Econometrics of Panel Data

Transcription

1 Econometrics of Panel Data Jakub Mućk Meeting # 4 Jakub Mućk Econometrics of Panel Data Meeting # 4 1 / 26

2 Outline 1 Two-way Error Component Model Fixed effects model Random effects model 2 Hausman-Taylor estimator Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 2 / 26

3 Two-way Error Component Model Two-way Error Component Model y it = α + X itβ + u it i {1,..., N }, t {1,..., T}, (1) the composite error component: u i,t = µ i + λ t + ε i,t, (2) where: µi the unobservable individual-specific effect; λt the unobservable time-specific effect; ui,t the remainder disturbance. Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 3 / 26

4 Fixed effects model Two-way fixed effects model: where y it = α i + λ t + β 1 x 1it β k x kit + u it, (3) αi - the individual-specific intercept; λi - the period-specific intercept; uit N ( 0, σu) 2. As in the case of the one-way fixed effect model, independent variables x 1,..., x k cannot be time invariant (for a given unit). The estimation: 1 the least squares dummy variable estimator (LSDV); 2 the within estimator; 3 combination of the above methods. Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 4 / 26

5 The LSDV estimator In the analogous fashion to the one-way fixed effect model, we define the following dummy variables: for the j-th unit: { 1 i = j D jit = 0 otherwise, (4) for the τ-th period: B τit = { 1 t = τ 0 otherwise. (5) Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 5 / 26

6 The LSDV estimator In the analogous fashion to the one-way fixed effect model, we define the following dummy variables: for the j-th unit: { 1 i = j D jit = 0 otherwise, (4) for the τ-th period: B τit = { 1 t = τ 0 otherwise. (5) The LSDV estimator can be obtained by estimating the following model: N N y it = α j D jit + λ τ B τit + β 1 x 1it β k x kit + u i,t. (6) j=1 τ=1 where u i,t N ( 0, σu) 2. The parameters in equation (7) can be estimated with the OLS. Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 5 / 26

7 The LSDV estimator: test for poolability The LSDV estimator: N N y it = α j D jit + λ τ B τit + β 1 x 1it β k x kit + u i,t, (7) j=1 where u i,t N ( 0, σ 2 u). τ=1 We can test whether the individual-specific and period-specific effects are significantly different: H 0 : α 1 =... = α k λ 1 =... = λ T = 0. (8) To verify the null described by (8) we compare the sum of squared errors from the pooled model with (with restrictions, SSE R ) with the sum of squared errors from the LSDV model (without restrictions, SSE R ). The test statistics F: F = (SSE R SSE U )/(N T 1) SSE U /(NT K T) if null is emph then F F (N T 1,NT K T). (9) Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 6 / 26

8 The LSDV estimator: test for poolability The LSDV estimator: N N y it = α j D jit + λ τ B τit + β 1 x 1it β k x kit + u i,t, (7) j=1 where u i,t N ( 0, σ 2 u). τ=1 We can test whether the individual-specific and period-specific effects are significantly different: H 0 : α 1 =... = α k λ 1 =... = λ T = 0. (8) To verify the null described by (8) we compare the sum of squared errors from the pooled model with (with restrictions, SSE R ) with the sum of squared errors from the LSDV model (without restrictions, SSE R ). The test statistics F: F = (SSE R SSE U )/(N T 1) SSE U /(NT K T) if null is emph then F F (N T 1,NT K T). One might test only individual or period effects: (9) H 0 : α 1 =... = α k, (10) H 0 : λ 1 =... = λ k. (11) The construction of test for poolability is the same but we use different degrees of freedom. Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 6 / 26

9 The within estimator Consider the following transformations: averaging over time (for each unit i): ȳi, averaging over unit (for each period t): ȳt, averaging over time and unit: ȳ, for the dependent variable y it : ȳ i = 1 T T y it, t ȳ t = 1 N N i y it, ȳ = 1 NT T t N y it. i Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 7 / 26

10 The within estimator Consider the following transformations: averaging over time (for each unit i): ȳi, averaging over unit (for each period t): ȳt, averaging over time and unit: ȳ, for the dependent variable y it : ȳ i = 1 T T y it, t ȳ t = 1 N N i y it, ȳ = 1 NT T t N y it. i Given the above transformations we can eliminate both the individual- and period specific effects: (y it ȳ i ȳ t + ȳ it ) = β (x it x i x t + x it ) + (u it ū i ū t + ū it ). (12) Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 7 / 26

