Spatial Econometrics Methods using Stata

Transcription

1 Introduction Spatial Econometrics Methods using Stata Marcos Herrera 1 1 CONICET-IELDE National University of Salta (Argentina) Luxembourg Institute of Socio-Economic Research Belval, 15th February 2017

2 General index 1 Introduction Introduction 2 Loading data and choropleth maps Matrices and spatial tests Spatial local analysis 3 Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation 4 Basic panel data models Static spatial panel models Dynamic spatial panel models

3 Introduction What is Spatial Econometrics? Denitions of Spatial econometrics: Jean Paelinck introduced the term Spatial Econometrics in 1974 to designate: a combination of economic theory, mathematical formalization and statistics with: role of spatial interdependence, importance of factors in other places, explicit modelling of space". Luc Anselin (1988) dened the Spatial Econometrics as: econometric branch dealing with spatial interaction and spatial structure in cross-sectional models and data panel (separating spatial dependence and spatial heterogeneity).

4 Introduction Why is Spatial Econometrics important nowadays? Theory-driven From individual decision to social-spatial interaction. Common shocks. Peer-eects, contextual eects, neighbourhood eects. Data-driven Geo-referenced information. Technology Geographical Information Systems. Capability of statistical software.

5 Introduction Types of spatial data and Objective of workshop Types of spatial data: Geostatistic data: continuous spatial eld (noise surface, pollution surface). Aerial (Lattice or Regional): discrete spatial data, xed polygons or points (counties, provinces, countries). Point pattern: location as a random event (crimes, accidents). This workshop proposes a data-driven analysis using lattice data: Specifying the structure of spatial dependence. Testing for the presence of spatial eects. Estimating and interpreting models with spatial eects.

6 Example used: Impact of net migration on unemployment Hot topic in economics. Competitive theories: Orthodox theory: net migration causes more unemployment (positive relationship). New Economic Geography theory: net migration causes less unemployment (negative relationship). Denition of variables: UNEMPLOYMENT RATE as the number of people unemployed as a percentage of the labour force. NET MIGRATION RATE as the ratio of net migration during the year to the average population in that year. The value is expressed per 1,000 persons. Net migration is the dierence between immigration into and emigration from the area during the year (net migration is therefore negative when the number of emigrants exceeds the number of immigrants). Level of analysis: NUTS 2 (Europe 15), 164 regions from 2007 to 2012.

7 General Index Introduction Loading data and choropleth maps Matrices and spatial tests Spatial local analysis 1 Introduction 2 Loading data and choropleth maps Matrices and spatial tests Spatial local analysis 3 Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation 4 Basic panel data models Static spatial panel models Dynamic spatial panel models

8 From shape to dta Introduction Loading data and choropleth maps Matrices and spatial tests Spatial local analysis First we need a le of administrative areas: Georeferencing information (lattice data) usually is stored in a shapele (a collection of les with a common lename with at least three connected les):.shp is the le that store geometric objects..shx is an index le of geometric objects..dbf is the database le, in dbase format, and contains information of attributes of the objects. Shape les cannot be read directly in Stata. However, shp2dta command can import shapeles and convert them in Stata format.

9 From shape to dta Introduction Loading data and choropleth maps Matrices and spatial tests Spatial local analysis First we need a le of administrative areas: Georeferencing information (lattice data) usually is stored in a shapele (a collection of les with a common lename with at least three connected les):.shp is the le that store geometric objects..shx is an index le of geometric objects..dbf is the database le, in dbase format, and contains information of attributes of the objects. Shape les cannot be read directly in Stata. However, shp2dta command can import shapeles and convert them in Stata format.

10 From shape to dta Introduction Loading data and choropleth maps Matrices and spatial tests Spatial local analysis First we need a le of administrative areas: Georeferencing information (lattice data) usually is stored in a shapele (a collection of les with a common lename with at least three connected les):.shp is the le that store geometric objects..shx is an index le of geometric objects..dbf is the database le, in dbase format, and contains information of attributes of the objects. Shape les cannot be read directly in Stata. However, shp2dta command can import shapeles and convert them in Stata format.

11 From shape to dta Introduction Loading data and choropleth maps Matrices and spatial tests Spatial local analysis First we need a le of administrative areas: Georeferencing information (lattice data) usually is stored in a shapele (a collection of les with a common lename with at least three connected les):.shp is the le that store geometric objects..shx is an index le of geometric objects..dbf is the database le, in dbase format, and contains information of attributes of the objects. Shape les cannot be read directly in Stata. However, shp2dta command can import shapeles and convert them in Stata format.

12 sh2dta command Introduction Loading data and choropleth maps Matrices and spatial tests Spatial local analysis Syntax: shp2dta using shp.lename, database(lename) coordinates(lename) [options] Example: shp2dta using nuts2_164, database(data_shp) coordinates(coord) genid(id) genc(c) The shp2dta command generates two les: data_shp.dta: contains information from.dbf le, id, latitude (y_c) y longitude (x_c). coord.dta: contains geometric information from.shp le.

