Comparing estimation methods. econometrics

Transcription

1 for spatial econometrics Recent Advances in Spatial Econometrics (in honor of James LeSage), ERSA 2012 Roger Bivand Gianfranco Piras NHH Norwegian School of Economics Regional Research Institute at West Virginia University Thursday, 23 August 2012

2 Outline Background Recent advances in spatial econometrics model fitting techniques have made it more desirable to be able to compare results

3 Outline Background Recent advances in spatial econometrics model fitting techniques have made it more desirable to be able to compare results Results should correspond between implementations using different applications

4 Outline Background Recent advances in spatial econometrics model fitting techniques have made it more desirable to be able to compare results Results should correspond between implementations using different applications A broad range of model fitting techniques are provided by the contributed R packages for spatial econometrics

5 Outline Background Recent advances in spatial econometrics model fitting techniques have made it more desirable to be able to compare results Results should correspond between implementations using different applications A broad range of model fitting techniques are provided by the contributed R packages for spatial econometrics These model fitting techniques are associated with methods for estimating impacts and some tests, which will also be presented and compared

6 Background The use of spatial econometrics tools was widened by the ease with which methods and examples presented in Anselin (1988) could be reproduced using SpaceStat TM, written in Gauss TM

7 Background The use of spatial econometrics tools was widened by the ease with which methods and examples presented in Anselin (1988) could be reproduced using SpaceStat TM, written in Gauss TM It was rapidly complemented by the Spatial Econometrics toolbox for Matlab TM, provided as source code together with extensive documentation

8 Background The use of spatial econometrics tools was widened by the ease with which methods and examples presented in Anselin (1988) could be reproduced using SpaceStat TM, written in Gauss TM It was rapidly complemented by the Spatial Econometrics toolbox for Matlab TM, provided as source code together with extensive documentation A suite of commands for spatial data analysis for use with Stata TM was provided by Maurizio Pisati, and macros for Minitab TM and SAS TM were also made available

9 Background The use of spatial econometrics tools was widened by the ease with which methods and examples presented in Anselin (1988) could be reproduced using SpaceStat TM, written in Gauss TM It was rapidly complemented by the Spatial Econometrics toolbox for Matlab TM, provided as source code together with extensive documentation A suite of commands for spatial data analysis for use with Stata TM was provided by Maurizio Pisati, and macros for Minitab TM and SAS TM were also made available The thrust of SpaceStat TM has largely been taken over by GeoDa (Anselin et al. 2006), and more recently by OpenGeoDa

10 Today s software There is now much more software available for spatial econometrics

11 Today s software There is now much more software available for spatial econometrics Stata TM with sppack and Matlab TM with Spatial Econometrics Toolbox are mainstream programmes; the Matlab TM toolbox remains in the public domain, and has a community of contributors

12 Today s software There is now much more software available for spatial econometrics Stata TM with sppack and Matlab TM with Spatial Econometrics Toolbox are mainstream programmes; the Matlab TM toolbox remains in the public domain, and has a community of contributors OpenGeoDa and PySAL are open source, with code hosted on Google, binary versions for common platforms, and a community of users

13 Today s software There is now much more software available for spatial econometrics Stata TM with sppack and Matlab TM with Spatial Econometrics Toolbox are mainstream programmes; the Matlab TM toolbox remains in the public domain, and has a community of contributors OpenGeoDa and PySAL are open source, with code hosted on Google, binary versions for common platforms, and a community of users R with spdep, sphet, McSpatial and other contributed packages is open source, and the packages are cross-platform; the packages also have a community of users and developers

14 Why compare? In the spirit of Rey (2009), this comparison will attempt to examine some features of the implementation of functions for fitting spatial econometrics models

15 Why compare? In the spirit of Rey (2009), this comparison will attempt to examine some features of the implementation of functions for fitting spatial econometrics models Firstly, it may be useful to show which kinds of functions for creating spatial weights, for diagnostics, and for model fitting are available

16 Why compare? In the spirit of Rey (2009), this comparison will attempt to examine some features of the implementation of functions for fitting spatial econometrics models Firstly, it may be useful to show which kinds of functions for creating spatial weights, for diagnostics, and for model fitting are available Next, it is comforting when one can show that fitting the same model on the same data using different implementations gives the same results

17 Why compare? In the spirit of Rey (2009), this comparison will attempt to examine some features of the implementation of functions for fitting spatial econometrics models Firstly, it may be useful to show which kinds of functions for creating spatial weights, for diagnostics, and for model fitting are available Next, it is comforting when one can show that fitting the same model on the same data using different implementations gives the same results Finally, if the results are not the same, it is helpful to be able to show why they vary, possibly because of different design choices in implementation

18 Data set: Ward and Gleditsch 2008 The data set used for comparison here is taken from a Sage volume Spatial Regression Models by Ward and Gleditsch (2008), with political science data for 158 countries

