Painel Espacial com R: reproduzindo Millo e Piras (2012)

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Citação

Sugestão para citação: Figueiredo, Adriano Marcos Rodrigues. Painel Espacial com R: reproduzindo Millo e Piras (2012). Campo Grande-MS,Brasil: RStudio/Rpubs, 2019. Disponível em http://rpubs.com/amrofi/spatial_panel_with_R.

1 Introdução

Este paper faz a reprodução do script do artigo de Millo and Piras (2012),

Giovanni Millo, Gianfranco Piras (2012). splm: Spatial Panel Data Models in
R. Journal of Statistical Software, 47(1), 1-38. URL
http://www.jstatsoft.org/v47/i01/.

Os scripts e arquivos de dados devem ser baixados no site do artigo e colocados no mesmo diretório de trabalho onde executará os presentes códigos.

3 Dados

Os dados utilizados serão do pacote Ecdat, um painel de 48 observações estaduais e anuais de 1970 a 1986, para os Estados Unidos, perfazendo um total de 816 observações.

O dataframe tem por base o painel de Rasmussen (2003):

state     - estado dos Estados Unidos;
year      - ano;
pcap      - estoque de capital privado;
hwy       - ruas e vias expressas;
water     - instalações de água e esgoto;
util      - outros edifícios e estruturas públicas; 
pc        - capital público;
gsp       - produtos estaduais brutos (PIB estadual);
emp       - insumo do trabalho medido pelo emprego nas folhas de pagamento não agrícolas; e,
unemp     - taxa de desemprego do estado.

Fonte:
Munnell, A. (1990) “Why has productivity growth declined? Productivity and public investment”, New England Economic Review, 3–22.
Baltagi, B. H. and N. Pinnoi (1995) “Public capital stock and state productivity growth: further evidence”, Empirical Economics, 20, 351–359.

4 O modelo do PIB estadual dos EUA

4.1 Regressão espacial e Matriz de vizinhança

Os modelos espaciais com dados de paineis são comumente classificados em efeitos aleatórios (random) e efeitos fixos (fixed). A fórmula a ser estimada, o modelo proveniente do modelo teórico é:

A matriz de vizinhança aqui é dada pelo pacote spdep.

4.2 Regressão espacial com efeitos aleatórios: Random Effects

Também chamado de Modelo SARARRE - Spatial Autorregressive model with Autorregressive Errors and Random Effects, a estimação de Máxima Verossimilhança (ML) do painel espacial com modelo de efeitos aleatórios (random effects) é realizada com a função spml e model = "random", para formula = fm e o comando listw = usaw indica a matriz de vizinhança. O lag = TRUE indica a inclusão da variável dependente defasada espacialmente (ou seja, neste caso, o lag da variável log(gsp). O comando spatial.error = "b" indica o tipo de correlação espacial do erro (spatial error correlation) na especificação, neste caso, de Baltagi. Outra alternativa seria com spatial.error = "kkp" para a especificação de Kapoor et al (2007) (Kapoor, Kelejian e Prucha), ou spatial.error = "none" para nenhuma.

## ML panel with spatial lag, random effects, spatial error correlation 
## 
## Call:
## spreml(formula = formula, data = data, index = index, w = listw2mat(listw), 
##     w2 = listw2mat(listw2), lag = lag, errors = errors, cl = cl)
## 
## Residuals:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -0.2477 -0.0411  0.0123  0.0191  0.0727  0.4841 
## 
## Error variance parameters:
##     Estimate Std. Error t-value  Pr(>|t|)    
## phi  7.53078    1.85638  4.0567 4.977e-05 ***
## rho  0.53683    0.05603  9.5811 < 2.2e-16 ***
## 
## Spatial autoregressive coefficient:
##         Estimate Std. Error t-value Pr(>|t|)
## lambda 0.0018204  0.0400679  0.0454   0.9638
## 
## Coefficients:
##               Estimate Std. Error t-value  Pr(>|t|)    
## (Intercept)  2.3735772  0.1394744 17.0180 < 2.2e-16 ***
## log(pcap)    0.0425017  0.0222146  1.9132  0.055719 .  
## log(pc)      0.2415075  0.0202970 11.8987 < 2.2e-16 ***
## log(emp)     0.7419063  0.0244212 30.3796 < 2.2e-16 ***
## unemp       -0.0034560  0.0010605 -3.2589  0.001118 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Agora com a mesma fórmula e função spml, alterando o argumento spatial.error para refletir os resultados de Kapoor et al. (2007):