13 sh2dta command Introduction Loading data and choropleth maps Matrices and spatial tests Spatial local analysis Syntax: shp2dta using shp.lename, database(lename) coordinates(lename) [options] Example: shp2dta using nuts2_164, database(data_shp) coordinates(coord) genid(id) genc(c) The shp2dta command generates two les: data_shp.dta: contains information from.dbf le, id, latitude (y_c) y longitude (x_c). coord.dta: contains geometric information from.shp le.

14 Merging data sets Introduction Loading data and choropleth maps Matrices and spatial tests Spatial local analysis The new database (data_shp.dta) does not contain information about economics variables. Using the index of geometric objects has been generated a excel le with variables from Eurostat: unemployment rate and net migration rate. Both dataset are easily jointed using POLY_ID as link variable: import excel "C:\.\.\.\nuts2_164.xls", firstrow save "C:\.\.\.\migr_unemp07_12.dta" use data_shp, clear merge 1:1 POLY_ID using migr_unemp, gen(union) force

15 Merging data sets Introduction Loading data and choropleth maps Matrices and spatial tests Spatial local analysis The new database (data_shp.dta) does not contain information about economics variables. Using the index of geometric objects has been generated a excel le with variables from Eurostat: unemployment rate and net migration rate. Both dataset are easily jointed using POLY_ID as link variable: import excel "C:\.\.\.\nuts2_164.xls", firstrow save "C:\.\.\.\migr_unemp07_12.dta" use data_shp, clear merge 1:1 POLY_ID using migr_unemp, gen(union) force

16 Merging data sets Introduction Loading data and choropleth maps Matrices and spatial tests Spatial local analysis The new database (data_shp.dta) does not contain information about economics variables. Using the index of geometric objects has been generated a excel le with variables from Eurostat: unemployment rate and net migration rate. Both dataset are easily jointed using POLY_ID as link variable: import excel "C:\.\.\.\nuts2_164.xls", firstrow save "C:\.\.\.\migr_unemp07_12.dta" use data_shp, clear merge 1:1 POLY_ID using migr_unemp, gen(union) force

17 Visualizing in maps Introduction Loading data and choropleth maps Matrices and spatial tests Spatial local analysis A choropleth is a map in which each area is coloured with an intensity proportional to the value of a quantitative variable. Some classical maps: Quantiles: class breaks correspond to quantiles of the distribution of variable (each class includes approximately the same number of polygons). Equal Intervals: class breaks correspond to values that divide the distribution of variable attribute into k equal-width intervals. Boxplot: the distribution of variable attribute is divided into 6 classes dened as follows: [min,p iqr], (p iqr,p25], (p25,p50],(p50,p75], (p75,p iqr] and (p iqr,max], where iqr is the interquartile range. Standard Deviates: the distribution of variable attribute is divided into k classes (2 k 9) whose width is dened as a fraction p of its standard deviation sd.

18 spmap command Introduction Loading data and choropleth maps Matrices and spatial tests Spatial local analysis Syntax: spmap [attribute] [if] [in] using basemap [,basemap_options] Details: basemap_options polygon(polygon_suboptions) line(line_suboptions) point(point_suboptions) diagram(diagram_suboptions) arrow(arrow_suboptions) label(label_suboptions) scalebar(scalebar_suboptions) graph_options]

19 spmap command Introduction Loading data and choropleth maps Matrices and spatial tests Spatial local analysis Syntax: spmap [attribute] [if] [in] using basemap [,basemap_options] Details: basemap_options polygon(polygon_suboptions) line(line_suboptions) point(point_suboptions) diagram(diagram_suboptions) arrow(arrow_suboptions) label(label_suboptions) scalebar(scalebar_suboptions) graph_options]

20 Quantile map spmap U2012 using coord, id(id) clmethod(q) title("unemployment rate") legend(size(medium) position(5)) fcolor(blues2) note("europe, 2012" "Source: Eurostat")

21 Equal intervals map spmap NM2012 using coord, id(id) clmethod(e) title("net migration rate") legend(size(medium) position(5)) fcolor(burd) note("europe, 2012" "Source: Eurostat")

22 Box map spmap U2012 using coord, id(id) clmethod(boxplot) title("unemployment rate") legend(size(medium) position(5)) fcolor(heat) note("europe, 2012" "Source: Eurostat")

23 Deviation map spmap NM2012 using coord, id(id) clmethod(s) title("net migration rate") legend(size(medium) position(5)) fcolor(burd) note("europa, 2012" "Source: Eurostat")

24 Combine map spmap U2012 using coord, id(id) fcolor(rdylbu) cln(8) point(data(migr_unemp_shp) xcoord(x_c) ycoord(y_c) deviation(nm2012) sh(t) fcolor(dknavy) size(*0.3)) legend(size(medium) position(5)) legt(unemployment) note(" " "Solid triangles indicate values over the mean of net-migration." "Europa, Source: Eurostat")

25 Combine map spmap NM2012 using coord, id(id) fcolor(rdylbu) cln(8) diagram(var(u2012) xcoord(x_c) ycoord(y_c) fcolor(gs2) size(1.7)) legend(size(medium) position(5)) legstyle(3) legt(net migration) note("boxes indicate values of unemployment." "Europe, Source: Eurostat")

26 General Index Introduction Loading data and choropleth maps Matrices and spatial tests Spatial local analysis 1 Introduction 2 Loading data and choropleth maps Matrices and spatial tests Spatial local analysis 3 Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation 4 Basic panel data models Static spatial panel models Dynamic spatial panel models

27 Introduction Centrality of spatial W Loading data and choropleth maps Matrices and spatial tests Spatial local analysis We show spatial concentration in previous maps, in a formal way: y i 0 α ij α ik y i u i y j = α ji 0 α jk y j + u j,(6) (1) y k α ki α kj 0 y k u k y = Ay + u, (2) Strategy of identication: 0 α ij α ik 0 w ij w ik A = α ji 0 α jk = ρ w ji 0 w jk = ρw. α ki α kj 0 w ki w kj 0 We transform a non-identied model in other that contains only one parameter: ρ. W captures `who is the neighbour of whom': must be EXOGENOUS!

