Panel Data

1
Panel Data

• Course Applied Econometrics
• Lecturer Zhigang Li

2
Outline

• Panel Data
• Fixed-effects vs. random-effects
• First-differencing or fixed-effects
• Strict Exogeneity Assumption

3
Panel Data (or Longitudinal Data)

• A typical panel data set has both a
  cross-sectional dimension and a time series
  dimension. In particular, the same
  cross-sectional units (e.g. individuals,
  families, firms, cities, states) are observed
  over time.
• Panel data is different from pooling
  independent cross sections across time (or
  pooled OLS). Estimating the latter is a simple
  extension of OLS.

4
Large N or Large T?

• N is the number of cross-sectional units and
  T is the number of time periods.
• Small N and small T (of little use)
• Large N and small T (Traditional Panel Data)
• N is large enough for the Law of Large Numbers to
  apply while T is not.
• Convenient to use if cross-sectional units are
  independent.
• Small N and Large T
• T is large enough for the Law of Large Numbers to
  apply while N is not.
• Autocorrelation has to be addressed.
• Large N and Large T (Still under exploration)

5
Fixed Effects Panel-data Model(individual-specifi
c intercepts)

• yitß0ßtß1xit1ß2xit2aiuit
• Strict Exogeneity Assumption
• Cov(Xit,uis)0 for all t and s
• Ruling out dynamic models, which have lagged
  dependent variables (e.g. yi,t-1) as explanatory
  variables. Models with the lags of dependent
  variables as ind. Var. are still fine.
• The effects of time-constant independent
  variables can not be directly estimated because
  they are mixed in ai
• ßt (time-specific intercepts) controls for common
  shocks to all agents at period t.

6
Names

• The individual-specific intercept ai may be
  called ai fixed effect or unobserved
  heterogenity.
• The term uit is called idiosyncratic error.
• The sum aiuit is often called the composite
  error.
• If Cov(Xit,ai) is nonzero but the pooled OLS
  method is used, estimates of all parameters might
  be biased.This bias can be called heterogeneity
  bias.
• Balanced Panel indicates panel data with
  observations for the same time periods for all
  individuals. Otherwise, the data are unbalanced.

7
Random Effects Models

• yitß0ßtß1xit1ß2xit2aiuit
• Key assumption
• ai is uncorrelated with each explanatory variable
  in all time periods.
• Difference between RE and FE estimators
• In FE, we effectively control for ai using dummy
  variables.
• In RE, ai is omitted and is part of the
  disturbance
• RE estimates are more efficient (or more precise)
  if the RE assumption is valid.

8
Random Effects Models (continued)

• Difference between RE and pooled OLS
• Since ai is in the error term, observations over
  time are correlated for the same individual i
• In RE approach, the correlation over time is
  eliminated using some sophisticated GLS
  (generalized least square) method.
• In pooled OLS, the GLS correction is not used.
• Hauman test
• Compare the RE and FE estimates, if the estimates
  are very different, then the RE assumption is
  probably invalid. In this case FE has to be used.
  Otherwise, RE is more efficient.

9
Estimation of the Fixed-effect Panel Data Model

• Fixed-effects (or Within) Estimator
• Each variable is demeaned (i.e. subtracted by its
  average)
• Dummy Variable Regression (i.e. put in a dummy
  variable for each cross-sectional unit, along
  with other explanatory variables.) This may cause
  estimation difficulty when N is large.
• First-difference Estimator
• Each variable is differenced once over time, so
  we are effectively estimating the relationship
  between changes of variables.

10
First Differencing or Fixed-Effect?

• Theoretically, when N is large and T is small but
  greater than 2, FE is more efficient when uit are
  serially uncorrelated while FD is more efficient
  when uit follows a random walk.
• When T is large and N is small
• FD has advantage for processes with large
  positive autocorrelation. FE is more sensitive to
  nonnormality, heteroskedasticity, and serial
  correlation in the idiosyncratic errors.
• On the other hand, FE is less sensitive to
  violation of the strict exogeneity assumption. So
  FE is preferred when the processes are weakly
  dependent over time.

11
With Classical Measurement Errors

• When T2, the measurement error bias using FE
  estimator may be smaller than that with FD
  approach but higher than that with OLS.
  (Griliches and Hausman, 1986)
• Natural IV for Measurement Error Lagged
  dependent variables

12
Violation of the Strict Exogeneity Assumption

• Parameter estimates are inconsistent, natural
  experiment approach (e.g. IV) is needed.

13
With Strict Exogeneity and Dependent Observations

• Parameter estimates are consistent
• Standard errors estimates could still be biased
• Cross-sectional correlation or serial correlation
  (over time) in error terms
• Heteroskedasticity

14
Possible Solutions (Need Large N and Zero
Cross-Sectional Correlation)

• Heteroskedasticity
• Use White robust standard errors
• Autocorrelation
• Group the sample time dimension into two periods
  and apply the first-difference estimator (need
  large N). (Perform the best with D-in-D approach
  by Bertrand et al. 2004)
• Clustered robust errors
• Newey-West standard errors (which also accounts
  for heteroskedasticity)
• Cross-sectional Correlations
• Clustered robust errors

15
Clustered Standard Errors

• Key Assumption
• Correlations within a cluster (a group of firms,
  a region, different years for the same firm,
  different years for the same region) are the same
  are the same for different observations.
• Procedure
• Identify clusters using economic theory
  (clustered by industry, year, industry and year)
• Let computer calculate clustered standard errors
• Try different ways of defining clusters and see
  how estimated standard errors are affected.

16
Unbalanced Panels

• If a panel data set is unbalanced for reasons
  uncorrelated with uit, estimation consistency
  using FE will not be affected
• The attrition problem If an unbalanced panel
  is a result of some selection process related to
  uit, then endogeneity problem is present and need
  to be dealt with using some correction methods.