Introduction
The general spatial static panel model takes the form:
\[\begin{equation}
\begin{aligned}
y_{t}&=\rho W y_{t} + X_{t} \beta + W X_{t} \theta + u_{t}, \\
u_{t} &=\lambda W u_{t}+\epsilon_{t}
\end{aligned} \label{eq:mod1}\tag{1}
\end{equation}\]
where the \(N \times 1\) vector
\(y_{t}\) is the dependent variable,
\(X_{t}\) is a \(N \times k\) matrix of \(k\) explanatory variables and \(W\) is a spatial weights matrix. \(N\) represents the number of units and
\(t=1,...,T\) are the time points. The
spatial lags for the vector of covariates is denoted with \(WX_t\). Spatial interaction in the error
term \(u_{it}\) is included with the
\(\lambda\) coefficient and it is
assumed that \(\epsilon_t\) is
independently and identically distributed error term for all \(t\) with zero mean and variance \(\sigma^2\). \(\rho\) is the spatial autoregressive
coefficient, \(\lambda\) the spatial
autocorrelation coefficient, \(\beta\)
is a vector of response parameters for the covariates.
Figure 1. The different spatial panel models and
their dependencies.
Note: SAC - spatial autoregressive combined,
SDM - spatial Durbin model, SDEM - spatial Durbin error model, SAR -
spatial autogregressive model (or Spatial lag model), SEM - spatial
error model, SLX - spatially lagged X model.
However, model (1) suffers from identification
problems (Elhorst 2010). Figure 1 shows the identifiable models, which are derived
by imposing some restrictions on model (1). If
all spatial coefficients are zero (\(\rho =
\theta = \lambda = 0\)), then the corresponding model will be the
ordinary least-squares model (OLS). If in each of the models in figure
(1) \(\eta W y_{t-1}
+\tau y_{t-1}\) is added, we get the corresponding dynamic panel
data models. \(y_{(t-1)}\) denotes the
time lag of the dependent variable and \(Wy_{(t-1)}\) is the time-space lag. \(\tau\) and \(\eta\) are the response parameters for the
lagged variable.
LeSage and Parent (2007) and LeSage (2014) suggest a Bayesian approach for
selecting an appropriate model. The calculation of the posterior
probabilities for 6 models (SAR, SEM, SDM, SDEM, SLX, OLS) is possible
with the function blmpSDPD.
The function SDPMm provides estimation of a SAR and SDM
model with the Lee-Yu transformation approach (Yu, De Jong, and Lee (2008), Lee and Yu (2010b), Lee
and Yu (2010a)).
Data
The Cigar data set (Baltagi and Levin
(1992)) from the plm package (Croissant and Millo (2008)) will be used for
describing the use of the blmpSDPD and SDPDm
functions. It contains panel data of cigarette consumption in 46 states
in the USA over the period from 1963 to 1992. The binary contiguity
matrix of the 46 states is included in the SDPDmod
package.
data("Cigar",package = "plm")
head(Cigar)
#> state year price pop pop16 cpi ndi sales pimin
#> 1 1 63 28.6 3383 2236.5 30.6 1558.305 93.9 26.1
#> 2 1 64 29.8 3431 2276.7 31.0 1684.073 95.4 27.5
#> 3 1 65 29.8 3486 2327.5 31.5 1809.842 98.5 28.9
#> 4 1 66 31.5 3524 2369.7 32.4 1915.160 96.4 29.5
#> 5 1 67 31.6 3533 2393.7 33.4 2023.546 95.5 29.6
#> 6 1 68 35.6 3522 2405.2 34.8 2202.486 88.4 32.0
data1<- Cigar
data1$logc<-log(data1$sales)
data1$logp<-log(data1$price/data1$cpi)
data1$logy<-log(data1$ndi/data1$cpi)
data1$lpm<-log(data1$pimin/data1$cpi)
data("usa46",package="SDPDmod") ## binary contiguity matrix of 46 USA states
str(usa46)
#> num [1:46, 1:46] 0 0 0 0 0 0 0 1 1 0 ...