## ML panel with , spatial RE (KKP), spatial error correlation 
## 
## Call:
## spreml(formula = formula, data = data, index = index, w = listw2mat(listw), 
##     w2 = listw2mat(listw2), lag = lag, errors = errors, cl = cl)
## 
## Residuals:
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## -0.27000 -0.06425 -0.01118 -0.00448  0.04889  0.46937 
## 
## Error variance parameters:
##     Estimate Std. Error t-value  Pr(>|t|)    
## phi 6.624775   1.549683  4.2749 1.912e-05 ***
## rho 0.526465   0.033338 15.7917 < 2.2e-16 ***
## 
## Coefficients:
##               Estimate Std. Error t-value  Pr(>|t|)    
## (Intercept)  2.3246707  0.1415894 16.4184 < 2.2e-16 ***
## log(pcap)    0.0445475  0.0220377  2.0214 0.0432362 *  
## log(pc)      0.2461124  0.0211341 11.6453 < 2.2e-16 ***
## log(emp)     0.7426319  0.0254663 29.1614 < 2.2e-16 ***
## unemp       -0.0036045  0.0010637 -3.3887 0.0007022 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

4.3 Regressão espacial com efeito fixos: Fixed Effects

Primeiro estima-se o modelo com efeitos within, ou seja, model = within para indicar que os efeitos fixos são transformados para eliminar os efeitos individuais. A função spml tem a opção de estimar um modelo com efeitos espaciais tanto lag como error, combinando os argumentos lag como spatial.error.

## Spatial panel fixed effects sarar model
##  
## 
## Call:
## spml(formula = fm, data = Produc, index = NULL, listw = usalw, 
##     na.action = na.fail, model = "within", effect = "individual", 
##     lag = TRUE, spatial.error = "b", method = "eigen", quiet = TRUE, 
##     zero.policy = NULL, tol.solve = 1e-10, control = list(), 
##     legacy = FALSE)
## 
## Residuals:
##       Min.    1st Qu.     Median    3rd Qu.       Max. 
## -0.1335552 -0.0220919 -0.0032048  0.0171787  0.1748911 
## 
## Spatial error parameter:
##     Estimate Std. Error t-value  Pr(>|t|)    
## rho 0.455312   0.042538  10.704 < 2.2e-16 ***
## 
## Spatial autoregressive coefficient:
##        Estimate Std. Error t-value  Pr(>|t|)    
## lambda 0.088576   0.026312  3.3663 0.0007618 ***
## 
## Coefficients:
##             Estimate Std. Error t-value  Pr(>|t|)    
## log(pcap) -0.0103497  0.0255345 -0.4053    0.6852    
## log(pc)    0.1905781  0.0242829  7.8483 4.219e-15 ***
## log(emp)   0.7552372  0.0290385 26.0081 < 2.2e-16 ***
## unemp     -0.0030613  0.0010315 -2.9678    0.0030 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Outra especificação indica o modelo apenas com efeitos fixos individuais (na cross-section) (effect = "individual"), sem dependência espacial nos erros e na variável dependente. Os efeitos fixos são obtidos com a função effects.

