Panel Data Linear Models
Fitting Panel Data Linear Models in Stata Gustavo Sanchez Senior Statistician StataCorp LP
Puebla, Mexico
Gustavo Sanchez (StataCorp)
June 22-23, 2012
1 / 42
Panel Data Linear Models Outline
Outline Brief introduction to panel data linear models Fixed and Random effects models
Fitting the model in Stata Specifying the panel structure Regression output
Testing and accounting for serial correlation and heteroskedasticity Panel Unit root tests - Model in first differences Dynamic panel linear models
Gustavo Sanchez (StataCorp)
June 22-23, 2012
2 / 42
Panel Data Linear Models Outline
Outline Brief introduction to panel data linear models Fixed and Random effects models
Fitting the model in Stata Specifying the panel structure Regression output
Testing and accounting for serial correlation and heteroskedasticity Panel Unit root tests - Model in first differences Dynamic panel linear models
Gustavo Sanchez (StataCorp)
June 22-23, 2012
2 / 42
Panel Data Linear Models Outline
Outline Brief introduction to panel data linear models Fixed and Random effects models
Fitting the model in Stata Specifying the panel structure Regression output
Testing and accounting for serial correlation and heteroskedasticity Panel Unit root tests - Model in first differences Dynamic panel linear models
Gustavo Sanchez (StataCorp)
June 22-23, 2012
2 / 42
Panel Data Linear Models Outline
Outline Brief introduction to panel data linear models Fixed and Random effects models
Fitting the model in Stata Specifying the panel structure Regression output
Testing and accounting for serial correlation and heteroskedasticity Panel Unit root tests - Model in first differences Dynamic panel linear models
Gustavo Sanchez (StataCorp)
June 22-23, 2012
2 / 42
Panel Data Linear Models Outline
Outline Brief introduction to panel data linear models Fixed and Random effects models
Fitting the model in Stata Specifying the panel structure Regression output
Testing and accounting for serial correlation and heteroskedasticity Panel Unit root tests - Model in first differences Dynamic panel linear models
Gustavo Sanchez (StataCorp)
June 22-23, 2012
2 / 42
Panel Data Linear Models Introduction to Panel Data Linear Models
Brief Introduction to Panel Data Linear Models
Gustavo Sanchez (StataCorp)
June 22-23, 2012
3 / 42
Panel Data Linear Models Introduction to Panel Data Linear Models One-way error component model
One-way error component model Yit = Îą + Xit1 ∗ β1 + ... + XitK ∗ βK + it Yit = Îą + Xit1 ∗ β1 + ... + XitK ∗ βK + Âľi + νit i = 1, ..., N j = 1, ..., T
Gustavo Sanchez (StataCorp)
June 22-23, 2012
4 / 42
Panel Data Linear Models Introduction to Panel Data Linear Models One-way error component model
One-way error component model Yit = Îą + Xit1 ∗ β1 + ... + XitK ∗ βK + it Yit = Îą + Xit1 ∗ β1 + ... + XitK ∗ βK + Âľi + νit i = 1, ..., N j = 1, ..., T
Gustavo Sanchez (StataCorp)
June 22-23, 2012
4 / 42
Panel Data Linear Models Introduction to Panel Data Linear Models One-way error component model - Fixed effects
Fixed Effects Models Yit = α + Xit1 ∗ β1 + ... + XitK ∗ βK + µi + νit
Some Relevant Assumptions The non-observable individual effects are represented by fixed parameters The explanatory variables in X are independent of the idiosyncratic error term but they are not independent of the individual fixed effects The idiosyncratic error term νit is iid(0, σν2 )
Gustavo Sanchez (StataCorp)
June 22-23, 2012
5 / 42
Panel Data Linear Models Introduction to Panel Data Linear Models One-way error component model - Fixed effects
Fixed Effects Models Yit = α + Xit1 ∗ β1 + ... + XitK ∗ βK + µi + νit
Some Relevant Assumptions The non-observable individual effects are represented by fixed parameters The explanatory variables in X are independent of the idiosyncratic error term but they are not independent of the individual fixed effects The idiosyncratic error term νit is iid(0, σν2 )
Gustavo Sanchez (StataCorp)
June 22-23, 2012
5 / 42
Panel Data Linear Models Introduction to Panel Data Linear Models One-way error component model - Fixed effects
Fixed Effects Models Yit = α + Xit1 ∗ β1 + ... + XitK ∗ βK + µi + νit
Some Relevant Assumptions The non-observable individual effects are represented by fixed parameters The explanatory variables in X are independent of the idiosyncratic error term but they are not independent of the individual fixed effects The idiosyncratic error term νit is iid(0, σν2 )
Gustavo Sanchez (StataCorp)
June 22-23, 2012
5 / 42
Panel Data Linear Models Introduction to Panel Data Linear Models One-way error component model - Fixed effects
Fixed Effects Models Let’s assume that we have only one explanatory variable: Yit = α + Xit ∗ β + µi + νit Let’s take the average in time: Y¯i. = α + X¯i. ∗ β + µi + ν¯i. Let’s take the difference between those two equations: Yit − Y¯i. = (Xit − X¯i. ) ∗ β + (νit − ν¯i. ) This ”within transformation” is the basis for the fixed effects estimator. The FE beta estimates could be obtained by using OLS in the latest equation Gustavo Sanchez (StataCorp)
June 22-23, 2012
6 / 42
Panel Data Linear Models Introduction to Panel Data Linear Models One-way error component model - Fixed effects
Fixed Effects Models Let’s assume that we have only one explanatory variable: Yit = α + Xit ∗ β + µi + νit Let’s take the average in time: Y¯i. = α + X¯i. ∗ β + µi + ν¯i. Let’s take the difference between those two equations: Yit − Y¯i. = (Xit − X¯i. ) ∗ β + (νit − ν¯i. ) This ”within transformation” is the basis for the fixed effects estimator. The FE beta estimates could be obtained by using OLS in the latest equation Gustavo Sanchez (StataCorp)
June 22-23, 2012
6 / 42
Panel Data Linear Models Introduction to Panel Data Linear Models One-way error component model - Fixed effects
Fixed Effects Models Let’s assume that we have only one explanatory variable: Yit = α + Xit ∗ β + µi + νit Let’s take the average in time: Y¯i. = α + X¯i. ∗ β + µi + ν¯i. Let’s take the difference between those two equations: Yit − Y¯i. = (Xit − X¯i. ) ∗ β + (νit − ν¯i. ) This ”within transformation” is the basis for the fixed effects estimator. The FE beta estimates could be obtained by using OLS in the latest equation Gustavo Sanchez (StataCorp)
June 22-23, 2012
6 / 42
Panel Data Linear Models Introduction to Panel Data Linear Models One-way error component model - Fixed effects
Fixed Effects Models Let’s assume that we have only one explanatory variable: Yit = α + Xit ∗ β + µi + νit Let’s take the average in time: Y¯i. = α + X¯i. ∗ β + µi + ν¯i. Let’s take the difference between those two equations: Yit − Y¯i. = (Xit − X¯i. ) ∗ β + (νit − ν¯i. ) This ”within transformation” is the basis for the fixed effects estimator. The FE beta estimates could be obtained by using OLS in the latest equation Gustavo Sanchez (StataCorp)
June 22-23, 2012
6 / 42
