Autocorrelation in Panel Data

Readings and References:

• B. H. Baltagi, Econopmetric Analysis of Panel Data, 4th. ed., John Wiley, 2008.
• B. H. Baltagi and Q. Li, "A Transformation That Will Circumvent the Problem of Autocorrelation in an Error Component Model," Journal of Econometrics 48, 385-393, 1991.
• B. H. Baltagi and P. X. Wu, "Unequally Spaced Panel Data Regression with AR(1) Disturbances," Econometric Theory 15, 814-823, 1999.
• C. Hsiao, Analysis of Panel Data, 2nd. ed., Cambridge University Press, 2003.
• W. H. Greene, Econometric Analysis, 5th ed., Prentice Hall, 2002.
• J. M. Wooldridge, Econometric Analysis of Cross Section and Panel Data, The MIT Press, 2002.

Y_it = X_itb + u_i + e_it
e_it = re_it-1 + v_it

where |r| < 0 and v_it ~ iid(0,s_v²). Both the fixed effects model and random effects model are possible.

Model Estimation

• Estimating r

  From a mean deviation regression, remove the individual effect u_i:

  Y_it-Y_i^m = X_it-X_i^mb + e_it-e_i^m

  where Y_i^m = å_t=1,...,TY_it, X_i^m = å_t=1,...,TX_it, e_i^m = å_t=1,...,Te_it.

  We write,

  Y_it^* = X_it^*b + e_it^*

  where Y_it^* = Y_it-Y_i^m, X_it^* = X_it-X_i^m, and e_it^* = e_it-e_i^m.

  The one-step estimator of r is obtained by

  å_i=1,...,Nå_t=2,...,Te_it^*e_it-1^* / å_i=1,...,Nå_t=1,...,Te_it^*2

  If the panel data is unbalanced, we have a sample of data in which each individual i has T_i observations over time. To deal with autocorrelation in panel data, a further complication of unequally spaced time periods may occurred. Specifically, the data may contain observations on individual i at time t_ij for j=1,2,...,N_i.

  For the first-order autocorrelation, set e_itij^* = 0 if t_ij-t_ij-1 > 0. Then the modified one-step estimator r

  (å_i=1,...,Nå_t=2,...,Te_it^*e_it-1^*)/m^* / (å_i=1,...,Nå_t=1,...,Te_it^*2)/n^*

  where n^* is the number of nonzero elements in e^* and m^* is the number of consecutive pairs of nonzeros e_it^*s.

  The efficiency of the estimator of r may be improved by using Prais-Winsten or Cochrane-Orcutt iterative method.

• Transforming Data

  To remove the AR(1) component in the data, the following transformation is used:

  Z_it^** = Z_it^* - rZ_it-1^* for t > 1, and
  Z_i1^** = (1-r²)^½Z_i1^*

  Denote Y_it^**, X_it^**, and e_it^** for such transformation of Y_it^*, X_it^*, and e_it^*, respectively.

  For unbalanced unequally spaced panel data, the transformation must be modified to reflect the missing data.

• Fixed Effects Model

  The within estimator of b can be obtained by mean deviation regression for the transformed model:

  Y_it^** = X_it^**b + e_it^**

• Random Effects Model (Baltagi-Wu GLS)

  Similar to the weighted mean deviation regression for estimating the classical random effects model, Baltagi and Wu (1999) developed a GLS method for unequally spaced panel data with AR(1) autocorrelation.

  (To Be Continued)

Hypothesis Testing: r = 0

LBI test statistic is described in Baltagi and Wu (1999).

(To Be Continued)