28 Introduction Criteria used to create W Loading data and choropleth maps Matrices and spatial tests Spatial local analysis Usually, the building of W is an ad-hoc procedure of the researcher. Common criteria are: 1 Geographical: Distance functions: inverse inverse with threshold Contiguity: Rook Queen K nearest neighbours. 2 Socio-economic: Similarity degree in economic dimensions (or social networks). 3 Combinations between both criteria.

29 Introduction Generating W using Stata Loading data and choropleth maps Matrices and spatial tests Spatial local analysis In Stata there are (at least) three commands to generate W: spatwmat: Distance criterion. Used for spatial univariate analysis. Format le no compatible with spmat. spwmatrix: spmat: Generate W using geographic criteria (no contiguity). Generate W under socio-economic criteria. Import, export and manipulate from GeoDa. Compatible format le with spatwmat. Generate W using geographic criteria (no under knn). Import, export and read matrices from GeoDa. Format le no compatible with spatwmat.

30 Introduction Generating W using Stata Loading data and choropleth maps Matrices and spatial tests Spatial local analysis We will use a geographic criterion: spwmatrix: for example 5-nn.. spwmatrix gecon y_c x_c, wn(w5st) knn(5) row connect Nearest neighbor (knn = 5) spatial weights matrix (164 x 164) calculated successfully and the following action(s) taken: - Spatial weights matrix created as Stata object(s): W5st. - Spatial weights matrix has been row-standardized. Connectivity Information for the Spatial Weights Matrix - Sparseness: 3.049% - Neighbors: Min : 5 Mean : 5 Median: 5 Max : 5 It is not advisable to work with units without neighbours. In addition, it is usual to standardize W (usually row-standardize).

31 Introduction Univariate spatial tests Loading data and choropleth maps Matrices and spatial tests Spatial local analysis The following statistics provide a measure of global spatial autocorrelation and allow us to know its signicance. Moran I test (1950): Geary c test (1954): Getis-Ord G test (1992): I = S n (y i y)w ij (y j y) i j 0 c = n 1 2S 0 N. (y i y) 2 i=1 n n w ij (y i y j ) 2 i=1j=1 n. (y i y) 2 i=1 n n w ij y i y j i j i G = n n. y i y j i j i Null hypotheses of tests: No spatial autocorrelation.

32 Introduction Global spatial tests in Stata Loading data and choropleth maps Matrices and spatial tests Spatial local analysis. spatgsa U2012, w(w5st) moran geary two Measures of global spatial autocorrelation Moran's I Variables I E(I) sd(i) z p-value* U Geary's c Variables c E(c) sd(c) z p-value* U *2-tail test. spatgsa U2012, w(w5bin) go two Measures of global spatial autocorrelation Getis & Ord's G Variables G E(G) sd(g) z p-value* U *2-tail test

33 Moran's I scatterplot splagvar U2012, wname(w5st) wfrom(stata) ind(u2012) order(1) plot(u2012) moran(u2012)

34 Moran's I scatterplot splagvar NM2012, wname(w5st) wfrom(stata) ind(nm2012) order(1) plot(nm2012) moran(nm2012)

35 General Index Introduction Loading data and choropleth maps Matrices and spatial tests Spatial local analysis 1 Introduction 2 Loading data and choropleth maps Matrices and spatial tests Spatial local analysis 3 Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation 4 Basic panel data models Static spatial panel models Dynamic spatial panel models

36 Introduction Local indicators of spatial association Loading data and choropleth maps Matrices and spatial tests Spatial local analysis A version of Moran I test is used to detect spatial clusters in local dimension: I i (d) = (x n i x) n w ij (d)(x j x), (3) 1 (x i x) 2 j=1,j i n i=1 where w ij (d) is a weighting distance. Null hypotheses is no spatial autocorrelation and the signicance of I i could be contrasted using normal distribution: z [I i ] = [I i E [I i ]] Var[Ii ]. This test allows grouping observations in 4 categories (see scatter Moran): High-High (H-H), Low-Low (L-L), Low-High (L-H) and High-Low (H-L).

