Introduction 1 Panel Data Analysis

And now for… Panel Data! Panel data has both a time series and cross- section component Observe same (eg) people over time You’ve already used it! Difference-in-differences is a panel (or pooled cross-section) data technique Panel data can be used to address some kinds of omitted variable bias E.g., use “yourself in a later period” as comparison group for yourself today If the omitted variables is fixed over time, this “fixed effect” approach removes bias

Unobserved Fixed Effects Initially consider having two periods of data (t=1, t=2), and suppose the population model is: y it =  0 +  0 d2 t +  1 x it1 +…+  k x itk + a i + u it a i = “person effect” (etc) has no “t” subscript u it = “idiosyncratic error” Person i… …in period t Dummy for t= 2 (intercept shift) a i = time-constant component of the composite error, third subscript: variable #

Unobserved Fixed Effects The population model is y it =  0 +  0 d2 t +  1 x it1 +…+  k x itk + a i + u it If a i is correlated with the x’s, OLS will be biased, since a i is part of the composite error term Aside: this also suffers from autocorrelation  Cov( i1, i2 ) = cov(a i,a i ) + 2cov(u it,a i ) + cov(u i2,u i1 ) = var(a i )  So OLS standard errors biased (downward) – more later. But supposing the u it are not correlated with the x’s – just the fixed part of the error is -- we can “difference out” the unobserved fixed effect… it

First differences Period 2: y i2 =  0 +  0 ∙1 +  1 x i21 +…+  k x i2k + a i + u i2 Period 1: y i1 =  0 +  0 ∙0 +  1 x i11 +…+  k x i1k + a i + u i1 Diff:  y i =  0 +  1  x i1 +…+  k  x ik +  u i  y i,  x i1,…,  x ik : “differenced data” – changes in y, x 1, x 2,…,x k from period 1 to period 2  Need to be careful about organization of the data to be sure compute correct change Model has no correlation between the x’s and the new error term (*just by assumption*), so no bias (Also, autocorrelation taken out)

Differencing w/ Multiple Periods Can extend this method to more periods Simply difference all adjacent periods So if 3 periods, then subtract period 1 from period 2, period 2 from period 3 and have 2 observations per individual; etc.  Also: include dummies for each period, so called “period dummies” or “period effects” Assuming the  u it are uncorrelated over time (and with  x’s) can estimate by OLS Otherwise, autocorrelation (and ov bias) remain

7 Two-period example from textbook Does higher unemployment rate raise crime? Data from:  46 U.S. cities (cross-sectional unit “i”)  in 1982, 1987 (the two years, “t”) Regress crmrte (crimes per 1000 population) on unem (unemployment rate) and a dummy for 1987 First, let’s see the data…

8 crmrteunemd ………

9 Pooled cross-section regression. reg crmrte unem d87, robust Linear regression Number of obs = 92 F( 2, 89) = 0.63 Prob > F = R-squared = Root MSE = | Robust crmrte | Coef. Std. Err. t P>|t| [95% Conf. Interval] unem | d87 | _cons | observations Nothing significant, magnitude of coefficients small

10 First difference regression c = “change” = . reg ccrmrte cunem, robust Linear regression Number of obs = 46 F( 1, 44) = 7.40 Prob > F = R-squared = Root MSE = | Robust ccrmrte | Coef. Std. Err. t P>|t| [95% Conf. Interval] cunem | _cons | Now only 46 observations (why?) Both intercept shift (-- now the constant) and unemployment rate are significant Also: magnitudes larger

11 crmrteccrmrteunemcunemd …………… Data convention: change is in later period observation

12 Why did coefficient estimates get larger and more significant? Perhaps cross-section regression suffered from omitted variables bias [cov(x it,a i ) ≠ 0] Third factors, fixed across the two periods, which raise unemployment rate and lower crime rate  (??) More generous unemployment benefits? … To be clear: taking differences can make omitted variables bias worse in some cases To oversimplify, depends which is larger:  cov(  x it,  u it ) or cov(x it,a i ) Possible example: crime and police

13 More police cause more crime?! (lpolpc = log police per capita). reg crmrte lpolpc d87, robust Linear regression Number of obs = 92 F( 2, 89) = 9.72 Prob > F = R-squared = Root MSE = | Robust crmrte | Coef. Std. Err. t P>|t| [95% Conf. Interval] lpolpc | d87 | _cons | A 100% increase in police officers per capita associated with 41 more crimes per 1,000 population Seems unlikely to be causal! (What’s going on?!)

14 In first differences. reg ccrmrte clpolpc, robust Linear regression Number of obs = 46 F( 1, 44) = 4.13 Prob > F = R-squared = Root MSE = | Robust ccrmrte | Coef. Std. Err. t P>|t| [95% Conf. Interval] clpolpc | _cons | % increase in police officer per capita now associated with 85 more crimes per 1,000 population!! Could it be that omitted variables bias is worse in changes in this case? On the other hand, confidence interval is wide

Bottom line Estimating in “differences” is not a panacea Though we usually trust this variation more than cross-sectional variation, it is not always the case it suffers from less bias  Another example: differencing also exacerbates bias from measurement error (soon!) Instead, as usual, a credible “natural experiment” is always what is really critical 15