Panel Data Linear Models Introduction to Panel Data Linear Models One-way error component model - Fixed effects
Fixed Effects Models Let’s assume that we have only one explanatory variable: Yit = α + Xit ∗ β + µi + νit Let’s take the average in time: Y¯i. = α + X¯i. ∗ β + µi + ν¯i. Let’s take the difference between those two equations: Yit − Y¯i. = (Xit − X¯i. ) ∗ β + (νit − ν¯i. ) This ”within transformation” is the basis for the fixed effects estimator. The FE beta estimates could be obtained by using OLS in the latest equation Gustavo Sanchez (StataCorp)
June 22-23, 2012
6 / 42
Panel Data Linear Models Introduction to Panel Data Linear Models One-way error component model - Random effects
Random Effects Models Yit = α + Xit1 ∗ β1 + ... + XitK ∗ βK + µi + νit
Some Relevant Assumptions The non-observable individual effects are iid(0,σµ2 ) The explanatory variables in X are independent of the idiosyncratic error term and they are also independent of the individual random effects (i.e. Cov(Xkit , µi ) = 0) The idiosyncratic error term νit is iid(0, σν2 )
Gustavo Sanchez (StataCorp)
June 22-23, 2012
7 / 42
Panel Data Linear Models Introduction to Panel Data Linear Models One-way error component model - Random effects
Random Effects Models Yit = α + Xit1 ∗ β1 + ... + XitK ∗ βK + µi + νit
Some Relevant Assumptions The non-observable individual effects are iid(0,σµ2 ) The explanatory variables in X are independent of the idiosyncratic error term and they are also independent of the individual random effects (i.e. Cov(Xkit , µi ) = 0) The idiosyncratic error term νit is iid(0, σν2 )
Gustavo Sanchez (StataCorp)
June 22-23, 2012
7 / 42
Panel Data Linear Models Introduction to Panel Data Linear Models One-way error component model - Random effects
Random Effects Models Yit = α + Xit1 ∗ β1 + ... + XitK ∗ βK + µi + νit
Some Relevant Assumptions The non-observable individual effects are iid(0,σµ2 ) The explanatory variables in X are independent of the idiosyncratic error term and they are also independent of the individual random effects (i.e. Cov(Xkit , µi ) = 0) The idiosyncratic error term νit is iid(0, σν2 )
Gustavo Sanchez (StataCorp)
June 22-23, 2012
7 / 42
Panel Data Linear Models Fitting the model in Stata
Fitting the model in Stata
Gustavo Sanchez (StataCorp)
June 22-23, 2012
8 / 42
Panel Data Linear Models Fitting the model in Stata Empirical example
Model for aggregate consumption consumoit = α + pibit ∗ β1 + pibit−1 ∗ β2 + irateit ∗ β3 + µi + νit Data World Bank public online data on: consumo: Final consumption expenditure (Y2000=100) pib: Gross domestic product (Y2000=100) irate deposit interest rate Example 1: 1980-2010 for 122 countries Example 2: 2003-2010 for 104-108 countries : Source:http://databank.worldbank.org/data/Home.aspx
Gustavo Sanchez (StataCorp)
June 22-23, 2012
9 / 42
Panel Data Linear Models Fitting the model in Stata Empirical example
Model for aggregate consumption consumoit = α + pibit ∗ β1 + pibit−1 ∗ β2 + irateit ∗ β3 + µi + νit Data World Bank public online data on: consumo: Final consumption expenditure (Y2000=100) pib: Gross domestic product (Y2000=100) irate deposit interest rate Example 1: 1980-2010 for 122 countries Example 2: 2003-2010 for 104-108 countries : Source:http://databank.worldbank.org/data/Home.aspx
Gustavo Sanchez (StataCorp)
June 22-23, 2012
9 / 42
Panel Data Linear Models Fitting the model in Stata Empirical example
Model for aggregate consumption consumoit = α + pibit ∗ β1 + pibit−1 ∗ β2 + irateit ∗ β3 + µi + νit Data World Bank public online data on: consumo: Final consumption expenditure (Y2000=100) pib: Gross domestic product (Y2000=100) irate deposit interest rate Example 1: 1980-2010 for 122 countries Example 2: 2003-2010 for 104-108 countries : Source:http://databank.worldbank.org/data/Home.aspx
Gustavo Sanchez (StataCorp)
June 22-23, 2012
9 / 42
Panel Data Linear Models Fitting the model in Stata Empirical example
Model for aggregate consumption consumoit = α + pibit ∗ β1 + pibit−1 ∗ β2 + irateit ∗ β3 + µi + νit Data World Bank public online data on: consumo: Final consumption expenditure (Y2000=100) pib: Gross domestic product (Y2000=100) irate deposit interest rate Example 1: 1980-2010 for 122 countries Example 2: 2003-2010 for 104-108 countries : Source:http://databank.worldbank.org/data/Home.aspx
Gustavo Sanchez (StataCorp)
June 22-23, 2012
9 / 42
Panel Data Linear Models Fitting the model in Stata Empirical example
Model for aggregate consumption consumoit = α + pibit ∗ β1 + pibit−1 ∗ β2 + irateit ∗ β3 + µi + νit Data World Bank public online data on: consumo: Final consumption expenditure (Y2000=100) pib: Gross domestic product (Y2000=100) irate deposit interest rate Example 1: 1980-2010 for 122 countries Example 2: 2003-2010 for 104-108 countries : Source:http://databank.worldbank.org/data/Home.aspx
Gustavo Sanchez (StataCorp)
June 22-23, 2012
9 / 42
Panel Data Linear Models Fitting the model in Stata Empirical example - Specifying the panel structure in Stata
Specifying the panel structure in Stata Assuming that the second dimension corresponds to time series, we use the -xtset- command to specify the panel structure with: Panel identifier variable (e.g. country) Time identifier variable (e.g. year)
. xtset country year panel variable: time variable: delta:
Gustavo Sanchez (StataCorp)
country (unbalanced) year, 1980 to 2010, but with gaps 1 unit
June 22-23, 2012
10 / 42
Panel Data Linear Models Fitting the model in Stata Empirical example - Specifying the panel structure in Stata
Specifying the panel structure in Stata Assuming that the second dimension corresponds to time series, we use the -xtset- command to specify the panel structure with: Panel identifier variable (e.g. country) Time identifier variable (e.g. year)
. xtset country year panel variable: time variable: delta:
Gustavo Sanchez (StataCorp)
country (unbalanced) year, 1980 to 2010, but with gaps 1 unit
June 22-23, 2012
10 / 42
Panel Data Linear Models Fitting the model in Stata Empirical example - Specifying the panel structure in Stata
Specifying the panel structure in Stata Assuming that the second dimension corresponds to time series, we use the -xtset- command to specify the panel structure with: Panel identifier variable (e.g. country) Time identifier variable (e.g. year)
. xtset country year panel variable: time variable: delta:
Gustavo Sanchez (StataCorp)
country (unbalanced) year, 1980 to 2010, but with gaps 1 unit
June 22-23, 2012
10 / 42
Panel Data Linear Models Fitting the model in Stata Empirical example - Fixed effects
Fixed effects linear model . xtreg lconsumo lpib lirate,fe Fixed-effects (within) regression Group variable: country R-sq: within = 0.9368 between = 0.9943 overall = 0.9929 corr(u_i, Xb)
Number of obs Number of groups Obs per group: min avg max F(2,2777) Prob > F
= 0.3537
lconsumo
Coef.
