spreg: GM estimation of a Cliff-Ord type model with Heteroskedastic Innovations

Description

Multi step GM/IV estimation of a linear Cliff and Ord -type of model of the form: $$y=\lambda W y + X \beta + u $$ $$u=\rho W u + e$$ with $$e ~ N(0,\sigma^2_i) $$

The model allows for spatial lag in the dependent variable and disturbances. The innovations in the disturbance process are assumed heteroskedastic of an unknown form.

Usage

spreg(formula, data = list(), listw, listw2 = NULL, 
        endog = NULL, instruments = NULL, 
        lag.instr = FALSE, initial.value = 0.2, q = 2,
        model = c("sarar", "lag", "error", "ivhac", "ols"),
        het = FALSE, verbose = FALSE, 
        na.action = na.fail,  HAC = FALSE, 
        distance = NULL, type =  c("Epanechnikov","Triangular",
                                   "Bisquare", "Parzen", "QS", "TH","Rectangular"), 
        bandwidth = "variable" , step1.c = FALSE,
        control = list(), Durbin = FALSE)

Value

A list object of class sphet

coefficients: Generalized Spatial two stage least squares coefficient estimates of $\delta$ and GM estimator for $\rho$.
var: variance-covariance matrix of the estimated coefficients
s2: GS2SLS residuals variance
residuals: GS2SLS residuals
yhat: difference between GS2SLS residuals and response variable
call: the call used to create this object
model: the model matrix of data
method: 'gs2slshac'

Arguments

formula: a description of the model to be fit
data: an object of class data.frame. An optional data frame containing the variables in the model
listw: an object of class listw, matrix, or Matrix
listw2: an object of class listw, matrix, or Matrix specified only when sarar is true
endog: additional endogenous variables. Default NULL. If not NULL should be specified as a formula with no dependent variable (endog = ~ x1 + x2). Note the ~ before the expression.
instruments: external instruments. Default NULL. If not NULL should be specified as a formula with no dependent variable (instruments = ~ x1 + x2). Note the ~ before the expression.
lag.instr: should the external instruments be spatially lagged?
initial.value: The initial value for $\rho$. It can be either numeric (default is 0.2) or set to 'SAR', in which case the optimization will start from the estimated coefficient of a regression of the 2SLS residuals over their spatial lag (i.e. a spatial AR model)
q: default equal 2, if 1 the only WX is considered in matrix of instruments
model: one of lag, error, sarar, ivhac, or ols. If HAC is TRUE, model should be one of ivhac, or ols.
het: default FALSE: if TRUE uses the methods developed for heteroskedasticity
verbose: print optimization details
na.action: a function which indicates what should happen when the data contains missing values. See lm for details.
HAC: perform the HAC estimator of Kelejian and Prucha, 2007.
distance: an object of class distance created for example by read.gwt2dist The object contains the specification of the distance measure to be employed in the estimation of the VC matrix. See Details.
type: One of c("Epanechnikov","Triangular","Bisquare","Parzen", "QS","TH","Rectangular"). The type of Kernel to be used. default = "Epanechnikov" See Details.
bandwidth: "variable" (default) - or numeric when a fixed bandwidth is specified by the user.
step1.c: Should step 1.c from Arraiz et al. 2012 be performed?
control: A list of control arguments. See nlminb
Durbin: Should (some of) the regressors be lagged? Default FALSE. If not FALSE should be specified as a formula with no dependent variable (Durbin = ~ x1 + x2) or set to TRUE. See details.

Author

Gianfranco Piras gpiras@mac.com

Details

The procedure consists of two steps alternating GM and IV estimators. Each step consists of sub-steps. In step one $\delta = [\beta',\lambda]'$ is estimated by 2SLS. The 2SLS residuals are first employed to obtain an consistent GM estimator of $\rho$.

In step two, the spatial Cochrane-Orcutt transformed model is estimated by 2SLS. This corresponds to a GS2SLS procedure. The GS2SLS residuals are used to obtain a consistent and efficient GM estimator for $\rho$.

The initial value for the optimization in step 1b is taken to be initial.value. The initial value for the optimization of step 2b is the optimal parameter of step 1b.

Internally, the object of class listw is transformed into a Matrix using the function listw2dgCMatrix.

