Lecture 6: Dynamic panel models 1
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1 Lecture 6: Dynamic panel models 1 Ragnar Nymoen Department of Economics, UiO 16 February 2010
2 Main issues and references Pre-determinedness and endogeneity of lagged regressors in FE model, and RE model Bias of within and between estimators, and GLS. In FE model: How small is large. Available instrumental variables (for IV estimation and other methods of moments ). GMM (principles) EB (book/kompendium): Ch 9.6 and Ch 10. EB (lecture notes) to No 8.and 12
3 The dynamic model I We now consider a model of the type: y it = k + α i + βx it + λy it 1 + u it, (1) u it IID(0, σ 2 ), which is a dynamic model, since time plays an essential role in the model (assuming that the parameter λ 0). A dynamic model is usually required to build relevant empirical models for time series data, because adjustments speeds in actual behaviour/decisions are not infinite as formally required by static models. Can also regard (1), with λ 0, as a general model, and the static model, with λ = 0, as a special case. The natural to test the validity of the static model formally by a test of H 0 λ = 0.
4 The dynamic model II In terms of econometric issues raised, model (1) is representative of a wider class of ADL model, with lagged x varaibles and higher order lags in y. The explanatory variable x it is strictly exogenous: Cov(x it, u it s ) = 0 for all t and s = 0, ±1, ±2,... The explanatory variable y it 1 is pre-determined with respect to the disturbances u it.. y it 1 is not, however, strictly exogenous, and may lead large sample bias (inconsistency) or small T sample bias in the OLS based estimators for the parameters of interest: β and λ.
5 Source of endogeneity: The solution of dynamic models I The statistical properties of y it, and y it 1, depend on the solution of the dynamic model (1): y it = k + α i + βx it + λy it 1 + u it Solutions to (1), and generalizations of that equation, exists under mild assumptions, but these solutions can be stable, unstable or explosive, and the (in)stability of the solution plays an important role in both dynamic economics and in economtrics Another, and related distinction is between causal and non-causal solutions.
6 Source of endogeneity: The solution of dynamic models II Causal solutions give y it as a function of past disturbances (and in (1), also of x variables ). The model is driven by shocks and causes that have already occurred. Models that have causal solutions are called causal models. In this course we consider causal models that have either stable or unstable solutions. Causal solutions of equation (1) are (asymptotically globally) stable if and only if 1 < λ < 1. y it = k + α i + βx it + λ{k + α i + βx it 1 + λy it 2 + u it 1 } + u it = (k + α i )(1 + λ) + β(x it + λx it 1 ) + u it + λu it 1 + λ 2 y it 2... J 1 J 1 J 1 = (k + α i ) λ j + β λ j x it j + λ j u it j + λ J y it J j=0 j=0 j=0
7 Source of endogeneity: The solution of dynamic models III Asymptotically when J, y it becomes independent of the initial condition y it J, if and only if 1 < λ < 1. In the stable case we can therefore write the asymptotic solution as y it = k + α i 1 λ + β λ j x it j + λ j u it j (2) j=0 But the same logic applies to y it 1 : j=0 j=0 y it 1 = k + α i 1 λ + β λ j x it j 1 + λ j u it j 1 (3) j=0 Hence: y it 1 is uncorrelated with current and future disturbances u it, u it+1,..., but is correlated with past disturbances u it 1, u it 2,..., (so y it is a predetermined variable).
8 Source of endogeneity: The solution of dynamic models IV As usual with panel data, the modelling assumptions for the individual effects α i are important: If we regard the α i s as parameters that can be estimated ( made observable ) by estimation as in the FE model, the endogeneity problem is limited to the consequences of having a predetermined regressor, rather than an exogenous regressor in the model. In the random effects (RE) model, the composite disturbance ɛ it = α i + u it will add a second dimension to the endogeneity problem.