lpib lirate _cons
.9399169 -.0041257 1.218756
.0052705 .002125 .1274414
sigma_u sigma_e rho
.16078074 .07814221 .80892226
(fraction of variance due to u_i)
F test that all u_i=0:
Std. Err.
t 178.34 -1.94 9.56
F(121, 2777) =
Gustavo Sanchez (StataCorp)
P>|t| 0.000 0.052 0.000
93.89
June 22-23, 2012
= = = = = = =
2901 122 11 23.8 31 20586.83 0.0000
[95% Conf. Interval] .9295824 -.0082925 .9688669
.9502514 .000041 1.468646
Prob > F = 0.0000
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Panel Data Linear Models Testing and accounting for serial correlation and heteroskedasticity
Testing and accounting for serial correlation and heteroskedasticity
http://www.stata.com/support/faqs/stat/panel.html Gustavo Sanchez (StataCorp)
June 22-23, 2012
12 / 42
Panel Data Linear Models Testing and accounting for serial correlation and heteroskedasticity Empirical example - Test for autocorrelation
Test for autocorrelation: Wooldridge (2002, pag. 2823) derives a simple test for autocorrelation in panel-data models. Regress the pooled (OLS) model in first difference and predict the residuals Regress the residuals on its first lag and test the coefficient on those lagged residuals
Drukker (2003) implements the test with the user-written command -xtserial. xtserial lconsumo lpib lirate if e(sample) Wooldridge test for autocorrelation in panel data H0: no first-order autocorrelation F( 1, 121) = 103.854 Prob > F = 0.0000
Gustavo Sanchez (StataCorp)
June 22-23, 2012
13 / 42
Panel Data Linear Models Testing and accounting for serial correlation and heteroskedasticity Empirical example - Test for autocorrelation
Test for autocorrelation: Wooldridge (2002, pag. 2823) derives a simple test for autocorrelation in panel-data models. Regress the pooled (OLS) model in first difference and predict the residuals Regress the residuals on its first lag and test the coefficient on those lagged residuals
Drukker (2003) implements the test with the user-written command -xtserial. xtserial lconsumo lpib lirate if e(sample) Wooldridge test for autocorrelation in panel data H0: no first-order autocorrelation F( 1, 121) = 103.854 Prob > F = 0.0000
Gustavo Sanchez (StataCorp)
June 22-23, 2012
13 / 42
Panel Data Linear Models Testing and accounting for serial correlation and heteroskedasticity Empirical example - Test for autocorrelation
Test for autocorrelation: Wooldridge (2002, pag. 2823) derives a simple test for autocorrelation in panel-data models. Regress the pooled (OLS) model in first difference and predict the residuals Regress the residuals on its first lag and test the coefficient on those lagged residuals
Drukker (2003) implements the test with the user-written command -xtserial. xtserial lconsumo lpib lirate if e(sample) Wooldridge test for autocorrelation in panel data H0: no first-order autocorrelation F( 1, 121) = 103.854 Prob > F = 0.0000
Gustavo Sanchez (StataCorp)
June 22-23, 2012
13 / 42
Panel Data Linear Models Testing and accounting for serial correlation and heteroskedasticity Empirical example - Test for heteroskedasticity
Test for heteroskedasticity: Poi and Wiggins (2001) suggest an LR test for panel-level heteroskedasticity: iterated GLS with heteroskedastic panels produces MLE. Thus, we can use a LR test with -xtgls, igls panels(heteroskdastic)- versus -xtgls, igls. quietly xtgls lconsumo . estimates store hetero . quietly xtgls lconsumo . estimates store homosk . local df = e(N_g) - 1 . . lrtest hetero homosk , Likelihood-ratio test (Assumption: homosk nested
lpib
lirate, panels(heterosk) igls
lpib
lirate, igls
df(`df´) in hetero)
Gustavo Sanchez (StataCorp)
June 22-23, 2012
LR chi2(121)= Prob > chi2 =
14 / 42
3428.91 0.0000
Panel Data Linear Models Testing and accounting for serial correlation and heteroskedasticity Empirical example - Test for heteroskedasticity
Test for heteroskedasticity: Poi and Wiggins (2001) suggest an LR test for panel-level heteroskedasticity: iterated GLS with heteroskedastic panels produces MLE. Thus, we can use a LR test with -xtgls, igls panels(heteroskdastic)- versus -xtgls, igls. quietly xtgls lconsumo . estimates store hetero . quietly xtgls lconsumo . estimates store homosk . local df = e(N_g) - 1 . . lrtest hetero homosk , Likelihood-ratio test (Assumption: homosk nested
lpib
lirate, panels(heterosk) igls
lpib
lirate, igls
df(`df´) in hetero)
Gustavo Sanchez (StataCorp)
June 22-23, 2012
LR chi2(121)= Prob > chi2 =
14 / 42
3428.91 0.0000
Panel Data Linear Models Testing and accounting for serial correlation and heteroskedasticity Empirical example - Test for heteroskedasticity
Test for heteroskedasticity: Poi and Wiggins (2001) suggest an LR test for panel-level heteroskedasticity: iterated GLS with heteroskedastic panels produces MLE. Thus, we can use a LR test with -xtgls, igls panels(heteroskdastic)- versus -xtgls, igls. quietly xtgls lconsumo . estimates store hetero . quietly xtgls lconsumo . estimates store homosk . local df = e(N_g) - 1 . . lrtest hetero homosk , Likelihood-ratio test (Assumption: homosk nested
lpib
lirate, panels(heterosk) igls
lpib
lirate, igls
df(`df´) in hetero)
Gustavo Sanchez (StataCorp)
June 22-23, 2012
LR chi2(121)= Prob > chi2 =
14 / 42
3428.91 0.0000
Panel Data Linear Models Testing and accounting for serial correlation and heteroskedasticity Empirical example - Test for heteroskedasticity
Test for heteroskedasticity: Poi and Wiggins (2001) suggest an LR test for panel-level heteroskedasticity: iterated GLS with heteroskedastic panels produces MLE. Thus, we can use a LR test with -xtgls, igls panels(heteroskdastic)- versus -xtgls, igls. quietly xtgls lconsumo . estimates store hetero . quietly xtgls lconsumo . estimates store homosk . local df = e(N_g) - 1 . . lrtest hetero homosk , Likelihood-ratio test (Assumption: homosk nested
lpib
lirate, panels(heterosk) igls
lpib
lirate, igls
df(`df´) in hetero)
Gustavo Sanchez (StataCorp)
June 22-23, 2012
LR chi2(121)= Prob > chi2 =
14 / 42
3428.91 0.0000
Panel Data Linear Models Testing and accounting for serial correlation and heteroskedasticity One-way linear AR(1) model
Linear model with first order autoregressive error term -xtregar-: Yit = α + Xit1 ∗ β1 + ... + XitK ∗ βK + µi + νit νit = ρ ∗ νit−1 + ηit
;
η is iid(0, ση2 )
Some Relevant Assumptions Fixed effects: The non-observable individual effects (µi ) are represented by fixed parameters and may be correlated with the covariates in X. Random effects: The non-observable individual effects are assumed to be independent of the idiosyncratic error term and they are also independent of the covariates in X. The µi are iid(0,σµ2 )
Gustavo Sanchez (StataCorp)
June 22-23, 2012
15 / 42
Panel Data Linear Models Testing and accounting for serial correlation and heteroskedasticity One-way linear AR(1) model
Linear model with first order autoregressive error term -xtregar-: Yit = α + Xit1 ∗ β1 + ... + XitK ∗ βK + µi + νit νit = ρ ∗ νit−1 + ηit
;
η is iid(0, ση2 )
Some Relevant Assumptions Fixed effects: The non-observable individual effects (µi ) are represented by fixed parameters and may be correlated with the covariates in X. Random effects: The non-observable individual effects are assumed to be independent of the idiosyncratic error term and they are also independent of the covariates in X. The µi are iid(0,σµ2 )
Gustavo Sanchez (StataCorp)
June 22-23, 2012
15 / 42
Panel Data Linear Models Testing and accounting for serial correlation and heteroskedasticity One-way linear AR(1) model
Linear model with first order autoregressive error term -xtregar-: Yit = α + Xit1 ∗ β1 + ... + XitK ∗ βK + µi + νit νit = ρ ∗ νit−1 + ηit
;
η is iid(0, ση2 )
Some Relevant Assumptions Fixed effects: The non-observable individual effects (µi ) are represented by fixed parameters and may be correlated with the covariates in X. Random effects: The non-observable individual effects are assumed to be independent of the idiosyncratic error term and they are also independent of the covariates in X. The µi are iid(0,σµ2 )
Gustavo Sanchez (StataCorp)
June 22-23, 2012
15 / 42
Panel Data Linear Models Testing and accounting for serial correlation and heteroskedasticity Empirical example - Fixed effects AR(1) model
Fit model accounting for autocorrelation . xtregar lconsumo lpib lirate, fe FE (within) regression with AR(1) disturbances Group variable: country R-sq: within = 0.9631 between = 0.9941 overall = 0.9930 corr(u_i, Xb)
= -0.2531
lconsumo
Coef.