For the HAC estimator (Kelejian and Prucha, 2007), there are four possibilities:

A model with only Wy
A model with Wy and additional endogenous
Additional endogenous variables but no Wy
No additional endogenous variables (A linear model with HAC estimation)

In the first two cases, the model should be "ivhac", in the last two cases, the model should be "ols".

Furthermore, the default sets the bandwidth for each observation to the maximum distance for that observation (i.e. the max of each element of the list of distances).

Six different kernel functions are implemented:

'Epanechnikov': $K(z) = 1-z^2$
'Triangular': $K(z) = 1-z$
'Bisquare': $K(z) = (1-z^2)^2$
'Parzen': $K(z) = 1-6z^2+6 |z|^3$ if $z \leq 0.5$ and $ K(z) = 2(1-|z|)^3$ if $0.5 < z \leq 1$
'TH' (Tukey - Hanning): $ K(z) = \frac{1+ \cos(\pi z)}{2}$
'Rectangular': $ K(z) = 1$
'QS' (Quadratic Spectral): $K(z) = \frac{25}{12\pi^2z^2} (\frac{\sin(6\pi z)/5)}{6\pi z/5} - \cos(6\pi z)/5)$).

If the kernel type is not one of the six implemented, the function will terminate with an error message.

References

Arraiz, I. and Drukker, M.D. and Kelejian, H.H. and Prucha, I.R. (2010) A spatial Cliff-Ord-type Model with Heteroskedastic Innovations: Small and Large Sample Results, Journal of Regional Sciences, 50, pages 592--614.

Drukker, D.M. and Egger, P. and Prucha, I.R. (2013) On Two-step Estimation of a Spatial Auto regressive Model with Autoregressive Disturbances and Endogenous Regressors, Econometric Review, 32, pages 686--733.

Kelejian, H.H. and Prucha, I.R. (2010) Specification and Estimation of Spatial Autoregressive Models with Autoregressive and Heteroskedastic Disturbances, Journal of Econometrics, 157, pages 53--67.

Kelejian, H.H. and Prucha, I.R. (1999) A Generalized Moments Estimator for the Autoregressive Parameter in a Spatial Model, International Economic Review, 40, pages 509--533.

Kelejian, H.H. and Prucha, I.R. (1998) A Generalized Spatial Two Stage Least Square Procedure for Estimating a Spatial Autoregressive Model with Autoregressive Disturbances, Journal of Real Estate Finance and Economics, 17, pages 99--121.

Gianfranco Piras (2010). sphet: Spatial Models with Heteroskedastic Innovations in R. Journal of Statistical Software, 35(1), 1-21. tools:::Rd_expr_doi("10.18637/jss.v035.i01").

Roger Bivand, Gianfranco Piras (2015). Comparing Implementations of Estimation Methods for Spatial Econometrics. Journal of Statistical Software, 63(18), 1-36. tools:::Rd_expr_doi("10.18637/jss.v063.i18").

Gianfranco Piras, Paolo Postiglione (2022). A deeper look at impacts in spatial Durbin model with sphet. Geographical Analysis, 54(3), 664-684.

Examples

Run this code

data(columbus, package="spdep")
listw <- spdep::nb2listw(col.gal.nb)
res <- spreg(CRIME ~ HOVAL + INC, data = columbus , listw = listw,
             het = TRUE, verbose = FALSE, model = "sarar")
summary(res)
Effects <- impacts(res, listw = listw, R = 1000)

library(spdep)
data("baltimore", package = "spData")
mat <- nb2listw(knn2nb(knearneigh(cbind(baltimore$X,baltimore$Y), 3)))

knb10 <- knn2nb(knearneigh(cbind(baltimore$X,baltimore$Y), k=5))
dists <- nbdists(knb10, cbind(baltimore$X,baltimore$Y))
k10lw <- nb2listw(knb10, glist=dists, style="B")
class(k10lw) <- "distance"

# OLS MODEL
res <- spreg(PRICE ~ NROOM  +AGE, data = baltimore, listw = mat, 
verbose = FALSE, model = "ols", Durbin = TRUE, HAC = TRUE, 
distance = k10lw, type = "Triangular")
summary(res)

# note model = "ols" but with endogenous variables 
res <- spreg(PRICE ~ NROOM  +AGE, data = baltimore, listw = mat, 
verbose = FALSE, model = "ols", Durbin = TRUE, HAC = TRUE, 
distance = k10lw, type = "Triangular", endog = ~SQFT, 
instruments = ~GAR + PATIO)
summary(res)