## Spatial panel fixed effects error model
##  
## 
## Call:
## spml(formula = fm, data = Produc, index = NULL, listw = usalw, 
##     na.action = na.fail, model = "within", effect = "individual", 
##     method = "eigen", quiet = TRUE, zero.policy = NULL, tol.solve = 1e-10, 
##     control = list(), legacy = FALSE)
## 
## Residuals:
##       Min.    1st Qu.     Median    3rd Qu.       Max. 
## -0.1246945 -0.0237699 -0.0034993  0.0170886  0.1882224 
## 
## Spatial error parameter:
##     Estimate Std. Error t-value  Pr(>|t|)    
## rho 0.557401   0.033075  16.853 < 2.2e-16 ***
## 
## Coefficients:
##             Estimate Std. Error t-value Pr(>|t|)    
## log(pcap)  0.0051438  0.0250109  0.2057  0.83705    
## log(pc)    0.2053026  0.0231427  8.8712  < 2e-16 ***
## log(emp)   0.7822540  0.0278057 28.1328  < 2e-16 ***
## unemp     -0.0022317  0.0010709 -2.0839  0.03717 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

O modelo pode ter efeitos fixos no tempo (effect = "time"). Os efeitos fixos no tempo podem ser extraídos pela função effects.

## Spatial panel fixed effects error model
##  
## 
## Call:
## spml(formula = fm, data = Produc, listw = usalw, na.action = na.fail, 
##     model = "within", effect = "time", method = "eigen", quiet = TRUE, 
##     zero.policy = NULL, tol.solve = 1e-10, control = list(), 
##     legacy = FALSE)
## 
## Residuals:
##        Min.     1st Qu.      Median     3rd Qu.        Max. 
## -0.21855239 -0.06447207 -0.00059156  0.05541491  0.31721113 
## 
## Spatial error parameter:
##     Estimate Std. Error t-value  Pr(>|t|)    
## rho 0.496230   0.035791  13.865 < 2.2e-16 ***
## 
## Coefficients:
##             Estimate Std. Error t-value  Pr(>|t|)    
## log(pcap)  0.1432725  0.0165720  8.6455 < 2.2e-16 ***
## log(pc)    0.3636539  0.0109631 33.1707 < 2.2e-16 ***
## log(emp)   0.5619649  0.0143684 39.1113 < 2.2e-16 ***
## unemp     -0.0078930  0.0018665 -4.2288 2.349e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Intercept:
##             Estimate Std. Error t-value  Pr(>|t|)    
## (Intercept) 1.412536   0.050965  27.716 < 2.2e-16 ***
## 
## 
## Time period fixed effects:
##       Estimate  Std. Error t-value Pr(>|t|)
## 1  -0.00515318  0.05167995 -0.0997   0.9206
## 2   0.00103556  0.05200686  0.0199   0.9841
## 3   0.01161188  0.05193737  0.2236   0.8231
## 4   0.02086866  0.05182860  0.4026   0.6872
## 5  -0.01243892  0.05194369 -0.2395   0.8107
## 6  -0.01638407  0.05254389 -0.3118   0.7552
## 7  -0.01602721  0.05238016 -0.3060   0.7596
## 8  -0.00817852  0.05217527 -0.1568   0.8754
## 9  -0.00108650  0.05184557 -0.0210   0.9833
## 10 -0.00714318  0.05177969 -0.1380   0.8903
## 11 -0.02071186  0.05204947 -0.3979   0.6907
## 12 -0.00791710  0.05222694 -0.1516   0.8795
## 13 -0.01409039  0.05284233 -0.2666   0.7897
## 14  0.00042906  0.05286077  0.0081   0.9935
## 15  0.01861529  0.05225588  0.3562   0.7217
## 16  0.02531034  0.05216326  0.4852   0.6275
## 17  0.03126013  0.05219134  0.5990   0.5492