lpib lirate _cons
.9887413 -.000825 .0431147
rho_ar sigma_u sigma_e rho_fov
.82953965 .15647357 .04443063 .92538831
F test that all u_i=0:
Std. Err. .0037579 .0021888 .015465
t
Number of obs Number of groups Obs per group: min avg max F(2,2655) Prob > F P>|t|
263.11 -0.38 2.79
0.000 0.706 0.005
= = = = = = =
2779 122 10 22.8 30 34634.76 0.0000
[95% Conf. Interval] .9813726 -.0051168 .01279
.99611 .0034668 .0734394
(fraction of variance because of u_i) F(121,2655) =
Gustavo Sanchez (StataCorp)
9.24
June 22-23, 2012
Prob > F = 0.0000
16 / 42
Panel Data Linear Models Testing and accounting for serial correlation and heteroskedasticity Feasible Generalized Least Squares (FGLS)
Feasible Generalized Least Squares (-xtgls-): Yit = Îą + Xit1 ∗ β1 + ... + XitK ∗ βK + it Where the variance matrix of the disturbances would be :   Ďƒ11 â„Ś11 Ďƒ12 â„Ś12 ¡ ¡ ¡ Ďƒ1m â„Ś1m  Ďƒ21 â„Ś21 Ďƒ22 â„Ś22 ¡ ¡ ¡ Ďƒ2m â„Ś2m    E [ ] = â„Ś =   .. .. .. .. ďŁ ďŁ¸ . . . . Ďƒm1 â„Śm1 Ďƒm2 â„Śm2 ¡ ¡ ¡ Ďƒmm â„Śmm Different variance-covariance structures Heteroskedasticity across panels Correlation across panels Autocorrelation within panels Gustavo Sanchez (StataCorp)
June 22-23, 2012
17 / 42
Panel Data Linear Models Testing and accounting for serial correlation and heteroskedasticity Feasible Generalized Least Squares (FGLS)
Feasible Generalized Least Squares (-xtgls-): Yit = Îą + Xit1 ∗ β1 + ... + XitK ∗ βK + it Where the variance matrix of the disturbances would be :   Ďƒ11 â„Ś11 Ďƒ12 â„Ś12 ¡ ¡ ¡ Ďƒ1m â„Ś1m  Ďƒ21 â„Ś21 Ďƒ22 â„Ś22 ¡ ¡ ¡ Ďƒ2m â„Ś2m    E [ ] = â„Ś =   .. .. .. .. ďŁ ďŁ¸ . . . . Ďƒm1 â„Śm1 Ďƒm2 â„Śm2 ¡ ¡ ¡ Ďƒmm â„Śmm Different variance-covariance structures Heteroskedasticity across panels Correlation across panels Autocorrelation within panels Gustavo Sanchez (StataCorp)
June 22-23, 2012
17 / 42
Panel Data Linear Models Testing and accounting for serial correlation and heteroskedasticity Feasible Generalized Least Squares (FGLS)
Feasible Generalized Least Squares (-xtgls-): Yit = Îą + Xit1 ∗ β1 + ... + XitK ∗ βK + it Where the variance matrix of the disturbances would be :   Ďƒ11 â„Ś11 Ďƒ12 â„Ś12 ¡ ¡ ¡ Ďƒ1m â„Ś1m  Ďƒ21 â„Ś21 Ďƒ22 â„Ś22 ¡ ¡ ¡ Ďƒ2m â„Ś2m    E [ ] = â„Ś =   .. .. .. .. ďŁ ďŁ¸ . . . . Ďƒm1 â„Śm1 Ďƒm2 â„Śm2 ¡ ¡ ¡ Ďƒmm â„Śmm Different variance-covariance structures Heteroskedasticity across panels Correlation across panels Autocorrelation within panels Gustavo Sanchez (StataCorp)
June 22-23, 2012
17 / 42
Panel Data Linear Models Testing and accounting for serial correlation and heteroskedasticity Feasible Generalized Least Squares (FGLS)
Feasible Generalized Least Squares (-xtgls-): Yit = Îą + Xit1 ∗ β1 + ... + XitK ∗ βK + it Where the variance matrix of the disturbances would be :   Ďƒ11 â„Ś11 Ďƒ12 â„Ś12 ¡ ¡ ¡ Ďƒ1m â„Ś1m  Ďƒ21 â„Ś21 Ďƒ22 â„Ś22 ¡ ¡ ¡ Ďƒ2m â„Ś2m    E [ ] = â„Ś =   .. .. .. .. ďŁ ďŁ¸ . . . . Ďƒm1 â„Śm1 Ďƒm2 â„Śm2 ¡ ¡ ¡ Ďƒmm â„Śmm Different variance-covariance structures Heteroskedasticity across panels Correlation across panels Autocorrelation within panels Gustavo Sanchez (StataCorp)
June 22-23, 2012
17 / 42
Panel Data Linear Models Testing and accounting for serial correlation and heteroskedasticity Feasible Generalized Least Squares (FGLS)
Feasible Generalized Least Squares -xtgls-: Yit = Îą + Xit1 ∗ β1 + ... + XitK ∗ βK + it Heteroskedasticity across panels -xtgls,panels(heteroskedastic)-:
   E [ ] = â„Ś =  ďŁ
Gustavo Sanchez (StataCorp)
Ďƒ1 I 0 .. .
0 Ďƒ2 I .. .
