S6: Spatial regression models: : OLS estimation and testing

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1 : Spatial regression models: OLS estimation and testing Course on Spatial Econometrics with Applications Profesora: Coro Chasco Yrigoyen Universidad Autónoma de Madrid Lugar: Universidad Politécnica de Barcelona -3, 8-0 de junio, , Coro Chasco Yrigoyen Course Index S: Introduction to spatial econometrics S: Spatial effects, spatial dependence S3: Spatial autocorrelation tests S4: Exploratory Spatial Data Analysis (ESDA) S5: Specification of spatial dependence models S6: Spatial regression models: : OLS estimation and testing PS: GeoDa: introduction and ESDA S7: Spatial dependence models: estimation and testing S8: Modelling strategies in spatial regression models PS: SpaceStat: confirmatory spatial data analysis S9: Specification of spatial heterogeneity models S0: Spatial heterogeneity models: estimation and testing PS3: Practical exercise and 007, Coro Chasco Yrigoyen. CHASCO, C. (003), Econometría espacial aplicada a la predicción-extrapolación de datos microterritoriales. Comunidad de Madrid; pp Overview and Goals The classical spatial regression model: specification and OLS estimation. Measures of fit and individual parameter tests Multicollinearity Normality of the errors Heteroskedasticity Spatial 007, Coro Chasco Yrigoyen 3

2 6.. The classical spatial regression model: specification and estimation The general purpose of linear regression analysis is to find out a (linear) relationship between a dependent variable and a set of explanatory variables. y X u u N ( ) = ( ) β( ) +,,, (,) ; ( 0, N N k k N ) Two objectives:. Good fit between Xb and y. Which explanatory variable contributes more OLS are 007, Coro Chasco Yrigoyen The classical spatial regression model: specification and estimation (II) OLS are BLUE if: OLS estimators: minimizes the sum of squared residuals of the regression Basic assumptions:. Linearity. Full rank (no multicollinearity). Good specification 3. Nonstochastic regressors Assumptions about u:. Mean zero. Constant variance (homoskedasticity) 3. No spatial autocorrelation , Coro Chasco Yrigoyen Measures of fit and individual parameter tests Measures of fit: S ( y) = S ( yˆ ) + S ( e). R : R = S ( e) S ( y). Adjusted R : k R = R ( R ) n k. Log-likelihood: n n e' e L = ln π ln 0, 5 3. AIC: f ( N, k) = k ln( N) IC = L + f ( k, N ) 4. SC: f ( N, k) = k 007, Coro Chasco Yrigoyen

3 6.. Measures of fit and individual parameter tests (II) Hypothesis tests over the significance of:. Individual regression coefficient.. t-test: b S bj = XX j t( bj) = { H0 ( β j = 0) } = S( b j ) ee ˆ = N k.. Sign H 0 (β=0) RSSc RSSu. Overall coefficients: F-test F = k RSSu N k ( ) [ 007, Coro Chasco Yrigoyen Multicollinearity Condition number test: GeoDa SpaceStat Square root of the ratio of the largest to the smallest eigenvalue of the matrix X'X, after standardization (each column sums ) As a rule of thumb, values of the condition number larger than 0 or 30 are considered to be suspect. A total lack of multicollinearity yields a condition number of. Consequences: though OLS are BLUE, S(b j ) augments leading to misleading t-student values and/or changes in coefficient signs. Solutions: ) changing the model, ) considering extra-sample data (space-time), 3) applying principal components or other synthesis 007, Coro Chasco Yrigoyen Normality of the errors Jarque-Bera is an asymptotic discrepancy test: N k JB S ( k 3 ) = GeoDa SpaceStat S: asymmetry K: kurtosis Consequences: Most hypothesis tests and a large number of regression diagnostics are based on the assumption of a normal error distribution: maximum-likelihood estimations, t-student, F-tests, Lagrange Multiplier tests (Breusch-Pagan, spatial autocorrelation)!!!!!! Solution: Logs or any other Box-Cox variable transformations, other estimation methods (IV, GMM), other tests (Koenker-Basset, 007, Coro Chasco Yrigoyen 9 3