4.4 Painel espacial com Momentos Generalizados (Spatial Panel with Generalized Moments - GM)

4.4.1 Painel Espacial GM, Random Effects, com erro espacial e sem lag espacial

##  
## 
## Call:
## spgm(formula = fm, data = Produc, listw = usaww, model = "random", 
##     spatial.error = TRUE, moments = "fullweights")
## 
## Residuals:
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## -0.26628 -0.06564 -0.00717 -0.00480  0.04855  0.45897 
## 
## Coefficients:
##               Estimate Std. Error t-value  Pr(>|t|)    
## (Intercept)  2.2273357  0.1350953 16.4871 < 2.2e-16 ***
## log(pcap)    0.0540212  0.0219722  2.4586 0.0139474 *  
## log(pc)      0.2565921  0.0209342 12.2571 < 2.2e-16 ***
## log(emp)     0.7278231  0.0252309 28.8464 < 2.2e-16 ***
## unemp       -0.0038108  0.0011004 -3.4630 0.0005341 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

4.4.2 Painel Espacial GM, Random Effects, com erro espacial e com lag espacial

##  
## 
## Call:
## spgm(formula = fm, data = Produc, listw = usaww, model = "random", 
##     lag = TRUE, spatial.error = TRUE, moments = "fullweights")
## 
## Residuals:
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## -0.27454 -0.06044 -0.00201 -0.00187  0.05255  0.47031 
## 
## Coefficients:
##                Estimate  Std. Error t-value  Pr(>|t|)    
## lambda       0.02232364  0.01348213  1.6558   0.09776 .  
## (Intercept)  2.01385829  0.16738225 12.0315 < 2.2e-16 ***
## log(pcap)    0.04629075  0.02245615  2.0614   0.03927 *  
## log(pc)      0.26657382  0.02033903 13.1065 < 2.2e-16 ***
## log(emp)     0.72124148  0.02472888 29.1660 < 2.2e-16 ***
## unemp       -0.00517520  0.00097115 -5.3289  9.88e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

4.4.3 Painel Espacial GM, Fixed Effects, com erro espacial e com lag espacial

## Spatial panel fixed effects GM model
##  
## 
## Call:
## spgm(formula = fm, data = Produc, listw = usaww, model = "within", 
##     lag = TRUE, spatial.error = TRUE)
## 
## Residuals:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    1.82    2.09    2.19    2.20    2.29    2.70 
## 
## Estimated spatial coefficient, variance components and theta:
##            Estimate
## rho       0.3254804
## sigma^2_v 0.0011306
## 
## Spatial autoregressive coefficient:
##        Estimate Std. Error t-value  Pr(>|t|)    
## lambda 0.132709   0.024593  5.3963 6.803e-08 ***
## 
## Coefficients:
##             Estimate Std. Error t-value  Pr(>|t|)    
## log(pcap) -0.0205827  0.0268688 -0.7660 0.4436500    
## log(pc)    0.1936870  0.0255383  7.5842 3.346e-14 ***
## log(emp)   0.7291745  0.0303750 24.0058 < 2.2e-16 ***
## unemp     -0.0037004  0.0010235 -3.6154 0.0002999 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

4.5 Testes no modelo

A função bsktest permite calcular os testes (conjunta, marginal ou condicional) de efeitos aleatórios e correlação espacial nos erros. São cinco opções correspondentes a Baltagi, Song, and Koh (2003), a saber: “LM1”, “LM2”, “LMJOINT”, “CLMlambda” e “CLMmu”.

