第4講空間計量経済学

ある地域の観測値が近隣地域の観測値で推測できる時、そこには空間的自己相関が存在する.
この空間自己相関構造をもつ地理空間データに対して、近隣地域の観測値を説明項に入れた回帰モデルを空間自己回帰モデルという.
このモデルは、時系列分析え発展した自己回帰モデルの考え方を空間に拡張したものである.

$$ y = \rho W_1 y + X \beta + u $$$$ u = \lambda W_2 u + \epsilon $$$$ \epsilon \sim N(0, \sigma^2I_n $$

・$W_1, W_2$:空間重み行列



In [2]:

    
library(spdep)









    



Loading required package: sp
Loading required package: Matrix



In [3]:

    
data(oldcol)



In [5]:

    
names(COL.OLD)









    Out[5]:





	'AREA_PL'
	'PERIMETER'
	'COLUMBUS.'
	'COLUMBUS.I'
	'POLYID'
	'NEIG'
	'HOVAL'
	'INC'
	'CRIME'
	'OPEN'
	'PLUMB'
	'DISCBD'
	'X'
	'Y'
	'AREA_SS'
	'NSA'
	'NSB'
	'EW'
	'CP'
	'THOUS'
	'NEIGNO'
	'PERIM'



In [13]:

    
COL.nb









    Out[13]:





Neighbour list object:
Number of regions: 49 
Number of nonzero links: 232 
Percentage nonzero weights: 9.662641 
Average number of links: 4.734694



In [12]:

    
COL.OLD









    Out[12]:





AREA_PL PERIMETER COLUMBUS. COLUMBUS.I POLYID NEIG HOVAL INC CRIME OPEN ellip.h X Y AREA_SS NSA NSB EW CP THOUS NEIGNO PERIM

