Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
Do spatial models help predicting population annual growth rates? Spatial Econometric Association Conference - Toulouse 2011
Thibault LAURENT Toulouse School of Economics
July 6, 2011
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
Introduction Description of the data set The data Spatial Exploratory Data Analysis The non spatial models OLS and WLS models GAM and wGAM models Moran test on the WLS residuals The Spatial Weight matrix Moran plot and Moran test The spatial models SAR and SEM models The GWR and wGWR model Conclusion Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
Introduction
Joint work (in progress) with C. Thomas-Agnan and A. Ruiz-Gazen (TSE) and J.-P. Lesne and H. Tranger (BVA). Context: the new French census for the small communes (less than 10000 inhabitants at the 1999 census) consists in dividing the set of small communes of each region in 5 rotation groups (in a balanced way) and survey each of these groups every 5 years. At the end of 2007, four fifth of the communes were censored and survey institutes were interested in predicting the population for the last fifth.
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
The case of the Midi-Pyr´en´ees region I
8 departements, 3020 communes
I
30 communes with more than 10 000 inhabitants
I
I
An european region mainly rural, with an economic hub Toulouse (aircraft and spatial industry, R&D, etc).
Urban/Rural area Urban 1 Urban 2 Urban 3 Rural
Lot Department Commune
Aveyron Tarn−et−Garonne
Tarn
Gers
Haute−Garonne
Hautes−Pyrénées Ariège 0
50 Km
The total population in the region has increased by 1.19% from 1999 to 2008
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
The census in Midi-Pyr´en´ees
2004 big communes
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●
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
SEA conference 2011
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
The census in Midi-Pyr´en´ees
2004 2005 big communes
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●
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
SEA conference 2011
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
The census in Midi-Pyr´en´ees
2004 2005 2006 big communes
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●
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
SEA conference 2011
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
The census in Midi-Pyr´en´ees
2004 2005 2006 2007 big communes
●
●
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
SEA conference 2011
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
The census in Midi-Pyr´en´ees
2008: communes to predict big communes
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●
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
SEA conference 2011
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
Aim of our study
I
Fitting models for explaining the annual mean growth rate of the small communes called evol using the four first years of the census (2004, 2005, 2006, 2007)
I
Predicting the dependent variable the last year of census 2008.
I
Comparing non spatial models and spatial models using the R packages project.
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
Methodology I
“Training” sample i.e. the sample used for modelling: the 2390 communes which have been sampled in 2004, 2005, 2006 and 2007.
I
“Test sample” or the sample on which we made our predictions: the 595 communes sampled in 2008.
I
Three explanatory variables used for each model.
I
Best linear unbiased prediction formula (BLUP) for each method.
I
Correcting for heteroscedasticity when possible: weighting by initial population counts.
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
The models
1. The non spatial models: I I
The OLS and WLS models The GAM and wGAM models
2. The spatial models I I
The GWR model and wGWR models The SAR and SEM models
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
Comparison criteria
I
The Mean Square Error criteria: 595 1 X (evolk − yˆk )2 MSE = 595 k=1
I
The relative MSE:
√ rMSE =
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
MSE ¯ evol
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
Introduction Description of the data set The data Spatial Exploratory Data Analysis The non spatial models OLS and WLS models GAM and wGAM models Moran test on the WLS residuals The Spatial Weight matrix Moran plot and Moran test The spatial models SAR and SEM models The GWR and wGWR model Conclusion Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
Introduction Description of the data set The data Spatial Exploratory Data Analysis The non spatial models OLS and WLS models GAM and wGAM models Moran test on the WLS residuals The Spatial Weight matrix Moran plot and Moran test The spatial models SAR and SEM models The GWR and wGWR model Conclusion Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
The data
The data The 2988 communes with less than 10 000 inhabitants of the Midi-Pyr´en´ees regions. I
Dependent variable: evol, annual mean growth rate of the communes. Computed from the population pop1999 in 1999 and the population pop1999+n in (1999 + n) by solving the following relationship : pop1999+n = pop1999 (1 + evol)n
I
Weight variable: pop 1999, population in 1999.
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
The data
The data Explanatory variables: (number of birth - number of death)1999−(1999+n) I accroi: . n This variable is given by the INSEE but might be obtained independently from the dependent variable. I
tx evol ff: increase rate for the number of fiscal families from 1999 to the year of census. It has been obtained on the site of the french tax administration http://www.impots.gouv.fr/.
I
HAU: equals to 1 if the commune is considered as rural and 0 if it is not. This variable is also given by the INSEE.
