Dynamic Panel Data Models
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1 Models Amjad Naveed, Nora Prean, Alexander Rabas 15th June 2011
2 Motivation Many economic issues are dynamic by nature. These dynamic relationships are characterized by the presence of a lagged dependent variable among the regressors, f.ex. Dynamic Wage equations Employment Models and many more
3 The Basic Model A one-way error component model: Y it = δy i,t 1 + x it β + u it ; i = 1,..., N t = 1,..., T where u it = µ i + ν it, with µ i IID(0, σ 2 µ) and ν it IID(0, σ 2 ν)
4 The Basic Problems Because of the lagged variable, OLS is biased and inconsistent even if the ν it are not serially correlated. Why: Since y it is a function of µ i, so is y i,t 1. FE is biased but consistent for T. Why: The within transformation wipes out the µ i, but we get problems because the y i,t 1 is still correlated with the ν i by construction (as this average contains ν i,t 1 ). The RE estimator is also biased, because (y i,t 1 θy i., 1 ) will be correlated with (u it θu i., 1 )
5 Suggested Solution Anderson and Hsiao (1981) suggested first differencing the model to get rid of the µ i, and then using an. This proposed method leads to consistent but not necessarily efficient estimates, because it does not make use of all available moment conditions it does not take into account the differenced structure on the residual disturbances ν it (1991) then proposed a more efficient estimation procedure.
6 Idea: IV-Estimation argue that additional instruments can be obtained if one utilizes the orthogonality conditions that exist between lagged values of y it and ν it. Idea: Take first differences to get rid of the individual effects and use all the past information of y it as instruments.
7 Illustration To illustrate, we use the model y it = δy i,t 1 + u it, where u it = µ i + ν it with µ i IID(0, σ 2 µ) and ν it IID(0, σ 2 ν). First, we difference to eliminate the individual effects: y it y i,t 1 = δ(y i,t 1 y i,t 2 ) + (ν it ν i,t 1 ). (1)
8 Instrumental variables The first period where we can use an instrumental variable is t = 3, where we have y i3 y i2 = δ(y i2 y i1 ) + (ν i3 ν i2 ). Here, y i1 is not correlated with the error and is therefore a valid instrument since it is correlated with (y i2 y i1 ) and not with (ν i3 ν i2 ). One period forward we have y i4 y i3 = δ(y i3 y i2 ) + (ν i4 ν i3 ) where y i1 and y i2 are valid instruments. Therefore, in period T, the set of valid instruments is (y i1,..., (y i,t 2 )).
9 Matrix of instruments We can define a matrix that contains all instruments of individual i: [y i1 ] 0 0 W i = 0 [y i1, y i2 ] [y i1,..., (y i,t 2 )]
10 Error term But we still need to account for the differenced error term (ν it ν i,t 1 ) in (1). The variance-covariance matrix of the error E[ v i v i ] = σ2 ν(i N G), where G = Since the instruments are orthogonal to the error by construction, we have the moment condition E[W i v i] = 0.
11 Consistent estimate Premultiplying the model (1) in vector form with the matrix of all instruments gives W y = W ( y 1 )δ + W ν. Performing GLS on this model, we get the Arellano and Bond (1991) one-step consistent estimator ˆδ 1 = [( y 1 ) W (W (I N G)W ) 1 W ( y 1 )] 1 x[( y 1 ) W (W (I N G)W ) 1 W ( y)]. (2)
12 Optimal GMM estimator The optimal GMM estimator (Hansen 1982) for this model for N and T fixed (using only the above moment restriction) is the same formula as (2) except replacing (W (I N G)W ) by V N = N i=1 W i ( v i)( v i ) W i, where the v are obtained from the residuals from the one-step estimation. The two-step (1991) estimator is then given by ˆδ 1 =[( y 1 ) W ( ˆV N ) 1 W ( y 1 )] 1 x[( y 1 ) W ( ˆV N ) 1 W ( y)].
13 Exogenous Parameters If there are additional strictly exogenous regressors x it with E(x it ν is ) = 0 (t, s), but where the x it are correlated with µ i, then the x it are valid instruments for the first-differenced equation. Therefore, [x i1, x i2,..., x it ] should be added to each diagonal element in W i.
14 Predetermined Parameters If x it are predetermined rather than strictly exogenous with E(x it ν is ) 0 for s < t and 0 otherwise, then only [x i1, x i2,..., x i(s 1) ] are valid instruments at period s. Then we get [y i1, x i1, x i2 ] 0 [y i1, y i2, x i1, x i2, x i3 ] W i =... 0 [y i1,..., (y i,t 2 ), x i1,..., x i,t 1 ]
15 Dynamic Demand for Cigarettes lnc it = α + β 1 lnc i,t 1 + β 2 lnp i,t + β 3 lnpn it + β 4 lny it + u it Notation with u it = µ i + λ t + ν it i = 1,..46 states t = 1,..30 years C it.. real p.c. sales of cigarettes (packs by persons of smoking age) P it.. average retail price (in real terms) Pn it.. minimum real price in neighboring state (proxy for smuggling) Y it.. real p.c. disposable income
16 FD-2SLS ols fe re FD-2SLS lnc_ *** 0.830*** 0.963*** (0.006) (0.013) (0.006) lnprice *** *** *** (0.015) (0.023) (0.015) lnpmin *** (0.013) (0.027) (0.013) lny *** 0.107*** (0.006) (0.023) (0.008) D.lnC_ ** (0.263) D.lnPrice *** (0.035) D.lnPmin (0.045) D.lnY 0.178*** (0.055) N Std. errors in parenthesis. FE, RE, and regressions include time dummies (jointly significant). * p<0.10, ** p<0.05, *** p<0.01
17 GMM one-step Arellano-Bond dynamic panel-data estimation Number of obs = 1288 Group variable: state Number of groups = 46 Time variable: yr Obs per group: min = 28 avg = 28 max = 28 Number of instruments = 437 Wald chi2(31) = Prob > chi2 = One-step results lnc Coef. Std. Err. z P> z [95% Conf. Interval] lnc L lnprice lnpmin lny Time dummies not shown. Sargan test of overidentifying restrictions H0: overidentifying restrictions are valid chi2(405) = Prob > chi2 =
18 GMM two-step Arellano-Bond dynamic panel-data estimation Number of obs = 1288 Group variable: state Number of groups = 46 Time variable: yr Obs per group: min = 28 avg = 28 max = 28 Number of instruments = 437 Wald chi2(31) = Prob > chi2 = Two-step results lnc Coef. Std. Err. z P> z [95% Conf. Interval] lnc L lnprice lnpmin lny Time dummies not shown. Sargan test of overidentifying restrictions H0: overidentifying restrictions are valid chi2(405) = Prob > chi2 =
19 Summary In this example the simple FE estimator does not seem to do such a bad job Stata is not able to exactly replicate the results in Baltagi Many estimators are implemented; but caution is necessary!
20 Thank you for your attention!
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