Pooled Cross Sections and Panel Data: Overview Econometrics 2 � Observations over individual units and time: Wooldridge chapters 13 and 14. � Pooling independent cross sections across time (13.1-2). � Panel data: Following the same individual units across time: � Two-period panel data (13.3-4) Pooled Cross Sections and Panel Data I � General case: Twor or more periods Fixed effects estimation (13.5, 14.1) � Random effects estimation (14.2) � � Four lectures to cover these chapters. � Exercises 2 and 3. Pooled Cross Sections and Panel Data 1 Pooled Cross Sections and Panel Data 2 1
Data structures and definitions Pooling independent cross sections across time Cross section (”tværsnit”): Observations on a set of variables in a � given period, t, for individual units i=1,2,…,n: � Independent cross sections for two periods: ( y , x , x ,..., x ) � Pooled (”sammenstykkede”) data: it it 1 it 2 itk = + + Usually think of the cross section as a random sample from some � ( y , x , x ,..., x ), i 1,2,..., n n , 1,...., n n it it 1 it 2 itk 1 1 1 2 population at time t = β + → β ˆ � One extreme: Estimating pooled model: y X u pooled Two period case: � Other extreme: Treat the data in each cross section � = ( y , x , x ,..., x ), i 1,2,..., n separately: Period 1 cross section: � = β + = → β ˆ i 1 i 11 i 12 i k 1 1 1 1 y X u , i 1,2,..., n = + + + Period 2 cross section: ( y , x , x ,..., x ), i n 1, n 2,..., n n � 1 i 2 i 21 i 22 i k 2 1 1 1 2 = β + = + + + → β ˆ 2 2 y X u , i n 1, n 2,..., n n 1 1 1 2 How are the period 1 and period 2 cross sections related? � � ”Partial pooling”: Combine the cross sections but allow the Independent cross sections : Two independently drawn random samples: � coefficients of some variables to change between cross (In general) different individual units in period 1 and period 2. sections. Panel data : Same n individuals appear in period 1 and in period 2. � Pooled Cross Sections and Panel Data 3 Pooled Cross Sections and Panel Data 4 2
Pooling independent cross sections Pooling independent cross sections: Testing β δ = Allow the coefficients of some of the variables to change over time: Testing: Is constant over time? Usual t- test for in 0 � � 1 1 A special case of structural change = β + δ + β + δ + β + β + y d 2 x d 2 x x ... x u Use dummy variables (W ch. 7): Time dummies (e.g. year dummies) i 0 0 i 1 i 1 1 i i 1 2 i 2 k ik i � Two periods: Need one dummy variable, usually for second period: � Allow all coefficients to change over time: No gain from pooling the � cross sections = d 2 1 if individual is in the period 2 sample i i Fully interacted regression: � = 0 if individual is not in the sample in period 2 i = β + δ + β + δ + β + δ y d 2 x d 2 x x d 2 x i 0 0 i 1 i 1 1 i i 1 2 i 2 2 i i 2 Usually: Allow intercept to change � + + β + δ + ... x d 2 x u = β + δ + β + + β + k ik k i ik i y d 2 x ... x u i 0 0 i 1 i 1 k ik i δ = δ = = δ = F- test for ... 0 � 0 1 k Other coefficients allowed to change as well: Interaction terms. Easy implementation of F -statistic: SSRs from pooled and separate � � regressions (”Chow test”) Pooled Cross Sections and Panel Data 5 Pooled Cross Sections and Panel Data 6 3
Pooling independent cross sections Policy analysis with pooled cross sections Wage regression: Example 13.2 � � Example 13.3: Effect of the location of a garbage Two independent cross sections: 1978-CPS, 1985-CPS � incinerator on house prices. Data on wage , educ , exper , expersq, union , female for 1,084 � workers. � Hypothesis: Having an incinerator nearby lowers the Define time dummy y85 . Use 1978-cross section as reference � price of a house. group. � Data: Prices and characteristics of houses in different Question: Has the return to education and/or the gender wage gap � changed between 1978 and 1985. distances to the incinerator. Include above variables and y85, y85*educ, y85*female � � Two cross-sections: 1978 and 1981. Data in CPS78_85.in7, analyze in PcGive. � � Before and ”after” the incinerator was built in 1981. Chow test of overall regression. Is it of interest in this case? Why � not? Pooled Cross Sections and Panel Data 7 Pooled Cross Sections and Panel Data 8 4
Policy analysis with pooled cross sections Policy analysis with pooled cross sections � Naive approach: Use 1981 cross section to estimate the � Difference-in-differences approach: model = γ + γ + price 1 nearinc u � House prices have gone up between 1978 and 1981 for 0 most houses. Whether nearby and far away from the � price is the price of a house, nearinc is a dummy variable that location of the incinerator. takes the value 1 if the house is located near the incinerator. � Relevant question: Has the change been bigger for houses � price = − � OLS estimates using 1981 cross section: 101 31 nearinc far from the incinerator? � Need to look at differences in space (nearby/far away) of � Is this a ”good” estimate of the causal effect on house prices differences in time (between 1978 and 1981): Diff-in-diff. of locating the incinerator nearby? � Regression implementation: � NO! Incinerator may have been located near houses that were already cheap in 1978. = β + δ + β + δ ⋅ + price y 81 nearinc y 81 nearinc u � price = − 0 0 1 1 � OLS estimates using 1978 cross section: 83 19 nearinc Pooled Cross Sections and Panel Data 9 Pooled Cross Sections and Panel Data 10 5
Policy analysis with pooled cross sections Policy analysis with pooled cross sections: Example 13.3 = β + δ + β + δ ⋅ + Model: price y 81 nearinc y 81 nearinc u � 0 0 1 1 = β + δ + β + δ ⋅ + price y 81 nearinc y 81 nearinc u 0 0 1 1 δ = Common change over time � 0 δ = δ < H : 0 vs. H : 0 β = Pre-incinerator difference in prices � 0 1 1 1 1 δ = Change in price due to incinerator � 1 Coefficient Standard 2 R Test of the hypothesis that nearby incinerator lowers house prices: δ ˆ � error 1 δ = δ < Model as -12 7.5 0.17 H : 0 vs. H : 0 0 1 1 1 above Full set of -14 5 0.67 ”controls” Pooled Cross Sections and Panel Data 11 Pooled Cross Sections and Panel Data 12 6
Quasi-experiments and natural experiments Next time � Mimic controlled experiments in science by finding � Panel data: Observations over time for the same something that happened ”naturally” to one group of individual units. people, but not to another. � W sec. 13.3-13.4: Two-period panels � Treated group: Houses nearby the location of the � No exercises this week! Will start next week. incinerator. � No Econometrics 2 lecture on Thursday. � Control group: Houses far away. � Comparing groups before and after the ”treatment”: � IV supplementary course starts Friday, 14-16, in Bisp Building the incinerator 214 Pooled Cross Sections and Panel Data 13 Pooled Cross Sections and Panel Data 14 7
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