¡¡¡ ¡¡¡ .. .
0 0 .. .
0
0
¡¡¡
Ďƒm I
June 22-23, 2012
    
18 / 42
Panel Data Linear Models Testing and accounting for serial correlation and heteroskedasticity Feasible Generalized Least Squares (FGLS)
Feasible Generalized Least Squares -xtgls-: Yit = Îą + Xit1 ∗ β1 + ... + XitK ∗ βK + it it = Ď âˆ— it−1 + Ρit
;
Ρ is iid(0, ĎƒÎˇ2 )
Heteroskedasticity across panels and autocorrelation within panels -xtgls,panels(heteroskedastic) corr(psar1)-:
   E [ ] = â„Ś =  ďŁ
Ďƒ1 â„Ś11 0 ¡¡¡ 0 Ďƒ2 â„Ś22 ¡ ¡ ¡ .. .. .. . . . 0 0 ¡¡¡
Gustavo Sanchez (StataCorp)
June 22-23, 2012
0 0 .. .
    
Ďƒm â„Śmm
19 / 42
Panel Data Linear Models Testing and accounting for serial correlation and heteroskedasticity Empirical example - Fit model accounting for heteroskedasticity
Fit model accounting for heteroskedasticity . xtgls lconsumo lpib lirate,panels(heterosk) Cross-sectional time-series FGLS regression Coefficients: generalized least squares Panels: heteroskedastic Correlation: no autocorrelation Estimated covariances = 122 Estimated autocorrelations = 0 Estimated coefficients = 3
lconsumo
Coef.
lpib lirate _cons
.9720376 .0193624 .4192768
Std. Err. .000853 .0013921 .0215707
z 1139.53 13.91 19.44
nolog
Number of obs Number of groups Obs per group: min avg max Wald chi2(2) Prob > chi2
= = = = = = =
2901 122 11 23.77869 31 1352635 0.0000
P>|z|
[95% Conf. Interval]
0.000 0.000 0.000
.9703657 .016634 .376999
.9737094 .0220908 .4615545
.
Gustavo Sanchez (StataCorp)
June 22-23, 2012
20 / 42
Panel Data Linear Models Testing and accounting for serial correlation and heteroskedasticity Empirical example - Fit model accounting for autocorrelation and heteroskedasticity
Fit model accounting for autocorrelation and heteroskedasticity . xtgls lconsumo lpib lirate,panels(heterosk) corr(psar1) nolog force Cross-sectional time-series FGLS regression Coefficients: generalized least squares Panels: heteroskedastic Correlation: panel-specific AR(1) Estimated covariances = 122 Number of obs = 2901 Estimated autocorrelations = 122 Number of groups = 122 Estimated coefficients = 3 Obs per group: min = 11 avg = 23.77869 max = 31 Wald chi2(2) = 255484.88 Prob > chi2 = 0.0000 lconsumo
Coef.
lpib lirate _cons
.957914 -.0009035 .7854506
Std. Err. .0019214 .0009393 .0472898
z 498.54 -0.96 16.61
P>|z| 0.000 0.336 0.000
[95% Conf. Interval] .9541481 -.0027444 .6927643
.96168 .0009375 .8781369
. Gustavo Sanchez (StataCorp)
June 22-23, 2012
21 / 42
Panel Data Linear Models Panel unit root tests - Model in first differences
Panel unit root tests - Model in first differences
Gustavo Sanchez (StataCorp)
June 22-23, 2012
22 / 42
Panel Data Linear Models Panel unit root tests - Model in first differences Empirical example - Plot variables in log-levels
Command lines to plot the the variables in levels (per panel) . > > > > > .
xtline lconsumo if country==9 | country==25 | country==28 | country==42 | country==44 | country==61 | country==87 | country==146 | country==176 | country==179 | country==238 | country==241, name(lconsumo) byopts(t1title("Log of Consumo for selected countries 1980-2010") note("Command: xtline lconsumo if country==**,[options]"))
Gustavo Sanchez (StataCorp)
June 22-23, 2012
/// /// /// /// ///
23 / 42
Panel Data Linear Models Panel unit root tests - Model in first differences Empirical example - Plot variables in log-levels
Gustavo Sanchez (StataCorp)
June 22-23, 2012
24 / 42
Panel Data Linear Models Panel unit root tests - Model in first differences Empirical example - Plot variables in log-levels
Gustavo Sanchez (StataCorp)
June 22-23, 2012
25 / 42
Panel Data Linear Models Panel unit root tests - Model in first differences Empirical example - Plot variables in log-levels
Gustavo Sanchez (StataCorp)
June 22-23, 2012
26 / 42
Panel Data Linear Models Panel unit root tests - Model in first differences Panel unit root - Fisher-type test
Fisher-type Test
(xtunitroot fisher)
Ho: All the panels have unit roots H1: At least one panel does not have unit roots (N finite), or some panels do not have unit roots (N → ∞) Allows unbalanced panels and gaps in any panel Performs Dickey-Fuller or Phillips-Perron test for each panel Combines p-values from the panel specific unit root tests Four different tests reported in the output. All tests are for T → ∞ P is for finite N Z, L*, and PM are are valid for N finite or infinite
Gustavo Sanchez (StataCorp)
June 22-23, 2012
27 / 42
Panel Data Linear Models Panel unit root tests - Model in first differences Panel unit root - Fisher-type test
Fisher-type Test
(xtunitroot fisher)
Ho: All the panels have unit roots H1: At least one panel does not have unit roots (N finite), or some panels do not have unit roots (N → ∞) Allows unbalanced panels and gaps in any panel Performs Dickey-Fuller or Phillips-Perron test for each panel Combines p-values from the panel specific unit root tests Four different tests reported in the output. All tests are for T → ∞ P is for finite N Z, L*, and PM are are valid for N finite or infinite
Gustavo Sanchez (StataCorp)
June 22-23, 2012
27 / 42
Panel Data Linear Models Panel unit root tests - Model in first differences Panel unit root - Fisher-type test
Fisher-type Test
(xtunitroot fisher)
Ho: All the panels have unit roots H1: At least one panel does not have unit roots (N finite), or some panels do not have unit roots (N → ∞) Allows unbalanced panels and gaps in any panel Performs Dickey-Fuller or Phillips-Perron test for each panel Combines p-values from the panel specific unit root tests Four different tests reported in the output. All tests are for T → ∞ P is for finite N Z, L*, and PM are are valid for N finite or infinite
Gustavo Sanchez (StataCorp)
June 22-23, 2012
27 / 42
Panel Data Linear Models Panel unit root tests - Model in first differences Panel unit root - Fisher-type test
Fisher-type Test
(xtunitroot fisher)
Ho: All the panels have unit roots H1: At least one panel does not have unit roots (N finite), or some panels do not have unit roots (N → ∞) Allows unbalanced panels and gaps in any panel Performs Dickey-Fuller or Phillips-Perron test for each panel Combines p-values from the panel specific unit root tests Four different tests reported in the output. All tests are for T → ∞ P is for finite N Z, L*, and PM are are valid for N finite or infinite
Gustavo Sanchez (StataCorp)
June 22-23, 2012
27 / 42
Panel Data Linear Models Panel unit root tests - Model in first differences Panel unit root - Fisher-type test
Fisher-type Test
(xtunitroot fisher)
Ho: All the panels have unit roots H1: At least one panel does not have unit roots (N finite), or some panels do not have unit roots (N → ∞) Allows unbalanced panels and gaps in any panel Performs Dickey-Fuller or Phillips-Perron test for each panel Combines p-values from the panel specific unit root tests Four different tests reported in the output. All tests are for T → ∞ P is for finite N Z, L*, and PM are are valid for N finite or infinite
Gustavo Sanchez (StataCorp)
June 22-23, 2012
27 / 42
Panel Data Linear Models Panel unit root tests - Model in first differences Empirical example - Panel unit root - Fisher-type test
Panel unit root Fisher type test for consumption . xtunitroot fisher lconsumo if e(sample),dfuller lags(1) Fisher-type unit-root test for lconsumo Based on augmented Dickey-Fuller tests Ho: All panels contain unit roots Ha: At least one panel is stationary AR parameter: Panel-specific Panel means: Included Time trend: Not included Drift term: Not included
Inverse chi-squared(244) Inverse normal Inverse logit t(614) Modified inv. chi-squared
P Z L* Pm
Number of panels = 122 Avg. number of periods = 22.68 Asymptotics: T -> Infinity
ADF regressions: 1 lag
Statistic
p-value
121.6417 12.7187 12.5789 -5.5389
1.0000 1.0000 1.0000 1.0000
P statistic requires number of panels to be finite. Other statistics are suitable for finite or infinite number of panels.
