ivregimes {hspm}

Estimation of regime models with endogenous variables

Description

The function ivregimes deals with the estimation of regime models. Most of the times the variable identifying the regimes reveals some spatial aspects of the data (e.g., administrative boundaries). The model includes exogenous as well as endogenous variables among the regressors.

Usage

ivregimes(formula, data, rgv = NULL, vc = c("homoskedastic", "robust", "OGMM"))

Arguments

formula

a symbolic description of the model of the form y ~ x_f | x_v | h_f | h_v where y is the dependent variable, x_f are the regressors that do not vary by regimes, x_v are the regressors that vary by regimes, h_f are the fixed instruments and h_v are the instruments that vary by regimes.

data

the data of class data.frame.

rgv

an object of class formula to identify the regime variables

vc

one of c("homoskedastic", "robust", "OGMM"). If "OGMM" an optimal weighted GMM is used to estimate the VC matrix.

Details

The basic (non spatial) model with endogenous variables can be written in a general way as:

y = \begin{bmatrix} X_1& 0 \\ 0 & X_2 \\ \end{bmatrix} \begin{bmatrix} \beta_1 \\ \beta_2 \\ \end{bmatrix} + X\beta + \begin{bmatrix} Y_1& 0 \\ 0 & Y_2 \\ \end{bmatrix} \begin{bmatrix} \pi_1 \\ \pi_2 \\ \end{bmatrix} + Y\pi + \varepsilon

where y = [y_1^\prime,y_2^\prime]^\prime, and the n_1 \times 1 vector y_1 contains the observations on the dependent variable for the first regime, and the n_2 \times 1 vector y_2 (with n_1 + n_2 = n) contains the observations on the dependent variable for the second regime. The n_1 \times k matrix X_1 and the n_2 \times k matrix X_2 are blocks of a block diagonal matrix, the vectors of parameters \beta_1 and \beta_2 have dimension k_1 \times 1 and k_2 \times 1, respectively, X is the n \times p matrix of regressors that do not vary by regime, \beta is a p\times 1 vector of parameters. The three matrices Y_1 (n_1 \times q), Y_2 (n_2 \times q) and Y (n \times r) with corresponding vectors of parameters \pi_1, \pi_2 and \pi, contain the endogenous variables. Finally, \varepsilon = [\varepsilon_1^\prime,\varepsilon_2^\prime]^\prime is the n\times 1 vector of innovations. The model is estimated by two stage least square. In particular:

• If vc = "homoskedastic", the variance-covariance matrix is estimated by \sigma^2(\hat Z^\prime \hat Z)^{-1}, where \hat Z= PZ, P= H(H^\prime H)^{-1}H^\prime, H is the matrix of instruments, and Z is the matrix of all exogenous and endogenous variables in the model.

• If vc = "robust", the variance-covariance matrix is estimated by (\hat Z^\prime \hat Z)^{-1}(\hat Z^\prime \hat\Sigma \hat Z) (\hat Z^\prime \hat Z)^{-1}, where \hat\Sigma is a diagonal matrix with diagonal elements \hat\sigma_i, for i=1,...,n.

• Finally, if vc = "OGMM", the model is estimated in two steps. In the first step, the model is estimated by 2SLS yielding the residuals \hat \varepsilon. With the residuals, the diagonal matrix \hat \Sigma is estimated and is used to construct the matrix \hat S = H^\prime \hat \Sigma H. Then \eta_{OWGMM}=(Z^\prime H\hat S^{-1}H^\prime Z)^{-1}Z^\prime H\hat S^{-1}H^\prime y, where \eta_{OWGMM} is the vector of all the parameters in the model, The variance-covariance matrix is: n(Z^\prime H\hat S^{-1}H^\prime Z)^{-1}.

Value

An object of class ivregimes. A list of five elements. The first element of the list contains the estimation results. The other elements are needed for printing the results.

Author(s)

Gianfranco Piras and Mauricio Sarrias

Examples

data("natreg")
form   <- HR90  ~ 0 | MA90 + PS90 + RD90 + UE90 | 0 | MA90 + PS90 + RD90 + FH90 + FP89 + GI89
split  <- ~ REGIONS
mod <- ivregimes(formula = form, data = natreg, rgv = split, vc = "robust")
summary(mod)
mod1 <- ivregimes(formula = form, data = natreg, rgv = split, vc = "OGMM")
summary(mod1)
form1   <- HR90  ~ MA90 + PS90 |  RD90 + UE90 -1 | MA90 + PS90 | RD90 + FH90 + FP89 + GI89 -1
mod2 <- ivregimes(formula = form1, data = natreg, rgv = split, vc = "homoskedastic")
summary(mod2)