#> - attr(*, "dimnames")=List of 2
#> ..$ : chr [1:46] "Alabama" "Arizona" "Arkansas" "California" ...
#> ..$ : chr [1:46] "Alabama" "Arizona" "Arkansas" "California" ...
W <- rownor(usa46) ## row-normalization
isrownor(W) ## check if W is row-normalized
#> [1] TRUE
Bayesian posterior probabilities
Static panel model with spatial fixed effects
If the option dynamic is not set, then the default model
is static. For spatial fixed effects effect should be set
to “individual”.
res1<-blmpSDPD(formula = logc ~ logp+logy, data = data1, W = W,
index = c("state","year"),
model = list("ols","sar","sdm","sem","sdem","slx"),
effect = "individual")
res1
#> ols sar sdm sem sdem slx
#> Log marginals 884.7551 938.6934 1046.4868 993.192 1039.6707 930.0585
#> Model probs 0.0000 0.0000 0.9989 0.000 0.0011 0.0000
Static panel model with time fixed effects
res2<-blmpSDPD(formula = logc ~ logp+logy, data = data1, W = W,
index = c("state","year"),
model = list("ols","sar","sdm","sem","sdem","slx"),
effect = "time")
Static panel model with both spatial and time fixed effects for 4
models (SAR, SDM, SEM, SDEM)
The default prior is uniform. With prior="beta" the beta
prior is used.
res3<-blmpSDPD(formula = logc ~ logp+logy, data = data1, W = W,
index = c("state","year"),
model = list("sar","sdm","sem","sdem"),
effect = "twoways",
prior = "beta")
Dynamic panel model with both spatial fixed effects with beta prior,
where the lagged dependent variable is included in the data
d2 <- plm::pdata.frame(data1, index=c('state', 'year'))
d2$llogc<-plm::lag(d2$logc) ## add lagged variable
data2<-d2[which(!is.na(d2$llogc)),]
rownames(data2)<-1:nrow(data2)
kk<-which(colnames(data2)=="llogc")
kk
#> [1] 14
res5<-blmpSDPD(formula = logc ~ logp+logy, data = data2, W = W,
index = c("state","year"),
model = list("sar","sdm","sem","sdem"),
effect = "individual",
ldet = "full",
dynamic = TRUE,
tlaginfo = list(ind=kk),
prior = "beta")
References
Baltagi, Badi H, and Dan Levin. 1992. “Cigarette Taxation: Raising
Revenues and Reducing Consumption.” Structural Change and
Economic Dynamics 3 (2): 321–35.
Croissant, Yves, and Giovanni Millo. 2008. “Panel Data
Econometrics in r: The Plm Package.” Journal of Statistical
Software 27 (2).
Elhorst, J Paul. 2010. “Applied Spatial Econometrics: Raising the
Bar.” Spatial Economic Analysis 5 (1): 9–28.
Lee, Lung-fei, and Jihai Yu. 2010a.
“A Spatial Dynamic Panel Data
Model with Both Time and Individual Fixed Effects.”
Econometric Theory 26 (2): 564–97.
https://doi.org/10.1017/S0266466609100099.
———. 2010b.
“Estimation of Spatial Autoregressive Panel Data
Models with Fixed Effects.” Journal of Econometrics 154
(2): 165–85.
https://doi.org/10.1016/j.jeconom.2009.08.001.
LeSage, James P. 2014.
“Spatial Econometric Panel Data Model
Specification: A Bayesian Approach.” Spatial Statistics
9: 122–45.
https://doi.org/10.1016/j.spasta.2014.02.002.
LeSage, James P, and Olivier Parent. 2007. “Bayesian Model
Averaging for Spatial Econometric Models.” Geographical
Analysis 39 (3): 241–67.
Yu, Jihai, Robert De Jong, and Lung-fei Lee. 2008.
“Quasi-Maximum
Likelihood Estimators for Spatial Dynamic Panel Data with Fixed Effects
When Both n and t Are Large.” Journal of Econometrics
146 (1): 118–34.
https://doi.org/10.1016/j.jeconom.2008.08.002.