19 Data set: Ward and Gleditsch 2008 The data set used for comparison here is taken from a Sage volume Spatial Regression Models by Ward and Gleditsch (2008), with political science data for 158 countries The model they explore is the relationship between democracy scores (POLITY IV indicators) and the logarithm of country GDP per capita in 2002

20 Data set: Ward and Gleditsch 2008 The data set used for comparison here is taken from a Sage volume Spatial Regression Models by Ward and Gleditsch (2008), with political science data for 158 countries The model they explore is the relationship between democracy scores (POLITY IV indicators) and the logarithm of country GDP per capita in 2002 They treat countries as neighbours with non-zero spatial weights if their borders are closer than 200km from each other; data and weights are available for download from their site

21 Data set: Ward and Gleditsch 2008 The data set used for comparison here is taken from a Sage volume Spatial Regression Models by Ward and Gleditsch (2008), with political science data for 158 countries The model they explore is the relationship between democracy scores (POLITY IV indicators) and the logarithm of country GDP per capita in 2002 They treat countries as neighbours with non-zero spatial weights if their borders are closer than 200km from each other; data and weights are available for download from their site They were among the first to examine the effects of feedback in spatial regression models

22 Spatial weights Creating spatial weights is a necessary step in using areal data, perhaps just to check that there is no remaining spatial patterning in residuals

23 Spatial weights Creating spatial weights is a necessary step in using areal data, perhaps just to check that there is no remaining spatial patterning in residuals The Matlab SE toolbox provides functions for creating nearest neighbour and triangulation contiguity weights, and for reading GAL files

24 Spatial weights Creating spatial weights is a necessary step in using areal data, perhaps just to check that there is no remaining spatial patterning in residuals The Matlab SE toolbox provides functions for creating nearest neighbour and triangulation contiguity weights, and for reading GAL files A number of functions are included in the R spdep package to create neighbours, and from them weights; GAL and GWT files may be read and written

25 Spatial weights Creating spatial weights is a necessary step in using areal data, perhaps just to check that there is no remaining spatial patterning in residuals The Matlab SE toolbox provides functions for creating nearest neighbour and triangulation contiguity weights, and for reading GAL files A number of functions are included in the R spdep package to create neighbours, and from them weights; GAL and GWT files may be read and written Weights may be constructed using functions in GeoDa, and in Pysal using Python programming; GAL and GWT files may be read and written

26 Spatial weights Creating spatial weights is a necessary step in using areal data, perhaps just to check that there is no remaining spatial patterning in residuals The Matlab SE toolbox provides functions for creating nearest neighbour and triangulation contiguity weights, and for reading GAL files A number of functions are included in the R spdep package to create neighbours, and from them weights; GAL and GWT files may be read and written Weights may be constructed using functions in GeoDa, and in Pysal using Python programming; GAL and GWT files may be read and written The Stata spmat command provides for the creation of a number of different kinds of weights, and for file import and export

27 The symptom residual autocorrelation A map of the residuals of a least squares regression may help us to see whether neighbouring observations appear to have residuals od similar value, indicating positive autocorrelation, or possibly dissimilar values, indicating negative autocorrelation: OLS residuals

28 Spatial lag and Durbin models The spatial lag model is not dissimilar to traditional time series models, with the autocorrelation process in the dependent variable controlled by the exogenous matrix of spatial weights W (Ord 1975): y = ρwy + Xβ + ε, where y is an (N 1) vector of observations on a dependent variable taken at each of N locations, X is an (N k) matrix of exogenous variables, β is an (k 1) vector of parameters, ε is an (N 1) vector of disturbances and ρ is a scalar spatial error parameter (but called λ in Stata)

29 Spatial lag and Durbin models The spatial lag model is not dissimilar to traditional time series models, with the autocorrelation process in the dependent variable controlled by the exogenous matrix of spatial weights W (Ord 1975): y = ρwy + Xβ + ε, where y is an (N 1) vector of observations on a dependent variable taken at each of N locations, X is an (N k) matrix of exogenous variables, β is an (k 1) vector of parameters, ε is an (N 1) vector of disturbances and ρ is a scalar spatial error parameter (but called λ in Stata) The spatial Durbin model is the lag model augmented by spatially lagged right hand side variables: y = ρwy + Xβ + WXθ + ε, where θ is an ((k 1) 1) vector of parameters where W is row-standardised, and a (k 1) vector otherwise

30 Spatial lag model log-likelihood function l(β, ρ, σ 2 ) = N 2 ln 2π N 2 ln σ2 + ln I ρw 1 2σ 2 [ y (I ρw) (I X(X X) 1 X )(I ρw)y ] and β = (X X) 1 (I ˆρW)y, where ˆρ is the ML estimate. Unlike the time series case, the logarithm of the determinant of the (N N) asymmetric matrix (I ρw) does not tend to zero with increasing sample size; it constrains the parameter values to their feasible range between the inverses of the smallest and largest eigenvalues of W, since for positive autocorrelation, as ρ 1, ln I ρw