4.5.1 LM1

## [1] "htest"
## 
##  Baltagi, Song and Koh SLM1 marginal test
## 
## data:  log(gsp) ~ log(pcap) + log(pc) + log(emp) + unemp
## LM1 = 64.304, p-value < 2.2e-16
## alternative hypothesis: Random effects

4.5.2 LM2

## 
##  Baltagi, Song and Koh LM2 marginal test
## 
## data:  log(gsp) ~ log(pcap) + log(pc) + log(emp) + unemp
## LM2 = 11.657, p-value < 2.2e-16
## alternative hypothesis: Spatial autocorrelation

4.5.3 CLM lambda

## 
##  Baltagi, Song and Koh LM*-lambda conditional LM test
##  (assuming sigma^2_mu >= 0)
## 
## data:  log(gsp) ~ log(pcap) + log(pc) + log(emp) + unemp
## LM*-lambda = 14.436, p-value < 2.2e-16
## alternative hypothesis: Spatial autocorrelation

4.6 Spatial Hausman Test (sphtest)

O teste de Hausman é usado para comparar os estimadores de efeitos fixos com os aleatórios para saber se a pressuposição de efeitos aleatórios é aceita pelos dados. Os estimadores do modelo “within” são comparados com o GLS espacial com uma estatística de teste do tipo \(\chi^2\) com k graus de liberdade em que k é o número de regressores do modelo.

4.6.1 sphtest1

## 
##  Hausman test for spatial models
## 
## data:  x
## chisq = 31.151, df = 4, p-value = 2.852e-06
## alternative hypothesis: one model is inconsistent

4.6.2 sphtest2

Esta opção evidencia que os dois modelos podem ser especificados entre os argumentos da função, caso os modelos tenham sido estimados separadamente.

## 
##  Hausman test for spatial models
## 
## data:  fm
## chisq = 41.897, df = 5, p-value = 6.18e-08
## alternative hypothesis: one model is inconsistent

4.7 Teste de significância dos coeficientes

## 
## z test of coefficients:
## 
##               Estimate Std. Error z value  Pr(>|z|)    
## (Intercept)  2.3735772  0.1394744 17.0180 < 2.2e-16 ***
## log(pcap)    0.0425017  0.0222146  1.9132  0.055719 .  
## log(pc)      0.2415075  0.0202970 11.8987 < 2.2e-16 ***
## log(emp)     0.7419063  0.0244212 30.3796 < 2.2e-16 ***
## unemp       -0.0034560  0.0010605 -3.2589  0.001118 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