	1001 0.259329 2.236939 3 1 2 1 44.567 21.232 18.80175 5.29672 ⋯ 35.62 42.38 8.621 1 1 0 0 1000 1001 2.236939
	1002 0.083841 1.427635 5 2 4 2 33.2 4.477 32.38776 0.394427 ⋯ 36.5 40.52 2.908 1 1 0 0 1000 1002 1.427635
	1003 0.204954 2.139524 9 3 8 3 37.125 11.337 38.42586 3.483478 ⋯ 36.71 38.71 6.439 1 1 0 0 1000 1003 2.139524
	1004 0.257084 2.554577 8 4 7 4 75 8.438 0.178269 0 ⋯ 33.36 38.41 16.22 1 1 0 0 1000 1004 2.554577
	1005 0.309441 2.440629 2 5 1 5 80.467 19.531 15.72598 2.850747 ⋯ 38.8 44.07 10.391 1 1 1 0 1000 1005 2.440629
	1006 0.192468 2.187547 4 6 3 6 26.35 15.956 30.62678 4.534649 ⋯ 39.82 41.18 6.981 1 1 1 0 1000 1006 2.187547
	1007 0.488888 2.997133 6 7 5 7 23.225 11.252 50.73151 0.405664 ⋯ 40.01 38 16.827 1 1 1 0 1000 1007 2.997133
	1008 0.283079 2.335634 7 8 6 8 28.75 16.029 26.06666 0.563075 ⋯ 43.75 39.28 8.929 1 1 1 0 1000 1008 2.335634
	1009 0.106653 1.437606 16 9 15 9 18 9.873 48.58549 0.174325 ⋯ 39.61 34.91 3.231 1 1 1 1 1000 1009 1.437606
	1010 0.246689 2.087235 11 10 10 10 96.4 13.598 34.00084 1.548348 ⋯ 47.61 36.42 7.268 1 1 1 0 1000 1010 2.087235
	1011 0.102087 1.382359 18 11 17 11 41.75 9.798 36.86877 0.448232 ⋯ 48.58 34.46 3.457 1 1 1 0 1000 1011 1.382359
	1012 0.24712 2.147528 24 12 23 12 47.733 21.155 20.0485 0.532632 ⋯ 49.61 32.65 6.901 0 0 1 0 1000 1012 2.147528
	1013 0.18558 2.108951 33 13 32 13 40.3 18.942 19.14559 2.221022 ⋯ 50.11 29.91 8.671 0 0 1 0 1000 1013 2.108951
	1014 0.137024 1.525097 42 14 41 14 42.1 22.207 18.90515 0.293317 ⋯ 51.24 27.8 4.632 0 0 1 0 1000 1014 1.525097
	1015 0.245249 2.079986 48 15 47 15 42.5 18.95 27.82286 0 ⋯ 50.89 25.24 7.989 0 0 1 0 1000 1015 2.079986
	1016 0.302908 2.285487 41 16 40 16 61.95 29.833 16.2413 6.45131 ⋯ 48.44 27.93 8.772 0 0 1 0 1000 1016 2.285486
	1017 0.444629 3.174601 21 17 20 17 81.267 31.07 0.223797 5.318607 ⋯ 46.73 31.91 14.453 0 1 1 0 1000 1017 3.174601
	1018 0.500755 3.169707 10 18 9 18 52.6 17.586 30.51592 0.527488 ⋯ 43.44 35.92 16.164 1 1 1 0 1000 1018 3.169707
	1019 0.192891 1.992717 23 19 22 19 30.45 11.709 33.70505 4.539754 ⋯ 43.37 33.46 6.84 1 1 1 1 1000 1019 1.992717
	1020 0.107298 1.406823 27 20 26 20 20.3 8.085 40.96974 1.238288 ⋯ 41.13 33.14 3.7 1 1 1 1 1000 1020 1.406823
	1021 0.137802 1.780751 28 21 27 21 34.1 10.822 52.79443 19.3681 ⋯ 43.95 31.61 5.7 0 0 1 1 1000 1021 1.780751
	1022 0.087472 1.507971 34 22 33 22 23.6 9.918 41.96816 0 ⋯ 44.1 30.4 2.029 0 0 1 1 1000 1022 1.507971
	1023 0.175453 1.974937 36 23 35 23 27 12.814 39.17505 4.220401 ⋯ 43.7 29.18 6.38 0 0 1 1 1000 1023 1.974937
	1024 0.053881 0.934509 39 24 38 24 22.7 11.107 53.71094 0 ⋯ 41.04 28.78 1.557 0 0 1 1 1000 1024 0.934508
	1025 0.173337 1.868044 45 25 44 25 33.5 16.961 25.96226 1.463993 ⋯ 43.23 27.31 6.305 0 0 1 0 1000 1025 1.868044
