Online Appendix to Grouped Coefficients to Reduce Bias in Heterogeneous Dynamic Panel Models with Small T

Transcription

1 Online Appendix to Grouped Coefficients to Reduce Bias in Heterogeneous Dynamic Panel Models with Small T Nathan P. Hendricks and Aaron Smith October 2014

2 A1 Bias Formulas for Large T The heterogeneous dynamic panel that we consider is written as follows: y it = γ i y i,t 1 + β i x it + α i + ε it, (A1) x it = µ i (1 ρ) + ρx i,t 1 + η it. We assume that the random coefficients α i, β i, γ i, and µ i are independent of each other and ε it and η it. We also assume that x it is strictly exogenous. We use formulas derived by Pesaran and Smith (1995) for the asymptotic bias of pooled fixed effects for the case where N and T to show how the bias from coefficient heterogeneity depends on various parameters. Bias formulas are cumbersome even when T, so to simplify the exposition we consider two cases of the model in equation (A1). Case 1. In the case of N and T, with a heterogeneous coefficient on an autocorrelated regressor ( σ 2 γ = 0, σ 2 β > 0, ρ > 0 ), the asymptotic bias of pooled fixed effects is as follows: where ˆγ F E γ ˆβ F E β p 1 Ψ 1 ρ (1 ργ) (1 γ 2 ) σ 2 β βρ 2 (1 γ 2 ) σ 2 β, (A2) Ψ 1 = σ2 ( ) ε 1 ρ 2 (1 γρ) 2 + ( 1 γ 2 ρ 2) σ 2 σx 2 β + ( 1 ρ 2) β 2. (A3) The asymptotic bias formulas in equations (A2)-(A3) are the same as equations (2.7)- (2.9) in Pesaran and Smith (1995), except that we think they have a minor typo. 1 The estimate of γ is biased up if ρ is positive and biased down if ρ is negative. Intuitively, ignoring the coefficient heterogeneity causes the coefficient on the lagged dependent variable to capture the autocorrelation in the independent variable in addition to the dynamics of the model. The estimate of β is biased towards zero, regardless of the sign of ρ. 1 For the last term in Ψ 1, they have ( 1 ρ 2) β, whereas we think this should be ( 1 ρ 2) β 2. A1

3 The magnitude of the asymptotic bias in case 1 depends on the following parameters: γ, β, ρ, σβ, 2 and σ2 ε/σ x. 2 As expected, the bias is increasing in σβ. 2 The bias is also increasing in the signal-to-noise ratio σ2 x/σ ε that 2 is, if x it explains a relatively large proportion of the variation in y it, then bias from ignoring the heterogeneity of β i increases. Case 2. In the case of N and T, with a heterogeneous coefficient the lagged dependent variable and an autocorrelated regressor ( σγ 2 > 0, σβ 2 = 0, ρ > 0 ), the asymptotic bias of pooled fixed effects is as follows: ˆγ F E γ ˆβ F E β p 1 Ψ 2 E [(γ i γ) Λ] ρβ ρβe [ ] [ ] γ i γ 1 ργ i EΛ ρβe 1 1 ργ i E [(γi γ) Λ], (A4) where and Λ = Ψ 2 = EΛ ρ 2 β 2 E [ 1 1 ργ i ] 2, (A5) β 2 (1 + ργ i ) (1 ργ i ) (1 γi 2 ) + σ2 ε (1 ρ 2 ) σx 2 (1 γi 2 ). (A6) The bias formulas in equations (A4)-(A6) are obtained from the derivations in the appendix of Pesaran and Smith (1995). If x it is not autocorrelated (ρ = 0), then the fixed effects estimate of the coefficient on the independent variable is not biased and the sign of [ ] the bias of ˆγ F E corresponds to the sign of E. γ i γ 1 γ 2 i The magnitude of the bias in case 2 depends on the following parameters: γ, β, ρ, σ 2 γ, and σ2 ε/σ 2 x. Although σ 2 γ does not appear directly in equations (A4)-(A6), terms that include some form of the expected value of 1 1 γ 2 i depend on σ 2 γ. A2 Additional Monte Carlo Results Figures A1 A6 display additional results from Monte Carlo simulations. In the main paper, we only show Monte Carlo results with σ x = 0.5 and T = 6. In the following figures we A2

4 show other combinations of the parameters including σ x = 2 and T = 10. Table A1 shows the probability of rejecting the null hypothesis of no autocorrelation of the errors for the additional combinations of parameter values. A3

5 Figure A1: Monte Carlo Results with γ = 0.5, σ x =2, and T = 6 Coefficient on Lagged Dependent Variable Coefficient on Independent Variable A4

6 Figure A2: Monte Carlo Results with γ = 0.8, σ x =2, and T = 6 Coefficient on Lagged Dependent Variable Coefficient on Independent Variable A5

7 Figure A3: Monte Carlo Results with γ = 0.5, σ x =0.5, and T = 10 Coefficient on Lagged Dependent Variable Coefficient on Independent Variable A6

8 Figure A4: Monte Carlo Results with γ = 0.8, σ x =0.5, and T = 10 Coefficient on Lagged Dependent Variable Coefficient on Independent Variable A7

9 Figure A5: Monte Carlo Results with γ = 0.5, σ x =2, and T = 10 Coefficient on Lagged Dependent Variable Coefficient on Independent Variable A8

10 Figure A6: Monte Carlo Results with γ = 0.8, σ x =2, and T = 10 Coefficient on Lagged Dependent Variable Coefficient on Independent Variable A9

11 Table A1: Probability of Rejecting Null Hypothesis of No Autocorrelation of Error for Additional Parameter Values Probability of σ γ σ β Rejecting Null γ = 0.5, σ x = 2, T = γ = 0.8, σ x = 2, T = γ = 0.5, σ x = 0.5, T = γ = 0.8, σ x = 0.5, T = γ = 0.5, σ x = 2, T = γ = 0.8, σ x = 2, T = A10

12 A3 Additional Details on Application to Labor Demand Table A2: Observations in each year of the original data from Arellano and Bond (1991) Year Observations A11

13 References Arellano, M., and S. Bond Some Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to Employment Equations. The Review of Economic Studies 58: Pesaran, M.H., and R. Smith Estimating Long-Run Relationships from Dynamic Heterogeneous Panels. Journal of Econometrics 68: A12