MODELS FOR PANEL DATA

PANEL DATA REGRESSION Double subscript on variables (observations) i… households, individuals, firms, countries t… period (time-series dimension) … scalar … vector K × 1 … vector of i,t th observation on K explanatory var.

ONE-WAY ERROR COMPONENT MODEL Utilized by most of the panel data applications … denotes unobservable individual specific effect time-invariant accounts for any individual-specific factor not included in the regression … the remainder disturbance term … varies both with individual and in time

Potential extension … denotes unobservable time effect individual-invariant accounts for any time-specific effect that is not included in the regression TWO-WAY ERROR COMPONENT MODEL

PANEL DATA REGRESSION vector form of the model … vector of ones of dimension NT stacked observations the slower index is index over INDIVIDUALS, the faster index is over TIME

PANEL DATA REGRESSION Vector of disturbances … matrix of individual dummies

Notes about matrices … square matrix of ones of dimension T matrix P – the projection matrix on – averages the observations across time for each individual – generates individual means

Notes about matrices matrix Q – obtains deviations from individual means Application of P and Q:

Properties of P and Q Symmetric, idempotent P and Q are othogonal They sum to the identity matrix

THE FIXED EFFECTS MODEL (FE)  i ’s … assumed to be fixed parameters to be estimated … assumed to be independent of the v it for all i and t FE model is an appropriate specification if we are focusing on a specific set of N individuals (firms, countries,…) Inference is restricted to the behavior of these sets of individuals

THE FIXED EFFECTS MODEL (FE) Model can be rewritten OLS can be used to obtain estimates of unknown parameters BUT! If N is large: – Too many individual dummies are included into the model – Matrix to be inverted by OLS is of dimension (N+K)

THE FIXED EFFECTS MODEL (FE) LSDV (least squares dummy variable) estimator – The model is premultiplied by Q Using the fact that: and – OLS performed on the resulting transformed model – matrix Q wipes out the individual specific effects – LSDV involves the inversion of a K × K matrix

THE FIXED EFFECTS MODEL (FE) LSDV (least squares dummy variable) estimator unbiased estimate of – residual sum of squares from LSDV regression divided by (NT-N-K) – Not by (NT-K)!

THE FIXED EFFECTS MODEL (FE) Dummy variable trap (perfect multicolinearity) Without additional restriction just (  +  i )’s are estimable, not  and  i ‘s separately Possible restrictions: 1. 2.Particular 3. Ad 3:

THE FIXED EFFECTS MODEL (FE) Limitations: – FE is not feasible for large panels (N is large) N-1 dummies included in the model large loss of degrees of freedom (extra N-1 parameters are to be estimated) Too many dummies may aggravate the problem of multicollinearity among regressors – (LSDV) estimator cannot estimate the effect of any time-invariant variable (race, religion, sex,…) Time-invariant variables are wiped out by the Q transformation

THE FIXED EFFECTS MODEL (FE) Properties of LSDV estimator: If is the true model: LSDV is the BLUE as long as as T  LSDV is consistent If T is fixed and N  : – LSDV estimator of  is consistent – Estimators of the individual-specific effects (  +  i ) are not consistent (the number of parameters increases as N increases) OLS on (pooled OLS estimator) yields biased and inconsistent estimates (due to omission variable bias)

THE FIXED EFFECTS MODEL (FE) Testing for fixed effects: – test of the joint significance of the individual dummies – H 0 : – F-test: Restricted model : model without individual dummies Unrestricted model : model with individual dummies (FE model)