Panel Data

Table of Contents

The Model

Model Estimation

• Fixed Effects Model
  • Dummy Variables Approach
  • Deviations Approach
  • Testing for Fixed Effects
• Random Effects Model
  • Partial Deviations Approach
  • Testing for Random Effects
• Hausman's Test for Fixed or Random Effects

Extensions

• Random Coefficients Model
• Seemingly Unrelated Regression Model

Example: Cost Function

• Data Files (Greene [1999], Chap. 14)
  • Cost.txt
  • Output.txt
• GAUSS Programs:
  1. Analyzing One-Way Effects
     • Fixed Effects Model: Dummy Variable Approach
     • Fixed and Random Effects Models: Deviation Approach
  2. Analyzing Two-Way Effects

The Model

Y_it = X_itb_it + e_it

Let b_it = b and assume e_it = u_i + v_t + e_it where u_i represents the individual or cross section difference in intercept and v_t is the time difference in intercept. Two-ways analysis includes both time and individual effects. For simplicity, we further assume v_t = 0. That is, there is no time effect. In other words, only the one-way individual effects will be analyzed in the following.

The component e_it is a classical error term, with zero mean, homogeneous variance, and there is no serial correlation and no contemporary correlation. Also, e_it is uncorrelated with the regressors X_it. That is,

• E(e_it) = 0
• E(e_it²) = s²_e
• E(e_ite_it) = 0, for t¹t
• E(e_ite_jt) = 0, for i¹j
• E(X_ite_it) = 0

Fixed Effects Model

Assume that the error component u_i, the individual difference, is fixed or nonstochastic (but it varies across individuals). Thus, the model error is simply e_it = e_it. The model is expressed as:

Y_it = (X_itb + u_i) + e_it

where u_i is interpreted as the change in the intercept. Therefore the individual effect is defined as u_i plus the intercept.

Random Effects Model

Assume that the error component u_i, the individual difference, is random and satisfies the following assumptions:

• E(u_i) = 0 (zero mean)
• E(u_i²) = s²_u (homoscedasticity)
• E(u_iu_j) = 0 for i¹j (no cross-section correlation)
• E(u_ie_it) = E(u_ie_jt) = 0 (independent from each e_it or e_jt)

• E(e_it) = E(u_i + e_it) = 0
• E(e_it²) = E[(u_i + e_it)²] = s²_u + s²_e
• E(e_ite_it) = E[(u_i + e_it)(u_i + e_it)] = s²_u, for t¹t
• E(e_ite_jt) = E[(u_i + e_it)(u_j + e_jt)] = 0, for i¹j

S =	é ê ê ë	s2e+s2u s2u .. s2u s2u s2e+s2u .. s2u : : : : s2u s2u .. s2e+s2u
ù ú ú û	= s2eI + s2u

Let e be a NT-element vector of the stacked errors e₁, e₂, ..., e_N, e = [e₁,e₂, ..., e_N]', then E(e) = 0 and E(ee') = SÄI, where I is an NxN identity matrix and S is the TxT variance-covariance matrix defined above.

Model Estimation

Fixed Effects Model

Y_it = (X_itb + u_i) + e_it (i=1,2,...,N; t=1,2,...,T).

Let Y_i = [Y_i1,Y_i2,...,Y_iT]', X_i = [X_i1,X_i2,...,X_iT]', and e_i = [e_i1,e_i2,...,e_iT]', then the pooled (stacked) model is

é ê ê ë	Y1 Y2 : YN	ù ú ú û	=	é ê ê ë	X1 X2 : XN
ù ú ú û	b +	é ê ê ë	e1 e2 : eN	ù ú ú û

or, Y = Xb + e

• Dummy Variables Approach

  For each i, define NT´1 vector D_i with the element:

  Dij =	1 if (i-1)´T+1 £ j £ i´T
  	0 otherwise

  Then D = [D₁, D₂, ..., D_N-1] is NT´(N-1) matrix of N-1 dummy variables. Ordinary least squares can be used to estimate the model with dummy variables as follows:

  Y = Xb + Dd +e

  Since X includes a constant term, one less dummy variables are included for estimation and the estimated d measures the individual change from the intercept.

• Deviation Approach
  

  Let Y^m_i = (S_t=1,2,...,TY_it)/T, X^m_i = (S_t=1,2,...,TX_it)/T, and e^m_i = (S_t=1,2,...,Te_it)/T. Then the within estimates of the model can be obtained by estimating the mean deviation model:

  (Y_it - Y^m_i) = (X_it - X^m_i)b + (e_it - e^m_i)

  Or, equivalently

  Y_it = X_itb + (Y^m_i - X^m_ib) + (e_it - e^m_i)

  Note that the constant term drops out due to mean deviation transformation. Therefore, the estimated individual effects of the model is u_i = Y^m_i - X^m_ib. The variance-covariance matrix of individual effects is estimated as follows:

  Var(u_i) = v/T + X^m_i [Var(b)] X^m_i'

  where v is the estimated variance of the mean deviation regression corrected for the degree of freedom NT-N-K (instead of NT-K). That is,

  v = S_i=1,2,...,NS_t=1,2,...,T (e_it - e^m_i)² / (NT-N-K).

