Lecture 7: Dynamic panel models 2 Ragnar Nymoen Department of Economics, UiO 25 February 2010
Main issues and references The Arellano and Bond method for GMM estimation of dynamic panel data models A stepwise procedure for dynamic models with RE effects. Unit roots and cointegration in panel data. EB (book/kompendium): Ch 9.6 and Ch 10. EB (lecture notes) to No 8.and 12. Baltagi, Ch 8 and 12
Arellano and Bond estimation generalized IV I We consider again y it = k + α i + βx it + λy it 1 + u it, (1) u it IID(0, σ 2 ), which is a relevant model if the parameters of interest is the impact coeffi cient of x it, β, the long-run multiplier β/(1 λ) and the associated dynamic multipliers. After differencing, we get: y it = βx it + λy it 1 + u it, t = 3,...T (2) for observations (t = 1, 2,..., T ).
Arellano and Bond estimation generalized IV II Instead of regarding (2) as one equation, regard it a system of T 2 equations. IV t = 3 y i3 = βx i3 + λy i2 + u i3 z i2 = (y i1,x i3 ) t = 4 y i4 = βx i4 + λy i3 + u i4 z i3 = (z i3, y i2,x i4 ) t = T y it = βx it + λy it 1 + u it z it 1 = (z it 2, y it 1,x it ) It is the use of different instruments for equations of different time periods that defines the A&B method relative to conventional IV estimation, which uses the same instrument set for all endogenous variables. Conventional instruments, w t, can be used. In both STATA and PcGIVE, there is a distinction between GMM instruments (y i1,...y T 2 ) and other, conventional instruments, w t. Note how the GMM instruments are cumulated. Number of instruments is increasing with T.
Arellano and Bond, matrix formulation I From the table we see that we formally have a system of T 2 equations with different IV s for each equation. z i2 0 0 0 0 z i3 Z i =..... (3). 0 0 z it 1 Vector formulation: ( y }{{} i = y i, 1 λ + x i β + u i = q i )( ) λ y i, 1. x i }{{} β }{{} W i δ + u i }{{} ɛ i (4) which can be stacked for all individuals into one matrix equation: q = Wδ + ɛ (5)
Arellano and Bond, matrix formulation II Similar stacking gives Z from the N matrices Z i, see Kompendium sec 10.2. Write the orthogonality conditions for GMM estimation of (4) as E [ ] Z i ɛ i = with sample counterpart: z i2 (y i3 βx i3 λy i2 ) z i3 (y i3 βx i3 λy i2 ) =. z it 1 (y it βx it λy it 1 ) }{{} Z i (q i W i δ) 0 0. 0 g N (δ) = 1 N N Z i (q i W i δ) i=1
Arellano and Bond 1-step and 2-step estimation I With reference to the principles of GMM, we choose δ so that the quadratic form based on a weighting matrix S N is minimized: δ= arg min g N (δ) S N g N (δ) δ When the estimated equation is in differences, as in this case with (4), the matrix S N reflects the induced first order Moving Average (MA(1)) in the disturbances. ɛ i = u i. This is as with weighted least squares (GLS), but of course use IV to form the raw moments here. The resulting IV estimator is usually denoted δ 1 for one-step A&B estimator, or preliminary A&B estimator.
Arellano and Bond 1-step and 2-step estimation II The two-step estimator uses δ 1 to construct û it. And then do a second GMM based on ŜN using those residuals: Ŝ N = [N 2 N Z i (û i )(û i ) Z i ] 1 i=1 This estimator is referred to as the two-step A&B estimator, and is denoted δ 2. In principle the difference between δ 1 and δ 2 is that δ 1 is based on a known S N, i.e. on the assumptions of the model namely that u it is IID(0, σ 2 ), while δ 2 is the GMM estimator which is consistent despite those assumptions not holding in the data. In practice the most cited rationale for considering δ 2 is that this estimator is more effi cient when u it is heteroscedastic.
Arellano and Bond, modifications, extensions and practical issues I Above the equation of interest is in terms of differences variables, and lagged levels is used as instruments. But as noted in Lecture 6 can use lagged differences as instruments instead. If we are explicit about the fixed effects model, then we can of course keep the equation of interest in its original levels form, and estimate that equation with GMM, since the main problem we are solving with IV is the finite T bias caused by y t 1 being pre-determined. δ1 will then be based on u it IID(0, σ 2 ), no weighting of the observation, equivalently: Ŝ N = [N 2 σ 2 N z j z j ] 1 j=1
Arellano and Bond, modifications, extensions and practical issues II in the GMM formula. In general the coeffi cient standard errors for 2-step estimation are downward biased, in particular in small samples. There is a small sample correction due to Windmeijer (2000) that may or may not be standard in the software you are using. In Stata 12: Offi cial xtabond does not contain this small sample correction, but xtbaond2 does In PcGive: The correction is provided when the option robust standard errors is chosen. Additional output from the software. STATA and PcGive both give test statistics for 1. and 2. order residual autocorrelation in the estimated equation. Why is this relevant information?