37 Local Moran's I scatterplot genmsp_v0 U2012, w(w5st) graph twoway (scatter Wstd_U2012 std_u2012 if pval_u2012>=0.05, msymbol(i) mlabel (id) mlabsize(*0.6) mlabpos(c)) (scatter Wstd_U2012 std_u2012 if pval_u2012<0.05, msymbol(i) mlabel (id) mlabsize(*0.6) mlabpos(c) mlabcol(red)) (lt Wstd_U2012 std_u2012), yline(0, lpattern(--)) xline(0, lpattern(--)) xlabel(-1.5(1)4.5, labsize(*0.8)) xtitle("{it:z}") ylabel(-1.5(1)3.5, angle(0) labsize(*0.8)) ytitle("{it:wz}") legend(o) scheme(s1color) title("local Moran I of Unemployment rate") splagvar NM2012, wname(w5st) wfrom(stata) ind(nm2012) order(1) plot(nm2012) moran(nm2012)

38 Local Moran's I map spmap msp_u2012 using coord, id(id) clmethod(unique) title("unemployment rate") legend(size(medium) position(4)) ndl("no signif.") fcolor(blue red) note("europe, 2012" "Source: Eurostat")

39 General Index Introduction Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation 1 Introduction 2 Loading data and choropleth maps Matrices and spatial tests Spatial local analysis 3 Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation 4 Basic panel data models Static spatial panel models Dynamic spatial panel models

40 Introduction Alternatives of specication Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation General Cli-Ord model (Manski model) y = λwy + X β + WX γ + u, u = ρwu + ε. Imposing restrictions in γ, ρ and λ we can obtain the following models: γ = 0, ρ = 0, λ 0 SLM. γ = 0, ρ 0, λ = 0 SEM. γ = 0, ρ 0, λ 0 SARAR. γ 0, ρ = 0, λ 0 SDM.

41 Alternatives of specication

42 General Index Introduction Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation 1 Introduction 2 Loading data and choropleth maps Matrices and spatial tests Spatial local analysis 3 Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation 4 Basic panel data models Static spatial panel models Dynamic spatial panel models

43 Residual tests Introduction Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation The rst step (under the specic to general strategy M8) is to estimate a regression model and obtain the residuals. In our case our initial model is This equation is estimated under OLS: U2012 = β 1 + β 2 NM u.. reg U2012 NM2012 Source SS df MS Number of obs = F( 1, 162) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = U2012 Coef. Std. Err. t P> t [95% Conf. Interval] NM _cons

44 Residual tests Introduction Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation There are a set of tests that allow the detection of spatial autocorrelation: Null hypotheses Parameters in H 1 Spatial lag Error lag Test ρ = 0 λ = 0 No spatial autocorrelation - yes LM ERROR yes yes LMERROR yes - LM LAG yes yes LMLAG Moran s I

45 Introduction Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation Spatial residual test in Stata reg U2012 NM2012 spatdiag, weights(w5st) Diagnostic tests for spatial dependence in OLS regression Diagnostics Test Statistic df p-value Spatial error: Moran's I Lagrange multiplier Robust Lagrange multiplier Spatial lag: Lagrange multiplier Robust Lagrange multiplier

46 Introduction Conclusion of spatial residual tests Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation Our rst step was a simple OLS: U20012 = β 1 + β 2 NM u. Now, according to the evidence reected in the tests, an SLM should be estimated: U20012 = λ (W U20012)+ β 1 + β 2 NM u. Problem: How do we estimate this model?

47 Introduction Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation Estimation methods from spatial models The estimation of models with spatial components can not be performed by OLS (except simple cases such as SLX: WX). The estimation needs alternative methods: ML or IV / GMM. Syntax: spreg estimator depvar [indepvars] [if] [in], id(varname) [options] Option estimator allows: ml: maximum likelihood (ML). gs2sls: generalized spatial two-stage least squares (IV/GMM).

48 Introduction Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation Estimation methods from spatial models The estimation of models with spatial components can not be performed by OLS (except simple cases such as SLX: WX). The estimation needs alternative methods: ML or IV / GMM. Syntax: spreg estimator depvar [indepvars] [if] [in], id(varname) [options] Option estimator allows: ml: maximum likelihood (ML). gs2sls: generalized spatial two-stage least squares (IV/GMM).

49 General Index Introduction Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation 1 Introduction 2 Loading data and choropleth maps Matrices and spatial tests Spatial local analysis 3 Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation 4 Basic panel data models Static spatial panel models Dynamic spatial panel models

50 Introduction Maximum Likelihood: spreg ml Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation SLM spreg ml U2012 NM2012, id(id) dlmat(w5_st) SEM spreg ml U2012 NM2012, id(id) elmat(w5_st) SARAR spreg ml U2012 NM2012, id(id) dlmat(w5_st) elmat(w5_st) SDM spreg ml U2012 NM2012 wx_nm2012, id(id) dlmat(w5_st) Cli-Ord spreg ml U2012 NM2012 wx_nm2012, id(id) dlmat(w5_st) elmat(w5_st)

51 Introduction Maximum Likelihood: spreg ml Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation SLM spreg ml U2012 NM2012, id(id) dlmat(w5_st) SEM spreg ml U2012 NM2012, id(id) elmat(w5_st) SARAR spreg ml U2012 NM2012, id(id) dlmat(w5_st) elmat(w5_st) SDM spreg ml U2012 NM2012 wx_nm2012, id(id) dlmat(w5_st) Cli-Ord spreg ml U2012 NM2012 wx_nm2012, id(id) dlmat(w5_st) elmat(w5_st)

