xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models

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1 xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models Sebastian Kripfganz University of Exeter Business School, Department of Economics, Exeter, UK UK Stata Users Group Meeting London, September 9, 2016 net install xtdpdqml, from( S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 1/27

2 Estimation of short-t linear dynamic panel models in Stata Least-squares estimation of dynamic models (i.e. models with a lagged dependent variable) with random or fixed effects (xtreg in Stata) yields biased coefficient estimates when the time horizon is short (Nickell, 1981). Predominent estimation technique in empirical research is the generalized method of moments (GMM): Arellano and Bond (1991) difference GMM : xtabond, Arellano and Bover (1995) and Blundell and Bond (1998) system GMM : xtdpdsys. Both Stata commands are wrappers for the more flexible command xtdpd. Alternative user-written command with full flexibility and many additional options by Roodman (2009): xtabond2. S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 2/27

3 Estimation of short-t linear dynamic panel models in Stata Other promising approaches that can be more efficient alternatives to GMM with potentially better finite-sample performance remain underrepresented in empirical work: Bias-correction procedures by Kiviet (1995), Bun and Kiviet (2003), and Everaert and Pozzi (2007): user-written implementations xtlsdvc (Bruno, 2005) and xtbcfe (De Vos, Everaert, and Ruyssen, 2015). Full-information maximum likelihood / structural equation modeling: xtdpdml command by Williams, Allison, and Moral-Benito (2015) as a wrapper for sem. Limited-information quasi-maximum likelihood (QML) estimation for dynamic random-effects models (Bhargava and Sargan, 1983) and dynamic fixed-effects models (Hsiao, Pesaran, and Tahmiscioglu, 2002): new xtdpdqml package. S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 3/27

4 Linear dynamic panel data model Linear panel model with first-order autoregressive dynamics: y it = λy i,t 1 + x itβ + f i γ + ǫ it, ǫ it = u i + e it, t = 1, 2,..., T i (potentially unbalanced but without gaps), and where e it iid (0, σ 2 e ). The regressors x it and f i are required to be strictly exogenous with respect to e it. The lagged dependent variable y i,t 1 is correlated by construction with the unit-specific error component u i. Dynamic random-effects model: The time-varying regressors x it and the time-invariant regressors f i are uncorrelated with u i. Dynamic fixed-effects model: All regressors are allowed to be correlated with u i. S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 4/27

5 Dynamic random-effects model y it = λy i,t 1 + x itβ + f i γ + ǫ it, ǫ it = u i + e it, Random-effects assumption: u i iid (0, σ 2 u ), uncorrelated with x it and f i. The classical random-effects estimator is a least-squares estimator treating the initial observations y i0 as exogenous. Consequently, it is biased when T is small due to the correlation of y i,t 1 (and therefore also y i0 ) with u i. To account for this correlation with a likelihood approach, the joint distribution of (y i0, y i1,..., y iti ) needs to be specified. S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 5/27

6 Dynamic random-effects model y it = λy i,t 1 + x itβ + f i γ + ǫ it, ǫ it = u i + e it, Bhargava and Sargan (1983) propose to model the initial observations as a function of the observed exogenous variables: T y i0 = x isπ x,s + f i π f + ν i0, s=0 with T = min(t i ), Var(ν i0 ) = σ 2 0, and Cov(ν i0, ǫ it ) = φσ 2 0. Implied restrictions under stationarity of all variables: φ = σ2 u (1 λ)σ0 2 γ in the presence of time-varying regressors x it, π f = 1 λ, σ2 0 = σ2 u (1 λ) + σ2 2 e 1 λ, and φ = 2 absence of time-varying regressors x it. σ2 u (1 λ)σ 2 0 in the S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 6/27

7 Dynamic fixed-effects model y it = λy i,t 1 + x itβ + f i γ + ǫ it, ǫ it = u i + e it, Fixed-effects assumption: u i allowed to be arbitrarily correlated with x it and f i. First-difference transformation to remove the fixed effects: y it = λ y i,t 1 + x itβ + e it, The lagged dependent variable y i,t 1 (and therefore also y i1 ) is correlated by construction with the transformed error term e it. Consequently, an estimator that treats y i1 as exogenous is biased. To account for this correlation with a likelihood approach, the joint distribution of ( y i1, y i2,..., y iti ) needs to be specified. S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 7/27

8 Dynamic fixed-effects model y it = λ y i,t 1 + x itβ + e it, Hsiao, Pesaran, and Tahmiscioglu (2002) justify the following representation for the initial observations: T y i1 = b + x isπ s + ν i1, s=1 with T = min(t i ), Var(ν i1 ) = ωσ 2 e, Cov(ν i0, e i2 ) = σ 2 e, and Cov(ν i0, e it ) = 0 for t > 2. Implied restrictions under stationarity of all (first-differenced) variables: b = 0 in the presence of regressors x it, b = 0 and ω = 2 1+λ in the absence of regressors x it. S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 8/27