Arellano and Bond, modifications, extensions and practical issues III Both programmes also gives test for instrument validity". STATA reports Hansen s test, which is also known as the J-test. PcGive reports a test due to Sargan. Heuristically, this statistic tests whether the information in the set of IV set is optimally used by treating them as instruments! The Sargan test may for example become significant if the structural equation has too simple dynamics, or if an extraneous instrument is an omitted variable in the equation. Dangers of over-instrumenting. It is important to keep in mind that A&B was invented to improve estimation properties when T is small (absolute and relative to N). The number of available A&B instruments is increasing in T, so choosing all available instruments will inflate the moments in ŜN.
Arellano and Bond, modifications, extensions and practical issues IV Known of problems of overfitting. Because of overfitting ŷ it 1 y it 1 and the 1-step estimator is driven towards the within group estimator, in GLS version if MA implication is used. The second step estimator may become impractical, both in terms of computation and numerical results. With enough instruments, the tests of instruments validity have very little power. Hides weak instrument problem Practical guidelines: Keep the maximum number of lags used as instruments to a reasonable level: 4 or 5 is still generous instrumentation in most cases. Judicious choice based on theory, existing studies and computer capacity, (A&B in their PcGive manual)
Arellano and Bond, modifications, extensions and practical issues V Report results for more than one estimator. Monte Carlo studies (e.g., Judson and Owen (1999)) show that already with T = 30, the LSDV estimator performs as well as the IV estimators (in terms of bias).
Dynamic model with RE I Consider now a dynamic version of the model discussed in Lecture 4 (see K 4.5 and 10.4): y it = k + α i + βx it + λy it 1 + δz i + u it, 1 < λ < 1 (6) u it IID(0, σ 2 ), α i IID(0, σ 2 a ), u it α i (x it, z i ) In addition to α i being uncorrelated with x it we include an individual specific explanatory variable. In Lecture 4 we learned that cannot be estimated δ under fixed effect. Kompendium Sec 10.4 motivates a 3-step procedure for estimation of β, λ and δ.
Dynamic model with RE II 1 GMM: Difference (6) and estimate β, λ by GMM. 2 Between: Treating β, λ as known from step 1, estimate k and δ by OLS on y i βx i λy i, 1 = k + δz i + u i + α i }{{} ε i Note: ε i = u i + α i has approximate IID properties since it only varies across i. 3 Standard errors. Define the composite residuals ɛ it by using the estimator from 1., and use residuals to estimate σ 2 and σ 2 a, as in the static version of this model, Kompendium, Ch 4.5.
Unit root and cointegration I In time series, the H 0 of λ = 1 is diffi cult to reject formally, for example for y t and x t. Given this, and if we want to test H 0 of β = 0 in y t = k + βx t + u t, (7) cannot use the usual F and t critical value, because then reject H 0 in 95 out of 100 tests, instead of in 5 out of 100 test (with 5% nominal size). This pitfall is known as spurious regression (which is different from spurious correlation although the statistician Yule worked with both problems), To do correct inference, need non-standard distribution theory, which is a central theme in a course in advanced time series econometrics.
Unit root and cointegration II Interesting to ponder why we should wish focus on λ = 1, when the stationary framework lets us use the whole line from 1 < λ < 1 (and more generally the whole unit circle)... But here only review the panel data counterparts to some of the popular unit-root and cointegration tests
Panel unit root test I Tests are with reference to or y it = γ z it + λ i y it 1 + u it (8) y it = γ z it + (λ i 1) }{{} ρ i y it 1 + u it (9) where γ z it represents deterministic terms. It can be a constant, or a constant and a trend. As with time series data, different distributions apply for the different specifications of the deterministic part.
Panel unit root test II Homogenous unit-root tests u it IID(0, σ 2 ) and λ i = λ, so homogenous DGP assumed in this test. Use t-statistics for H 0 : ρ = 0 against ρ < 0 based on the relevant estimation of (8) or (9), for example to account of fixed individual effects. These t-statistics are asymptotically N(0, 1) when T, and N. See Baltagi Ch 12. Entails that these tests have larger power than the corresponding Dickey-Fuller test for pure time series. When T is fixed, the t-statistics are still asymptotically normal but their variance depend on T.
Panel unit root test III Tests for individual unit roots. against H 1 : H 0 : ρ i = 0 for all i { ρi = 0, for i = 1, 2, 3,.., N 1. ρ i < 0, for i = N 1 + 1,...,N N Dickey-Fuller (DF) statistics are calculated, and then averaged for all N. A computer program like EViews gives these statistics with approximate critical values, also for the case when (9) is augmented by y t j (j > 0) terms. In the simplest case, they are N(0, 1) under H 0.
Panel cointegration tests I Like the first generation of cointegration test, the H 0 of no cointegration in the static panel model y it = k + α i + βx it + u it where both y it and x it have unit roots, can be based on the residuals û it. The tests are based on û it = λ i û it 1 + v it (10) where the H 0 of no cointegration translates to H 0 : λ i = 1 (or ρ i λ i 1 = 0). Like in the unit root-test there are two branches of tests:
Panel cointegration tests II Homogenous λ i = λ Gives Dickey-Fuller type tests, for example: Heterogenous λ i DF t = 1.25t p + 1.875N N(0, 1) ass, NT Averaging from N individual tests, see e.g. Baltagi ch 12.4.2 for details.