9 Predeterminedness and endogeneity in the FE model I With reference to the solution above: E [y it 1 u it+j ] = 0, j = 0, 1, 2... [( ) ] E [y it 1 u it 1 ] = E λ j u it j 1 u it 1 = σ 2 j=0 and generally: [( ) ] E [y it 1 u it j 1 ] = E λ j u it j 1 u it j 1 = λ j σ 2, j = 0, 1,.. j=0 The correlations are declining with decreasing j if and only if 1 < λ < 1.
10 The Random Effects Model I We have (1) y it = k + α i + βx it + λy it 1 + u it, with u it IID(0, σ 2 ), but the model for the individual effects is now: α i IID(0, σ 2 α ) and, ɛ it = α i + u it, α i u it The solution for y it 1 : y it 1 = k 1 λ + β λ j x it j 1 + λ j it j 1 ɛ it 1 j=0 j=0 show that y it 1 is correlated with ɛ it :
11 The Random Effects Model II [( ) ] E [y it 1 ɛ it ] = E λ j (α i + u it j 1 (α i + u it ) j=0 = 1 1 λ σ 2 α Note the difference from the FE model, where the corresponding correlation is zero. In fact, all the forward correlations are non-zero, given the assumptions of the RE model: E [y it 1 ɛ it+s ] = 1 1 λ σ 2 α, for s = 0, 1, 2, (4)
12 The Random Effects Model III While the current and lagged correlations are and, in general E [y it 1 ɛ it 1 ] = 1 1 λ σ 2 α + σ 2 E [y it 1 ɛ it j 1 ] = 1 1 λ σ 2 α + λj σ 2, for j = 0, 1, 2, (5) We see that the endogeneity of y it 1 is more pervasive in the RE model than in the FE model, and that y it 1 is not pre-determined with respect to the composite dummy ɛ it. One way to obtain intuition is to remember back to your first econometrics course, where we learned that the OLS estimator for the parameters in a model with lagged regressors are:
13 The Random Effects Model IV 1 consistent, but biased in small samples (small T ), if the disturbances are without autocorrelation. 2 Inconsistent if the disturbances are autocorrelated. Simply writing the RE model as: y it = k + βx it + λy it 1 + ɛ it, with (6) E [ɛ it ɛ it 1 ] = E [(α i + u it )(α i + u it 1 )] = σ 2 α shows the RE model is an example of case 2! In fact, lecture 3 showed that for t s: Covar[ɛ it, ɛ is ] = E [(α i + u it )(α i + u is )] = E [α 2 i ] = σ 2 α, a form of autocorrelation called equi-correlation. β OLS and β GLS estimators in (6) become inconsistent. Stochastic individual effects are the reason why the autocorrelation never dies in this model!
14 Inconsistency of the within estimator in the FE model I Since the source of the problem is the lagged regressor, we analyse first the model without exogenous regressors (but remember that any bias in the estimation of λ also affects the estimation of β). The model is now: y it = α i + λy it 1 + u it, u it IID(0, σ 2 ) (7) where α i are stochastic individual effects that we model by N dummies.
15 Inconsistency of the within estimator in the FE model II The LSDV estimator is identical to λ W : N T i=1 t=1 λ W = (y it 1 ȳ i 1 )y it N T i=1 t=1 (y it 1 ȳ i 1 ) 2 Insertion for y it from (7) gives λ W = λ + 1 N T TN i=1 t=1 (y it 1 ȳ i 1 )u it T t=1 (y it 1 ȳ i 1 ) 2 1 TN N i=1 def = C D λ W is consistent if plim(c) = 0 (assuming plim(d) > 0). Can evaluate plim both in the N and T dimension. Note that we have C = 1 N T TN u i=1 t=1 ity it 1 1 N N ȳi 1u i=1 i 1
16 Inconsistency of the within estimator in the FE model III Look first at N, with T fixed plim(c) = 1 T T plim 1 N t=1 N u i=1 ity it 1 }{{} =0 T t=1 plim 1 N N i=1 ȳi 1u i 1 }{{} 0 Because the terms ȳ i 1 and u i 1 are correlated with reference to the solution of y it, while y it 1 does not contain u it. plim (C) 0 plim N N λ W λ, when T is fixed
17 Inconsistency of the within estimator in the FE model IV Hence, since the most relevant asymptotics for panel data is N large, we conclude that the Within estimator is inconsistent when the model is dynamic. The individual effect plays no role in the inconsistency problem in the FE model.