4 6.5. Heteroskedasticity Definition: Regression disturbance do not have a constant variance over all observations (i.e., is not homoskedastic). Causes in spatial data analysis: ) When using data for irregular spatial units (with different area). ) When there are systematic regional differences in the relationships you model (i.e., spatial regimes). 3) When there is a continuous spatial drift in the parameters of the model (i.e., spatial expansion). Consequences: ) A standard regression model that ignores this will be misspecified. ) OLS estimates are unbiased, but they will no longer be most efficient. 3) Inference based on the usual t and F statistics will be misleading. 4) R measure of goodness-of-fit will be wrong (based on e e, instead e Ω - 007, Coro Chasco Yrigoyen Heteroskedasticity (II) All tests start from the null hypothesis of homoskedasticity: [ ] = H 0 : E u i The alternative hypothesis is that each observation's error term has a different variance. A common approach is to relate the variability in the error variance to a number of variables, via a functional form that includes a few parameters (say P parameters): Linear = Additive (SpaceStat) H: i = f ( α0 + z p piα p) Exponential = Multiplicative z-variables = squares of explanatory variables x j (SpaceStat default), area of the spatial unit, or any other variable that relates to its size 007, Coro Chasco Yrigoyen 6.5. Heteroskedasticity (III) Known H specification: ) Normal errors, no small samples: Breusch-Pagan LM-test (979) ) Non-normal errors: Koenker- Basset (98) Both tests are asymptotic and achieve a χ distribution, with P d.f. (P = # z vars) Unknown H specification: White test Asymptotic, achieve a χ distrib., d.f. = # vars, except the constant. BP = / the explained sum of squares in a regression of ( e i ) on a constant and the z-variables. KB = studentized version of BP, in that the is replaced by a more robust estimate of the fourth moment. W = N x R (in an auxiliary regression of the squared OLS residuals on all cross products between the explanatory variables and/or the squares of 007, Coro Chasco Yrigoyen 4

5 6.5. Heteroskedasticity (IV) Solutions to heteroskedasticity:. Known heteroskedasticity:.. ( p ) H f z : i α0 piαp = + FGLS.. Systematic regional differences in relationships spatial regimes.3. Spatial drift in the parameters spatial expansion, 007, Coro Chasco Yrigoyen Heteroskedasticity (V) Solutions to 007, Coro Chasco Yrigoyen 4. Unknown heteroskedasticity robust OLS inference OLS are not the most efficient:.. White solution: ( ) = [ ] ' Ω [ ] Var b XX X X XX ( ) = [ ] ' [ ] Var b X X X SX X X { ( ) } ei Sii = ; kii = diag X XX X.. Efron s Jacknife solution: kii Resampling approach: each obs in turn is dropped from the data set. The empirical distribution of the OLS estimates obtained for all M replications provides the basis for V(b) N N N M V( b) = bi () b( j) bi ( ) b( j) M i= M j= M j=! 6.6. Spatial autocorrelation Concept: Spatial autocorrelation, or more generally, spatial dependence, is the situation where the dependent variable or error term at each location is correlated with observations on the dependent variable or values for the error term at other locations. The general case is formally: Spatial-lag ) y in location i is correlated with y in location j : ) e in location i iscorrelated with i in location j : W * 007, Coro Chasco Yrigoyen 5 5

