Econometrics of Panel Data

Transcription

1 Econometrics of Panel Data Jakub Mućk Meeting # 3 Jakub Mućk Econometrics of Panel Data Meeting # 3 1 / 21

2 Outline 1 Fixed or Random Hausman Test 2 Between Estimator 3 Coefficient of determination (R 2 ) 4 Testing linear hypotheses Jakub Mućk Econometrics of Panel Data Fixed or Random Meeting # 3 2 / 21

3 Fixed or Random Random effects Fixed effects Individual µ i N ( ) 0, σµ 2 α i effects drawn from the random sample α i are assumed to be constant = we can estimate the parameter over time of distribution, i.e., σµ 2 Assumptions: (i) E (µ i ε it) = 0 (i) E (α i ε it) = 0 (ii) E (µ i x it) = 0 individual effects are independent of the explanatory variable x it Estimation GLS OLS (within or LSDV) Efficiency higher lower Additional: impossible to use time invariant regressors (collinearity with α i) Jakub Mućk Econometrics of Panel Data Fixed or Random Meeting # 3 3 / 21

4 Fixed or Random Random and Fixed effects models: RE: y it = α + β 1 x 1it β 1 x kit + µ i + ε it, FE: y it = α i + β 1 x 1it β 1 x kit + u it where µ i N (0, σµ), 2 ε i,t N (0, σε) 2 and u i,t N (0, σu). 2 We can test significance of individual effects: The random effect model: the Langrange multiplier statistic: H 0 : σµ 2 = 0 The fixed effect model: test for poolability: H 0 : α = α 1 =... = α N. Can we compare the FE and RE models? Jakub Mućk Econometrics of Panel Data Fixed or Random Meeting # 3 4 / 21

5 Classical example Cobb-Douglas production function. Consider the following model: y it = βx it + u it + η i (1) yit the log output. xit the log of a variable input; ηi an firm input that is constant over time, e.g., managerial skills. uit a stochastic input that is outside framer s control. β - the technological parameter. η i is the unobserved individual effect. Let s assume that the input (x it ) depends on the quality of managerial skills (individuals effects η i ) = E (η i x it ) 0 = the RE estimator is not consistent. Jakub Mućk Econometrics of Panel Data Fixed or Random Meeting # 3 5 / 21

6 Hausman Test The null H 0 in the Hausman test is that both the random and fixed effect estimates are consistent. If the alternative hypothesis H 1 holds then the random effect estimates are inconsistent. Test statistics: H = [ ˆβFE ˆβ RE] ( Var ˆβ FE Var ˆβ RE) 1 [ ˆβFE ˆβ RE] (2) The statistics H is distributed χ 2 with degrees of freedom determined by K, i.e., the dimension of the coefficient vector β. Jakub Mućk Econometrics of Panel Data Fixed or Random Meeting # 3 6 / 21

7 Hausman Test- some remarks In general, the Hausman test asks whether the fixed effects and random effects estimates of β are significantly different. We can test only models with the same set of explanatory variables: We cannot compare the random effects estimates corresponding to time-invariant regressors due to their collinearity with individual intercept. The rejection of the null hypothesis indicates that the random effect estimates of β are not consistent or that the model is wrongly specified (misspecification error). It is assumed that the fixed effect model is consistent under both null and alternative. What if regressors are not strictly exogenous? The Hausman may be used in more general context. Jakub Mućk Econometrics of Panel Data Fixed or Random Meeting # 3 7 / 21

8 Empirical example Pooled OLS FE RE α (9.512) (12.454) (28.899) [0.000] [0.000] [0.045] β (0.025) (0.017) (0.012) [0.000] [0.000] [0.000] β (0.006) (0.012) (0.010) [0.000] [0.000] [0.000] Note: The expressions in round and squared brackets stand for standard errors and probability values corresponding to the null about parameter s insignificance, respectively. I it = α + β 1 K it + β 2 F it + u it Jakub Mućk Econometrics of Panel Data Fixed or Random Meeting # 3 8 / 21

9 Empirical example The RE and FE estimates: [ ˆβ FE = ], ˆβRE = [ ] = ˆβ FE ˆβ RE = [ ] The estimates of the variance-covariance matrix: ( ) [ Var ˆβFE = ], Var ( ˆβRE ) = [ ], but diag The test statistics: The probability value: ( ( ) ( )) [ Var ˆβFE Var ˆβRE = ] Jakub Mućk Econometrics of Panel Data Fixed or Random Meeting # 3 9 / 21

10 Outline 1 Fixed or Random Hausman Test 2 Between Estimator 3 Coefficient of determination (R 2 ) 4 Testing linear hypotheses Jakub Mućk Econometrics of Panel Data Between Estimator Meeting # 3 10 / 21

11 Between Estimator The between estimator uses just cross-sectional variation. Averaging over time yields: ȳ i = α Between + β Between 1 x 1i β Between k x ki + ū i, (3) where ȳ i = T 1 t y it, x 1i = T 1 t x 1it,..., x ki = T 1 t x kit, ū i = T 1 t u it. or in the matrix form: ȳ = α Between + xβ Between + u. (4) The parameters α Between, β1 Between,..., βk Between OLS estimator. can be estimated with the Jakub Mućk Econometrics of Panel Data Between Estimator Meeting # 3 11 / 21