4.7.1 Teste da igualdade da elasticidade de crescimento do capital público (pcap) e capital privado (pc)

5 Exemplo Cigarros

5.2 Modelos em painel e com efeitos espaciais

## Spatial panel fixed effects lag model
##  
## 
## Call:
## spml(formula = fm, data = cigar, listw = lwcig, model = "within", 
##     effect = "individual", lag = TRUE, spatial.error = "none")
## 
## Residuals:
##        Min.     1st Qu.      Median     3rd Qu.        Max. 
## -0.11014656 -0.02370245 -0.00094553  0.02382706  0.15797342 
## 
## Spatial autoregressive coefficient:
##        Estimate Std. Error t-value Pr(>|t|)   
## lambda 0.198648   0.067391  2.9477 0.003202 **
## 
## Coefficients:
##        Estimate Std. Error  t-value  Pr(>|t|)    
## logp  -0.608614   0.048101 -12.6529 < 2.2e-16 ***
## logpn  0.232903   0.065469   3.5575 0.0003745 ***
## logy   0.294722   0.038226   7.7099 1.259e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Spatial panel fixed effects error model
##  
## 
## Call:
## spml(formula = fm, data = cigar, listw = lwcig, model = "within", 
##     effect = "individual", lag = FALSE, spatial.error = "b")
## 
## Residuals:
##        Min.     1st Qu.      Median     3rd Qu.        Max. 
## -0.09893082 -0.02681578 -0.00099112  0.02422114  0.14963824 
## 
## Spatial error parameter:
##     Estimate Std. Error t-value  Pr(>|t|)    
## rho 0.302676   0.070229  4.3099 1.634e-05 ***
## 
## Coefficients:
##        Estimate Std. Error  t-value  Pr(>|t|)    
## logp  -0.618338   0.046913 -13.1806 < 2.2e-16 ***
## logpn  0.128986   0.064097   2.0124   0.04418 *  
## logy   0.335879   0.044932   7.4753 7.702e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## ML panel with spatial lag, random effects 
## 
## Call:
## spreml(formula = formula, data = data, index = index, w = listw2mat(listw), 
##     w2 = listw2mat(listw2), lag = lag, errors = errors, cl = cl)
## 
## Residuals:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   0.261   0.771   0.847   0.870   0.932   1.650 
## 
## Error variance parameters:
##     Estimate Std. Error t-value  Pr(>|t|)    
## phi  20.3341     4.7178  4.3101 1.632e-05 ***
## 
## Spatial autoregressive coefficient:
##        Estimate Std. Error t-value Pr(>|t|)   
## lambda 0.181275   0.061912  2.9279 0.003412 **
## 
## Coefficients:
##              Estimate Std. Error  t-value  Pr(>|t|)    
## (Intercept)  2.521162   0.170026  14.8281 < 2.2e-16 ***
## logp        -0.618952   0.052152 -11.8683 < 2.2e-16 ***
## logpn        0.228368   0.065218   3.5016 0.0004624 ***
## logy         0.313567   0.038295   8.1882 2.651e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## ML panel with , random effects, spatial error correlation 
## 
## Call:
## spreml(formula = formula, data = data, index = index, w = listw2mat(listw), 
##     w2 = listw2mat(listw2), lag = lag, errors = errors, cl = cl)
## 
## Residuals:
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## -0.60011 -0.09489 -0.01757 -0.00001  0.05193  0.78684 
## 
## Error variance parameters:
##      Estimate Std. Error t-value  Pr(>|t|)    
## phi 21.269291   4.948478  4.2981 1.722e-05 ***
## rho  0.310914   0.073819  4.2119 2.533e-05 ***
## 
## Coefficients:
##              Estimate Std. Error  t-value  Pr(>|t|)    
## (Intercept)  3.157798   0.212812  14.8385 < 2.2e-16 ***
## logp        -0.629792   0.050629 -12.4393 < 2.2e-16 ***
## logpn        0.123491   0.068410   1.8052   0.07105 .  
## logy         0.361601   0.047713   7.5787  3.49e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

