Spatial Regression. 11. Spatial Two Stage Least Squares. Luc Anselin. Copyright 2017 by Luc Anselin, All Rights Reserved

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1 Spatial Regression 11. Spatial Two Stage Least Squares Luc Anselin 1

2 endogeneity and instruments spatial 2SLS best and optimal estimators HAC standard errors 2

3 Endogeneity and Instruments 3

4 Spatial Lag Model y = ρwy + Xβ + u Wy endogenous = simultaneous equation bias y = Zθ + u Z = [ Wy X ] θ = [ ρ β ] 4

5 Endogeneity of Wy reduced form: y = (I - ρw) -1 Xβ + (I - ρw) -1 u Wy = W(I - ρw) -1 Xβ + W(I - ρw) -1 u E[(Wy) u] = E[y W u] 0 because E[u (I -ρw ) -1 W u] = tr W(I -ρw) -1 E[uu ] 0 5

6 Instruments from Reduced Form basis for instruments is E[ Wy X ] E[ y X ] = (I - ρw) -1 Xβ, therefore E[ Wy X ] = W(I - ρw) -1 Xβ series expansion yields: (I-ρW) -1 Xβ = Xβ+ ρwxβ+ ρ 2 W 2 Xβ+... 6

7 Instruments - Kelejian and Prucha (1999) selection of instruments from higher order terms E[ Wy X ] = WXβ + W(ρWXβ) +... instruments Q = [ X, WX, W 2 X,... ] in practice first order spatial lags, sometimes second order as well potential problems with multicollinearity 7

8 Spatial 2SLS 8

9 Assumptions same regularity conditions as 2SLS but requires triangular array central limit theorem to prove consistency (due to Kelejian and Prucha) plim (1/n)Q Q = H QQ finite, non-singular WX and X not linearly dependent excludes situations where Wx k = xk, such as constant or certain dummy variables 9

10 Assumptions (2) plim (1/n)Q Z = H QZ finite and full column rank requires at least one exogenous x k other than constant S2SLS does not work for pure SAR model identifiability condition precludes H0: β=0 10

11 Spatial 2SLS Estimation standard result, with instruments Q = [ X, WX, W 2 X,... ] for homoskedastic, uncorrelated errors θ2sls = [Z Q(Q Q) -1 Q Z] -1 Z Q(Q Q) -1 Q y Var[θ2sls] = σ 2 [Z Q(Q Q) -1 Q Z] -1 consistent, but not most efficient 11

12 spatial 2SLS estimation - WX as instruments 12

13 spatial 2SLS estimation - WX and W 2 X as instruments 13

14 estimates with different spatial lag orders for the instruments 14

15 Spatial 2SLS with Additional Endogenous Variables not only Wy endogenous but also some of the Z instruments for the other endogenous variables instruments and their spatial lags 15

16 additional instruments for endogenous variable UE90 16

17 Best and Optimal Estimators 17

18 Optimal Instrument Matrix (Lee 2003) Q = [X, W(I - ρw) -1 Xβ ] using consistent estimates for ρ and β from first stage estimation requires inverse of n by n matrix yields Best 2SLS 18

19 Best 2SLS (Kelejian et al 2004) avoids inverse matrix exploits the usual series expansion Q = [ X, Σs=0 ρ s W s+1 Xβ ] ρ and β from first stage estimation approximation up to s = r = o(n1/2 ) 19

20 Optimal GMM (Lee 2006) general set of moment conditions E[Q u] = 0 E[u Pu] = σ 2 tr P = 0 matrices P 1 with trace 0 and matrices P2 with diagonal zero 20

21 Optimal GMM (2) examples of matrix P W: diagonal zero W 2 - [trw 2 )/2]IN : trace zero optimal GMM is consistent and asymptotically normal for normal error terms OGMM has same limiting distribution as MLE 21

22 HAC 22

23 Consistent Covariance Estimate 23

24 General Covariance Structure y = Xβ + u, E[uu ] = Σ both heteroskedasticity and autocorrelation Var[βOLS] = (1/n)(1/n X X) -1 (1/n X ΣX)(1/n X X) -1 develop estimator for (1/n)X ΣX (k by k) but NOT an estimator for Σ (n by n) all cross products: plim (1/n) ΣiΣj σijxixj 24

25 Heteroskedastic-Consistent Covariance Estimation (White 1980) Σ = diag(σi 2 ) different variance for each i plim (1/n)X ΣX = (1/n) Σi σi 2 xixi no separate estimator for each σi 2 estimator S = (1/n) Σi ei 2 xixi Var[βOLS] = n(x X) -1 [(1/n)Σi ei 2 xixi ](X X) -1 25

26 Spatial Lag with Heteroskedasticity White (1980) correction for 2SLS estimation coefficient variance for general error covariance structure Q ΣQ estimated by Q SQ (S as squared residuals) 26

27 Temporal Correlation Newey-West (87), Andrews (91) plim (1/n) X ΣX = (1/n) Σi Σj σij xixj too many terms to estimate; average is over n but there are n 2 covariance terms impose structure no temporal correlation beyond a given time decay in the correlation as the time lag is larger estimate must yield a positive definite covariance matrix 27

28 Heteroskedastic and Temporal Consistent heteroskedastic part: S 0 = (1/n) ei 2 xixi off-diagonal terms sums of sample covariances, zero for t-h > L weights to ensure positive definiteness S = S0 + (1/n) Σh L Σt=h+1 whetet-h(xtxt-h + xt-hxt ) w h = 1 - h / (L + 1) Bartlett weights 28

29 Spatial HAC 29

30 Spatial Covariance Estimator same principle as for temporal correlation average of sample spatial covariances up to a distance cut-off zero covariance beyond cut-off S = S0 + (1/n) Σdij<δ eiejxixj 30

31 Implementation on a Grid Conley (1999) observations arranged on M by N grid Bartlett window analogue of time series K(j,k) = [1 - (j/lm)][1 - (k/ln)] a proper choice of L M and LN, weights ensures that variance covariance matrix is positive definite 31

32 Conley Spatial Covariance Estimator 32

33 Generalization - Kelejian and Prucha (2006) error terms u = Rε, R unknown GMM setup with instrument matrix H Ψ = VC(n -1/2 H u) = (1/n) H ΣH using Kernel function K(dij/δ) δ = bandwidth, no covariance beyond δ 33

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35 Spatial HAC Estimator elements of k by k covariance matrix Ψrs = (1/n) Σi Σj K(dij/δ) hirhjseiej Ψ = (1/n) Σi Σj K(dij/δ) eiejhihj HAC covariance matrix Φ = n(z Z) -1 Z H(H H) -1 Ψ(H H) -1 H Z(Z Z) -1 35

36 White and HAC standard errors 36