# ERROR MODEL
res <- spreg(PRICE ~ NROOM  +AGE, data = baltimore, listw = mat, 
het = TRUE, verbose = FALSE, model = "error", Durbin = FALSE)
summary(res)

res <- spreg(PRICE ~ NROOM + AGE + SQFT + NBATH, data = baltimore, listw = mat, 
het = TRUE, verbose = FALSE, model = "error", Durbin = TRUE)
summary(res)

res <- spreg(PRICE ~ NROOM + AGE + SQFT + NBATH, data = baltimore, listw = mat, 
het = TRUE, verbose = FALSE, model = "error", Durbin = ~SQFT + NBATH)
summary(res)

res <- spreg(PRICE ~ NROOM + AGE +  NBATH, data = baltimore, listw = mat, 
het = TRUE, verbose = FALSE, model = "error", Durbin = ~SQFT + NBATH)
summary(res)

res <- spreg(PRICE ~ NROOM + AGE, data = baltimore, listw = mat, 
het = TRUE, verbose = FALSE, model = "error", Durbin = ~SQFT + NBATH)
summary(res)

res <- spreg(PRICE ~ NROOM + AGE -1, data = baltimore, listw = mat, 
het = TRUE, verbose = FALSE, model = "error", Durbin = ~SQFT + NBATH)
summary(res)

# LAG MODEL
res <- spreg(PRICE ~ NROOM  +AGE, data = baltimore, listw = mat, 
het = TRUE, verbose = FALSE, model = "lag", Durbin = FALSE)
summary(res)

res <- spreg(PRICE ~ NROOM + AGE + SQFT + NBATH, data = baltimore, listw = mat, 
het = TRUE, verbose = FALSE, model = "lag", Durbin = TRUE)
summary(res)

res <- spreg(PRICE ~ NROOM + AGE + SQFT + NBATH, data = baltimore, listw = mat, 
het = TRUE, verbose = FALSE, model = "lag", Durbin = ~SQFT + NBATH)
summary(res)

res <- spreg(PRICE ~ NROOM + AGE +  NBATH, data = baltimore, listw = mat, 
het = TRUE, verbose = FALSE, model = "lag", Durbin = ~SQFT + NBATH)
summary(res)

res <- spreg(PRICE ~ NROOM + AGE, data = baltimore, listw = mat, 
het = TRUE, verbose = FALSE, model = "lag", Durbin = ~SQFT + NBATH)
summary(res)

res <- spreg(PRICE ~ NROOM + AGE -1, data = baltimore, listw = mat, 
het = TRUE, verbose = FALSE, model = "lag", Durbin = ~SQFT + NBATH)
summary(res)


# IVHAC MODEL
res <- spreg(PRICE ~ NROOM  +AGE, data = baltimore, listw = mat, 
het = TRUE, verbose = FALSE, model = "ivhac", Durbin = FALSE,
HAC = TRUE, distance = k10lw, type = "Triangular", endog = ~SQFT, 
instruments = ~GAR + PATIO)

# SARAR MODEL
res <- spreg(PRICE ~ NROOM  +AGE, data = baltimore, listw = mat, 
het = TRUE, verbose = FALSE, model = "sarar", Durbin = FALSE, q = 1)
summary(res)

res <- spreg(PRICE ~ NROOM + AGE + SQFT + NBATH, data = baltimore, listw = mat, 
het = TRUE, verbose = FALSE, model = "sarar", Durbin = TRUE, q = 1)
summary(res)

res <- spreg(PRICE ~ NROOM + AGE + SQFT + NBATH, data = baltimore, listw = mat, 
het = TRUE, verbose = FALSE, model = "sarar", Durbin = ~SQFT + NBATH, q =2)
summary(res)

res <- spreg(PRICE ~ NROOM + AGE +  NBATH, data = baltimore, listw = mat, 
het = TRUE, verbose = FALSE, model = "sarar", Durbin = ~SQFT + NBATH)
summary(res)

res <- spreg(PRICE ~ NROOM + AGE, data = baltimore, listw = mat, 
het = TRUE, verbose = FALSE, model = "sarar", Durbin = ~SQFT + NBATH)
summary(res)

res <- spreg(PRICE ~ NROOM + AGE -1, data = baltimore, listw = mat, 
het = TRUE, verbose = FALSE, model = "sarar", Durbin = ~SQFT + NBATH)
summary(res)

summary(res)

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