9 Quasi-maximum likelihood estimation Given the assumptions on the error components and treating all of them as if they were normally distributed, the log-likelihood function for the system of equations can be maximized with a gradient-based optimization technique. This iterative procedure needs appropriate starting values: 1 By default, xtdpdqml obtains initial estimates for the model coefficients from a consistent GMM estimator (xtdpd). Initial estimates for the initial-observations coefficients are obtained from a separate least-squares estimation. The initial variance parameter estimates are computed from the respective residuals. Alternative initial estimates for the model coefficients and variance parameters can be specified by the user. Analytical first-order and second-order derivatives largely speed up the computations. 1 See the paper and the online appendix at for details. S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 9/27

10 Stata syntax of the xtdpdqml command xtdpdqml depvar [indepvars ] [if ] [in ] [, options ] Selected options: fe: uses the fixed-effects estimator, the default, re: uses the random-effects estimator, projection(varlist [, leads(#) nodifference omit]): specifies the initial-observations projection, stationary: imposes restrictions valid under stationarity, vce(robust): uses the sandwich VC estimator for valid inference under cross-sectional heteroskedasticity (Hayakawa and Pesaran, 2015), mlparams: reports all ML parameter estimates, from(init specs ) and initval(numlist ): specify alternative starting values, additional display options, maximize options,... Selected postestimation commands: predict: similar to xtreg plus equation-level scores, estat, hausman, lrtest, nlcom, suest, test,... S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 10/27

11 Estimation of an employment equation for 140 UK companies, , based on the Arellano and Bond (1991) data set:. webuse abdata Dependent variable: Logarithm of the number of employees (n). Strictly exogenous explanatory variables: Real wage (w), Gross capital stock (k), Time dummies (yr1978-yr1984). S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 11/27

12 QML estimation of the dynamic fixed-effects model:. xtdpdqml n w k yr1978-yr1984, nolog Quasi-maximum likelihood estimation Group variable: id Number of obs = 891 Time variable: year Number of groups = 140 Fixed effects Obs per group: min = 6 avg = (Estimation in first differences) max = n Coef. Std. Err. z P>z [95% Conf. Interval] n L w k yr yr yr yr yr yr yr _cons S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 12/27

13 Reporting of all parameter estimates:. xtdpdqml n w k yr1978-yr1984, mlparams nolog Quasi-maximum likelihood estimation Group variable: id Number of obs = 891 Time variable: year Number of groups = 140 Fixed effects Obs per group: min = 6 avg = max = D.n Coef. Std. Err. z P>z [95% Conf. Interval] _model n LD w D k D yr1978 D (Continued on next page) S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 13/27

14 yr1979 D yr1980 D yr1981 D yr1982 D yr1983 D yr1984 D _initobs w D FD F2D F3D F4D F5D (Continued on next page) S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 14/27

15 k D FD F2D F3D F4D F5D yr1978 D FD _cons /_sigma2e /_omega estimates store fe S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 15/27

16 Restricted model versions:. xtdpdqml n w k yr1978-yr1984, stationary mlparams nolog (Output omitted). lrtest fe Likelihood-ratio test LR chi2(1) = 0.03 (Assumption:. nested in fe) Prob > chi2 = xtdpdqml n w k yr1978-yr1984, stationary projection(yr*, omit) mlparams nolog (Output omitted). lrtest fe Likelihood-ratio test LR chi2(3) = 6.29 (Assumption:. nested in fe) Prob > chi2 = estimates restore fe (results fe are active now). test [_initobs]: D.yr1978 FD.yr1978 _cons ( 1) [_initobs]d.yr1978 = 0 ( 2) [_initobs]fd.yr1978 = 0 ( 3) [_initobs]_cons = 0 chi2( 3) = 6.36 Prob > chi2 = S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 16/27

17 . xtdpdqml n w k yr1978-yr1984, stationary projection(w k, leads(0)) mlparams nolog (Output omitted). lrtest fe Likelihood-ratio test LR chi2(11) = (Assumption:. nested in fe) Prob > chi2 = Alternative starting values from system GMM estimator (default starting values are from difference GMM estimator):. quietly xtdpdsys n w k yr1978-yr1984, twostep. matrix b = e(b). xtdpdqml n w k yr1978-yr1984, stationary from(b, skip) (Output omitted). estimates store fe S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 17/27

18 QML estimation of the dynamic random-effects model:. xtdpdqml n w k yr1978-yr1984, re nolog Quasi-maximum likelihood estimation initial values not feasible Feasible starting values for the variance parameters (σu, 2 σe, 2 σ0 2, φ) need to satisfy the restriction (σ 2 u φ 2 σ 2 0) max(t i ) > σ 2 e.. xtdpdqml n w k yr1978-yr1984, re initval( ) nolog (Output omitted). estimates store re S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 18/27

19 Traditional Hausman test:. hausman fe re, df(3) ---- Coefficients ---- (b) (B) (b-b) sqrt(diag(v_b-v_b)) fe_eq1 re_eq1 Difference S.E. n L w k yr yr yr yr yr yr yr b = consistent under Ho and Ha; obtained from xtdpdqml B = inconsistent under Ha, efficient under Ho; obtained from xtdpdqml Test: Ho: difference in coefficients not systematic chi2(3) = (b-b) [(V_b-V_B)ˆ(-1)](b-B) = Prob>chi2 = (V_b-V_B is not positive definite) S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 19/27