11 The within estimator The within estimator: where ỹ it = β 1 x 1it β k x kit + ũ it (13) ỹ it = y it ȳ i ȳ t + ȳ it x 1it = x 1it x 1i x 1t + x 1it... x kit = x kit x ki x kt + x kit ũ it = u it ū i ū t + ū it. The parameters β 1,..., β k can be estimated by the OLS. The independent variables, i.e., x 1it,..., x kit, cannot be time (unit) invariant. Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 8 / 26

12 Random effects model Two-way random effects model: where the error component is as follows: where y it = α + β 1 x 1it β k x kit + u it, (14) u it = µ i + λ t + ε it, (15) µi is the individual-specific error component and µ i N ( 0, σ 2 µ) ; λt is the period-specific error component and λ t N ( 0, σ 2 λ) ; εit is the idiosyncratic error component and ε it N ( 0, σ 2 ε). Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 9 / 26

13 Random effects model Two-way random effects model: where the error component is as follows: where y it = α + β 1 x 1it β k x kit + u it, (14) u it = µ i + λ t + ε it, (15) µi is the individual-specific error component and µ i N ( 0, σµ) 2 ; λt is the period-specific error component and λ t N ( 0, σλ) 2 ; εit is the idiosyncratic error component and ε it N ( 0, σε) 2. The independent variables can be time invariant. Individual-specific and period-specific effects are independent: E (µ i, µ j ) = 0 if i j, E (λ s, λ t ) = 0 if s t E (µ i, λ t ) = 0 Estimation method: GLS (generalized least squares). Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 9 / 26

14 The two-way RE model - the variance-covariance matrix of error term I It is assumed that the error term in the two-way RE model can be described as follow: u it = µ i + λ t + ε it, (16) where µ i N ( 0, σ 2 µ), λt N ( 0, σ 2 λ) and εit N ( 0, σ 2 ε). The variance-covariance matrix of the error term is not diagonal!. The diagonal elements of the variance covariance matrix of the error term: E ( ) uit 2 = E (µ i + λ t + ε it) 2 = E ( ) µ 2 i + E ( ) λ 2 t + E ( ) ε 2 it + 2cov (µ i, λ t) + 2cov (µ i, ε it) + 2cov (λ t, ε it) }{{}}{{}}{{}}{{}}{{}}{{} =0 =0 =0 =σ 2 µ =σ 2 λ = σ 2 µ + σ 2 λ + σ 2 ε. =σ 2 ε Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 10 / 26

15 The two-way RE model - the variance-covariance matrix of error term II For a given unit, non-diagonal elements of the variance covariance matrix of the error term (t s): cov (u it, u is ) = E [(µ ( i + λ t + ε it ) (µ i + λ s + ε is )], ) = E µ 2 i + E (λ t λ s) + E (ε it ε is ) + COV }{{}}{{}}{{}}{{} 1 =0 =0 =0 = σ 2 µ, =σ 2 µ where COV 1 = cov(λ t, µ i )+cov(λ s, µ i )+cov(λ t, ε it )+cov(λ s, ε it )+2cov(µ i, ε it ). Similarly, for a a given period, non-diagonal elements of the variance covariance matrix of the error term (i j): cov (u it, u jt ) = E [(µ i + λ t + ( ε it (µ j + λ t + ε jt )], ) = E (µ i µ j ) + E λ }{{} 2 t + E (ε it ε jt ) + COV }{{}}{{}}{{} 2 =0 =0 =0 = σ 2 λ. where COV 2 = cov(µ i, λ t )+cov(µ i, λ t )+cov(µ i, ε it )+cov(µ j, ε it )+2cov(λ t, ε it ). =σ 2 λ Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 11 / 26

16 The two-way RE model - the variance-covariance matrix of error term III Finally, the variance-covariance matrix of the error term can be described: σµ 2 + σλ 2 + σ2 ε if i = j, t = s, σµ E (u it, u js ) = 2 if i = j, t s, σλ 2 (17) if i j, t = s, 0 if i j, t s. Unlike the one-way RE model, the variance-covariance matrix E (u it, u js ) is not block-diagonal because there is correlation between units. This correlation is caused by the the period-specific effects and equals σ 2 λ /(σ2 µ + σ 2 λ + σ2 ε). Akin to the one-way RE model there is equicorrelation which is implied by the unit-specific effects and equals σ 2 µ/(σ 2 µ + σ 2 λ + σ2 ε). Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 12 / 26