31 ML spatial lag model results The SE toolbox uses a pre-computed grid of log determinant values, choosing the nearest rather than computing exactly at each call to the log likelihood function; this accounts for slightly different coefficient estimates compated to other implementations. There are two implementations of the spatial lag in R in spdep and McSpatial respectively: SE toolbox R spdep R McSpatial Stata spatreg OpenGeoDa (Intercept) log GDP pc ρ SE ρ σ LL

32 ML spatial lag model results The SE toolbox uses a pre-computed grid of log determinant values, choosing the nearest rather than computing exactly at each call to the log likelihood function; this accounts for slightly different coefficient estimates compated to other implementations. There are two implementations of the spatial lag in R in spdep and McSpatial respectively: SE toolbox R spdep R McSpatial Stata spatreg OpenGeoDa (Intercept) log GDP pc ρ SE ρ σ LL

33 ML spatial lag model differences I There are two major discrepancies in the table of results: the first is that the log-likelihood values at the optimimum differ between R McSpatial and the SE toolbox and the rest. SE toolbox R spdep R McSpatial Stata spatreg OpenGeoDa (Intercept) log GDP pc ρ SE ρ σ LL The reason appears to be that π in the log likelihood calculation is not multiplied by 2 in these cases, but is in the remainder. If we convert the R McSpatial value of by subtracting n 2 log(π) (line 65 in file McSpatial/R/sarml.R), and adding n 2 log(2π), we get Similarly, correcting the SE toolbox value of , we get (line 453 file spatial/sar models/sar.m). The same kind of difference appears in other reported SE toolbox log likelihood values

34 ML spatial lag model differences II The other discrepancy is in the coefficient standard errors, given here in full: SE toolbox R spdep R McSpatial Stata spatreg OpenGeoDa SE (Intercept) SE log GDP pc SE ρ We see that R spdep and OpenGeoDa agree, with the SE toolbox function very close the code in spatial/sar models/sar.m following line 202 indicates that for N 500, asymptotic calculations will be used based on Anselin, (1980, 1988). The same source is used in R spdep and OpenGeoDa (probably after line 861, Regression/smile2.cpp), but what about the others?

35 ML spatial lag model differences II The other discrepancy is in the coefficient standard errors, given here in full: SE toolbox R spdep R McSpatial Stata spatreg OpenGeoDa SE (Intercept) SE log GDP pc SE ρ We see that R spdep and OpenGeoDa agree, with the SE toolbox function very close the code in spatial/sar models/sar.m following line 202 indicates that for N 500, asymptotic calculations will be used based on Anselin, (1980, 1988). The same source is used in R spdep and OpenGeoDa (probably after line 861, Regression/smile2.cpp), but what about the others?

36 ML spatial lag model differences II more The standard errors reported by R McSpatial are taken from the Hessian returned by the optimization function nlm. The R spdep function lagsarlm can return a similar numerical Hessian, computed by as a finite difference Hessian by fdhess in nlme, or as the Hessian output by numerical optimization function optim, using by default a quasi-newton method due to Broyden, Fletcher, Goldfarb and Shanno (BFGS). Stata spreg ml can also use a bfgs technique, but the default is a modified Newton-Raphson method nr; spatreg uses optimization method lf for easy fitting of maximum likelihood models. Since the R and Stata BFGS standard errors agree, it is confirmed that Stata is reporting standard errors taken from the Hessian returned by the optimization function, rather than the analytical calculations even for small n. R nlm R fdhess R BFGS Stata BFGS Stata NR Stata lf SE (Intercept) SE log GDP pc SE ρ

37 ML spatial lag model differences II more The standard errors reported by R McSpatial are taken from the Hessian returned by the optimization function nlm. The R spdep function lagsarlm can return a similar numerical Hessian, computed by as a finite difference Hessian by fdhess in nlme, or as the Hessian output by numerical optimization function optim, using by default a quasi-newton method due to Broyden, Fletcher, Goldfarb and Shanno (BFGS). Stata spreg ml can also use a bfgs technique, but the default is a modified Newton-Raphson method nr; spatreg uses optimization method lf for easy fitting of maximum likelihood models. Since the R and Stata BFGS standard errors agree, it is confirmed that Stata is reporting standard errors taken from the Hessian returned by the optimization function, rather than the analytical calculations even for small n. R nlm R fdhess R BFGS Stata BFGS Stata NR Stata lf SE (Intercept) SE log GDP pc SE ρ

38 ML spatial lag model differences II more The standard errors reported by R McSpatial are taken from the Hessian returned by the optimization function nlm. The R spdep function lagsarlm can return a similar numerical Hessian, computed by as a finite difference Hessian by fdhess in nlme, or as the Hessian output by numerical optimization function optim, using by default a quasi-newton method due to Broyden, Fletcher, Goldfarb and Shanno (BFGS). Stata spreg ml can also use a bfgs technique, but the default is a modified Newton-Raphson method nr; spatreg uses optimization method lf for easy fitting of maximum likelihood models. Since the R and Stata BFGS standard errors agree, it is confirmed that Stata is reporting standard errors taken from the Hessian returned by the optimization function, rather than the analytical calculations even for small n. R nlm R fdhess R BFGS Stata BFGS Stata NR Stata lf SE (Intercept) SE log GDP pc SE ρ

39 Maximum likelihood fitting differences The differences identified in the spatial lag case follow through for the other model specifications examined here.