12 Empirical example Pooled OLS FE RE BE α (9.512) (12.454) (28.899) (47.515) [0.000] [0.000] [0.045] [0.863] β (0.025) (0.017) (0.012) (0.191) [0.000] [0.000] [0.000] [0.872] β (0.006) (0.012) (0.010) (0.029) [0.000] [0.000] [0.000] [0.002] Note: The expressions in round and squared brackets stand for standard errors and probability values corresponding to the null about parameter s insignificance, respectively. I it = α + β 1 K it + β 2 F it + u it Jakub Mućk Econometrics of Panel Data Between Estimator Meeting # 3 12 / 21

13 Outline 1 Fixed or Random Hausman Test 2 Between Estimator 3 Coefficient of determination (R 2 ) 4 Testing linear hypotheses Jakub Mućk Econometrics of Panel Data Coefficient of determination (R 2 ) Meeting # 3 13 / 21

14 Coefficient of determination (R 2 ) Coefficient of determination (R 2 ): is an indicator showing how well data fit a statistical model. General definition: R 2 = 1 ESS TSS, (5) where ESS - is the residuals sum of squares: ESS = (ŷ i y) 2, (6) i TSS - is the total (observed) sum of square, TSS = (y i ȳ) 2. (7) i R 2 is between 0 and 1. In the context of panel data, the conventional coefficient of determination is maximized by the pooled estimator. Jakub Mućk Econometrics of Panel Data Coefficient of determination (R 2 ) Meeting # 3 14 / 21

15 The within R 2 The within R 2 : where ESSWithin - is the residuals sum of squares: R 2 Within = 1 ESS Within TSS Within, (8) ESS Within = N i=1 T t=1 ( yˆ it y it) 2, (9) TSSWithin - is the total (observed) sum of square, TSS Within = N i=1 T (y it ȳ i) 2. (10) t=1 The within R 2 is maximized by the within estimator. Jakub Mućk Econometrics of Panel Data Coefficient of determination (R 2 ) Meeting # 3 15 / 21

16 The between R 2 The between R 2 : where R 2 Between = 1 ESS Between TSS Between, (11) ESSBetween - is the residuals sum of squares: ESS Between = N ( ) 2 ˆȳ i ȳ i, (12) i=1 TSSBetween - is the total (observed) sum of square, TSS Between = N (ȳ i ȳ) 2. (13) i=1 The between R 2 is maximized by the between estimator. Jakub Mućk Econometrics of Panel Data Coefficient of determination (R 2 ) Meeting # 3 16 / 21

17 Empirical example Table: The comparison of the coefficients of determination R 2 Pooled FE RE BE the overall R the within R the between R Jakub Mućk Econometrics of Panel Data Coefficient of determination (R 2 ) Meeting # 3 17 / 21

18 Outline 1 Fixed or Random Hausman Test 2 Between Estimator 3 Coefficient of determination (R 2 ) 4 Testing linear hypotheses Jakub Mućk Econometrics of Panel Data Testing linear hypotheses Meeting # 3 18 / 21

19 Testing linear hypotheses Statistical models are usually formulated in order to verify testable implications of the economic theory. A set of linear hypotheses can be described as follows: Rβ = q, (14) where β is the vector of parameters, the matrix R and vector q describe the linear constraints. R is m (K + 1) and q is m 1 when m is the number of linear constraints. Example. Consider the case when k = 4 and the following three hypotheses: β 1 + β 3 = 1 β 2 β 4 = 0 β β 4 = 2 then the matrix R and vector q: R = and q = Jakub Mućk Econometrics of Panel Data Testing linear hypotheses Meeting # 3 19 /

20 Testing linear hypotheses Restricted Least Squares (RLS) estimator: ˆβ RLS = ˆβ OLS + ( X X ) ( 1 R R ( X X ) 1 R ) 1 ( ) q R ˆβOLS, (15) where ˆβ OLS is the OLS estimator. Jakub Mućk Econometrics of Panel Data Testing linear hypotheses Meeting # 3 20 / 21

21 Testing linear hypotheses Restricted Least Squares (RLS) estimator: ˆβ RLS = ˆβ OLS + ( X X ) ( 1 R R ( X X ) 1 R ) 1 ( ) q R ˆβOLS, (15) where ˆβ OLS is the OLS estimator. Wald test: Test statistics for m linear hypotheses: H 0 : Rβ = q H 1 : Rβ q F = (Rβ q) ( R (X X) 1 R ) 1 (Rβ q) m if true then F F(m, NT K) where K is the number of regressors and NT refers to the number of observations. Alternatively: W = (Rβ q) ( R ( X X ) 1 R ) 1 (Rβ q) = mf (17) The W is χ 2 distributed with m degree of freedoms. Jakub Mućk Econometrics of Panel Data Testing linear hypotheses Meeting # 3 20 / 21 (16)

22 Testing linear hypotheses general remarks The rank of the matrix R should equal the number of restrictions (m). The test statistics F is preferable in smaller samples. The test statistics, both F and W, are sometimes very sensitive to the estimates of the variance-covariance of the error terms. Viewed from the empirical perspective, it s essentially to investigate whether the obtained results are not sensitive to a choice of the variance-covariance estimator. Alternative procedures can be used here, e.g. the Lagrange multiplier test (score test) or the likelihood-ratio test. Engle (1983) shows that these tests (Wald, score & LR) are asymptotically equivalent but in finite samples they can lead to different conclusions. Jakub Mućk Econometrics of Panel Data Testing linear hypotheses Meeting # 3 21 / 21