5.3 Modelos com dados empilhados (apenas efeitos espaciais)

## ML panel with spatial lag and iid errors 
## 
## Call:
## spreml(formula = formula, data = data, index = index, w = listw2mat(listw), 
##     w2 = listw2mat(listw2), lag = lag, errors = errors, cl = cl)
## 
## Residuals:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  -0.209   0.299   0.386   0.395   0.467   1.105 
## 
## Spatial autoregressive coefficient:
##        Estimate Std. Error t-value Pr(>|t|)
## lambda 0.082250   0.058619  1.4031   0.1606
## 
## Coefficients:
##              Estimate Std. Error t-value  Pr(>|t|)    
## (Intercept)  1.304801   0.291914  4.4698 7.829e-06 ***
## logp        -1.038347   0.119610 -8.6811 < 2.2e-16 ***
## logpn        0.180146   0.122931  1.4654    0.1428    
## logy         0.683452   0.065246 10.4750 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## ML panel with , spatial error correlation 
## 
## Call:
## spreml(formula = formula, data = data, index = index, w = listw2mat(listw), 
##     w2 = listw2mat(listw2), lag = lag, errors = errors, cl = cl)
## 
## Residuals:
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## -0.60762 -0.09700 -0.00801  0.00024  0.07633  0.70739 
## 
## Error variance parameters:
##     Estimate Std. Error t-value Pr(>|t|)  
## rho 0.147554   0.067699  2.1796  0.02929 *
## 
## Coefficients:
##              Estimate Std. Error t-value  Pr(>|t|)    
## (Intercept)  1.484186   0.312829  4.7444 2.091e-06 ***
## logp        -1.060385   0.118173 -8.9732 < 2.2e-16 ***
## logpn        0.150483   0.125306  1.2009    0.2298    
## logy         0.730092   0.069817 10.4572 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Characteristics of weights list object:
## Neighbour list object:
## Number of regions: 276 
## Number of nonzero links: 1128 
## Percentage nonzero weights: 1.480781 
## Average number of links: 4.086957 
## Link number distribution:
## 
##  1  2  3  4  5  6  7  8 
## 18 24 54 66 72 24 12  6 
## 18 least connected regions:
## 17 36 43 63 82 89 109 128 135 155 174 181 201 220 227 247 266 273 with 1 link
## 6 most connected regions:
## 23 69 115 161 207 253 with 8 links
## 
## Weights style: M 
## Weights constants summary:
##     n    nn  S0       S1       S2
## M 276 76176 276 154.3387 1148.968
## 
## Call:lagsarlm(formula = fm, data = cigar, listw = pool.lwcig)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -0.6124792 -0.0951589 -0.0060833  0.0732741  0.7047066 
## 
## Type: lag 
## Coefficients: (asymptotic standard errors) 
##              Estimate Std. Error z value  Pr(>|z|)
## (Intercept)  1.304801   0.360331  3.6211 0.0002933
## logp        -1.038347   0.119958 -8.6559 < 2.2e-16
## logpn        0.180146   0.125899  1.4309 0.1524658
## logy         0.683452   0.070304  9.7213 < 2.2e-16
## 
## Rho: 0.08225, LR test value: 1.9474, p-value: 0.16286
## Asymptotic standard error: 0.069537
##     z-value: 1.1828, p-value: 0.23688
## Wald statistic: 1.3991, p-value: 0.23688
## 
## Log likelihood: 86.52832 for lag model
## ML residual variance (sigma squared): 0.031223, (sigma: 0.1767)
## Number of observations: 276 
## Number of parameters estimated: 6 
## AIC: -161.06, (AIC for lm: -161.11)
## LM test for residual autocorrelation
## test value: 3.1563, p-value: 0.075633
## 
## Call:errorsarlm(formula = fm, data = cigar, listw = pool.lwcig)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -0.5998877 -0.0916386 -0.0081973  0.0740753  0.6904247 
## 
## Type: error 
## Coefficients: (asymptotic standard errors) 
##              Estimate Std. Error z value  Pr(>|z|)
## (Intercept)  1.484185   0.312829  4.7444 2.091e-06
## logp        -1.060385   0.118173 -8.9732 < 2.2e-16
## logpn        0.150483   0.125306  1.2009    0.2298
## logy         0.730092   0.069817 10.4572 < 2.2e-16
## 
## Lambda: 0.14756, LR test value: 4.6572, p-value: 0.030924
## Asymptotic standard error: 0.076541
##     z-value: 1.9278, p-value: 0.053882
## Wald statistic: 3.7163, p-value: 0.053882
## 
## Log likelihood: 87.88319 for error model
## ML residual variance (sigma squared): 0.030797, (sigma: 0.17549)
## Number of observations: 276 
## Number of parameters estimated: 6 
## AIC: -163.77, (AIC for lm: -161.11)

Referências

Baltagi, Badi H, Seuck Heun Song, and Won Koh. 2003. “Testing Panel Data Regression Models with Spatial Error Correlation.” Journal of Econometrics 117 (1). Elsevier: 123–50.

Millo, Giovanni, and Gianfranco Piras. 2012. “splm: Spatial Panel Data Models in R.” Journal of Statistical Software 47 (1): 1–38. http://www.jstatsoft.org/v47/i01/.

Rasmussen, Poul Nørregaard. 2003. Econometric Analysis of Panel Data. John Wiley; Sons. http://www.wiley.com/legacy/wileychi/baltagi/.

Adriano Marcos Rodrigues Figueiredo, *e-mail: adriano.figueiredo@ufms.br*

18 março 2019