	1026 0.205964 2.199169 50 26 49 26 35.8 18.796 22.54149 0.259826 ⋯ 42.67 24.96 7.005 0 0 1 0 1000 1026 2.199169
	1027 0.069762 1.102032 49 27 48 27 26.8 11.813 26.64527 4.884389 ⋯ 41.21 25.9 2.662 0 0 1 1 1000 1027 1.102032
	1028 0.256431 2.193039 46 28 45 28 27.733 14.135 29.02849 1.006118 ⋯ 39.32 25.85 8.785 0 0 1 1 1000 1028 2.193039
	1029 0.060241 0.967793 44 29 43 29 25.7 13.38 36.66361 0 ⋯ 41.09 27.49 2.274 0 0 1 1 1000 1029 0.967793
	1030 0.121154 1.402252 38 30 37 30 43.3 17.017 42.44508 4.839273 ⋯ 38.32 28.82 7.573 0 0 1 1 1000 1030 1.402253
	1031 0.174773 1.637148 29 31 28 31 22.85 7.856 56.91978 0.509305 ⋯ 41.31 30.9 5.167 0 0 1 1 1000 1031 1.637148
	1032 0.17168 1.666489 26 32 25 32 17.9 8.461 61.29917 0 ⋯ 39.36 32.88 5.554 1 1 1 1 1000 1032 1.666489
	1033 0.085972 1.312158 30 33 29 33 32.5 8.681 60.75045 0 ⋯ 39.72 30.64 2.218 0 0 1 1 1000 1033 1.312158
	1034 0.104355 1.524931 31 34 30 34 22.5 13.906 68.89204 1.63878 ⋯ 38.29 30.35 3.193 0 0 0 1 1000 1034 1.524931
	1035 0.192226 2.240392 25 35 24 35 53.2 14.236 38.29787 0.62622 ⋯ 36.6 32.09 6.0287 1 1 0 1 1000 1035 2.240392
	1036 0.093154 1.340061 17 36 16 36 18.8 7.625 54.83871 0.533737 ⋯ 37.6 34.08 3.59 1 1 0 1 1000 1036 1.34006
	1037 0.035769 0.902125 13 37 12 37 19.9 10.048 56.70567 3.157895 ⋯ 37.13 36.12 1.472 1 1 0 1 1000 1037 0.902125
	1038 0.041012 0.919488 12 38 11 38 19.7 7.467 62.27545 0 ⋯ 37.85 36.3 1.285 1 1 0 1 1000 1038 0.919488
	1039 0.034377 0.93659 14 39 13 39 41.7 9.549 46.71613 0 ⋯ 35.95 36.4 1.093 1 1 0 1 1000 1039 0.93659
	1040 0.060884 1.128424 15 40 14 40 42.9 9.963 57.06613 0.477104 ⋯ 35.72 35.6 2.077 1 1 0 1 1000 1040 1.128423
	1041 0.061342 1.249247 20 41 19 41 30.6 11.618 54.52197 0.111111 ⋯ 35.76 34.66 1.369 1 1 0 1 1000 1041 1.249247
	1042 0.055494 1.183352 19 42 18 42 60 13.185 43.96249 24.99807 ⋯ 36.15 33.92 2.691 1 1 0 1 1000 1042 1.183352
	1043 0.699258 5.07749 22 43 21 43 19.975 10.655 40.07407 1.643756 ⋯ 34.08 30.42 21.282 0 0 0 1 1000 1043 5.077491
	1044 0.226594 2.519132 35 44 34 44 28.45 14.948 23.97403 3.029087 ⋯ 30.32 28.26 7.608 0 0 0 0 1000 1044 2.519132
	1045 0.117409 1.716047 32 45 31 45 31.8 16.94 17.67721 3.936443 ⋯ 27.94 29.85 3.812 1 1 0 0 1000 1045 1.716046
	1046 0.17813 1.790058 37 46 36 46 36.3 18.739 14.30556 6.773331 ⋯ 27.27 28.21 5.191 0 0 0 0 1000 1046 1.790058
	1047 0.174823 2.335402 40 47 39 47 39.6 18.477 19.10086 0 ⋯ 24.25 26.69 6.402 0 0 0 0 1000 1047 2.335402
	1048 0.124728 1.841029 47 48 46 48 76.1 18.324 16.53053 9.683953 ⋯ 25.47 25.71 3.822 0 0 0 0 1000 1048 1.841029
	1049 0.266541 2.176543 43 49 42 49 44.333 25.873 16.49189 1.792993 ⋯ 29.02 26.58 8.695 0 0 0 0 1000 1049 2.176543