40 Maximum likelihood fitting differences The differences identified in the spatial lag case follow through for the other model specifications examined here. The Matlab SE toolbox uses a grid rather than a line search/optimization to fit the spatial coefficient(s), so they usually agree only for the first few digits.

41 Maximum likelihood fitting differences The differences identified in the spatial lag case follow through for the other model specifications examined here. The Matlab SE toolbox uses a grid rather than a line search/optimization to fit the spatial coefficient(s), so they usually agree only for the first few digits. The Matlab SE toolbox also reports a log likelihood value using π rather than 2π.

42 Maximum likelihood fitting differences The differences identified in the spatial lag case follow through for the other model specifications examined here. The Matlab SE toolbox uses a grid rather than a line search/optimization to fit the spatial coefficient(s), so they usually agree only for the first few digits. The Matlab SE toolbox also reports a log likelihood value using π rather than 2π. Stata (both spreg ml and spatreg) reports coefficient standard errors taken from the coefficient covariance matrix (Hessian) used in optimization, rather than analytical values reported for small n by R spdep functions, Matlab SE toolbox functions, and OpenGeoDa.

43 ML spatial Durbin model results Only R and the Matlab SE toolbox provide the spatial Durbin model directly, Stata and OpenGeoDa can fit it after creating the lagged right-hand side variable(s) by hand. Here, all applications agree except SE toolbox, again because of the gridded log determinant value causing the numerical optimization to exit before finding the exact optimum. This is justified because the SE toolbox is intended to provide for Bayesian methods, and in that context the distribution of values is more important than point estimates: SE toolbox R spdep Stata spatreg OpenGeoDa (Intercept) log GDP pc lag log GDP pc ρ SE ρ σ LL

44 Spatial error model There are a number of alternative forms of spatial regression models; here we will also consider the spatial error model (also known as the simultaneous autoregressive (SAR) model); the model may be written as (Ord 1975): y = Xβ + u, u = λwu + ε, where y is an (N 1) vector of observations on a dependent variable taken at each of N locations, X is an (N k) matrix of exogenous variables, β is an (k 1) vector of parameters, ε is an (N 1) vector of disturbances and λ is a scalar spatial error parameter (except by Kelejian and Prucha, who term it ρ),

45 Spatial error model There are a number of alternative forms of spatial regression models; here we will also consider the spatial error model (also known as the simultaneous autoregressive (SAR) model); the model may be written as (Ord 1975): y = Xβ + u, u = λwu + ε, where y is an (N 1) vector of observations on a dependent variable taken at each of N locations, X is an (N k) matrix of exogenous variables, β is an (k 1) vector of parameters, ε is an (N 1) vector of disturbances and λ is a scalar spatial error parameter (except by Kelejian and Prucha, who term it ρ), and u is a spatially autocorrelated disturbance vector with constant variance and covariance terms specified by a fixed spatial weights matrix and a single coefficient λ: u N(0, σ 2 (I λw) 1 (I λw ) 1 )

46 Spatial error model log-likelihood function The log-likelihood function for the spatial error model: l(β, λ, σ 2 ) = n 2 ln(2π) n 2 ln(σ2 ) + ln( I λw ) 1 2σ 2 [ (y Xβ) (I λw) (I λw)(y Xβ) ]

47 Spatial error model log-likelihood function The log-likelihood function for the spatial error model: l(β, λ, σ 2 ) = n 2 ln(2π) n 2 ln(σ2 ) + ln( I λw ) 1 2σ 2 [ (y Xβ) (I λw) (I λw)(y Xβ) ] As we can see, the problem is one of balancing the log determinant term ln( I λw ) against the sum of squares term. When λ approaches the ends of its feasible range, the log determinant term may swamp the sum of squares term

48 ML spatial error model results Once again, the line search with exact calculation of the log determinant for R, Stata and OpenGeoDa agrees fully. There are minor differences in the standard errors between R and OpenGeoDa on the one hand and Stata on the other, because of the use of analytical standard errors for small n in OpenGeoDa and R, with Stata using a numerical Hessian. The SE toolbox estimates differ somewhat, with λ truncated to its gridded value: SE toolbox R spdep Stata spatreg OpenGeoDa (Intercept) log GDP pc λ SE λ σ LL

49 ML general spatial model results The general model includes two spatial processes and faces identification issues: y = ρwy + Xβ + u, u = λwu + ε, It is however used in some analyses, often in GM estimators, so the ML version is useful for comparison, but does need care in selecting starting values for numerical optimization; here R spdep and Stata spreg ml agree in all but coefficient standard errors (spdep and SE toolbox use analytical calculations, spreg uses the optimization Hessian): SE toolbox R spdep Stata (Intercept) log GDP pc ρ SE ρ λ SE λ σ LL