6 6.6. Spatial autocorrelation (II) Consequencies The consequences of ignoring spatial autocorrelation in a regression model, when it is in fact present, depend on the form for the No spatial dependence Sustantive spatial dependence Error spatial dependence alternative hypothesis, H : H 0 :ρ=λ=0 If H :ρ 0(spatial-lag) If H :λ 0(spatial-error) y = ρwy+ Xβ + u y = Xβ + u u = λwu + ε y = Xβ + u u = ε + λwε b: Biased and inefficient Specification error: omitting a significant explanatory 007, Coro Chasco Yrigoyen b: Unbiased but inefficient t-student, F tests are misleading R invalid 6. CLIFF, A. y J. ORD (98), Spatial processes, models and applications. London: Pion 6.6. Spatial autocorrelation (III) There are 7 spatial autocorrelation tests: 4 (error) + (lag) + (SARMA) I. H :λ 0(spatial-error) ) Moran s I: u =ε +λwε ewe I = w e e e = ee N N N ij i j i i j i = = = Inference is based on a standardized z-value ~ N(0,) (asymptotically). Require normality for the error terms The theoretical E(I), SD(I) are more complex. The interpretation of the statistic is the same as for the general case. It is by far the most familiar test, it is fairly unreliable: - This statistic picks up a range of misspecification errors: from nonnormality and heteroskedasticity to spatial lag dependence. - It does not provide any guidance for H in terms of which of the substantive or error dependence is the most likely 007, Coro Chasco Yrigoyen 7. ANSELIN, L & R FLORAX (995), Small sample properties of tests for spatial 6.6. Spatial autocorrelation (IV) dependence in regression models: Some further results. In Anselin A and R. Florax, New directions in spatial econometrics, pp. 74. Berlin: Springer-Verlag. I. H :λ 0(spatial-error) ) Burridge s Lagrange Multiplier test (980) Asymptotic test, which follows a χ distribution with degree of freedom. It requires normality in the errors. The test is the same for both H of SAR and SMA in the 007, Coro Chasco Yrigoyen u =ε +λwε 8 LM ERR ewe ' = tr W ' W + W 3) Anselin & Florax s Lagrange Multiplier Asymptotic test χ, d.f. test on errors, robust to ignored spatial lag Requires normality in the errors (995) ewe ' ewy ' ( WXb )' M( WXb ) T ( RJ ρβ ) ( RJ ρβ ) = T + LM EL = T T ( RJ ρβ ) ' T = tr WW + W = ee N ( ) 6

7 . KELEJIAN H & D ROBINSON (99), Spatial autocorrelation: a new computationally simple test with an application to per capita county policy expenditures. Regional Science and Urban Economics ; pp Spatial autocorrelation (IV) I. H :λ 0(spatial-error) 4) Kelejian-Robinson test (99) KR is obtained from an auxiliary regression of cross products of residuals and cross products of the explanatory variables (collected in a matrix Z with P columns). The cross products are for all pairs of observations for which a nonzero correlation is postulated (but each pair is only entered once), for a total of hn pairs. γ is the coefficient vector in this auxiliary regression. α is the resulting residual vector. u =ε +λwε γ ' Z ' Zγ KR = α' α Asymptotic test, which follows a χ distribution with P degrees of freedom. It does not require normality for the error terms. It also is applicable to both linear and nonlinear regressions Only for large sample test (may not have much power for small data sets). It s not known its performance with linearity and normality in the errors h 007, Coro Chasco Yrigoyen 9. ANSELIN, L (988), Lagrange Multiplier test diagnostics for spatial dependence and spatial heterogeneity, Geographical Analysis 0, Spatial autocorrelation (V) M = I X[X X] - X II. H :ρ 0 (spatial lag) ewy ' 5) Anselin s Lagrange Multiplier LM = LAG test for spatial lag (988) [ WXb] ' MWXb + tr W ' W + W Asymptotic test, which follows a χ distribution with degree of freedom. It is only valid under the assumption of normality in the error terms. Only for large sample test (may not have much power for small data sets). 6) Anselin & Florax s Lagrange Multiplier Asymptotic test χ, d.f. test for spatial lags, robust to ignored Requires normality in the errors spatial error (995) ( WXb )' M( WXb ) ewy ' ewe ' ( RJ ρβ ) = T + LM LE = ' RJ ρβ T T ( ) = tr WW + W = ee 007, Coro Chasco Yrigoyen 0. ANSELIN, L (988), Lagrange Multiplier test diagnostics for spatial dependence and spatial heterogeneity, Geographical Analysis 0, Spatial autocorrelation (VI) III. H : λ 0 ; ρ 0 (SARMA) 7) Anselin s Lagrange Multiplier SARMA test ewy ' ewe ' ewe ' SARMA = + RJ T T ρ β Asymptotic test, which follows a χ distribution with degrees of freedom. Only valid in linear models under the assumption of normality in the 007, Coro Chasco Yrigoyen 7

8 6.6. Spatial autocorrelation (VII) Consider 7 spatial autocorrelation tests: 4 (error) + (lag) + (SARMA) I. H :λ 0 (spatial-error) ) Moran s I unreliable ) LM-ERR SAR/SMA 3) Robust LM-ERR omitted lag 4) Kelejian-Robinson non-norm. II. H :ρ 0 (spatial-lag) 5) LM-LAG 6) Robust LM-LAG omitted err. III. H : λ 0 ; ρ 0 (SARMA) 7) Lagrange 007, Coro Chasco Yrigoyen 8