18 Reference case: T large consistency in FE model I Then consider T, with N fixed C = 1 N N i=1 1 T T u t=1 ity it 1 1 N N ȳi 1u i=1 i 1 plim(c) = 1 N N plim 1 T i=1 T u t=1 ity it 1 }{{} =0 1 N N i=1 plim(ȳ i 1u i 1 ) }{{} =0 1 plim(ȳ i 1 u i 1 ) = plim T T T t=1 y it 1 1 T plim T T u t=1 it }{{} =0
19 Reference case: T large consistency in FE model II Hence plim (C) = 0 plim λ W = λ, iff 1 < λ < 1 T T REMARK 1: Two assumptions about model properties drives the asymptotic result about consistency of λ W for fixed N and Stability of solution: 1 < λ < 1 No autocorrelation in the u it disturbances. If this fails, plim 1 T T t=1 u ity it 1 0. REMARK 2: The analysis and results extend to models with explanatory variables, x it and lags x it j. Specifically: plim β W = β, but plim β W β. N T
20 How small is large in the FE model? I How small can T be before the bias λ W λ becomes so little that other concerns get priority (for example possible misspecification of the model that we estimate). There are some results from time series that gives insight. Consider a time series version of (7): y t = λy t 1 + u t, u IIN(0, σ 2 ) (8) The OLS estimator λ (the ML estimator conditional on u 0 ) has the properties: Function Asymptotic Finite sample E [ λ] λ λ 2λ/(T 1) Var[ λ] 0 (1 λ 2 )/T
21 How small is large in the FE model? II Examples of E [ λ] λ 2λ/(T 1) : E [ λ T = 4] /(4 1) = 0.43 E [ λ T = 40] /(40 1) = 0.93 E [ λ T = 10] /(40 1) = 0.76 See that not only the nominal sample size T, but also the true value of λ plays a role. Define the effective sample size, T as: T = T (1 λ 2 ) With λ = 0, each new observation is simply the new disturbance u t. Conversely, with high λ, each new observation adds only little new information about the parameter. Hence, bias is large for a given nominal sample size.
22 How small is large in the FE model? III A more general formula, shown in EB : plim N λ W λ = 1 T = 2 gives plim ( λ W λ) = 1+λ 2, N λ = 0 gives plim( λ W λ) = 1 T 1+λ 1 λt T 1 (1 T (1 λ) ) 2λ 1 λt (1 λ)(t (1 1) T (1 λ) ) Thus confirming, qualitatively, the time series approxinmations above.
23 Avoid wrong response to small T bias! Note that if the true model (the DGP) is y it = α i + βx it + λy it 1 + u it, u it IID(0, σ 2 ) as in (1), and the parameters of interest is β and B = β/(1 λ), worry about bias due to pre-determinedness of y it 1 should not lead to estimation of a static model y it = α i + βx it + u it (9) because β from (9) gives plim β β B, because of omitted variables bias, which can be huge relative to andy induced bias caused by lagged regressor.
24 How can we improve on RE or FE model for dynamic panels? I The most popular approach has become IV estimation on the differenced model, but see also EB(K) 10.1.c, p 151 for a transformation that gives consistent OLS estimation. To motivate the differencing, note: y it y }{{ it 1 = α } i α i + β(x it x it 1 ) + λ(y }{{} it 1 y it 2 ) + (u }{{} it u it 1 ) }{{} y it x it y it 1 u it (10) which is an example of data differencing prior to estimation, rather than a 1-1 transformation of the original model, for example equation (7). This removes α i, so in RE model that source of autocorrelation has been removed.