Gustavo Sanchez (StataCorp)
June 22-23, 2012
28 / 42
Panel Data Linear Models Panel unit root tests - Model in first differences Empirical example - Panel unit root - Fisher-type test
Panel unit root Fisher type test for gross domestic product . xtunitroot fisher lpib if e(sample),dfuller lags(1) Fisher-type unit-root test for lpib Based on augmented Dickey-Fuller tests Ho: All panels contain unit roots Ha: At least one panel is stationary AR parameter: Panel-specific Panel means: Included Time trend: Not included Drift term: Not included
Inverse chi-squared(244) Inverse normal Inverse logit t(604) Modified inv. chi-squared
P Z L* Pm
Number of panels = 122 Avg. number of periods = 22.68 Asymptotics: T -> Infinity
ADF regressions: 1 lag
Statistic
p-value
97.1014 11.9780 12.5444 -6.6498
1.0000 1.0000 1.0000 1.0000
P statistic requires number of panels to be finite. Other statistics are suitable for finite or infinite number of panels.
Gustavo Sanchez (StataCorp)
June 22-23, 2012
29 / 42
Panel Data Linear Models Panel unit root tests - Model in first differences Empirical example - Panel unit root - Fisher-type test
Panel unit root Fisher type test for interest rate . xtunitroot fisher lirate if e(sample),dfuller lags(1) Fisher-type unit-root test for lirate Based on augmented Dickey-Fuller tests Ho: All panels contain unit roots Ha: At least one panel is stationary AR parameter: Panel-specific Panel means: Included Time trend: Not included Drift term: Not included
Inverse chi-squared(244) Inverse normal Inverse logit t(609) Modified inv. chi-squared
P Z L* Pm
Number of panels = 122 Avg. number of periods = 22.68 Asymptotics: T -> Infinity
ADF regressions: 1 lag
Statistic
p-value
256.2905 2.9384 2.2199 0.5564
0.2819 0.9984 0.9866 0.2890
P statistic requires number of panels to be finite. Other statistics are suitable for finite or infinite number of panels.
Gustavo Sanchez (StataCorp)
June 22-23, 2012
30 / 42
Panel Data Linear Models Panel unit root tests - Model in first differences Empirical example - Model in first difference
Model in first difference . xtreg D.lconsumo D.lpib D.lirate, fe vsquish Fixed-effects (within) regression Number of obs Group variable: country Number of groups R-sq: within = 0.3177 Obs per group: min between = 0.7577 avg overall = 0.3552 max F(2,2513) corr(u_i, Xb) = -0.0126 Prob > F D.lconsumo
Coef.
Std. Err.
t
= = = = = = =
2637 122 9 21.6 29 585.16 0.0000
P>|t|
[95% Conf. Interval]
lpib D1. lirate D1. _cons
.8089838
.0236943
34.14
0.000
.7625214
.8554462
-.0034948 .0051172
.0022693 .0012508
-1.54 4.09
0.124 0.000
-.0079447 .0026646
.0009551 .0075699
sigma_u sigma_e rho
.00820494 .04523059 .03185848
(fraction of variance due to u_i)
F test that all u_i=0:
F(121, 2513) =
Gustavo Sanchez (StataCorp)
0.60
June 22-23, 2012
Prob > F = 0.9998
31 / 42
Panel Data Linear Models Panel unit root tests - Model in first differences Empirical example - Serial correlation and heteroskedasticity tests for the model in first difference
Serial correlation test for the model in first difference . xtserial dlconsumo dlpib dlirate Wooldridge test for autocorrelation in panel data H0: no first-order autocorrelation F( 1, 121) = 0.333 Prob > F = 0.5647
Heteroskedasticity test for the model in first difference . lrtest hetero homosk , df(`df´) Likelihood-ratio test (Assumption: homosk nested in hetero)
Gustavo Sanchez (StataCorp)
LR chi2(121)= Prob > chi2 =
June 22-23, 2012
2582.04 0.0000
32 / 42
Panel Data Linear Models Panel unit root tests - Model in first differences Empirical example - Serial correlation and heteroskedasticity tests for the model in first difference
Serial correlation test for the model in first difference . xtserial dlconsumo dlpib dlirate Wooldridge test for autocorrelation in panel data H0: no first-order autocorrelation F( 1, 121) = 0.333 Prob > F = 0.5647
Heteroskedasticity test for the model in first difference . lrtest hetero homosk , df(`df´) Likelihood-ratio test (Assumption: homosk nested in hetero)
Gustavo Sanchez (StataCorp)
LR chi2(121)= Prob > chi2 =
June 22-23, 2012
2582.04 0.0000
32 / 42
Panel Data Linear Models Panel unit root tests - Model in first differences Empirical example - FE model in first difference with robust standard errors
FE model in first difference with robust standard errors . xtreg D.lconsumo D.lpib D.lirate, fe robust vsquish Fixed-effects (within) regression Number of obs Group variable: country Number of groups R-sq: within = 0.3177 Obs per group: min between = 0.7577 avg overall = 0.3552 max F(2,121) corr(u_i, Xb) = -0.0126 Prob > F (Std. Err. adjusted for 122 clusters
D.lconsumo
Coef.
Robust Std. Err.
t
= 2637 = 122 = 9 = 21.6 = 29 = 241.80 = 0.0000 in country)
P>|t|
[95% Conf. Interval]
lpib D1. lirate D1. _cons
.8089838
.0371275
21.79
0.000
.7354802
.8824874
-.0034948 .0051172
.001886 .0013755
-1.85 3.72
0.066 0.000
-.0072286 .0023942
.000239 .0078403
sigma_u sigma_e rho
.00820494 .04523059 .03185848
(fraction of variance due to u_i)
Gustavo Sanchez (StataCorp)
June 22-23, 2012
33 / 42
Panel Data Linear Models Panel unit root tests - Model in first differences Empirical example - FGLS Model in first difference accounting for heteroskedasticity
FGLS model in first difference accounting for heteroskedasticity . xtgls D.lconsumo D.lpib D.lirate, panels(heterosk) nolog Cross-sectional time-series FGLS regression Coefficients: generalized least squares Panels: heteroskedastic Correlation: no autocorrelation Estimated covariances = 122 Number of obs Estimated autocorrelations = 0 Number of groups Estimated coefficients = 3 Obs per group: min avg max Wald chi2(2) Prob > chi2 D.lconsumo
Coef.