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
SEDA
Choropleth map of the dependent variable
annual mean groth rate 3.684 Urban
0
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
50 Km
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
SEDA
Identification of outliers
●
15
5.453
10
● ● ● ● ●
Urban
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
−5
0
5
annual mean groth rate
● ● ● ● ● ● ● ● ● ●
−10
● ● ●
0
50 Km
●
annual mean groth rate
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
SEDA
Scatterplot matrix −10
−5
0
5
10
15 ●
100
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15
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evol Frequency
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tx_evol_ff
●
●
Frequency
5
x
●
●
0
● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ●● ● ● ● ●● ● ●● ● ● ● ●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
−5
●
10
−10
−5
0
5
10
x
●
●
0
●
●●
●
−50
50
Frequency
●
●
●
●
0
●
−50
accroi
●
50
100
−5
0
5
10
x
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
SEDA
Parallel boxplot
15
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5
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0
annual mean groth rate
10
● ● ● ● ●
−5
● ● ●
● ● ● ● ● ● ● ● ●
−10
● ● ●
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urban
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
rural
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
Introduction Description of the data set The data Spatial Exploratory Data Analysis The non spatial models OLS and WLS models GAM and wGAM models Moran test on the WLS residuals The Spatial Weight matrix Moran plot and Moran test The spatial models SAR and SEM models The GWR and wGWR model Conclusion Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
OLS and WLS
OLS and WLS models
evol = β0 + β1 accroi + β2 (tx evol ff ) + β3 HAU + 1. OLS: with ∼ N (0, σ 2 I ) in the homoscedastic case 2. WLS: with ∼ N (0, σ 2 Φ) for a given diagonal matrix of heteroscedasticity weights Φ. How choosing Φ?
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
OLS and WLS
Residuals of the OLS model
15
●
5 −5
0
OLS residuals
10
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−10
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2000
4000
6000
8000
10000
Population in 1999
=⇒ WLS with the population 1999 used as the weights Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
OLS and WLS
Results of the OLS and WLS models
(Intercept) accroi tx evol ff HAU R-square MSE rel.MSE
OLS 1.0363 0.0289 0.3962 -0.6266 0.2177 2.2064 1.4103
WLS 0.4244 0.0097 0.6788 -0.3535 0.4995 2.1226 1.3832
=⇒ the fact to consider a WLS model instead of a simple OLS model improves the result. Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
GAM and wGAM
Presentation of the Generalized Additive Model Hastie and Tibshirani (1990) yi = Xi∗ β +
p X
fj (xij ) + εi
j=1
with possibly heterosckedastic errors and where in our case Xi∗ is a row vector with two components corresponding to the constant and the categorical variable HAU and the fj are smooth functions estimated non parametrically. package mgcv with default options
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
GAM and wGAM
6 4 2 0
s(tx_evol_ff,6.26)
−6
−4
−2
0 −6
−4
−2
s(accroi,8.8)
2
4
6
Smooth function components
−50
0
50
100
accroi
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
−5
0
5
10
tx_evol_ff
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
GAM and wGAM
Results of the non spatial models
MSE MSE/mean(evol)
OLS 2.2064 1.4103
WLS 2.1226 1.3832
GAM 2.1918 1.4056
wGAM 2.1117 1.3797
It seems fundamental to take into account the heterosckedasticity of the data. However, it does not seem necessary to consider a non parametric model.
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
Introduction Description of the data set The data Spatial Exploratory Data Analysis The non spatial models OLS and WLS models GAM and wGAM models Moran test on the WLS residuals The Spatial Weight matrix Moran plot and Moran test The spatial models SAR and SEM models The GWR and wGWR model Conclusion Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
The spatial weight matrix
The Spatial Weight matrix
I
Matrix W based on the ten nearest neighbors.
I
We decompose the matrix W as: WS WSO W = WOS WO I
I
S (S for “Sampled”) corresponds to the communes sampled in 2004, 2005, 2006 and 2007 O (O for “Out-Of-Sample”) corresponds to the communes sampled in 2008.
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
The spatial weight matrix
Representation of WS ● ●
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Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
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Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
Moran plot and Moran test
Spatial autocorrelation in the residuals
I
Moran scatterplot (Anselin, 1993) of the residuals,
I
Moran test of H0 : no spatial autocorrelation using the Gaussian or the randomization assumption.
Let z be a spatial sample of dimension n, ¯z the vector of the sample mean, W a spatial structure matrix: scatterplot of the variable W × (z − ¯z) against (z − ¯z). where W is spatial weight matrix or spatial structure matrix n × n which specifies the neighborhood information.
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
Moran plot and Moran test
4
Moran plot of the WLS residuals
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0
1
●
−3
−2
−1
W x WLS residuals
2
3
H−H H−L L−L L−H
−10
−5
0
5
10
15
WLS residuals
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
Moran plot and Moran test
Limitations of the WLS model We reject the assumption of no spatial autocorrelation in the residuals of the WLS model (under the randomization assumption) =⇒ spatial dynamics not explained (mean process due to inherent heterogeneity or spatial autocorrelation not taken into account). =⇒ models allowing for heterogeneity or spatial autocorrelation. Remark: with only one realization of the process, not possible to decide between these two alternatives. Attention restricted to models that can be adjusted using the R environment.
