Sensitivity of GLS estimators in random effects models

Transcription

1 of GLS estimators in random effects models Andrey L. Vasnev (University of Sydney) Tokyo, August 4, / 19

2 Plan Plan Simulation studies and estimators 2 / 19

3 Simulation studies Plan Simulation studies Several units (such as households, individuals, firms, countries, etc.) are followed through several time periods. Response variable y it can be modeled as y it = x itβ + µ i + ν it, i = 1,...N, t = 1,...T, where x it - observable characteristics, µ i - unobservable (random) individual specific effect and ν it - remainder disturbance. Maddala and Mount (1973) report that the improvement in the variance estimator does not influence much the second step estimator. Baltagi (1981) finds the similar result for the two-way error model. Taylor (1980) gives a theoretical example of this effect. This paper suggests an answer in terms of sensitivity. 3 / 19

4 Plan Simulation studies Magnus and Vasnev (2007) define sensitivity s(θ 0 ) of the second step estimator ˆβ to the first step estimator θ from the Taylor expansion around the true value θ 0 ˆβ( θ) = ˆβ(θ0) + s(θ0)( θ θ0) + r. Properties of s(θ 0 ) are important. Under certain conditions sensitivity statistic and diagnostic tests (LM, LR, W) are asymptotically independent. 4 / 19

5 (cont) Estimators Figure 1 where y y = Xβ + Zµ + ν E(y) = Xβ var(y) = σ 2 µzz + σ 2 νi NT, is an NT 1 vector of the dependent variable, X is an NT K matrix of the explanatory variables, Z = I N ı T is an NT N matrix of zeros and ones ( I N ı T represents a Kronecker product), is an N N identity matrix, is a T 1 vector of ones, ν is an NT 1 vector of the unknown stochastic components, β is a K 1 vector of unknown parameters of interest, µ is an N 1 vector of individual unobservable effects. We assume that the data are generated with the true unknown parameter values β 0, σ 2 ν0, σ2 µ0, and θ 0. 5 / 19

6 (cont) (cont) Estimators Figure 1 The matrices of projection into the space of dummy variables Z and its complement, P = Z(Z Z) 1 Z and Q = I NT P. Also W xy = X Qy, B xy = X Py, T xy = W xy + B xy = X y and analogously W xx, B xx, T xx, W yy, B yy, and T yy. It is convenient to consider the parameters (σν, 2 θ) rather than (σν, 2 σµ) 2 with θ = σ 2 ν/(σ 2 ν + Tσ 2 µ). Then the variance var(y) = σ 2 ν(q + θ 1 P) 6 / 19

7 Estimators (cont) Estimators Figure 1 1. The true generalized least squared (GLS) ˆβ GLS = (W xx + θ 0 B xx ) 1 (W xy + θ 0 B xy ), var( ˆβ GLS ) = σ 2 ν0 (W xx + θ 0 B xx ) The ordinary least squared (OLS) estimator ˆβ OLS = Txx 1 T xy, var( ˆβ OLS ) = σν0 2 T xx 1 (B xx /θ 0 + W xx )Txx The least squared with dummy variables (LSDV) / within ˆβ LSDV = W 1 xx W xy, var( ˆβ LSDV ) = σ 2 ν0 W 1 xx. 4. The Least Squared Between Groups (LSBG) ˆβ LSBG = B 1 xx B xy, var( ˆβ LSBG ) = σ 2 ν0 (θ 0B xx ) Maximum Likelihood (ML), var( ˆβ ML ) = var( ˆβ GLS ). 7 / 19

8 Figure 1 (cont) Estimators Figure 1 var GLS OLS LSDV LSBG Figure 1: Variance of different estimators θ 0 8 / 19

9 Estimators (cont) (cont) Estimators Figure 1 We need ˆσ 2 ν and ˆσ2 µ to construct feasible GLS estimators 6. Wallace and Hussain s estimator (WH): With the OLS residuals instead of the true errors Zµ + ν one can construct the biased, but consistent estimators 7. Amemiya s estimator (AM): use within residuals. 8. Nerlove s estimator (NER): Another way to estimate σµ 2 is to use estimators of µ from the within regression. 9. Analysis of covariance (ANOVA): From the within and between sums of squared residuals, the unbiased estimators can be constructed 9 / 19