Std. Err.
lpib D1.
.7983589
.0118107
lirate D1.
-.0043501
_cons
.0042671
2637 122 9 21.61475 29 4586.70 0.0000
P>|z|
[95% Conf. Interval]
67.60
0.000
.7752104
.8215073
.0008465
-5.14
0.000
-.0060092
-.002691
.0004889
8.73
0.000
.0033089
.0052253
Gustavo Sanchez (StataCorp)
z
= = = = = = =
June 22-23, 2012
34 / 42
Panel Data Linear Models Dynamic panel linear models
Dynamic Panel Linear Models
Gustavo Sanchez (StataCorp)
June 22-23, 2012
35 / 42
Panel Data Linear Models Dynamic panel linear models
Dynamic Panel Linear Model Yit =
p X
δj ∗ Yit−j +
j=1
K X
Xitk ∗ βk + it
;
it = ¾i + νit
k=1
Basic problems Yit is a function of Âľi ⇒ Yit−1 is a function of Âľi . Thus, OLS is biased because corr ( it , Yit−1 ) 6= 0. The within transformation for FE wipes out Âľi . However, ( ÂŻi ) contains it−1 , which is in turn correlated with Yit−1 . The fixed effects estimator is then biased of order O(1/T) and inconsistent for large N and fixed T
Gustavo Sanchez (StataCorp)
June 22-23, 2012
36 / 42
Panel Data Linear Models Dynamic panel linear models
Dynamic Panel Linear Model Yit =
p X
δj ∗ Yit−j +
j=1
K X
Xitk ∗ βk + it
;
it = ¾i + νit
k=1
Basic problems Yit is a function of Âľi ⇒ Yit−1 is a function of Âľi . Thus, OLS is biased because corr ( it , Yit−1 ) 6= 0. The within transformation for FE wipes out Âľi . However, ( ÂŻi ) contains it−1 , which is in turn correlated with Yit−1 . The fixed effects estimator is then biased of order O(1/T) and inconsistent for large N and fixed T
Gustavo Sanchez (StataCorp)
June 22-23, 2012
36 / 42
Panel Data Linear Models Dynamic panel linear models
Dynamic Panel Linear Model Yit =
p X
δj ∗ Yit−j +
j=1
K X
Xitk ∗ βk + it
;
it = ¾i + νit
k=1
Basic problems Yit is a function of Âľi ⇒ Yit−1 is a function of Âľi . Thus, OLS is biased because corr ( it , Yit−1 ) 6= 0. The within transformation for FE wipes out Âľi . However, ( ÂŻi ) contains it−1 , which is in turn correlated with Yit−1 . The fixed effects estimator is then biased of order O(1/T) and inconsistent for large N and fixed T
Gustavo Sanchez (StataCorp)
June 22-23, 2012
36 / 42
Panel Data Linear Models Dynamic panel linear models Arellano-Bond Arellano-Bover/Blundell-Bond
Arellano-Bond GMM-IV estimator The formulation departs from a fixed effects specification but the model is fitted in first differences and, therefore, the fixed effects are removed from the estimation. The estimator is based on a matrix of instruments constructed with the exogenous variables, and also with subsets of the lagged values of the levels of dependent variable and of the levels of the endogenous or predetermined variables.
Arellano-Bover/Blundell-Bond GMM system estimator They extend the model to a system containing the equation in first differences and also an equation in levels. The new matrix of instruments is constructed including instruments in levels for the equation in differences and instruments in first differences for the equation in levels.
Gustavo Sanchez (StataCorp)
June 22-23, 2012
37 / 42
Panel Data Linear Models Dynamic panel linear models Arellano-Bond Arellano-Bover/Blundell-Bond
Arellano-Bond GMM-IV estimator The formulation departs from a fixed effects specification but the model is fitted in first differences and, therefore, the fixed effects are removed from the estimation. The estimator is based on a matrix of instruments constructed with the exogenous variables, and also with subsets of the lagged values of the levels of dependent variable and of the levels of the endogenous or predetermined variables.
Arellano-Bover/Blundell-Bond GMM system estimator They extend the model to a system containing the equation in first differences and also an equation in levels. The new matrix of instruments is constructed including instruments in levels for the equation in differences and instruments in first differences for the equation in levels.
Gustavo Sanchez (StataCorp)
June 22-23, 2012
37 / 42
Panel Data Linear Models Dynamic panel linear models Arellano-Bond Arellano-Bover/Blundell-Bond
Arellano-Bond GMM-IV estimator The formulation departs from a fixed effects specification but the model is fitted in first differences and, therefore, the fixed effects are removed from the estimation. The estimator is based on a matrix of instruments constructed with the exogenous variables, and also with subsets of the lagged values of the levels of dependent variable and of the levels of the endogenous or predetermined variables.
Arellano-Bover/Blundell-Bond GMM system estimator They extend the model to a system containing the equation in first differences and also an equation in levels. The new matrix of instruments is constructed including instruments in levels for the equation in differences and instruments in first differences for the equation in levels.
Gustavo Sanchez (StataCorp)
June 22-23, 2012
37 / 42
Panel Data Linear Models Dynamic panel linear models Arellano-Bond Arellano-Bover/Blundell-Bond
Arellano-Bond GMM-IV estimator The formulation departs from a fixed effects specification but the model is fitted in first differences and, therefore, the fixed effects are removed from the estimation. The estimator is based on a matrix of instruments constructed with the exogenous variables, and also with subsets of the lagged values of the levels of dependent variable and of the levels of the endogenous or predetermined variables.
Arellano-Bover/Blundell-Bond GMM system estimator They extend the model to a system containing the equation in first differences and also an equation in levels. The new matrix of instruments is constructed including instruments in levels for the equation in differences and instruments in first differences for the equation in levels.
Gustavo Sanchez (StataCorp)
June 22-23, 2012
37 / 42
Panel Data Linear Models Dynamic panel linear models Arellano-Bond Arellano-Bover/Blundell-Bond
Arellano-Bond GMM-IV estimator The formulation departs from a fixed effects specification but the model is fitted in first differences and, therefore, the fixed effects are removed from the estimation. The estimator is based on a matrix of instruments constructed with the exogenous variables, and also with subsets of the lagged values of the levels of dependent variable and of the levels of the endogenous or predetermined variables.
Arellano-Bover/Blundell-Bond GMM system estimator They extend the model to a system containing the equation in first differences and also an equation in levels. The new matrix of instruments is constructed including instruments in levels for the equation in differences and instruments in first differences for the equation in levels.