17 The GLS estimation of the two-way RE model: example Consider the following transformation: zit = z it θ 1 z i θ 2 z t + θ 3 z it, (18) where z it {y it, x 1it,..., x kit, u it } and z i = 1 T T z it, t ȳ t = 1 N N i z it, z = 1 NT T t N z it, i and θ 1 = 1 θ 2 = 1 σ ε Tσ 2 µ + σ 2 ε σ ε N σ 2 λ + σ 2 ε. θ 3 = θ 1 + θ 2 + σ ε Tσ 2 µ + N σ 2 λ + σ2 ε 1. Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 13 / 26

18 The GLS estimation of the two-way RE model: Swamy and Aurora (1972) Swamy and Aurora (1972) propose running three least squares regressions to estimate σ 2 ε, σ 2 µ and σ 2 λ which allow us to calculate θ 1, θ 2 and θ 3. 1 The within regression allows to estimate the variance of the idiosyncratic component, i.e., ˆσ 2 ε. 2 The between (individuals) regression allows to estimate the σ µ which is the estimated variance of the error term from the following regression: ȳ i ȳ = β 1 ( x 1i x 1) β k ( x ki x k ) + ū i ū, and ˆσ I 2 is the estimated variance of the error term from the above regression. Then, the estimated variance of the individual specific-error term can be described as follow ˆσ µ 2 = ˆσ2 I ˆσ ε 2. T 3 The between (periods) regression allows to estimate the σ λ which is the estimated variance of the error term from the following regression: ȳ t ȳ = β 1 ( x 1t x 1) β k ( x kt x k ) + ū t ū, and ˆσ T 2 is the estimated variance of the error term from the above regression. Then, the estimated variance of the individual period-error term can be described as follow ˆσ λ 2 = ˆσ2 T ˆσ ε 2. N Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 14 / 26

19 The GLS estimation of the two-way RE model: general remarks Alternative estimators of σε, 2 σµ 2 and σλ 2 are proposed by Wallace and Hussain (1969), Amemiya (1971), Fuller and Battese (1974) and Nerlove (1971). In general, estimates of σ 2 ε, σ 2 µ and σ 2 λ allow to calculate θ 1, θ 2 and θ 3 and estimate model for transformed variables: y it = β 1 x 1it β k x kit + u it (19) where z it = z it θ 1 z i θ 2 z t + θ 3 z it and z it {y it, x 1it,..., x kit, u 1it }. Computational warning: sometimes estimates of σµ 2 or σλ 2 could be negative. This is due to the fact that we use two-step strategy rather than jointly estimation. Jakub Mućk Econometrics of Panel Data Two-way Error Component Model Meeting # 4 15 / 26

20 Two-way Error Component Model y it = α + β 1 x 1it + β k x kit + µ i + λ t + ε it (20) Random ( effects ) Individual µ i N ( 0, σµ 2 ) time λ t N 0, σλ 2 effects drawn from the random sample = we can estimate the parameters of distribution, i.e., σµ 2 and Fixed effects µ i = individual intercepts α i λ t α i and λ t are assumed to be constant over time σλ 2 Assumptions: (i) E (µ i ε it ) = 0 E (λ t ε it ) = 0 (i) E (µ i ε it ) = 0 E (λ t ε it ) = 0 (ii) E (µ i x it ) = 0 E (λ t x it ) = 0 individual-specific and periodspecific 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 Two-way Error Component Model Meeting # 4 16 / 26

21 Outline 1 Two-way Error Component Model Fixed effects model Random effects model 2 Hausman-Taylor estimator Jakub Mućk Econometrics of Panel Data Hausman-Taylor estimator Meeting # 4 17 / 26

22 Hausman-Taylor estimator Let s consider the following one-way RE model: y it = x 1it β 1 + x 2it β 2 + z 1i γ 1 + z 2i γ 2 + µ i + u it (21) where: x 1it are time-varying variables; not correlated with µ i x 2it are time-varying variables; correlated with µ i z 1i are time-invariant variables; not correlated with µ i z 2i are time-invariant variables; correlated with µ i The RE model estimates on γ 2 are inconsistent. The estimator proposed by Hausman and Taylor (1981) takes into account the above correlation. Jakub Mućk Econometrics of Panel Data Hausman-Taylor estimator Meeting # 4 18 / 26