25 How can we improve on RE or FE model for dynamic panels? II But still: Cov [(y it 1 y it 2 ), (u it u it 1 )] = E [y it 1 u it 1 ] 0 so OLS on the transformed equation (10) gives inconsistent estimators, for example λ W. In addition, u it is a MA(1) variable if u it is white noise as assumed (affects t-values and inference). Solution: Find instrumental variables for (y it 1 y it 2 ).
26 IV estimation of the equation in differences I For simplicity omit x it have (7): and by setting β = 0 in the DGP. We then y it = α i + λy it 1 + u it, u it IID(0, σ 2 ) (11) with the use of transformed data. We have, from the solution for y it, that y it = λ y it 1 + u it (12) y it = y it 1 = α i 1 λ + j=0 α i 1 λ + j=0 λ j u it j λ j u it j 1
27 IV estimation of the equation in differences II So, for y it : Hence: y it = u it + λ j u it j λ j u it j 1 j=1 j=1 j=0 = u it + λ( λ j 1 u it j ) λ j 1 u it j j=1 = u it (1 λ)( λ j 1 u it j ) j=1 y it 2, y t 3,... are uncorrelated with u it in (12), and all these lags can be IVs for y it 1. Note: Strength of instrument may suffer if λ is large (close to 1).
28 Case of only one IV I The status of y it 2 as a valid IV can be summarized in the orthogonality condition: E(y it 2, u it ) = 0 (13) λ IV = = N i=1 N i=1 N i=1 N i=1 T t=1 y it 2 y it T t=1 y it 2 y it 1 T t=1 y it 2 (λ y it 1 + u it ) T t=1 y it 2 y it 1 = λ + N i=1 N i=1 T t=1 y it 2 u it T t=1 y it 2 y it 1 Since 1 N T plim N(T 1) y i=1 t=1 it 2 u it = 0 N
29 Case of only one IV II is implied by the solution for y it 2, we have that Extensions/modifications: plim λ IV = λ. N 1 Keep (11) in level and use y it 2 as instrument, se EB (K) 10.2.b 2 In (12) use y it 2 as IV instead, or together with, y it 2, see EB (K).10.1.c 3 Use longer lags of y it j and y t j (j > 0) to form a longer list of IVs. 4 Increase effi ciency and improve inference by taking account of departure from IID assumption for u it in (12). Use General Methods of Moments (GMM) to tackle 3. and 4.
30 Eksurs to basic of GMM (see LN 8) I We have the theoretical equation: where β is (K 1) and x is 1 K. y and ɛ are scalar stochastic variables y = xβ + ɛ (14) ɛ and x are correlated, but there exists a 1 G vector z that satisfy the orthogonality condition: E [z ɛ] = E [z (y xβ)] = 0 G,1 (15) Both (14) and (15) hold in the population.
31 Eksurs to basic of GMM (see LN 8) II In the data we have N observations (y j, x j, z j ) and can formulate the empirical counterpart to E [z (y xβ)]: g N (y, x, z; β) = 1 N N z j (y j x j β) j=1 The idea is to choose β GMM so that g N (y, x, z; β) = 0 G,1 or as close to 0 G,1 as possible. If G = K: β GLS = β IV. If G > K: Use a linear combination to obtain G optimal instruments.
32 Eksurs to basic of GMM (see LN 8) III We find the optimal instruments by minimizing a quadratic form based on a matrix S N : Formally: Q(S N ) = g N (y, x, z; β) S N g N (y, x, z; β) βgmm = arg min β g N (y, x, z; β) S N g N (y, x, z; β) If ɛ have classical properties, S N = [N 2 σ 2 ɛ N j=1 z j z j ] 1 and βgmm = β 2SLS. If S N is unknown: use S N = [N 2 σ 2 ɛ Nj=1 z j z j ] 1 to obtain a first-step GMM estimator, then estimate S N from the first-step residuals. Using ŜN back in the GMM fomulae gives a second-step GMM estimator.
33 Eksurs to basic of GMM (see LN 8) IV GMM is highly relevant for dynamic panels model, since IV estimation is required for N consistency, and the disturbance in the structural equation do not have classical properties. Arellano & Bond estimator has become much used in practice.
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