KoulMde (version 3.2.1)

Koul2StageMde: Two-stage minimum distance estimation in linear regression model with autoregressive error.

Description

Estimates both regression and autoregressive coefficients in the model \(Y=X\beta + \epsilon\) where \(\epsilon\) is autoregressive process of known order q

Usage

Koul2StageMde(
  Y,
  X,
  D,
  b0,
  RegIntMeasure,
  AR_Order,
  ArIntMeasure,
  TuningConst = 1.345
)

Arguments

Y

- Vector of response variables in linear regression model.

X

- Design matrix of explanatory variables in linear regression model.

D

- Weight Matrix. Dimension of D should match that of X. Default value is XA where A=(X'X)^(-1/2).

b0

- Initial value for beta.

RegIntMeasure

- Symmetric and \(\sigma\)-finite measure used for estimating \(\beta\): Lebesgue, Degenerate or Robust.

AR_Order

- Order of the autoregressive error.

ArIntMeasure

- Symmetric and \(\sigma\)-finite measure used for estimating autoregressive coefficients of the error: Lebesgue, Degenerate or Robust.

TuningConst

- Used only for Robust measure.

Value

MDE1stage - The list of the first stage minimum distance estimation result. It contains betahat1stage, residual1stage, and rho1stage.

• betahat1stage - The first stage minimum distance estimators of regression coefficients.

• residual1stage - Residuals after the first stage minimum distance estimation.

• rho1stage - The first stage minimum distance estimators of autoregressive coefficients of the error.

MDE2stage - The list of the second stage minimum distance estimation result. It contains betahat2stage, residual2stage, and rho2stage.

• betahat2stage - The second stage minimum distance estimators of regression coefficients.

• residual2stage - Residuals after the second stage minimum distance estimation.

• rho2stage - The second stage minimum distance estimators of autoregressive coefficients of the error.

References

[1] Kim, J. (2018). A fast algorithm for the coordinate-wise minimum distance estimation. J. Stat. Comput. Simul., 3: 482 - 497

[2] Kim, J. (2020). Minimum distance estimation in linear regression model with strong mixing errors. Commun. Stat. - Theory Methods., 49(6): 1475 - 1494

[3] Koul, H. L (1985). Minimum distance estimation in linear regression with unknown error distributions. Statist. Probab. Lett., 3: 1-8.

[4] Koul, H. L (1986). Minimum distance estimation and goodness-of-fit tests in first-order autoregression. Ann. Statist., 14 1194-1213.

[5] Koul, H. L (2002). Weighted empirical process in nonlinear dynamic models. Springer, Berlin, Vol. 166

See Also

KoulArMde() and KoulLrMde()

Examples

# NOT RUN {
####################
n <- 10
p <- 3
X <- matrix(runif(n*p, 0,50), nrow=n, ncol=p)  #### Generate n-by-p design matrix X
beta <- c(-2, 0.3, 1.5)                        #### Generate true beta = (-2, 0.3, 1.5)'
rho  <- 0.4                                    #### True rho = 0.4
eps <- vector(length=n)
xi <- rnorm(n, 0,1)                            #### Generate innovation from N(0,1)
                                              #### Generate autoregressive process of order 1
for(i in 1:n){
 if(i==1){eps[i] <- xi[i]}
 else{eps[i] <- rho*eps[i-1] + xi[i]}
}
Y <- X%*%beta + eps
#####################
D <- "default"                                  #### Use the default weight matrix
b0 <- solve(t(X)%*%X)%*%(t(X)%*%Y)              #### Set initial value for beta

IntMeasure <- "Lebesgue"                                ##### Define Lebesgue measure
MDEResult <- Koul2StageMde(Y,X, "default", b0, IntMeasure, 1, IntMeasure, TuningConst = 1.345)
MDE1stageResult <- MDEResult[[1]]
MDE2stageResult <- MDEResult[[2]]

beta1 <- MDE1stageResult$betahat1stage
residual1 <- MDE1stageResult$residual1stage
rho1 <- MDE1stageResult$rhohat1stage

beta2 <- MDE2stageResult$betahat2stage
residual2 <- MDE1stageResult$residual2stage
rho2 <- MDE2stageResult$rhohat2stage
# }