Introduction to Dynamic Panel Data: Autoregressive Models with Fixed Effects

Transcription

1 Introduction to Dynamic Panel Data: Autoregressive Models with Fixed Effects Eric Zivot Winter, 2013

2 Dynamic Panel Data (without covariates) Typical assumptions = = 1 (individuals) = 1 (time periods) 1. Stationarity: 1 2. [ 0 1 ]=0 3a.Noserialcorrelation: (0 2 ) 3b. Homoskedasticity: (0 2 )

3 Stationary Model Representation = By recursive substitution 1 = = = [ ]+ + 2 = 2 0 +(1+ ) = X + 1 X

4 Now For large [ ] = [ 0 ]+ + 1 X [ ] = [ 0 ]+ 1 X 1 X 1 X 0 1 1

5 so that (for large ) [ ] 1 var[ ] = X var[ ]

6 Estimation = = X + 1 X Can t use RE estimation because [ 1 ] = X + 2 X 6= 0 What about FE estimation?

7 Write the model in FE matrix notation as y = y η 1 =. y 1 = 1 y η = 1. Define Q = I P 1 so that Q y = ỹ = 1. = 1

8 Then, the transformed model is Q y = Q y 1 + Q 1 + Q η ỹ = ỹ 1 + η The FE estimator of is the pooled OLS estimator on the transformed model ˆ = = 1 ỹ 1ỹ 1 0 ỹ 1ỹ 0 1 y 0 1 Q y 1 y 0 1 Q y

9 For consistency, consider ˆ = 1 ˆ will be consistent if y 0 1 Q y y 0 1 Q η lim 1 y 0 1 Q η = [y 0 1 Q η ]=0 Note: In the asymptotic analysis, the cross section dimension but the time series dimension is held fixed!

10 Now y 0 1 Q η = h 0 1 i = = 1 1 ( ) 1 = X + 2 X

11 Using 1 = it follows that 2 X + 2 X because [y 0 1 Q η ]= [ 1 ]= h 1 ( ) i 6= = 0

12 Result: ˆ 9 as for fixed due to correlation between 1 and Remark: If both and then it can be shown that ˆ because = 1 0 [ 1 ]=0 This is an example of double asymptotic analysis that is common in the analysis of panel data models.

13 Bias of Fixed Effects Estimator In the stationary model with iid (0 2 ) Nickell (1981, Ecta) showed that for fixed and [y 1 0 Q η ] = 2 ( ) " 1 ( ) = 1 1 à ˆ is downward biased Further, Nickell showed that for fixed as ˆ (1 2 ) 1 Notice that as and sequentially à ˆ ( ) 1!!#

14 Example of Bias in FE estimation of dynamic panel model FE bias: ˆ / Remarks 1. If 0 the bias is always negative, and massive for very small values of 2. As increases, the bias decreases but even with =15the bias is still substantial for large.

15 IV Estimation Anderson and Hsiao (1981) suggested the following approach 1. First eliminate the fixed effect by taking first differences Note 1 = ( 1 2 )+ 1 = 1 + =2 [ 1 ] 6= 0 due to correlation between 1 and 1 terms.

16 Stacking across time gives y ( 1) 1 = y 1 + η 2. Do IV estimation using 2 as an instrument for 1 [ 2 1 ] = [ 2 ( 1 2 )] 6= 0 ³ [ 2 ] = [ 2 1 ] =0 since is iid.

17 The IV estimator is then 1 ˆ = y 2 0 y 1 y 2 0 y P P =2 2 = P P =2 2 1 whichisconsistentbyconstruction. Remark: The Anderson-Hsiao estimator does not exploit all the relevant moment conditions so it is not the most efficient GMM estimator.

