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MODELS FOR PANEL DATA

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.

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MODELS FOR PANEL DATA

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  1. MODELS FOR PANEL DATA

  2. 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,tth observation on K explanatory var.

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

  4. TWO-WAY ERROR COMPONENT MODEL • Potentialextension … denotesunobservabletimeeffect individual-invariant accounts for any time-specific effect that is not included in the regression

  5. 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

  6. PANEL DATA REGRESSION • Vectorofdisturbances … matrix ofindividualdummies

  7. Notes aboutmatrices … square matrix of ones of dimension T • matrix P • theprojection matrix on • averages the observations across time for each individual • generates individual means

  8. Notes aboutmatrices • matrix Q • obtains deviations from individual means • Application of P and Q:

  9. PropertiesofPandQ • Symmetric, idempotent • P and Q are othogonal • They sum to the identity matrix

  10. THE FIXED EFFECTS MODEL (FE) • mi’s … assumed to be fixed parameters to be estimated • … assumed to be independent of the vit 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

  11. 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)

  12. 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

  13. 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)!

  14. THE FIXED EFFECTS MODEL (FE) • Dummy variable trap (perfect multicolinearity) Without additional restriction just (a+mi)’s are estimable, not a and mi‘s separately • Possible restrictions: • Particular • Ad 3:

  15. 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

  16. 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 b is consistent • Estimators of the individual-specific effects (a+mi) 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)

  17. THE FIXED EFFECTS MODEL (FE) • Testing for fixed effects: • test of the joint significance of the individual dummies • H0: • F-test: • Restricted model : model without individual dummies • Unrestricted model : model with individual dummies (FE model)

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