20 Generalized (robust) Hausman test:. quietly xtdpdqml n w k yr1978-yr1984, mlparams. estimates store fe. quietly xtdpdqml n w k yr1978-yr1984, re initval( ) mlparams. estimates store re. suest fe re, vce(cluster id) Simultaneous results for fe, re Number of obs = 1,031 (Std. Err. adjusted for 140 clusters in id) Robust Coef. Std. Err. z P>z [95% Conf. Interval] fe model n LD w D (Continued on next page) S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 20/27

21 k D yr1978 D yr1979 D yr1980 D yr1981 D yr1982 D yr1983 D yr1984 D (Continued on next page) S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 21/27

22 fe initobs w D FD F2D F3D F4D F5D k D FD F2D F3D F4D F5D yr1978 D FD _cons fe sigma2e _cons fe omega _cons (Continued on next page) S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 22/27

23 re model n L w k yr yr yr yr yr yr yr _cons re initobs w F F F F F F (Continued on next page) S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 23/27

24 k F F F F F F yr1978 F yr1979 F _cons re sigma2u _cons re sigma2e _cons re sigma2e0 _cons re phi _cons S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 24/27

25 . test ([fe model]ld.n = [re model]l.n) ([fe model]d.w = [re model]w) ([fe model]d.k = [re model]k) ( 1) [fe model]ld.n - [re model]l.n = 0 ( 2) [fe model]d.w - [re model]w = 0 ( 3) [fe model]d.k - [re model]k = 0 chi2( 3) = 5.97 Prob > chi2 = Computation of long-run effects:. xtdpdqml n w k yr1978-yr1984, stationary vce(robust) (Output omitted). nlcom (_b[w] / (1 - _b[l.n])) (_b[k] / (1 - _b[l.n])) _nl_1: _nl_2: _b[w] / (1 - _b[l.n]) _b[k] / (1 - _b[l.n]) n Coef. Std. Err. z P>z [95% Conf. Interval] _nl_ _nl_ S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 25/27

26 Summary: the new xtdpdqml package for Stata (Quasi-)maximum likelihood estimation can be an attractive alternative to widely used GMM estimators with potential efficiency gains and better finite-sample performance. The xtdpdqml implements the Bhargava and Sargan (1983) random-effects QML estimator and the Hsiao, Pesaran, and Tahmiscioglu (2002) fixed-effects QML estimator for linear dynamic panel data models. It provides a complement to Stata s existing estimation toolbox for dynamic panel models that can be valuable to assess the robustness of estimates obtained with different methods. Kripfganz, S. (forthcoming). xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models. Stata Journal (accepted manuscript). net install xtdpdqml, from( help xtdpdqml help xtdpdqml postestimation S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 26/27

27 References Arellano, M., and S. R. Bond (1991). Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations. Review of Economic Studies 58(2): Arellano, M., and O. Bover (1995). Another look at the instrumental variable estimation of error-components models. Journal of Econometrics 68(1): Bhargava, A., and J. D. Sargan (1983). Estimating dynamic random effects models from panel data covering short time periods. Econometrica 51(6): Blundell, R., and S. R. Bond (1991). Initial conditions and moment restrictions in dynamic panel data models. Journal of Econometrics 87(1): Bruno, G. S. F. (2005). Estimation and inference in dynamic unbalanced panel-data models with a small number of individuals. Stata Journal 5(4): Bun, M. J. G., and J. F. Kiviet (2003). On the diminishing returns of higher-order terms in asymptotic expansions of bias. Economics Letters 79(2): De Vos, I., G. Everaert, and I. Ruyssen (2015). Bootstrap-based bias correction and inference for dynamic panels with fixed effects. Stata Journal 15(4): Everaert, G., and L. Pozzi (2007). Bootstrap-based bias correction for dynamic panels. Journal of Economic Dynamics and Control 31(4): Hayakawa, K., and M. H. Pesaran (2015). Robust standard errors in transformed likelihood estimation of dynamic panel data models with cross-sectional heteroskedasticity. Journal of Econometrics 188(1): Hsiao, C., M. H. Pesaran, and A. K. Tahmiscioglu (2002). Maximum likelihood estimation of fixed effects dynamic panel data models covering short time periods. Journal of Econometrics 109(1): Kiviet, J. F. (1995). On bias, inconsistency, and efficiency of various estimators in dynamic panel data models. Journal of Econometrics 68(1): Roodman, D. (2009). How to do xtabond2: An introduction to difference and system GMM in Stata. Stata Journal 9(1): Williams, R., P. D. Allison, and E. Moral-Benito (2015). Linear dynamic panel-data estimation using maximum likelihood and structural equation modeling. Presented July 30, 2015 at the Stata Conference 2015, Columbus, Ohio. S. Kripfganz (2016) xtdpdqml: Quasi-maximum likelihood estimation of linear dynamic short-t panel data models 27/27