ON THE NEGATION OF THE UNIFORMITY OF SPACE RESEARCH ANNOUNCEMENT

Transcription

1 MIDWEST STUDENT SUMMIT ON SPACE, HEALTH AND POPULATION ECONOMICS APRIL 18-19, 2007 PURDUE UNIVERSITY ON THE NEGATION OF THE UNIFORMITY OF SPACE RESEARCH ANNOUNCEMENT Benoit Delbecq Agricultural Economics

2 2 Introduction Stationarity in Space Basic assumption in spatial statistics literature: stationary random field [Yaglom (1957, 1961,1962)] Homogeneity (vs. Heterogeneity) Stationarityti it under translation ti Random field stucture does not change systematically from one place to another Isotropy (vs. Anisotropy) Stationarity under rotations around a fixed point Random field structure does not change systematically along different directions

3 Anisotropy in Spatial Econometrics Quick overview 3 Basic spatial econometrics models Spatial lag model - y =ρwy + Xβ + ε Spatial error model - y = Xβ + u / u = λmu + ε SARMA (1,1) - Spatial lag and error combined Isotropy can be incorporated into the spatial autocorrelation parameter ρ or λ (Deng, 2007) the spatial weight matrices W (Arbia and Piras, upcoming) and/or M

4 Anisotropy in Spatial Econometrics Messing with ρ 4 Anisotropic spatial lag model - Deng (2007) y = [F(Z ρ) W]y + Xβ + ε [F(Z ρ) W] ij = (ρ 0 Z 0 + ρ 1 Z 1,ij + + ρ Q Z Q,ij )w ij Spatial autocorrelation depends on a set of Q variables Z thought to generate anisotropy between location i and location j Examples of Z s: dummy indicating similarity of poverty levels or upstream/downstream location Anisotropic spatial error model Model estimated by Maximum Likelihood

5 5 Anisotropy in Spatial Econometrics Messing with W Non-isotropic Spatial Lag Model of order 2 NISLM(2) - Arbia and Piras (upcoming) y = (ρ 1 W 1 + ρ 2 W 2 )y + Xβ + ε How to incorporate anisotropy? W 1 and W 2 are non-overlapping binary weight matrices and W 1 + W 2 = W - the full weight matrix Isotropy if f X1,X 2,,X n (x 1,x 2,,x n ;q;w 1 )=f X1,X 2,,X n (x 1,x 2,,x n ;q;w 2 ). Assumption of isotropy relaxed i.e. W 1 W 2 but no additional restriction Estimation by Maximum Likelihood or Instrumental Variables

6 Anisotropy in Econometrics Omitted variables bias 6 Cameron (2005) If distance and direction are correlated and this is neglected, then there is an omitted variable bias Fig 1. One possible source of bias Fig 2. One other possible source of bias

7 7 Anisotropy in Econometrics Spatial trend models Spatial level curves for E[Y i ] are circular under isotropy elliptical under anisotropy with main axis following direction of dominant spatial effect Cameron s model allows for orientation of main axis to vary freely Y i = α + (β + γ 1 cos θ i + γ 2 sin θ i )f(d i ) + ε i Distance effect depends upon direction Maximum distance effect in direction: θ* = arctan(γ 2 /γ 1 )

8 Testing for anisotropy Directional Moran s I Simon (1997) 8 where θ is the angle measured with respect to a reference direction. and Distribution: r(θ) is maximized for and Significance of r 2 max can be tested by

9 What spatial structure? Specification tests for non-nested hypotheses 9 Non-nested vs. Nested : The null hypothesis is not a more general version of the alternatives Nested: can use the traditional F-, LM-, Wald- or Likelihood ratio tests Non-nested: need to use J-,P-,C-,CPD-,PA-tests Spatial J-test for model specification (Kelejian, 2007)

10 What spatial structure? Kelejian s J-test for dummies 10 Test null hypothesis SARMA(1,1) model against G alternative SARMA(1,1) models Special case: G=1 and W=M both in the null and in the alternative H 0 : y n = X n β + λw n ny n + u n / u n = ρw n u n + ε n H a : y n = X n,1 β + λw n,1 y n,1 + u n,1 / u n,1 = ρ 1 W n,1 u n,1 + ε n,1 5-step procedure based on 2SLS and GMM estimation of an augmented model combining both H 0 and H a Y n (ρ) = Z n (ρ)γ + α 1 Z n,1 (ρ)γ 1 + ε n Small sample inferences based on approximate normal distribution of estimates and Wald test of α 1 =0

11 Research Idea Directional Technology Spillovers across countries 11 Main idea: technology spreads along latitude and not longitude Total Factor Productivity growth tfpshp GRTFP Data source: Abreu, de Groot and Florax (2004)

12 Testing for Anisotropy First attempt of application 12 North R max = *** West 70 Moran si= 0.145*** East South Directional Moran s I based on distance Moran s I with inverse distance weight matrix

13 Testing for Model Specification A Simple Model for Productivity Growth 13 Base model and data: Abreu, de Groot and Florax (2004) gr_tfp2 = growth rate of TFP lnav25 = log of average schooling years of people over ( proxy for human capital) gap2 = Nelson-Phelps term (proxy for technology gap) Two alternative spatial weight matrices (binary) latitudinal neighbors / 4.5 longitudinal neighbors / 23.7

14 Testing for Model Specification Latitude vs. Longitude 14 Moran s I Residual Moran s I RLM test Spatial lag RLM test Spatial Error LM test SARMA Wlatitude 0.216*** 0.291*** ** *** Wlongitude 0.010*** 0.216*** 6.812*** *** *** SARMA(1,1) is most appropriate model specification for both spatial weight matrices J-test with H 0 : Wlatitude against H a :Wlongitude Wald test statistic = 28.54*** Reject the null hypothesis, hence Wlatitude (which would corroborate initial hypotheses but caution!!!)

15 Model Estimation Generalized Spatial 2SLS w/ Wlat 15 Explanatory Variable Estimate lnav gap ρ λ GS2SLS Estimation Results of SARMA(1,1) with latitudinal W No inference/significance tests needs to be implemented in the future Outliers/heteroskedasticity not dealt with

16 Research Announcement Three Essays on Anisotropy 16 Research announcement three essays Anisotropy Investigate incorporation of trigonometry in W Technology spillovers using TFP data at the country level Build TFP data for agricultural sector Investigate further directionality Impact of controlled drainage on corn yields using geocoded harvest data at the field level Spatial panel data on large datasets with anisotropy

17 Thank you! Questions? Comments? 17