52 Introduction Maximum Likelihood: spreg ml Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation SLM spreg ml U2012 NM2012, id(id) dlmat(w5_st) SEM spreg ml U2012 NM2012, id(id) elmat(w5_st) SARAR spreg ml U2012 NM2012, id(id) dlmat(w5_st) elmat(w5_st) SDM spreg ml U2012 NM2012 wx_nm2012, id(id) dlmat(w5_st) Cli-Ord spreg ml U2012 NM2012 wx_nm2012, id(id) dlmat(w5_st) elmat(w5_st)

53 Introduction Maximum Likelihood: spreg ml Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation SLM spreg ml U2012 NM2012, id(id) dlmat(w5_st) SEM spreg ml U2012 NM2012, id(id) elmat(w5_st) SARAR spreg ml U2012 NM2012, id(id) dlmat(w5_st) elmat(w5_st) SDM spreg ml U2012 NM2012 wx_nm2012, id(id) dlmat(w5_st) Cli-Ord spreg ml U2012 NM2012 wx_nm2012, id(id) dlmat(w5_st) elmat(w5_st)

54 Introduction Maximum Likelihood: spreg ml Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation SLM spreg ml U2012 NM2012, id(id) dlmat(w5_st) SEM spreg ml U2012 NM2012, id(id) elmat(w5_st) SARAR spreg ml U2012 NM2012, id(id) dlmat(w5_st) elmat(w5_st) SDM spreg ml U2012 NM2012 wx_nm2012, id(id) dlmat(w5_st) Cli-Ord spreg ml U2012 NM2012 wx_nm2012, id(id) dlmat(w5_st) elmat(w5_st)

55 Introduction Maximum Likelihood: spreg ml Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation Variable SLM SEM SARAR SDM CLIFF-ORD NM W NM const λ ρ Nota: p < What is the best model?

56 Introduction Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation Maximum Likelihood: selecting the best model From specic to general strategy: Using LM tests: spatial lag model (SLM). But there are other posible models: SDM, SARAR and Cli-Ord. Between SDM and SEM: LR COMFAC. From general to specic strategy: Start using Cli-Ord model (M1) and to eliminate sequentially non-signicant variables.

57 Maximum Likelihood: Common factor test Assuming the SDM model has been estimated: y = λwy + X β + WX γ + u. H The null and alternative hypotheses are: 0 : γ + λβ = 0, H 1 : γ + λβ 0. Under H 0, γ = λβ, and replacing into the SDM model: y = λwy + X β + WX ( λβ) + u = λwy + X β λwx β + u, (I λw )y = (I λw )X β + u. The last expression is summarized in SEM: y = X β + (I ρw ) 1 ε, where λ has been replaced by ρ. Under null hypothesis, we have an SEM and, under alternative hypothesis, an SDM: ] LR COMFAC = 2 [l H1 l H0 χ 2 as q

58 Introduction Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation Maximum Likelihood: Alternative models Variable SLM SEM SARAR SDM CLIFF-ORD NM W NM const λ ρ LR COMFAC Nota: p < The nal model is the SLM. 6.81

59 Introduction IV-GMM: spivreg command Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation SLM spivreg U2012 NM2012, dl(w5_st) id(id) SEM spivreg U2012 NM2012, el(w5_st) id(id) SARAR spivreg U2012 NM2012, dl(w5_st) el(w5_st) id(id) SDM spivreg U2012 NM2012 wx_nm2012, dl(w5_st) id(id) Cli-Ord spivreg U2012 NM2012 wx_nm2012, dl(w5_st) el(w5_st) id(id)

60 Introduction IV-GMM: spivreg command Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation SLM spivreg U2012 NM2012, dl(w5_st) id(id) SEM spivreg U2012 NM2012, el(w5_st) id(id) SARAR spivreg U2012 NM2012, dl(w5_st) el(w5_st) id(id) SDM spivreg U2012 NM2012 wx_nm2012, dl(w5_st) id(id) Cli-Ord spivreg U2012 NM2012 wx_nm2012, dl(w5_st) el(w5_st) id(id)

61 Introduction IV-GMM: spivreg command Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation SLM spivreg U2012 NM2012, dl(w5_st) id(id) SEM spivreg U2012 NM2012, el(w5_st) id(id) SARAR spivreg U2012 NM2012, dl(w5_st) el(w5_st) id(id) SDM spivreg U2012 NM2012 wx_nm2012, dl(w5_st) id(id) Cli-Ord spivreg U2012 NM2012 wx_nm2012, dl(w5_st) el(w5_st) id(id)

62 Introduction IV-GMM: spivreg command Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation SLM spivreg U2012 NM2012, dl(w5_st) id(id) SEM spivreg U2012 NM2012, el(w5_st) id(id) SARAR spivreg U2012 NM2012, dl(w5_st) el(w5_st) id(id) SDM spivreg U2012 NM2012 wx_nm2012, dl(w5_st) id(id) Cli-Ord spivreg U2012 NM2012 wx_nm2012, dl(w5_st) el(w5_st) id(id)

63 Introduction IV-GMM: spivreg command Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation SLM spivreg U2012 NM2012, dl(w5_st) id(id) SEM spivreg U2012 NM2012, el(w5_st) id(id) SARAR spivreg U2012 NM2012, dl(w5_st) el(w5_st) id(id) SDM spivreg U2012 NM2012 wx_nm2012, dl(w5_st) id(id) Cli-Ord spivreg U2012 NM2012 wx_nm2012, dl(w5_st) el(w5_st) id(id)

64 Introduction IV-GMM: Alternative models Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation Variable SLM SEM SARAR SDM CLIFF-ORD NM W NM const λ ρ Nota: p < 0.05.