23 The anatomy of the Hausman-Taylor estimator First step: Within regression for the model including only time-variable regressors, both x 1it and x 2it. Here, the usual differences from the temporal mean are used: (y it ȳ i ) = β 1 (x 1i, x 1i ) + β 2 (x 2it x 2i ) + (u it ū i ) (22) Based on the expression above we can estimate variance of the idiosyncratic error, i.e., ˆσ 2 ε. Jakub Mućk Econometrics of Panel Data Hausman-Taylor estimator Meeting # 4 19 / 26

24 The anatomy of the Hausman-Taylor estimator Second step: construct the intra-temporal mean of the residuals from (22): ē = [(ē 1, ē 1,..., ē 1 ),..., (ē N, ē N,..., ē N )] (23) }{{}}{{} T T Then make TSLS for ē i using: variables: z 1it (time invariant, not correlated with µ i ), z 2it (time invariant, correlated with µ i ) instruments: z 1it, x 1it (time invariant, not correlated with µ i ) Specifically, 1 Regress z 2it on z 1it as well as x 1it. 2 Use the predicted value from the above regression and create new matrix, i.e., Z = [z 1it, ẑ 21t]. 3 Regress ē i on Z to get estimates of γ 1 and γ 2. 4 Calculate σtsls,ē 2 the variance of the error components from the above regression. Now, we can calculate the variation of the individual-specific error component: σ 2 µ = σ 2 TSLS,ē σ2 ε T. (24) Jakub Mućk Econometrics of Panel Data Hausman-Taylor estimator Meeting # 4 20 / 26

25 The anatomy of the Hausman-Taylor estimator Based on the estimates of σµ 2 and σε 2 calculate the conventional in the FGLS regression scale parameter θ: θ = σ 2 ε σ 2 ε + T 1 σ 2 µ (25) Finally, do a TSLS regression of y on X with instruments described by V : more specifically: y = y it θy it, (26) X = [x 1it, x 2it, z 1i, z 2i ] θ[x 1it, x 2it, z 1i, z 2i ], (27) V = [(x 1it x 1i ), (x 2it x 2i ), z 1i, x 1i ], (28) 1 Regress X on the instruments (V ) and obtain fitted values, i.e., ˆX, 2 Regress y on the predicted values from the previous step, i.e, ˆX, in order to get the estimates of [β, γ]. The estimates of the variance-covariance of the structural parameters are a little bit more complicated. Jakub Mućk Econometrics of Panel Data Hausman-Taylor estimator Meeting # 4 21 / 26

26 Empirical example: PSID Consider the following model: lwage it = α + β 1occ it + β 2south it + β 3smsa it + β 4ind it + (29) +β 5exp it + β 6exp2 it + β 7wks it + β 8ms it + (30) +β 9union it + β 10fem i + β 11blk i + β 12ed i + u it (31) Variable lwage exp it wks it occ it ind it south it smsa it ms it fem i union it ed i blk i Description log wage years of full-time work experience weeks worked occupation; occ it = 1 if in a blue-collar occupation industry; ind it = 1 if working in a manufacturing industry residence; south it = 1 if in the South area smsa it = 1 if in the Standard metropolitan statistical area marital status female or male if wage set be a union contract years of education black Jakub Mućk Econometrics of Panel Data Hausman-Taylor estimator Meeting # 4 22 / 26

27 Empirical example: PSID FE RE occ south smsa ind exp exp wks ms union fem blk ed cons σ µ σ ε ρ Note: Note: the superscripts, and denote the rejection of null about parameters insignificance at 1%, 5% and 10% significance level, respectively. Jakub Mućk Econometrics of Panel Data Hausman-Taylor estimator Meeting # 4 23 / 26

28 Empirical example: PSID Correlation between the estimated random unit-specific effect and independent variables x jit ρ(ˆµ i, x jit ) p-value occ south smsa ind exp exp wks ms union fem blk ed Jakub Mućk Econometrics of Panel Data Hausman-Taylor estimator Meeting # 4 24 / 26

29 Empirical example: PSID Classification of the explanatory variables uncorrelated with µ i correlated with µ i time-variant wks, south, ms, smsa occ, exp, exp2, ind, union time-invariant fem, blk ed Jakub Mućk Econometrics of Panel Data Hausman-Taylor estimator Meeting # 4 25 / 26

30 Empirical example: PSID FE RE HT occ south smsa ind exp exp wks ms union fem blk ed cons σ µ σ ε ρ Note: Note: the superscripts, and denote the rejection of null about parameters insignificance at 1%, 5% and 10% significance level, respectively. Jakub Mućk Econometrics of Panel Data Hausman-Taylor estimator Meeting # 4 26 / 26