10 Estimators (cont) (cont) Estimators Figure Henderson s method III (H3): It is also possible to construct the unbiased estimators with the OLS residuals instead of the between residuals 11. Minimum norm quadratic unbiased estimator (MINQUE): Uniqueness of unbiased estimator y Ay can be achieved with an additional restriction. 10 / 19

11 results results results (cont) Figure 2 Figure 3 ˆβ( θ) = ˆβ(θ 0 ) + s(θ 0 )( θ θ 0 ) + r. Theorem 1 (): The sensitivity of any feasible GLS estimator to the first-step estimator of θ is given by s(θ 0 ) = (W xx + θ 0 B xx ) 1 B xx ( ˆβ LSBG ˆβ GLS ), its expectation is E(s(θ 0 )) = 0 and its variance is var(s(θ 0 )) = σν0(w 2 xx + θ 0 B xx ) 1 B xx [(θ 0 B xx ) 1 (W xx + θ 0 B xx ) 1 ]B xx (W xx + θ 0 B xx ) 1. In addition, if µ and ν are normally distributed, the sensitivity is also normally distributed. 11 / 19

12 results (cont) results results (cont) Figure 2 Figure 3 Theorem 2 (Independency): If µ and ν are normally distributed, the sensitivity statistics, s(θ 0 ), and any of the first-step estimators, θ WH, θ AM, θ NER, θ ANOVA, θ H3, θ MINQUE, are independent. Theorem 3 (Unbiasedness): When the sensitivity statistic, s(θ 0 ), and the first-step estimator, θ, are uncorrelated, then the second-step estimator, β( θ), is unbiased (up to the higher order terms) regardless the bias of θ. Theorem 4 (Variance decomposition): The variance of any feasible GLS estimator can be decomposed (up to the higher order terms) as var( ˆβ( θ)) = var( ˆβ(θ 0 )) + var(s(θ 0 )) E( θ θ 0 ) / 19

13 Figure 2 2 θ results results (cont) Figure 2 Figure 3 1 var(s(θ 0 )) (1 θ 0 ) var(ˆθ ML ) θ 0 Figure 2: Variance of components in the second term 13 / 19

14 Figure 3 results results (cont) Figure 2 Figure θ 2 0 var(s(θ 0 )) 0.02 var(s(θ 0 ))var(ˆθ ML ) (1 θ 0 ) 2 var(s(θ 0 )) θ 0 Figure 3: Variance of the second term 14 / 19

15 Design Design Figure 4 Results Results(cont) Similar to Maddala and Mount (1973). N = 25, T = 20, β 0 = 1, σ 2 ν0 = 100, and we vary θ 0 between 0 and 1: σ 2 µ θ The vector of the exogenous variable is generated from x it = 0.1t + 0.5x it 1 + ω it, where the iid-innovations, ω it, are uniformly distributed on the interval [ 1 2, 1 2 ] and the initial condition is x i0 = ω i0. For each set of parameters, in R = repetitions the individual effects, µ, and the vector of stochastic errors, ν, are generated independently from each other: µ N(0, σ 2 µ0i N ) and ν N(0, σ 2 ν0i NT ). The regression equation generates the dependent variable y = xβ 0 + Zµ + ν. 15 / 19

16 Figure 4 Design Figure MINQUE GLS ANOVA OLS Results Results(cont) ˆβ( θ) 1 AM NER WH ML H3 LSDV θ 1 Figure 4: Geometry of random effects estimators 16 / 19

17 Results θ 0 var(ˆβ) 1E-03 Design Figure 4 Results Results(cont) GLS OLS LSDV LSBG ML var( θ) 1E-04 WH AM NER ANOVA H3 MINQUE / 19

18 Results(cont) Design Figure 4 Results Results(cont) Big differences in θ small differences in β (as expected) Now the results are explained with sensitivity analysis: θ var(s(θ 0 )) The variance decomposition is accurate. 18 / 19

19 Final remarks sensitivity helps to explain evidence from simulation sensitivity is important by itself Final remarks 19 / 19