50 ML general spatial model results The general model includes two spatial processes and faces identification issues: y = ρwy + Xβ + u, u = λwu + ε, It is however used in some analyses, often in GM estimators, so the ML version is useful for comparison, but does need care in selecting starting values for numerical optimization; here R spdep and Stata spreg ml agree in all but coefficient standard errors (spdep and SE toolbox use analytical calculations, spreg uses the optimization Hessian): SE toolbox R spdep Stata (Intercept) log GDP pc ρ SE ρ λ SE λ σ LL

51 ML general model standard error differences The R spdep function sacsarlm can return a similar numerical Hessian, computed by as a finite difference Hessian by fdhess in nlme. Stata spreg ml can use a bfgs technique, but the default is a modified Newton-Raphson method nr. Since the R finite difference values and Stata BFGS standard errors agree closely, it appears that Stata is reporting standard errors taken from the Hessian returned by the optimization function, rather than the analytical calculations even for small n. R asymptotic R fdhess Stata BFGS Stata NR SE (Intercept) SE log GDP pc SE ρ SE λ

52 ML general model standard error differences The R spdep function sacsarlm can return a similar numerical Hessian, computed by as a finite difference Hessian by fdhess in nlme. Stata spreg ml can use a bfgs technique, but the default is a modified Newton-Raphson method nr. Since the R finite difference values and Stata BFGS standard errors agree closely, it appears that Stata is reporting standard errors taken from the Hessian returned by the optimization function, rather than the analytical calculations even for small n. R asymptotic R fdhess Stata BFGS Stata NR SE (Intercept) SE log GDP pc SE ρ SE λ

53 The recent introduction of Stata TM and GeoDaSpace functions makes it helpful to compare them with SE toolbox and R functions

54 The recent introduction of Stata TM and GeoDaSpace functions makes it helpful to compare them with SE toolbox and R functions Within R, some functions have been contributed to spdep by Luc Anselin, and modified by the authors, and others are in sphet, which now uses the function wrapper spreg

55 The recent introduction of Stata TM and GeoDaSpace functions makes it helpful to compare them with SE toolbox and R functions Within R, some functions have been contributed to spdep by Luc Anselin, and modified by the authors, and others are in sphet, which now uses the function wrapper spreg The functions use different parts of the literature as bases for implementation, and the consequences of these choices will be made clear here

56 The recent introduction of Stata TM and GeoDaSpace functions makes it helpful to compare them with SE toolbox and R functions Within R, some functions have been contributed to spdep by Luc Anselin, and modified by the authors, and others are in sphet, which now uses the function wrapper spreg The functions use different parts of the literature as bases for implementation, and the consequences of these choices will be made clear here Once again, we examine spatial lag, spatial error, and general spatial models

57 GMM spatial lag models Using two stage least squares with Wy instrumented by [WX, WWX], all the functions yield the same coefficient estimates. In the two R and SE toolbox functions, the error variance is calculated as σ 2 = e e n k, while in the other two implementations is simply calculated as e e n : SE toolbox R spdep R sphet spreg Stata GeoDaSpace (Intercept) (2.4576) (2.4576) (2.4576) (2.4341) (2.4341) log GDP pc (0.3783) (0.3783) (0.3783) (0.3747) (0.3747) ρ (0.2283) (0.2283) (0.2283) (0.2261) (0.2261)

58 GMM spatial lag models Using two stage least squares with Wy instrumented by [WX, WWX], all the functions yield the same coefficient estimates. In the two R and SE toolbox functions, the error variance is calculated as σ 2 = e e n k, while in the other two implementations is simply calculated as e e n : SE toolbox R spdep R sphet spreg Stata GeoDaSpace (Intercept) (2.4576) (2.4576) (2.4576) (2.4341) (2.4341) log GDP pc (0.3783) (0.3783) (0.3783) (0.3747) (0.3747) ρ (0.2283) (0.2283) (0.2283) (0.2261) (0.2261)

59 The heteroskedastic error case I White standard errors may be calculated in most of the functions directly, in which the asymptotic VC matrix can be estimated consistently by the sandwich form: (Ẑ Ẑ) 1 (Ẑ ˆΣẐ)(Ẑ Ẑ) 1, where ˆΣ is a diagonal matrix whose elements are the ei 2 ; the results are the same in all cases: R spdep R sphet Stata GeoDaSpace (Intercept) log GDP pc ρ

60 The heteroskedastic error case II GeoDaSpace and sphet also implement the Kelejian and Prucha (2007) HAC estimator of the variance covariance matrix. Here we compare standard error estimates using a Triangular kernel with a variable bandwidth of the six nearest neighbours. The available options for the kernel function in R are the Epanechnikov, Triangular, Bisquare, Parzen, Tukey-Hanning and Quadratic Spectral. The options available in GeoDa space are the Uniform, Triangular, Epanechnikov, Quartic and Gaussian. GeoDa space only allows for the implementation of adaptive kernel. R sphet spreg GeoDaSpace (Intercept) log GDP pc ρ