  Note that K is the number of explanatory variables not counting the constant term.

  It may be of interest to estimate the between parameters of the model by estimating

  Y^m_i = X^m_ib + u_i + e^m_i

  which is related to the estimated individual effects from the within estimates.

• Testing for Fixed Effects

  Based on the dummy variable approach, this is a Wald F-test for the joint significance of the parameters associated with dummy variables representing the individual effects. If the null hypothesis d = 0 can not be rejected, then there is no fixed effects in the model.

  Based on the deviation approach, the equivalent test statistic is computed from the restricted (pooled model) and unrestricted (mean deviation model) sum of squared residuals. That is,

  RSSR-RSSU N-1       RSSU NT-N-K

  ~ F(N-1, NT-N-K)

Random Effects Model

Y = Xb + e

where e = [e₁,e₂,...,e_N]', e_i = [e_i1,e_i2,...,e_iT]', and the random error components e_it = u_i + e_it. By assumptions, E(e) = 0, and E(ee') = SÄI. The Generalized Least Squares estimates of b is

b = [X'(S^-1ÄI)X]^-1X'(S^-1ÄI)Y

Since S^-1 can be derived from the estimated variance components s²_e and s²_u, in practice the model is estimated using the following partial deviation approach.

• Partial Deviation Approach
  1. From the dummy variable approach for estimating the fixed effect model, the estimated variance s²_e is obtained.

  2. Assuming the randomness of u_i, estimate the between parameters of the model:

     Y^m_i = X^m_ib + (u_i + e^m_i)

     where the error structure of u_i + e^m_i satisfies:

     E(u_i + e^m_i) = 0
     E((u_i + e^m_i)²) = s²_u + s²_e/T
     E((u_i + e^m_i)(u_j + e^m_j)) = 0, for i¹j

     Let v = s²_e and v₁ = T s²_u + s²_e. Define w = 1 - (v/v₁)^½.

  3. Using w to transform (partial deviations) all the data series as follows:

     Y^*_it = Y_it - w Y^m_i
     X^*_it = X_it - w X^m_i

     Then the model for estimation is:

     Y^*_it = X^*_itb + e^*_it

     where e^*_it = (1-w) u_i + e_it - w e^m_i.

     Or, equivalently

     Y_it = X_itb + w (Y^m_i - X^m_ib) + e^*_it

     It is easy to validate that

     E(e^*_it) = 0
     E(e^*2_it) = s²_e
     E(e^*_ite^*_it) = 0 for t¹t
     E(e^*_ite^*_jt) = 0 for i¹j

     The least squares estimate of [w (Y^m_i - X^m_ib)] is interpreted as the change of individual effects.

• Testing for Random Effects

  To test for no correlation relationship of the error terms u_i + e_it and u_i + e_it, the following Breusch-Pagan LM test statistic based on the estimated residuals of the restricted (pooled) model, e_it (i=1,2,...N, t=1,2,...,T), is distributed as Chi-square with one degree of freedom:

  LM =

  NT 2(T-1)

  æ ç è

  Si=1,2,...N (St=1,2,...Teit)2 Si=1,2,...,N St=1,2,...,Teit2

  - 1

  ö ÷ ø

  2

  =

  NT 2(T-1)

  æ ç è

  Si=1,2,...N (Temi)2 Si=1,2,...,N St=1,2,...,Teit2

  - 1

  ö ÷ ø

  2

  Note that e^m_i = S_t=1,2,...,Te_it/T.

Hausman's Test for Fixed or Random Effects

H = (b_random-b_fixed)'[Var(b_random)-Var(b_fixed)]^-1(b_random-b_fixed)

Extensions

Unbalanced Panel Data

Random Coefficients Model

Y_i = X_ib_i + e_i
b_i = b + u_i

where Y_i = [Y_i1,Y_i2,...,Y_iT]', X_i = [X_i1,X_i2,...,X_iT]', and e_i = [e_i1,e_i2,...,e_iT]'. We note that not only the intercept but also the slope parameters are random across individuals. The assumptions of the model are:

• E(e_i) = 0_Nx1
  
• Var(e_i) = E(e_ie_i') = s_i²I_NxN
  
• Cov(e_i,e_j) = 0_NxN, i¹j

• E(u_i) = 0_Kx1
  
• Var(u_i) = E(u_iu_i') = G_KxK
  
• Cov(u_i,u_j) = 0_KxK, i¹j
  
• Cov(u_i,e_i) = 0_Kx1

The model for estimation is

Y_i = X_ib + (X_iu_i + e_i), or
Y_i = X_ib + w_i where w_i = X_iu_i + e_i, and

• E(w_i) = 0_Nx1
• Var(w_i) = E(w_iw_i') = E(X_iuu'X_i'+X_iu_ie+e_iu_i'X_i+e_ie_i')
  = s_i²I_NxN + X_iGX_i' = W_i

