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1. MODULE OBJECTIVE
This module attempts to explore the possibilities of panel data modeling. The model gives high stress on data structure. The modelling on such problems usually increases model accuracy. In this section, we highlight the followings for the panel data modeling:
- WHAT IS PANEL DATA?
- MODELLING STRUCTURE
- FIXED EFFECT PANEL DATA MODEL
- RANDOM EFFECT PANEL DATA MODEL
- BALANCED/ UNBALANCED PANEL DATA MODEL WHAT IS PANEL DATA OR LONGITUDINAL DATA? In general, data are usually fitted in the modeling in four different ways. This includes Times series data Cross sectional data Pooled data Panel data Panel data is the cross sectional data but they observed periodically; or else, it is time series data but observed as a cross sectional set up. Panel data or pooled or longitudinal data is nothing but an admixture of cross section and time series data that is data acquired from repeated observation
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over a certain period of time. 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. MODEL REPRESENTATION: In general, we have the following modelling structure for data analysis: Case 1; Model with Cross Sectional Data: Yi = Β 0 + Β 1 xi + Ui For i = 1, 2,….N Case 2; Model with Time Series Data: Yt = Β 0 + Β 1 xt + Ut For t = 1, 2,….T Case 3; Model with Pooled Data: Yit = Β 0 + Β 1 xit + Uit For i = 1, 2,….N & t = 1, 2,….T Case 4; Model with Panel Data: Yit = Β0i + Β 1 xit + Uit For i = 1, 2,….N & t = 1, 2,….T Where, α (= α 1 , α 2 , α 3 , …..αn) is the intercept parameters β = β 1 , β 2 , β 3 , …..βn is the slope parameters Let Yit = βXit + u (^) it where uit = αi + ξit
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Case 1: Independent Regression Yit = α + βXit + u (^) it where i = 1, 2, … N & t = 1, 2, …… T If cov (uit, ujt) = 0; cov (uit, uit-1 ) = 0; E (uit) = 0 and Var (uit) = σ^2
Then we can estimate the model by separating its time component so that we have T regressions each having N observations.
This can be as follows: Yi1 = α + βXi1 + u (^) i1 where i = 1, 2, … N ……………………………………………………….. YiT = α + βXiT + u (^) iT where i = 1, 2, … N Or else , we can have N regressions and each having T observations
This can be as follows: Y1t = α + βX1t + u (^) 1t where t = 1, 2, … T ……………………………………………………….. YNt = α + βXNt + u (^) Nt where t = 1, 2, … T
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Case 2: Pooled Data Approach Yit = α + βXit + u (^) it where i = 1, 2, … N & t = 1, 2, …… T Here, we assume that α (intercept) and the residuals are constant over cross sectional units and time series units. It is, however, very rare in reality. So, we should consider model where intercepts or residuals change over time and across individual. Case 3: Fixed Effect Model Let Yit = αi + βXit + u (^) it Here, variations of individuals and over time are captured in the intercepts. The structure is as follows: Yit = α + … +ηN WNt + … + δT ZiT + βXit + u (^) it Where Wit and Zit are dummy Wit = 1 for i = 1,2,… N & 0, for otherwise Zit = 1 for t = 1,2,… T & 0, for otherwise OLS can be applied to estimate the parameters When i = 1 and t = 1, Y 11 = α + βX 11 + u (^11) No of parameters: N +T η = N – 1 numbers; δ = T-1 numbers; α and β
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Case 4: Random Effect Model Here, variations of individuals and over time are captured in the residuals. Here, random error is composed of error of individual component, error of time component and error of both. REM is represented as follows: Let Yit = α + βXit + u (^) it Where uit = u (^) i + v (^) t + wit ui is error for cross section; vt is error for time series; wit is error for both The assumptions is that ui ~ N (o, σ^2 u ); vt ~ N (o, σ^2 v ); wit ~ N (o, σ^2 w) So for REM, var (uit) = σ^2 u + σ^2 v + σ^2 w ; But for OLS (pooled data), var (uit) = σ^2 w REM can be estimated by OLS, if σ^2 u = σ^2 v = 0 Otherwise, REM is estimated using GLS, which is of two steps Step 1: Estimate REM by OLS and calculate RSS to estimate sample variance Step 2: by using sample variance estimated at the step 1, use GLS to estimate the parameters of the model. If errors are normally distributed (by assumption), MLE can be used Thumb rule: If T > N, use FEM and if N > T, use REM To know the FEM or REM, Hausman Specification Test can be applied
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DIFFERENCE BETWEEN RANDOM EFFECT AND FIXED EFFECT ESTIMATORS
RE estimates are more efficient (or more precise) if αi is uncorrelated with the explanatory variables. Otherwise, the FE estimate is consistent, while the RE estimates are inconsistent.
DIFFERENCE BETWEEN BALANCED AND UNBANACED PANEL Balanced Panel indicates panel data with observations for the same time periods for all individuals. Otherwise, the data are unbalanced_._ 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. This problem cannot be solved by just deleting the units that have missing observations for some time periods.