61 GMM spatial error models In the GMM spatial error model, we depend on the first stage residuals, the implementation of the moment coment conditions, and the tuning of the optimiser finding the spatial parameter λ, as well as defintions for finding the standard error of λ. We see that there are three cases, the first for the SE toolbox, spdep, and GeoDaSpace-2, using the Kelejian and Prucha (1999) moment conditions (OLS first stage), and the standard error of λ from Pruch (2004): SE toolbox R spdep R sphet Stata GeoDaSpace-1 GeoDaSpace-2 (Intercept) (3.0403) (3.0403) (3.0319) (3.2603) (3.0788) (3.0403) log GDP pc (0.3767) (0.3767) (0.3758) (0.4072) (0.3796) (0.3767) λ (0.1820) (0.1820) (0.0661) (0.0668) (0.0660)

62 GMM spatial error models The next case is for the sphet and GeoDaSpace-1 implementations, using Drukker et al., which use a TSLS first stage. The results are close, and differences seem to come from numerical optimization. The final case is Stata, which appears to use a restricted instrument set work is continuing to establish why this is chosen: SE toolbox R spdep R sphet Stata GeoDaSpace-1 GeoDaSpace-2 (Intercept) (3.0403) (3.0403) (3.0319) (3.2603) (3.0788) (3.0403) log GDP pc (0.3767) (0.3767) (0.3758) (0.4072) (0.3796) (0.3767) λ (0.1820) (0.1820) (0.0661) (0.0668) (0.0660)

63 GMM spatial error models The next case is for the sphet and GeoDaSpace-1 implementations, using Drukker et al., which use a TSLS first stage. The results are close, and differences seem to come from numerical optimization. The final case is Stata, which appears to use a restricted instrument set work is continuing to establish why this is chosen: SE toolbox R spdep R sphet Stata GeoDaSpace-1 GeoDaSpace-2 (Intercept) (3.0403) (3.0403) (3.0319) (3.2603) (3.0788) (3.0403) log GDP pc (0.3767) (0.3767) (0.3758) (0.4072) (0.3796) (0.3767) λ (0.1820) (0.1820) (0.0661) (0.0668) (0.0660)

64 Heteroskedasticity Results from sphet and GeoDaSpace are quite similar and the very minor differences (in the estimated value of the spatial parameter) seem to be due to differences in the optimizers. This confirms our intuition on the error model with homoskedastic errors. The standard error results can be made closer by making the same implementation choices, which differ slightly with regard to simplifications. Stata differs as before: R sphet-spreg Stata GeoDaSpace (Intercept) (2.9878) (3.1150) (2.9888) log GDP pc (0.3613) (0.3664) (0.3614) ρ (0.0743) (0.0754) (0.0742)

65 GMM implementations of the general (SARAR) model There are various implementations of the GMM general model. Some of them are based on the Kelejian and Prucha (1999) moment conditions (SE toolbox, gstsls in spdep and GeoDaSpace-2), the others are based on the Drukker, Egger and Prucha moments conditions (sphet, Stata and GeoDaSpace-1) with big differences in λ (ML: ρ 0.78, λ 0.45): SE toolbox R spdep R sphet Stata GeoDaSpace-1 GeoDaSpace-2 (Intercept) (2.6631) (2.3185) (2.2101) (2.2101) (2.0800) (2.2963) log GDP pc (0.3556) (0.3600) (0.3314) (0.3314) (0.3255) (0.3566) ρ (0.2033) (0.2072) (0.1814) (0.1814) (0.1763) (0.2053) λ (0.0363) (0.2502) (0.2502) (0.2224)

66 The general model under heteroskedasticity Three implementation are available: one from sphet, one from Stata, and the one from GeoDaSpace. The results are the same for all of the implementations. Again, sphet and GeDaSpace have the option of performing step 1.c from Arraiz et al. Even in this case, the results match, and, therefore, are not reported. R sphet-spreg Stata GeoDaSpace (Intercept) (2.2119) (2.2119) (2.2119) log GDP pc (0.3527) (0.3527) (0.3527) ρ (0.1845) (0.1845) (0.1845) λ (0.2562) (0.2562) (0.2562)

67 Interpreting spatial lag, Durbin and general models It has emerged over time, however, that the spatial dependence in the parameter ρ feeds back

68 Interpreting spatial lag, Durbin and general models It has emerged over time, however, that the spatial dependence in the parameter ρ feeds back This feedback comes from the fact that the reduced form model is y = (I ρw) 1 Xβ + (I ρw) 1 ε

69 Interpreting spatial lag, Durbin and general models It has emerged over time, however, that the spatial dependence in the parameter ρ feeds back This feedback comes from the fact that the reduced form model is y = (I ρw) 1 Xβ + (I ρw) 1 ε In the spatial lag model, y i / x jr = ((I ρw) 1 Iβ r ) ij, where I is the N N identity matrix, and (I ρw) 1 is known to be dense

