Spatial Regression 11. Spatial Two Stage Least Squares Luc Anselin http://spatial.uchicago.edu 1
endogeneity and instruments spatial 2SLS best and optimal estimators HAC standard errors 2
Endogeneity and Instruments 3
Spatial Lag Model y = ρwy + Xβ + u Wy endogenous = simultaneous equation bias y = Zθ + u Z = [ Wy X ] θ = [ ρ β ] 4
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
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
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
Spatial 2SLS 8
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
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
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
spatial 2SLS estimation - WX as instruments 12
spatial 2SLS estimation - WX and W 2 X as instruments 13
estimates with different spatial lag orders for the instruments 14
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
additional instruments for endogenous variable UE90 16
Best and Optimal Estimators 17
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
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
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
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
HAC 22
Consistent Covariance Estimate 23
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
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
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
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
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
Spatial HAC 29
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
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
Conley Spatial Covariance Estimator 32
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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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
White and HAC standard errors 36