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
Introduction Description of the data set The data Spatial Exploratory Data Analysis The non spatial models OLS and WLS models GAM and wGAM models Moran test on the WLS residuals The Spatial Weight matrix Moran plot and Moran test The spatial models SAR and SEM models The GWR and wGWR model Conclusion Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
SAR and SEM
Presentation of the SAR and SEM (Anselin, 1988) I
Model SAR Y = ρWY + X β + ,
(1)
with ∼ N (0, σ 2 I ) in the homoscedastic case I
Models SEM Y
= Xβ +
(2)
= λW + U,
(3)
where U ∼ N (0, σ 2 I ) in the homoscedastic case (package spdep)
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
SAR and SEM
Formula predictions for the SAR model I
implemented in the package spdep: YˆOTS = XO βˆ + ρˆWOS YS
I
(4)
implemented in Matlab: ˆO YˆOTC = [(I − ρˆW )−1 X β]
I
(5)
the BLUP prediction: YˆOBP = YˆOTC − QO−1 QOS × (YS − YˆSTC )
(6)
with: Q=I−
ρˆ(W 0
+ W) +
ρˆ2 W 0 W
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
=
QS QOS
QSO QO
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
SAR and SEM
Formula predictions for the SEM model
I
implemented in the package spdep: YˆOTC = XO βˆ
I
(7)
the BLUP prediction: YˆOBP = YˆOTC − QO−1 QOS × (YS − YˆSTC ) 0 + W) + λ ˆ ˆ 2 W 0 W = QS QSO with: Q = I − λ(W QOS QO
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
(8)
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
SAR and SEM
SAR and SEM model estimates
(Intercept) accroi tx evol ff HAU autocor. coef.
SAR coeff 0.6380 0.0240 0.3700 -0.4240 0.2520
t-stat 6.7890 4.9750 19.2060 -4.7610 7.7160
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
SEM coeff 1.0440 0.0270 0.3740 -0.5980 0.2280
t-stat 11.4230 5.2360 19.0830 -6.0040 6.1810
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
SAR and SEM
Cumulative effects scalar summary estimates for the SAR
accroi tx evol ff HAU
direct 0.0244 (***) 0.3733 (***) -0.4272 (***)
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
indirect 0.0080 (***) 0.1219 (***) -0.1395 (***)
total 0.0324 (***) 0.4952 (***) -0.5667 (***)
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
GWR and wGWR
Presentation of the GWR model (Fortheringham et al., 2002) Aim: to account for spatial heterogeneity in the regression relationship.
yi = β0 (ui , vi ) +
k X
βj (ui , vi ) × xij + εi
j=1
with βj continuous functions of the location coordinates (ui , vi ), i = 1, . . . , n. Local weighted least squares estimates with the geographical distance as the argument of the weighting scheme. Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
GWR and wGWR
Presentation of the GWR model n X
yi − β0 (ui , vi ) −
i=1
k X
2 βj (ui , vi ) × xij K
j=1
ks − si k h
ˆ = (X 0 W (s)X )−1 X 0 W (s)Y . β(s) where, if the observed locations are denoted by si = (ui , vi ), the weights are given by ks − si k W (s) = diag K h for a kernel K and a bandwidth h. We choose a gaussian kernel and a bandwidth equal to h = 10 km. Thibault LAURENT et Al.
(package spgwr).
Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
GWR and wGWR
Representation of the wGWR
1820000
accroi
1810000
0.053
1780000
1790000
1800000
●
0
400000
410000
420000
430000
440000
50 Km
450000
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
GWR and wGWR
Representation of the wGWR (2)
tx_evol_ff
HAU
0.454
−0.057
0
50 Km
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
0
50 Km
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
GWR and wGWR
Comparison of the spatial methods
MSE MSE/mean(evol)
GWR 2.1349 1.3872
wGWR 2.0166 1.3482
MSE MSE/mean(evol)
SAR.TC 2.1859 1.4037
SEM.TC 2.2272 1.4169
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
SAR.BP 2.0997 1.3757
SAR.TS 2.1101 1.3791
SEM.BP 2.1202 1.3824
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
Parallel boxplot of the residuals
WLS
●
wGWR wGAM
●
●
●
SEM−TC
SAR−TS
●
SAR−TC
●
OLS
●
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−10
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SAR−BP
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SEM−BP
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−5
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
0
●
●
●
●
●
●●
5
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
Conclusion
I
by looking at the MSE criteria, we can assert that the use of spatial models helps predicting population annual growth rates
I
Non weighted version: the SAR model gives the best result by considering the BLUP prediction
I
heteroscedasticity in the model: the GWR gives the best result (SAR and SEM not included)
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
Introduction
The Data
The non spatial models
Moran test
The spatial models
Conclusion
Perspectives
I
SAR and SEM weighted versions
I
SDM and CAR models
I
spatio-temporal information
I
add some economic variables among the explanatory variables
Thibault LAURENT et Al. Do spatial models help predicting population annual growth rates?
Toulouse School of Economics