70 Interpreting spatial lag, Durbin and general models It has emerged over time, however, that the spatial dependence in the parameter ρ feeds back This feedback comes from the fact that the reduced form model is y = (I ρw) 1 Xβ + (I ρw) 1 ε In the spatial lag model, y i / x jr = ((I ρw) 1 Iβ r ) ij, where I is the N N identity matrix, and (I ρw) 1 is known to be dense In the spatial Durbin model, y i / x jr = ((I ρw) 1 Iβ r Wθ r ) ij

71 Implementing impact measures The awkward S r (W) = ((I ρw) 1 Iβ r ) matrix term needed to calculate impact measures for the lag model, and S r (W) = ((I ρw) 1 (Iβ r Wθ r )) for the spatial Durbin model, may be approximated using traces of powers of the spatial weights matrix as well as analytically

72 Implementing impact measures The awkward S r (W) = ((I ρw) 1 Iβ r ) matrix term needed to calculate impact measures for the lag model, and S r (W) = ((I ρw) 1 (Iβ r Wθ r )) for the spatial Durbin model, may be approximated using traces of powers of the spatial weights matrix as well as analytically The average direct impacts are represented by the sum of the diagonal elements of the matrix divided by N for each exogenous variable

73 Implementing impact measures The awkward S r (W) = ((I ρw) 1 Iβ r ) matrix term needed to calculate impact measures for the lag model, and S r (W) = ((I ρw) 1 (Iβ r Wθ r )) for the spatial Durbin model, may be approximated using traces of powers of the spatial weights matrix as well as analytically The average direct impacts are represented by the sum of the diagonal elements of the matrix divided by N for each exogenous variable The average total impacts are the sum of all matrix elements divided by N for each exogenous variable

74 Implementing impact measures The awkward S r (W) = ((I ρw) 1 Iβ r ) matrix term needed to calculate impact measures for the lag model, and S r (W) = ((I ρw) 1 (Iβ r Wθ r )) for the spatial Durbin model, may be approximated using traces of powers of the spatial weights matrix as well as analytically The average direct impacts are represented by the sum of the diagonal elements of the matrix divided by N for each exogenous variable The average total impacts are the sum of all matrix elements divided by N for each exogenous variable The average indirect impacts are the differences between the direct and total impact vectors

75 Total impacts Total impacts are defined as the sum of the elements of S r (W) divided by N. Here we only have a single variable, so the table of total impacts is simple. The SE toolbox implementation uses Monte Carlo simulation to provide a measure of the significance of the impacts, and by default reports the mean of the simulated values, which may differ a little from the computed value. SE toolbox (simulated) SE toolbox (computed) R spdep Stata Spatial lag Spatial Durbin General These are only headline values, and do not do justice to the possibilities for interpretation offered by this analytical advance.

76 Finding total impacts in Stata In Stata, we use the difference in predictions from the reduced form model when incrementing a chosen right-hand side variable, or even a single observation on that variable:. spreg ml democracy x, id(id) dlmat(w). predict y0. generate x_orig = x. quietly replace x = x + 1. predict y1. generate deltay = y1-y0. mean deltay. quietly replace x = x_orig We can do the same in R, using EXP to increment the variable in the scope of the objects: > EXP <- exp(0) > form <- formula(democracy ~ + log((gdp_2002/population) * + EXP)) > sldv.lag <- lagsarlm(form, data = sldv, + listw = lw) > p0 <- predict(sldv.lag, newdata = sldv, + listw = lw) > EXP <- exp(1) > p1 <- predict(sldv.lag, newdata = sldv, + listw = lw) > d <- p1 - p0 > mean(d) [1]

77 Spatial lag model impacts We use Monte Carlo methods to infer from the impacts, by drawing from the fitted model using the variance-covariance matrix of fitted coefficients: > set.seed(120823) > imp.lag <- impacts(sldv.lag, tr = trmat, R = 1999) > summary(imp.lag, short = TRUE, zstats = TRUE) Impact measures (lag, trace): Direct Indirect Total log((gdp_2002/population) * EXP) ======================================================== Simulation results (asymptotic variance matrix): ======================================================== Simulated z-values: Direct Indirect Total log((gdp_2002/population) * EXP) Simulated p-values: Direct Indirect Total log((gdp_2002/population) * EXP) > invirw <- invirw(lw, rho = coef(sldv.lag)[1]) > N <- nrow(sldv) > Sr <- invirw %*% (diag(n) * coef(sldv.lag)[3]) > c(direct = sum(diag(sr))/n, indirect = sum(sr)/n - sum(diag(sr))/n, + total = sum(sr)/n) direct indirect total