Gustavo Sanchez (StataCorp)
June 22-23, 2012
37 / 42
Panel Data Linear Models Dynamic panel linear models Empirical example
Model for aggregate consumption consumoit = α+consumoit−1 ∗β2 +pibit ∗β1 +irateit ∗β3 +µi +νit Data World Bank public online data on: consumo: Final consumption expenditure (Y2000=100) pib: Gross domestic product (Y2000=100) irate deposit interest rate Example 2: 2003-2010 for 104-108 countries : Source:http://databank.worldbank.org/data/Home.aspx
Gustavo Sanchez (StataCorp)
June 22-23, 2012
38 / 42
Panel Data Linear Models Dynamic panel linear models Empirical example
Model for aggregate consumption consumoit = α+consumoit−1 ∗β2 +pibit ∗β1 +irateit ∗β3 +µi +νit The presence of lags of the dependent variable among the regressors requires fitting the model with dynamic panel data estimators. -xtivreg,fd-: Anderson and Hsiao (1981) first difference estimator. -xtabond-: Arellano Bond (1991) GMM estimator. -xtdpdsys-: Arellano-Bover (1995) - Blundell-Bond (1998) GMM estimator. -xtdpd- generalizes -xtabond- and -xtdpdsys-.
Gustavo Sanchez (StataCorp)
June 22-23, 2012
39 / 42
Panel Data Linear Models Dynamic panel linear models Empirical example
Model for aggregate consumption consumoit = α+consumoit−1 ∗β2 +pibit ∗β1 +irateit ∗β3 +µi +νit The presence of lags of the dependent variable among the regressors requires fitting the model with dynamic panel data estimators. -xtivreg,fd-: Anderson and Hsiao (1981) first difference estimator. -xtabond-: Arellano Bond (1991) GMM estimator. -xtdpdsys-: Arellano-Bover (1995) - Blundell-Bond (1998) GMM estimator. -xtdpd- generalizes -xtabond- and -xtdpdsys-.
Gustavo Sanchez (StataCorp)
June 22-23, 2012
39 / 42
Panel Data Linear Models Dynamic panel linear models Empirical example
Model for aggregate consumption consumoit = α+consumoit−1 ∗β2 +pibit ∗β1 +irateit ∗β3 +µi +νit The presence of lags of the dependent variable among the regressors requires fitting the model with dynamic panel data estimators. -xtivreg,fd-: Anderson and Hsiao (1981) first difference estimator. -xtabond-: Arellano Bond (1991) GMM estimator. -xtdpdsys-: Arellano-Bover (1995) - Blundell-Bond (1998) GMM estimator. -xtdpd- generalizes -xtabond- and -xtdpdsys-.
Gustavo Sanchez (StataCorp)
June 22-23, 2012
39 / 42
Panel Data Linear Models Dynamic panel linear models Empirical example
Model for aggregate consumption consumoit = α+consumoit−1 ∗β2 +pibit ∗β1 +irateit ∗β3 +µi +νit The presence of lags of the dependent variable among the regressors requires fitting the model with dynamic panel data estimators. -xtivreg,fd-: Anderson and Hsiao (1981) first difference estimator. -xtabond-: Arellano Bond (1991) GMM estimator. -xtdpdsys-: Arellano-Bover (1995) - Blundell-Bond (1998) GMM estimator. -xtdpd- generalizes -xtabond- and -xtdpdsys-.
Gustavo Sanchez (StataCorp)
June 22-23, 2012
39 / 42
Panel Data Linear Models Dynamic panel linear models Empirical example
Model for aggregate consumption consumoit = α+consumoit−1 ∗β2 +pibit ∗β1 +irateit ∗β3 +µi +νit The presence of lags of the dependent variable among the regressors requires fitting the model with dynamic panel data estimators. -xtivreg,fd-: Anderson and Hsiao (1981) first difference estimator. -xtabond-: Arellano Bond (1991) GMM estimator. -xtdpdsys-: Arellano-Bover (1995) - Blundell-Bond (1998) GMM estimator. -xtdpd- generalizes -xtabond- and -xtdpdsys-.
Gustavo Sanchez (StataCorp)
June 22-23, 2012
39 / 42
Panel Data Linear Models Dynamic panel linear models Empirical example - Arellano-Bond
Arellano-Bond . xtabond lconsumo lpib lirate ,lags(1) maxldep(4) vsquish Arellano-Bond dynamic panel-data estimation Number of obs Group variable: country Number of groups Time variable: year Obs per group:
Number of instruments =
21
Wald chi2(3) Prob > chi2
= =
550 104
min = avg = max = = =
1 5.288462 6 5583.65 0.0000
One-step results lconsumo
Coef.
lconsumo L1. lpib lirate _cons
.2003175 .7843574 -.0109575 .2193715
Std. Err.
.036051 .0394084 .0032787 .3215974
z
5.56 19.90 -3.34 0.68
P>|z|
0.000 0.000 0.001 0.495
[95% Conf. Interval]
.1296588 .7071183 -.0173837 -.4109479
.2709762 .8615965 -.0045313 .8496908
Instruments for differenced equation GMM-type: L(2/5).lconsumo Standard: D.lpib D.lirate Instruments for level equation Standard: _cons Gustavo Sanchez (StataCorp)
June 22-23, 2012
40 / 42
Panel Data Linear Models Dynamic panel linear models Empirical example - Arellano-Bover/Blundell-Bond
Arellano-Bover/Blundell-Bond . xtdpdsys lconsumo lpib lirate ,lags(1) maxldep(4) vsquish System dynamic panel-data estimation Number of obs Group variable: country Number of groups Time variable: year Obs per group:
Number of instruments =
27
Wald chi2(3) Prob > chi2
= =
658 108
min = avg = max = = =
1 6.092593 7 16278.38 0.0000
One-step results lconsumo
Coef.
lconsumo L1. lpib lirate _cons
.3169481 .6010134 .0010375 1.834053
Std. Err.
.0290153 .0288126 .003142 .17602
z
10.92 20.86 0.33 10.42
P>|z|
0.000 0.000 0.741 0.000
[95% Conf. Interval]
.2600791 .5445417 -.0051206 1.48906
.373817 .6574852 .0071957 2.179046
Instruments for differenced equation GMM-type: L(2/5).lconsumo Standard: D.lpib D.lirate Instruments for level equation GMM-type: LD.lconsumo Standard: _cons Gustavo Sanchez (StataCorp)
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Panel Data Linear Models Summary
Summary Brief introduction to panel data linear models Fitting the model in Stata Testing and accounting for serial correlation and heteroskedasticity Panel Unit root tests - Model in first differences Dynamic panel linear models
Gustavo Sanchez (StataCorp)
June 22-23, 2012
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Panel Data Linear Models References
References Anderson and Hsiao 1981. Estimation of dynamic models with error components. Journal of the American Statistical Association 76:598—606 Arellano, M. and S. Bond. 1991. Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations. The Review of Economic Studies 58: 277—297. Arellano, M. and O. Bover. 1995. Another look at the instrumental variable estimation of error-components models. Journal of Econometrics 68: 29—51. Blundell, R. and S. Bond. 1998. Initial conditions and moment restrictions in dynamic panel data models. Journal of Econometrics 87: 115—43. Drukker, D. M. 2003. Testing for serial correlation in linear panel-data models. Stata Journal 3: 168—177. Poi and Wiggins 2001. http://www.stata.com/support/faqs/stat/panel.html Wooldridge, J. M. 2002. Econometric Analysis of Cross Section and Panel Data. Cambridge, MA: MIT Press World Bank DataBank http://databank.worldbank.org/data/Home.aspx Gustavo Sanchez (StataCorp)
June 22-23, 2012
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