α version (only brief introduction so far)

Transcription

1 Econometrics I KS Module 8: Panel Data Econometrics Alexander Ahammer Department of Economics Johannes Kepler University of Linz This version: June 18, 2018 α version (only brief introduction so far) Alexander Ahammer (JKU) Module 8: Panel data 1 / 10

2 Now we consider different data dimensions, cont d Finally, we shall give a very brief introduction into panel data econometrics. Time restrictions prevent us from going into too much detail. Panel econometrics are a crucial component of the subsequent courses, Econometrics II (treatment evaluation) and Microeconometrics. Here we discuss some very basic ideas, What is time series data? What is pooled OLS, what are fixed effects and random effects? How can we estimate these models? We cannot cover a detailed discussion of (1) the underlying assumptions behind these models, and (2) their advantages and disadvantages. In case you have to use panel methods before the subsequent courses start, make sure to read Wooldridge (ch. 13 & 14) in great detail. Alexander Ahammer (JKU) Module 8: Panel data 2 / 10

3 The 2-period model Panel data have a cross-sectional and a time series component you observe the same individuals across time. Consider the simplest framework; you follow each individual i = 1,..., n for T = 2 periods; t = 1 and t = 2. You are interested in the relationship between y and x. Simply ignoring the panel component would lead to the pooled OLS model, y it = β 0 + β 1 x it + u it, i = 1,..., n; t = 1, 2. (1) where, ideally, you would adjust your standard errors to account for the autocorrelation amongst the u it s (see Wooldridge, p. 483; Computing Standard Errors Robust to Serial Correlation and Heteroskedasticity of Unknown Form ). Mechanics and coefficient interpretation are similar to cross-sectional OLS. Alexander Ahammer (JKU) Module 8: Panel data 3 / 10

4 The 2-period model The beauty of panel data is that it allows you to account for unobserved heterogeneity that stays constant over time among individuals. Consider a model where unobserved factors affecting the dependent variable consist of two types: those that are constant and those that vary over time, y it = β 0 + β 1 x it + ψ i + u it (2) xit and u it vary over time, ψ i is time-invariant. Additionally, you may want to allow for different intercepts depending on whether you are in period 1 or period 2, y it = β 0 + δ1{t = 2} + β 1x it + ψ i + u it (3) In (2), the variable ψ i captures all unobserved, time-constant factors that affect y it. In applied work, ψ i is typically referred to as a fixed effect. Alexander Ahammer (JKU) Module 8: Panel data 4 / 10

5 The 2-period model First-differencing How do we estimate the model in 2? Using pooled OLS is inappropriate, because we have to assume that Cov(ψ i, u it ) = 0. Otherwise, we obtain inconsistent estimates. We can easily allow for Cov(ψ i, u it ) 0. For a cross-sectional observation i n, we can write the two periods as y i2 = (β 0 + δ) + β 1 x i2 + ψ i + u i2 for t = 2 (4) y i1 = β 0 + β 1 x i1 + ψ i + u i1 for t = 1 (5) Subtracting (4) from (5), we get (y i2 y i1 ) = δ + β 1 (x i2 x i1 ) + (u i2 u i1 ) (6) or y i = δ + β 1 x i + u i (7) Alexander Ahammer (JKU) Module 8: Panel data 5 / 10

6 The 2-period model First-differencing First-differenced equation y i = δ + β 1 x i + u i (7) In (7), is the change from period 1 to period 2. Importantly, ψ i does not appear in this equation anymore. This is called first-differencing. = We differenced-out the unobserved heterogeneity. Key identifying assumption: Cov( u i, x i ) = 0. This assumption holds if the idiosyncratic error at each time t, u it, is uncorrelated with the explanatory variable in both time periods. This is a variant of the strict exogenenity assumption read Wooldridge (ch. 13, esp. pp ) for more information, also on other underlying assumptions. Alexander Ahammer (JKU) Module 8: Panel data 6 / 10

7 The 2-period model First-differencing Advantages of first-differencing Suppose you want to estimate returns to schooling. The following model for t = 1, 2 holds in the population, wage it = β 0 + β 1 educ it + β 2 exper it + β 3 female i + β 4 ability i + u it (8) If you first-difference the model, you get wage i = δ + β 1 educ i + β 2 exper i + u it (9) where δ is the difference in intercepts between t 1 und t 2. Notice that ability i disappears, because it doesn t have a t subscript? This applies to all time-invariant variables. In such a model, unobserved variables x that are correlated with u only cause problems if they are time-variant. All x you can safely assume to be time-invariant are differenced-out. Disadvantage: You cannot identify parameters of observable, time-invariant variables such as female i. Alexander Ahammer (JKU) Module 8: Panel data 7 / 10

8 Fixed effects model Consider a general fixed effects model, y it = β 1 x it + ψ i + u it, i = 1,..., n; t = 1,..., T (10) First differencing works great with T = 2. The easiest method to get rid of ψ i with large T is to demean. For each i, define the average of equation (10) over time as ȳ i = β 1 x i + ψ i + ū i (11) where, e.g., ȳ i = T 1 T t=1 x it, and so on. Because ψ i is fixed over time, it appears in both (10) and (11). If we subtract one from the other, we obtain (y it ȳ i ) = β 1 (x it x i ) + (u it ū i ) (12) which is called demeaning or within transforming data. You can show that demeaning leads to an equivalent estimate of ˆβ 1 as including a dummy variable for each i in model (10). Alexander Ahammer (JKU) Module 8: Panel data 8 / 10

9 Random effects model Consider the same unobserved effects model as before, y it = β 0 + β 1 x it + ψ i + u it (13) where we include an intercept in order to be able to assume that E(ψ i ) = 0, without loss of generality. The goal of fixed effects estimation is to get rid of ψ i, because we assume it to be correlated with the x it. If we assume ψ i is uncorrelated with each explanatory variable, Cov(x itk, ψ i ) = 0 for all t = 1,..., T and all variables k = 1,..., K, we arrive at the random effects model. It has the advantage of allowing to estimate also coefficients of time-varying variables, but requires a different stricter set of assumptions. Thus, it is hardly used in applied Econometrics nowadays. Check Wooldridge (ch. 14) for more information. Alexander Ahammer (JKU) Module 8: Panel data 9 / 10

10 Literature Main reference: Wooldridge, J. M. (2015). Introductory Econometrics: A Modern Approach, 5th ed., South Western College Publishing. Alexander Ahammer (JKU) Module 8: Panel data 10 / 10