65 Introduction Correction by heteroskedasticity Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation Variable SLM SEM SARAR SDM CLIFF-ORD NM W NM const λ ρ Nota: p < 0.05.

66 Introduction Interpretation of estimated parameters Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation In SLM, SARAR or SDM models, a change of the variable x k in region i will aect the region itself and aects potentially the other regions indirectly through the spatial multiplier mechanism ((I λw ) 1 ). In a linear model, the marginal eect is: E (y i ) x ik = β k E (y j ) x ik = 0 but in spatial models with Wy and/or Wx, the second eect is not zero.

67 Introduction SLM. Direct and indirect eects Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation The marginal eect of the explanatory variable x k on the dependent variable is: y 1 y 1 [ ] x 1k x nk y y... = x 1k x nk , y n y n x 1k x 1k β k β k 0 = (I n λw ) β k, = (I n λw ) 1 [β k I n ], (4) Direct eect: average of the elements of principal diagonal of (I n λw ) 1 [β k I n ]. Indirect eect: (spatial spillover) average of sum of rows, without of elements of principal diagonal of (I n λw ) 1 [β k I n ].

68 Example under SLM in Stata If we apply the above expression to our example (SLM using MLE):. mata: mata (type end to exit) : b = st_matrix("e(b)") : b : lambda = b[1,3] : lambda : S = luinv(i(rows(w))-lambda*w) : end * Total effects. mata: (b[1,1]/rows(w))*sum(s) * Direct effects. mata: (b[1,1]/rows(w))*trace(s) * Indirect effects (spatial spillovers). mata: (b[1,1]/rows(w))*sum(s) - (b[1,1]/rows(w))*trace(s)

69 General Index Introduction Basic panel data models Static spatial panel models Dynamic spatial panel models 1 Introduction 2 Loading data and choropleth maps Matrices and spatial tests Spatial local analysis 3 Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation 4 Basic panel data models Static spatial panel models Dynamic spatial panel models

70 Introduction Fixed eect and random eects Basic panel data models Static spatial panel models Dynamic spatial panel models Consider a linear model with k independent variables x it : y it = x it β + u it, (5) where i = 1,...,n, t = 1,...,T and u it is a random error term. This model doesn't control by heterogeneity: specic temporal variables could be aect on dependent variable. Solution: u it = µ i + φ t + ε it, where µ i is a common region-specic eect and φ t is a common time-specic eect for all regions. These eects could be treated as xed or random.

71 Introduction Fixed or random eects Basic panel data models Static spatial panel models Dynamic spatial panel models Hausman test (1978) is computed as: H = (β fe β re ) (V fe V re ) 1 (β fe β re ), where β fe is the vector of coecients of the consistent estimator fe, β re is the vector of coecients of the ecient estimator re, with V fe and V re as the variance-covariance matrix of fe and re, respectively. This statistic is distributed as χq 2, with q degrees (number of common coecients in both models). Hausman test can be consider as a statistic of validation of re estimator, null hypotheses.

72 Detection of spatial dependence To incorporate spatial eects we must have some evidence of their presence. A possible test that can be used is CD test (Pesaran, 2004): CD = 2T n(n 1) ( n 1 n ρ ij ), i=1j=i+1 where ρ ij is the correlation coecient in the residuals between i and j: ρ ij = ρ ji = ( T T û it û jt t=1 ( T ûit)1/2 2 ûjt 2 t=1 t=1 Null hypothesis: no autocorrelation in cross-section dimension. In Stata:. xtreg U NM, fe (ommitted product). xtcsd, pes abs,(1) (6) )1/2 Pesaran's test of cross sectional independence = , Pr = Average absolute value of the off-diagonal elements = 0.464

73 General Index Introduction Basic panel data models Static spatial panel models Dynamic spatial panel models 1 Introduction 2 Loading data and choropleth maps Matrices and spatial tests Spatial local analysis 3 Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation 4 Basic panel data models Static spatial panel models Dynamic spatial panel models

74 Spatial lag model Introduction Basic panel data models Static spatial panel models Dynamic spatial panel models The SLM with xed eects is: where y t = y 1t y 2t. y nt y t = ρwy t + X t β + µ + ε t, ε t N [ 0,σ 2 ε I n ], (7) x 11t x 21t x k1t, X x 12t x 22t x k2t t = , µ = x 1nt x 2nt x knt Under random eects, this model can be written as: µ 1 µ 2.. µ n. y t = ρwy t + X t β + µ + ε }{{} t, u t ε t N [ ] [ ] (8) 0,σε 2 I n, µ N 0,σ 2 µ I n.