78 Spatial Durbin model impacts The spatial Durbin impacts are less significant than those of the spatial lag model: > EXP <- exp(0) > sldv.sd <- lagsarlm(form, data = sldv, listw = lw, type = "mixed") > imp.sd <- impacts(sldv.sd, tr = trmat, R = 1999) > summary(imp.sd, short = TRUE, zstats = TRUE) Impact measures (mixed, trace): Direct Indirect Total log((gdp_2002/population) * EXP) ======================================================== Simulation results (asymptotic variance matrix): ======================================================== Simulated z-values: Direct Indirect Total log((gdp_2002/population) * EXP) Simulated p-values: Direct Indirect Total log((gdp_2002/population) * EXP) > invirw <- invirw(lw, rho = coef(sldv.sd)[1]) > Sr <- invirw %*% ((diag(n) * coef(sldv.sd)[3]) + (W * coef(sldv.sd)[4])) > c(direct = sum(diag(sr))/n, indirect = sum(sr)/n - sum(diag(sr))/n, + total = sum(sr)/n) direct indirect total

79 General model impacts The general model impacts are similar to those of the spatial lag model: > EXP <- exp(0) > sldv.sac <- sacsarlm(form, data = sldv, listw = lw) > imp.sac <- impacts(sldv.sac, tr = trmat, R = 1999) > summary(imp.sac, short = TRUE, zstats = TRUE) Impact measures (sac, trace): Direct Indirect Total log((gdp_2002/population) * EXP) ======================================================== Simulation results (asymptotic variance matrix): ======================================================== Simulated z-values: Direct Indirect Total log((gdp_2002/population) * EXP) Simulated p-values: Direct Indirect Total log((gdp_2002/population) * EXP) > invirw <- invirw(lw, rho = coef(sldv.sac)[1]) > Sr <- invirw %*% ((diag(n) * coef(sldv.sac)[4])) > c(direct = sum(diag(sr))/n, indirect = sum(sr)/n - sum(diag(sr))/n, + total = sum(sr)/n) direct indirect total

80 Impact measures At present, there is no provision for measures of impact in OpenGeoDa or Pysal

81 Impact measures At present, there is no provision for measures of impact in OpenGeoDa or Pysal The total impact (emanating effect, equilibrium effect) can be calculated in Stata, but not broken down into direct and indirect

82 Impact measures At present, there is no provision for measures of impact in OpenGeoDa or Pysal The total impact (emanating effect, equilibrium effect) can be calculated in Stata, but not broken down into direct and indirect Only SE toolbox and R provide full support with Monte Carlo simulation for inference

83 Impact measures At present, there is no provision for measures of impact in OpenGeoDa or Pysal The total impact (emanating effect, equilibrium effect) can be calculated in Stata, but not broken down into direct and indirect Only SE toolbox and R provide full support with Monte Carlo simulation for inference They draw samples from the fitted model using the coefficient values and covariance matrix, and present summaries of the sample values

84 Distributions of general model impact measures Once one has the samples, it is possible to show how the distributions shift. In this case, the direct impacts lie further from zero than the coefficient, followed by indirect impacts even further from zero, with the total impacts shifted substantially beyond the shape of the distribution of the coefficient: Direct Indirect Total Coefficient Log GDP pc

85 Conclusions Background In this case, impact measures were not needed, because the LR and Hausman tests pointed to the spatial error specification; a recent paper by Pace and Zhu (2012) points to a enhanced error Durbin model as being of promise (it didn t help here).

86 Conclusions Background In this case, impact measures were not needed, because the LR and Hausman tests pointed to the spatial error specification; a recent paper by Pace and Zhu (2012) points to a enhanced error Durbin model as being of promise (it didn t help here). We have not considered Bayesian estimation methods, which will be covered in a separate study, where the SE toolbox is the only alternative so far, but an R GSoC project has been carried out in 2012

87 Conclusions Background In this case, impact measures were not needed, because the LR and Hausman tests pointed to the spatial error specification; a recent paper by Pace and Zhu (2012) points to a enhanced error Durbin model as being of promise (it didn t help here). We have not considered Bayesian estimation methods, which will be covered in a separate study, where the SE toolbox is the only alternative so far, but an R GSoC project has been carried out in 2012 The arrival of Stata s sppack opens up the alternatives a lot, but its spatial weights are dense or banded, limiting maximum likelihood estimation to smaller data sets

88 Conclusions Background In this case, impact measures were not needed, because the LR and Hausman tests pointed to the spatial error specification; a recent paper by Pace and Zhu (2012) points to a enhanced error Durbin model as being of promise (it didn t help here). We have not considered Bayesian estimation methods, which will be covered in a separate study, where the SE toolbox is the only alternative so far, but an R GSoC project has been carried out in 2012 The arrival of Stata s sppack opens up the alternatives a lot, but its spatial weights are dense or banded, limiting maximum likelihood estimation to smaller data sets Estimating models with maximum likelihood for large data sets is possible in the SE toolbox, OpenGeoDa and R using sparse matrix methods; GM models are not as limited by the size of data sets given care in avoiding handling n n matrices