18 Arellano-Bond GMM Estimator Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to Employment Equations, Review of Economic Studies, 58, 1991 Arellano and Bond (AB) derived all of the relevant moment conditions from the dynamic panel data model to be used in GMM estimation. The moment condtions are based on the first differenced model = 1 + =2 They showed that the number of moment conditions depends on (number of time periods)

19 Example: =4 6 moment condtions For GMM estimation, define g 4 ( ) = = 2 : [ 2 0 ]=0 [ = 3 : 3 0 ]=0 [ 3 1 ]=0 = 4 : [ 4 0 ]=0 [ 4 1 ]=0 [ 4 2 ]=0 = ( 2 1 ) 0 ( 3 3 ) 0 ( 3 2 ) 1 ( 4 3 ) 0 ( 4 3 ) 1 ( 4 3 ) 2

20 Notice that g 4 ( ) is a linear function of Itmaybere-expressedinmatrix form as g 4 ( ) = = X 40 [ y4 y4 1 ]

21 where y 4 = X 4 = y 1 4 =

22 ThesamplemomentsusedforGMMestimationarethen g 4 ( ) = 1 S 4 = 1 S 4 1 = 1 X 40 [ y4 y4 1 ] = S 4 S4 1 X 40 y4 X 40 y4 1

23 Because is randomly sampled over it follows that S = 6 6 [g4 ( )g4 ( )0 ]= [X 40 η η0 X4 ] Under conditional heteroskedasticity, a consistent estimate is Ŝ = 1 X 40 ˆη ˆη0 X4 ˆη = [ y 4 ˆ y4 1 ] ˆ For example, can use the Anderson and Hsiao IV estimate of

24 Under conditional homoskedasticity where S = [X 40 η η0 X4 ]= [ [X40 C0 η η 0 CX4 X4 ]] = = [X 40 C0 [η η 0 X4 ]CX4 ]= 2 [X 40 C0 CX 4 ] C 0 ( 1) = So a consistent estimate of S is Ŝ = X 40 C0 CX 4 Estimation of 2 is not required for GMM estimation because it cancels out in the resulting estimator.

25 The efficient GMM estimator solves min ( Ŝ 1 )= g ( )Ŝ 4 1 g ( ) 4 = [S 4 S4 1 ] 0 Ŝ 1 [S 4 S4 1 ] Since ( Ŝ 1 ) is linear in the analytic solution is ˆ (Ŝ 1 )=(S 40 1 Ŝ 1 S 4 1 )S 40 1 Ŝ 1 S 4 This estimator is known as the Arellano-Bond GMM estimator.

26 Example: International Difference in Output Growth Rates (Hayashi, Section 5.4) Q: Do poor countries grow faster than rich countries? If so how much faster? Neoclassical growth theory foundations ( ) = output per effective labor at time for a country = steady state The log-linear approximation around gives the adjustment equation ln( ( )) = [ln( ) ln( ( )] = speed of convergence 0

27 Between any two time periods 1 and, the log-linear adjustment equation implies ln( ( )) = (1 ) ln( )+ ln( ( 1 )) = exp[ ( 1 )] Define ( ) = ( ) ( ) ( ) ( ) = aggregate output ( ) = aggregate hours worked ( ) = level of labor augmenting technical progress

28 Assume ( ) grows at a constant rate Then ( ) = (0) exp( ) and Ã! ( ) ln ( ) =ln ln( (0)) ( ) Substituting into the output equation gives Ã! Ã! ( ) ( 1 ) ln = ln +(1 )[ln( ) ln( (0))] + ( ) ( 1 ) = ( 1 ) Subtracting ln ln à ( ) ( )! µ ( 1 ) ( 1 ) =( 1) ln from both sides gives the growth equation à ( 1 ) ( 1 )! +(1 )[ln( ) ln( (0)] + Because 1, the level of per capita output has a negative effect on growth. Hence, poor countries should grow faster than rich countries.

29 Turning the output equation into a dynamic panel data model Assume Ã! Ã! ( ) ( 1 ) ln = ln +(1 )[ln( ) ln( (0)] + ( ) ( 1 ) = ( 1 ) holds for every country where and are the same for every country but that and might differ. Then we can write = = ln( ( ) ( )) = log per capita output at time = (1 ){ln( ) ln( (0))} for country = country and time specific shock (e.g. business cycle)