本講義で扱うデータを読み込む.
米国オハイオ州コロンバス市の犯罪データセットである.
近隣に住む1000世帯あたりの不法行為目的侵入および自動車窃盗(CRIME)、家の評価額(HOVAL、単位は\$1000)、世帯収入(INC、単位は\$1000)が含まれる。



In [8]:

    
COL.ols <- lm(CRIME ~ INC + HOVAL, data = COL.OLD)



In [9]:

    
summary(COL.ols)









    Out[9]:





Call:
lm(formula = CRIME ~ INC + HOVAL, data = COL.OLD)

Residuals:
    Min      1Q  Median      3Q     Max 
-34.418  -6.388  -1.580   9.052  28.649 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  68.6190     4.7355  14.490  < 2e-16 ***
INC          -1.5973     0.3341  -4.780 1.83e-05 ***
HOVAL        -0.2739     0.1032  -2.654   0.0109 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 11.43 on 46 degrees of freedom
Multiple R-squared:  0.5524,	Adjusted R-squared:  0.5329 
F-statistic: 28.39 on 2 and 46 DF,  p-value: 9.341e-09



In [14]:

    
W <- nb2listw(COL.nb, style = "W")



In [18]:

    
moran.test(COL.OLD$CRIME, W)









    Out[18]:





	Moran I test under randomisation

data:  COL.OLD$CRIME  
weights: W  

Moran I statistic standard deviate = 5.6341, p-value = 8.797e-09
alternative hypothesis: greater
sample estimates:
Moran I statistic       Expectation          Variance 
      0.510951264      -0.020833333       0.008908762



In [15]:

    
lm.LMtests(COL.ols, listw = W, test = c("LMlag", "LMerr"))









    Out[15]:





	Lagrange multiplier diagnostics for spatial dependence

data:  
model: lm(formula = CRIME ~ INC + HOVAL, data = COL.OLD)
weights: W

LMlag = 9.3637, df = 1, p-value = 0.002213


	Lagrange multiplier diagnostics for spatial dependence

data:  
model: lm(formula = CRIME ~ INC + HOVAL, data = COL.OLD)
weights: W

LMerr = 5.7231, df = 1, p-value = 0.01674



In [ ]:

	AREA_PL	PERIMETER	COLUMBUS.	COLUMBUS.I	POLYID	NEIG	HOVAL	INC	CRIME	OPEN	ellip.h	X	Y	AREA_SS	NSA	NSB	EW	CP	THOUS	NEIGNO	PERIM
1001	0.259329	2.236939	3	1	2	1	44.567	21.232	18.80175	5.29672	⋯	35.62	42.38	8.621	1	1	0	0	1000	1001	2.236939
1002	0.083841	1.427635	5	2	4	2	33.2	4.477	32.38776	0.394427	⋯	36.5	40.52	2.908	1	1	0	0	1000	1002	1.427635
1003	0.204954	2.139524	9	3	8	3	37.125	11.337	38.42586	3.483478	⋯	36.71	38.71	6.439	1	1	0	0	1000	1003	2.139524
1004	0.257084	2.554577	8	4	7	4	75	8.438	0.178269	0	⋯	33.36	38.41	16.22	1	1	0	0	1000	1004	2.554577
1005	0.309441	2.440629	2	5	1	5	80.467	19.531	15.72598	2.850747	⋯	38.8	44.07	10.391	1	1	1	0	1000	1005	2.440629
1006	0.192468	2.187547	4	6	3	6	26.35	15.956	30.62678	4.534649	⋯	39.82	41.18	6.981	1	1	1	0	1000	1006	2.187547
1007	0.488888	2.997133	6	7	5	7	23.225	11.252	50.73151	0.405664	⋯	40.01	38	16.827	1	1	1	0	1000	1007	2.997133
1008	0.283079	2.335634	7	8	6	8	28.75	16.029	26.06666	0.563075	⋯	43.75	39.28	8.929	1	1	1	0	1000	1008	2.335634
1009	0.106653	1.437606	16	9	15	9	18	9.873	48.58549	0.174325	⋯	39.61	34.91	3.231	1	1	1	1	1000	1009	1.437606
1010	0.246689	2.087235	11	10	10	10	96.4	13.598	34.00084	1.548348	⋯	47.61	36.42	7.268	1	1	1	0	1000	1010	2.087235
1011	0.102087	1.382359	18	11	17	11	41.75	9.798	36.86877	0.448232	⋯	48.58	34.46	3.457	1	1	1	0	1000	1011	1.382359
1012	0.24712	2.147528	24	12	23	12	47.733	21.155	20.0485	0.532632	⋯	49.61	32.65	6.901	0	0	1	0	1000	1012	2.147528
1013	0.18558	2.108951	33	13	32	13	40.3	18.942	19.14559	2.221022	⋯	50.11	29.91	8.671	0	0	1	0	1000	1013	2.108951
1014	0.137024	1.525097	42	14	41	14	42.1	22.207	18.90515	0.293317	⋯	51.24	27.8	4.632	0	0	1	0	1000	1014	1.525097
1015	0.245249	2.079986	48	15	47	15	42.5	18.95	27.82286	0	⋯	50.89	25.24	7.989	0	0	1	0	1000	1015	2.079986
1016	0.302908	2.285487	41	16	40	16	61.95	29.833	16.2413	6.45131	⋯	48.44	27.93	8.772	0	0	1	0	1000	1016	2.285486
1017	0.444629	3.174601	21	17	20	17	81.267	31.07	0.223797	5.318607	⋯	46.73	31.91	14.453	0	1	1	0	1000	1017	3.174601
1018	0.500755	3.169707	10	18	9	18	52.6	17.586	30.51592	0.527488	⋯	43.44	35.92	16.164	1	1	1	0	1000	1018	3.169707
1019	0.192891	1.992717	23	19	22	19	30.45	11.709	33.70505	4.539754	⋯	43.37	33.46	6.84	1	1	1	1	1000	1019	1.992717
1020	0.107298	1.406823	27	20	26	20	20.3	8.085	40.96974	1.238288	⋯	41.13	33.14	3.7	1	1	1	1	1000	1020	1.406823
1021	0.137802	1.780751	28	21	27	21	34.1	10.822	52.79443	19.3681	⋯	43.95	31.61	5.7	0	0	1	1	1000	1021	1.780751
1022	0.087472	1.507971	34	22	33	22	23.6	9.918	41.96816	0	⋯	44.1	30.4	2.029	0	0	1	1	1000	1022	1.507971
1023	0.175453	1.974937	36	23	35	23	27	12.814	39.17505	4.220401	⋯	43.7	29.18	6.38	0	0	1	1	1000	1023	1.974937
1024	0.053881	0.934509	39	24	38	24	22.7	11.107	53.71094	0	⋯	41.04	28.78	1.557	0	0	1	1	1000	1024	0.934508
1025	0.173337	1.868044	45	25	44	25	33.5	16.961	25.96226	1.463993	⋯	43.23	27.31	6.305	0	0	1	0	1000	1025	1.868044
1026	0.205964	2.199169	50	26	49	26	35.8	18.796	22.54149	0.259826	⋯	42.67	24.96	7.005	0	0	1	0	1000	1026	2.199169
1027	0.069762	1.102032	49	27	48	27	26.8	11.813	26.64527	4.884389	⋯	41.21	25.9	2.662	0	0	1	1	1000	1027	1.102032
1028	0.256431	2.193039	46	28	45	28	27.733	14.135	29.02849	1.006118	⋯	39.32	25.85	8.785	0	0	1	1	1000	1028	2.193039
1029	0.060241	0.967793	44	29	43	29	25.7	13.38	36.66361	0	⋯	41.09	27.49	2.274	0	0	1	1	1000	1029	0.967793
1030	0.121154	1.402252	38	30	37	30	43.3	17.017	42.44508	4.839273	⋯	38.32	28.82	7.573	0	0	1	1	1000	1030	1.402253
1031	0.174773	1.637148	29	31	28	31	22.85	7.856	56.91978	0.509305	⋯	41.31	30.9	5.167	0	0	1	1	1000	1031	1.637148
1032	0.17168	1.666489	26	32	25	32	17.9	8.461	61.29917	0	⋯	39.36	32.88	5.554	1	1	1	1	1000	1032	1.666489
1033	0.085972	1.312158	30	33	29	33	32.5	8.681	60.75045	0	⋯	39.72	30.64	2.218	0	0	1	1	1000	1033	1.312158
1034	0.104355	1.524931	31	34	30	34	22.5	13.906	68.89204	1.63878	⋯	38.29	30.35	3.193	0	0	0	1	1000	1034	1.524931
1035	0.192226	2.240392	25	35	24	35	53.2	14.236	38.29787	0.62622	⋯	36.6	32.09	6.0287	1	1	0	1	1000	1035	2.240392
1036	0.093154	1.340061	17	36	16	36	18.8	7.625	54.83871	0.533737	⋯	37.6	34.08	3.59	1	1	0	1	1000	1036	1.34006
1037	0.035769	0.902125	13	37	12	37	19.9	10.048	56.70567	3.157895	⋯	37.13	36.12	1.472	1	1	0	1	1000	1037	0.902125
1038	0.041012	0.919488	12	38	11	38	19.7	7.467	62.27545	0	⋯	37.85	36.3	1.285	1	1	0	1	1000	1038	0.919488
1039	0.034377	0.93659	14	39	13	39	41.7	9.549	46.71613	0	⋯	35.95	36.4	1.093	1	1	0	1	1000	1039	0.93659
1040	0.060884	1.128424	15	40	14	40	42.9	9.963	57.06613	0.477104	⋯	35.72	35.6	2.077	1	1	0	1	1000	1040	1.128423
1041	0.061342	1.249247	20	41	19	41	30.6	11.618	54.52197	0.111111	⋯	35.76	34.66	1.369	1	1	0	1	1000	1041	1.249247
1042	0.055494	1.183352	19	42	18	42	60	13.185	43.96249	24.99807	⋯	36.15	33.92	2.691	1	1	0	1	1000	1042	1.183352
1043	0.699258	5.07749	22	43	21	43	19.975	10.655	40.07407	1.643756	⋯	34.08	30.42	21.282	0	0	0	1	1000	1043	5.077491
1044	0.226594	2.519132	35	44	34	44	28.45	14.948	23.97403	3.029087	⋯	30.32	28.26	7.608	0	0	0	0	1000	1044	2.519132
1045	0.117409	1.716047	32	45	31	45	31.8	16.94	17.67721	3.936443	⋯	27.94	29.85	3.812	1	1	0	0	1000	1045	1.716046
1046	0.17813	1.790058	37	46	36	46	36.3	18.739	14.30556	6.773331	⋯	27.27	28.21	5.191	0	0	0	0	1000	1046	1.790058
1047	0.174823	2.335402	40	47	39	47	39.6	18.477	19.10086	0	⋯	24.25	26.69	6.402	0	0	0	0	1000	1047	2.335402
1048	0.124728	1.841029	47	48	46	48	76.1	18.324	16.53053	9.683953	⋯	25.47	25.71	3.822	0	0	0	0	1000	1048	1.841029
1049	0.266541	2.176543	43	49	42	49	44.333	25.873	16.49189	1.792993	⋯	29.02	26.58	8.695	0	0	0	0	1000	1049	2.176543

第4講 空間計量経済学

第4講空間計量経済学