75 Introduction SLM. Direct and indirect eects Basic panel data models Static spatial panel models Dynamic spatial panel models The partial eect of one unit increases on the SLM model is as follows: y 1 y 1 [ ] x 1k x nk y y... = x 1k x nk , y n y n x 1k x 1k β k 0 0 = (I n ρw ) 1 0 β k , 0 0 β k = (I n ρw ) 1 [β k I n ], (9) Direct eect: average of the elements of principal diagonal of (I n ρw ) 1 [β k I n ]. Indirect eect: (spatial spillover) average of sum of rows, without of elements of principal diagonal of (I n ρw ) 1 [β k I n ].

76 Introduction Spatial Error Model Basic panel data models Static spatial panel models Dynamic spatial panel models The SEM model with xed eects is: y t = X t β + µ + ε t ε t = ρw ε t + η t η t N [ 0,σ 2 ηi n ] (10) and the version of SEM model with xed eects is: y t = X t β + µ + ε }{{} t, u t ε t = ρw ε t + η t, η t N [ ] [ ] 0,σηI 2 n, µ N 0,σ 2 µ I n,

77 Introduction Spatial Durbin Model Basic panel data models Static spatial panel models Dynamic spatial panel models SDM specication: with direct-indirect eects: [ y x 1k... ] y x nk y t = ρwy t + X t β + WX t γ + ε t, (11) = y 1 y 1 x nk x 1k y n x 1k y n x 1k, β k w 12 γ k w 1n γ k = (I n ρw ) 1 w 21 γ k β k w 2n γ k , w n1 γ k w n2 γ k β k = (I n ρw ) 1 [β k I n + γ k W ], (12)

78 command xsmle Introduction Basic panel data models Static spatial panel models Dynamic spatial panel models SLM xsmle U NM, fe wmat(w5_st) mod(sar) hausman SEM xsmle U NM, fe emat(w5_st) mod(sem) hausman SARAR xsmle U NM, fe wmat(w5_st) emat(w5_st) vce(r) mod(sac) SDM xsmle U NM, fe wmat(w5_st) mod(sdm) hausman

79 command xsmle Introduction Basic panel data models Static spatial panel models Dynamic spatial panel models SLM xsmle U NM, fe wmat(w5_st) mod(sar) hausman SEM xsmle U NM, fe emat(w5_st) mod(sem) hausman SARAR xsmle U NM, fe wmat(w5_st) emat(w5_st) vce(r) mod(sac) SDM xsmle U NM, fe wmat(w5_st) mod(sdm) hausman

80 command xsmle Introduction Basic panel data models Static spatial panel models Dynamic spatial panel models SLM xsmle U NM, fe wmat(w5_st) mod(sar) hausman SEM xsmle U NM, fe emat(w5_st) mod(sem) hausman SARAR xsmle U NM, fe wmat(w5_st) emat(w5_st) vce(r) mod(sac) SDM xsmle U NM, fe wmat(w5_st) mod(sdm) hausman

81 command xsmle Introduction Basic panel data models Static spatial panel models Dynamic spatial panel models SLM xsmle U NM, fe wmat(w5_st) mod(sar) hausman SEM xsmle U NM, fe emat(w5_st) mod(sem) hausman SARAR xsmle U NM, fe wmat(w5_st) emat(w5_st) vce(r) mod(sac) SDM xsmle U NM, fe wmat(w5_st) mod(sdm) hausman

82 Alternative Models Introduction Basic panel data models Static spatial panel models Dynamic spatial panel models Variable SLM SEM SARAR SDM NM W NM 0.02 ρ λ COMFAC Spatial effects (long run) Directs Indirects Totals AIC Nota: p < 0.05.

83 General Index Introduction Basic panel data models Static spatial panel models Dynamic spatial panel models 1 Introduction 2 Loading data and choropleth maps Matrices and spatial tests Spatial local analysis 3 Taxonomy of spatial models Spatial detection using OLS residual Methods of estimation 4 Basic panel data models Static spatial panel models Dynamic spatial panel models

84 Introduction Dynamic SLM models Basic panel data models Static spatial panel models Dynamic spatial panel models Dynamic versions of Spatial Lag Model (dynslm): y t = τy t 1 + ρwy t + X t β + µ + ε t, (13) y t = ψwy t 1 + ρwy t + X t β + µ + ε t, (14) y y = τy t 1 + ψwy t 1 + ρwy t + X t β + µ + ε t, (15) These model give us the option to obtain direct and indirect eects in short and long run: Short run (assuming τ = ψ = 0,): [ y x 1k... y x nk ]t = (I n ρw ) 1 [β k I n ]. (16) Long run (assuming y t = y t 1 = y ): [ y y x... 1k x nk ]t = [(1 τ)i n (ρ + ψ)w ] 1 [β k I n ]. (17)

85 Introduction Dynamic SDM models Basic panel data models Static spatial panel models Dynamic spatial panel models Dynamic versions of Spatial Durbin Model (dynsdm): y t = τy t 1 + ρwy t + X t β + WX t γ + µ + ε t, (18) y t = ψwy t 1 + ρwy t + X t β + WX t γ + µ + ε t, (19) y t = τy t 1 + ψwy t 1 + ρwy t + X t β + WX t γ + µ + ε t, (20) These model give us the option to obtain direct and indirect eects in short and long run: Short run (assuming τ = ψ = 0,): [ y x 1k... Long run (assuming y t = y t 1 = y ): y x nk ]t = (I n ρw ) 1 [β k I n + γ k W ]. (21) [ y x 1k... y x nk ]t = [(1 τ)i n (ρ + ψ)w ] 1 [β k I n + γ k W ]. (22)

