lmtestspsur: Testing for the presence of spatial effects in Seemingly Unrelated Regressions

Description

The function spsurml reports a collection of Lagrange Multipliers designed to test for the presence of different forms of spatial dependence in a SUR model of the "sim" type. That is, the approach of this function is from 'specific to general'. As said, the model of the null hypothesis is the "sim" model whereas the model of the alternative depends on the effect whose omission we want to test.

The collection of Lagrange Multipliers obtained by lmtestspsur are standard in the literature and take into account the multivariate nature of the SUR model. As a limitation, note that each Multiplier tests for the omission of the same spatial effects in all the cross-sections of the G equations.

Usage

lmtestspsur(
  Form = NULL,
  data = NULL,
  W = NULL,
  X = NULL,
  Y = NULL,
  time = NULL,
  G = NULL,
  N = NULL,
  Tm = NULL,
  print_table = TRUE
)

Arguments

Form

An object created with the package Formula that describes the model to be estimated. This model may contain several responses (explained variables) and a varying number of regressors in each equation.

data

An object of class data.frame or a matrix.

A spatial weighting matrix of order (NxN), assumed to be the same for all equations and time periods.

A data matrix of order (NTmGxp) with the observations of the regressors The number of covariates in the SUR model is p = $sum(p_{g})$ where $p_{g}$ is the number of regressors (including the intercept) in the g-th equation, g = 1,...,G). The specification of X is only necessary if not available a Formula and a data frame. Default = NULL.

A column vector of order (NTmGx1), with the observations of the explained variables. The ordering of the data must be (first) equation, (second) time dimension and (third) Cross-sectional/spatial units. The specification of Y is only necessary if not available a Formula and a data frame. Default = NULL.

time

Time variable.

Number of equations.

Number of cross-section or spatial units

Number of time periods.

print_table

Logical value to print the output. Default = TRUE

Value

A list including:

`stat_names`	Name of Lagrange Multiplier.
`stat`	Value of the corresponding Lagrange Multiplier.
`df`	Degrees of freedom for each Multiplier.

Details

lmtestspsur tests for the omission of spatial effects in the "sim" version of the SUR model:

$$y_{tg} = X_{tg} \beta_{g} + u_{tg}$$ $$E[u_{tg}u_{th}']= \sigma_{gh}I_{N} \quad E[u_{tg}u_{sh}']= 0 \mbox{ if } t ne s$$

where $y_{tg}$ and $u_{tg}$ are (Nx1) vectors, corresponding to the g-th equation and time period t; $X_{tg}$ is the matrix of exogenous variables, of order$(Nxp_{g})$. Moreover, $\beta_{g}$ is an unknown $(p_{g}x1)$ vector of coefficients and $\sigma_{gh}I_{N}$ the covariance between equations g and h, being $\sigma_{gh}$ and scalar and $I_{N}$ the identity matrix of orden N.

The Lagrange Multipliers reported by this function are the followings:

LM-SUR-LAG: Tests for the omission of a spatial lag of the explained variable in the right hand side of the "sim" equation. The model of the alternative is:

$y_{tg} = \lambda_{g}Wy_{tg}+X_{tg} \beta_{g} + u_{tg}$

The null and alternative hypotheses are:

$H_{0}: \lambda_{g}=0 (forall g)$ vs $H_{A}: \lambda_{g} ne 0 (exist g)$
LM-SUR-ERR: Tests for the omission of spatial dependence in the equation of the errors of the "sim" model. The model of the alternative is:

$y_{tg} = X_{tg} \beta_{g} + u_{tg}$; $u_{tg}= \rho_{g}Wu_{tg}+\epsilon_{tg}$

The null and alternative hypotheses are:

$H_{0}: \rho_{g}=0 (forall g)$ vs $H_{A}: \rho_{g} ne 0 (exist g)$
LM-SUR-SARAR: Tests for the simultaneous omission of a spatial lag of the explained variable in the right hand side of the "sim" equation and spatial dependence in the equation of the errors. The model of the alternative is:
$y_{tg} = \lambda_{g}Wy_{tg}+X_{tg} \beta_{g} + u_{tg}$; $u_{tg}= \rho_{g}Wu_{tg}+\epsilon_{tg}$

The null and alternative hypotheses are:

$H_{0}: \lambda_{g}=\rho_{g}=0 (forall g)$ vs $H_{A}: \lambda_{g} ne 0 or \rho_{g} ne 0 (exist g)$
LM*-SUR-SLM and LM*-SUR-SEM: These two test are the robustifyed version of the original, raw Multipliers, LM-SUR-SLM and LM-SUR-SEM, which can be severely oversized if the respective alternative hypothesis is misspeficied (this would be the case if, for example, we are testing for omitted lags of the explained variable whereas the problem is that there is spatial dependence in the errors, or viceversa). The null and alternative hypotheses of both test are totally analogous to their twin non robust Multipliers.

References

Mur, J., L<U+00F3>pez, F., and Herrera, M. (2010). Testing for spatial effects in seemingly unrelated regressions. Spatial Economic Analysis, 5(4), 399-440.
L<U+00F3>pez, F.A., Mur, J., and Angulo, A. (2014). Spatial model selection strategies in a SUR framework. The case of regional productivity in EU. Annals of Regional Science, 53(1), 197-220.

Examples

Run this code

# NOT RUN {
#################################################
######## CROSS SECTION DATA (G>1; Tm=1) # #######
#################################################

#### Example 1: Spatial Phillips-Curve. Anselin (1988, p. 203)
rm(list = ls()) # Clean memory
data("spc")
Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA
LMs <- lmtestspsur(Form = Tformula, data = spc, W = Wspc)

# }
# NOT RUN {
#################################################
######## PANEL DATA (G>1; Tm>1)          ########
#################################################

#### Example 2: Homicides & Socio-Economics (1960-90)
# Homicides and selected socio-economic characteristics for
# continental U.S. counties.
# Data for four decennial census years: 1960, 1970, 1980 and 1990.
# https://geodacenter.github.io/data-and-lab/ncovr/

data("NCOVR")
# With different number of exogenous variables in each equation
Tformula <- HR70 | HR80  | HR90 ~ PS70 + UE70 | PS80 + UE80 +RD80 |
            PS90 + UE90 + RD90 + PO90
LMs <- lmtestspsur(Form = Tformula, data = NCOVR, W = W)
# }
# NOT RUN {
################################################################
######## PANEL DATA: TEMPORAL CORRELATIONS (G=1; Tm>1) ########
################################################################
# }
# NOT RUN {
#### Example 3: NCOVR in panel data form
## Takes less than 1 minute
data("NCOVR")
Year <- as.numeric(kronecker(c(1960,1970,1980,1990),matrix(1,nrow = dim(NCOVR)[1])))
HR <- c(NCOVR$HR60,NCOVR$HR70,NCOVR$HR80,NCOVR$HR90)
PS <- c(NCOVR$PS60,NCOVR$PS70,NCOVR$PS80,NCOVR$PS90)
UE <- c(NCOVR$UE60,NCOVR$UE70,NCOVR$UE80,NCOVR$UE90)
NCOVRpanel <- as.data.frame(cbind(Year,HR,PS,UE))
Tformula <- HR ~ PS + UE
LM_time <- lmtestspsur(Form = Tformula, data = NCOVRpanel, time = Year, W = W)
# }
# NOT RUN {
# }

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