (Continued) Figure: Effect of Enterprise Zones on Unemployment Claims 29 / 30

30 Announcement Homework 1 due on March 11th, Homework 2 due on March 25th, First midterm is scheduled on April 2nd, No class on March 18th, / 30

Econometrics. Week 6. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague

Econometrics. Week 6. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Econometrics Week 6 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 21 Recommended Reading For the today Advanced Panel Data Methods. Chapter 14 (pp.