86 Introduction Basic panel data models Static spatial panel models Dynamic spatial panel models Example of unemployment-migration. Serial correlation Wooldridge (2002) develops a simple statistic for autocorrelation detection. Drukker (2003) implements it in Stata, xtserial command:. xtserial U NM Wooldridge test for autocorrelation in panel data H0: no first-order autocorrelation F( 1, 163) = Prob > F =

87 xsmle command Introduction Basic panel data models Static spatial panel models Dynamic spatial panel models SLM 1 (eq. 13) xsmle U NM, dlag(1) fe wmat(w5_st) type(ind) mod(sar) effects nsim(499) SLM 2 (eq. 14) xsmle U NM, dlag(2) fe wmat(w5_st) type(ind) mod(sar) effects nsim(499) SLM 3 (eq. 15) xsmle U NM, dlag(1) fe wmat(w5_st) type(ind) mod(sar) effects nsim(499)

88 xsmle command Introduction Basic panel data models Static spatial panel models Dynamic spatial panel models SLM 1 (eq. 13) xsmle U NM, dlag(1) fe wmat(w5_st) type(ind) mod(sar) effects nsim(499) SLM 2 (eq. 14) xsmle U NM, dlag(2) fe wmat(w5_st) type(ind) mod(sar) effects nsim(499) SLM 3 (eq. 15) xsmle U NM, dlag(1) fe wmat(w5_st) type(ind) mod(sar) effects nsim(499)

89 xsmle command Introduction Basic panel data models Static spatial panel models Dynamic spatial panel models SLM 1 (eq. 13) xsmle U NM, dlag(1) fe wmat(w5_st) type(ind) mod(sar) effects nsim(499) SLM 2 (eq. 14) xsmle U NM, dlag(2) fe wmat(w5_st) type(ind) mod(sar) effects nsim(499) SLM 3 (eq. 15) xsmle U NM, dlag(1) fe wmat(w5_st) type(ind) mod(sar) effects nsim(499)

90 Alternative models of dynamic SLM Variable SLM 1 SLM 2 SLM 3 U(t 1) NM W U W U(t 1) Spatial effects (short run) Directs Indirects Totals Spatial effects (long run) Directs Indirects Totals AIC Nota: p < 0.05.

91 Introduction Summing up Stata has incorporated tools for spatial analysis. ESDA can be carried out completely, as in others software. Also, for cross-section data, the most common spatial specications can be estimated by ML and/or IV/GMM. For panel data, recent developments provide alternatives for estimating static and dynamic models. Main results of the impact of net migration: Cross-section model: SLM shows a negative impact in unemployment (long run eect). Panel-data: Dynamic SLM shows a positive impact in short run (orthodox theory) and negative impact in long run, total eect (NEG theory).

92 Introduction Summing up Stata has incorporated tools for spatial analysis. ESDA can be carried out completely, as in others software. Also, for cross-section data, the most common spatial specications can be estimated by ML and/or IV/GMM. For panel data, recent developments provide alternatives for estimating static and dynamic models. Main results of the impact of net migration: Cross-section model: SLM shows a negative impact in unemployment (long run eect). Panel-data: Dynamic SLM shows a positive impact in short run (orthodox theory) and negative impact in long run, total eect (NEG theory).

93 Some references Theoretical references: Introduction Anselin, L. and A. Bera (1998). Spatial dependence in linear regression models with an Introduction to Spatial Econometrics, Handbook of Applied Economic Statistics, pp Brueckner, J. (2003). Strategic interaction among governments: An overview of empirical studies, International Regional Science Review, 26(2). Elhorst, J. P. (2013). Spatial Panel Models, Chapter 82, en Fischer y Nijkamp (eds.) Handbook of Regional Science. Heidelberg: Springer. Applied references: Ertur, C. and W. Koch (2007). Growth, technological interdependence and spatial externalities: Theory and evidence, Journal of Applied Econometrics, 22. Le Gallo, J. and C. Ertur (2003). Exploratory spatial data analysis of the distribution of regional per capita GDP in Europe, 1980= 1995, Papers in regional science, 82(2). Fingleton, B. (2006). A cross-sectional analysis of residential property prices: the eects of income, commuting, schooling, the housing stock and spatial interaction in the English regions, Papers in Regional Science, 85(3).

94 Some references Introduction Books: Stata: Anselin, L. (1988). Spatial Econometrics: Methods and Models. LeSage, J. and Pace (2009). An introduction to spatial econometrics. Elhorst, J. P. (2014). Spatial econometrics: from cross-sectional data to spatial panels. Belotti, F. et al (2016). XSMLE: Stata module for spatial panel data models estimation. Statistical Software Components. Drukker, D. M. et al. (2011). A command for estimating spatial-autoregressive models with spatial-autoregressive disturbances and additional endogenous variables. Econometric Reviews, 32, Drukker, D. M. et al. (2013). Creating and managing spatial-weighting matrices with the spmat command. Stata Journal, 13(2), Pisati, M. (2008). SPMAP: Stata module to visualize spatial data. Statistical Software Components.