The stacked (pooled) model is

Y = Xb + w

where w = [w₁,...,w_N]', and

E(w) = 0_NTx1

Var(w) = E(ww') = V =	é ê ê ë	W1 0 .. 0 0 W2 .. 0 : : : : 0 0 .. WN
ù ú ú û

GLS is used to estimate the model. That is,

b^* = (X'V^-1X)^-1X'V^-1Y
Var(b^*) = (X'V^-1X)^-1

The computation is based on the following steps (Swamy, 1971):

1. For each regression equation i, Y_i = X_ib_i + e_i, obtain the OLS estimator of b_i:
   b_i = (X_i'X_i)^-1X_i'Y_i
   Var(b_i) = (X_i'X_i)^-1(X_i'W_iX_i)(X_i'X_i)^-1 = s_i²(X_i'X_i)^-1+G = V_i+G
   (Taking account of heteroscedasticity, where V_i = s_i²(X_i'X_i)^-1)
   Note that s_i² is estimated by s²_i = e_i'e_i/(N-K), where e_i = Y_i - X_ib_i.
   Then, V_i = s_i²(X_i'X_i)^-1.

2. For the random coefficients equation, b_i = b + u_i, the variance of b_i (estimator of b_i) is estimated by å_i=1,...,G(b_i-b^m)(b_i-b^m)'/(G-1) = å_i=1,...,G(b_ib_i'-G b^mb^m')/(G-1), where b^m = å_i=1,...,Gb_i/G.
   Therefore, G = å_i=1,...,G(b_ib_i'-G b^mb^m')/(G-1) - å_i=1,...,GV_i/G
   Concerning the possibility that G may be nonpositive definite, we use
   G = å_i=1,...,G(b_ib_i'-G b^mb^m')/(G-1).

3. Write the GLS estimator of b as:
   b^* = (X'V^-1X)^-1X'V^-1Y
   = [å_i=1,...,GX_i'W_iX_i]^-1 [å_i=1,...,GX_i'W_iY_i]
   = [å_i=1,...,GX_i'W_iX_i]^-1 [å_i=1,...,GX_i'W_iX_ib_i]
   = [å_i=1,...,G(G+V_i)^-1]^-1 [(G+V_i)^-1b_i]
   = å_i=1,...,GW_ib_i, where W_i = [å_i=1,...,G(G+V_i)^-1]^-1 [(G+V_i)^-1].
   Similarly,
   Var(b^*) = (X'V^-1X)^-1 = [å_i=1,...,G(G+V_i)^-1]^-1

The individual parameter vectors may be predicted as follows:

b_i^* = (G+V_i)^-1[G^-1b^*+V_i^-1b_i] = A_ib^* + (I-A_i)b_i,
where A_i = (G+V_i)^-1G^-1.

Var(bi*) = [Ai		I-Ai]	é ë	åi=1,2,...,GWi(G+Vi)Wi'    Wi(G+Vi) (G+Vi)Wi'    (G+Vi)
ù û	é ë	Ai I-Ai	ù û

Seemingly Unrelated System Model

Y_it = X_itb_i + e_it (i=1,2,...,N; t=1,2,...,T).

Let Y_i = [Y_i1,Y_i2,...,Y_iT]', X_i = [X_i1,X_i2,...,X_iT]', and e_i = [e_i1,e_i2,...,e_iT]', the stacked N equations (T observations each) system is Y = Xb + e, or

é ê ê ë	Y1 Y2 : YN	ù ú ú û	=	é ê ê ë	X1 0 .. 0 0 X2 .. 0 : : : : 0 0 .. XN
ù ú ú û	é ê ê ë	b1 b2 : bN	ù ú ú û	+	é ê ê ë	e1 e2 : eN	ù ú ú û

Notice that not only the intercept but also the slope terms of the estimated parameters are different across individuals. The error structure of the model is summarized as follows:

• E(e) = 0
• E(X'e) = 0
• E(ee') = SÄI, where S = [s_ij, i,j=1,2,...N] is the NxN variance-covariance matrix and I is a TxT identity matrix. Notice that contemporary correlation across individuals is assumed although there is no time serial correlation. The error structure of this model is different than that of random effects model described above.

The model is estimated using techniques for systems of regression equations.

The system estimation techniques such as 3SLS and FIML should be used for parameter estimation. It is called the Seemingly Unrelated Regression Estimation (SURE) in the current context. Denote b and S as the estimated b and S, respectively. Then,

b = [X'(S^-1ÄI)X]^-1X'(S^-1ÄI)Y
Var(b) = [X'(S^-1ÄI)X]^-1, and
S = ee'/T, where e = Y-Xb is the estimated error e.

More Examples: Cost of Production for Airline Services