# Two-stage minimum distance estimation in linear regression model with autoregressive error. Estimates both regression and autoregressive coefficients in the model $Y=X\beta + \epsilon$ where $\epsilon$ is autoregressive process of known order `q` ```r 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. ## Returns 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. ## Examples ```r #################### 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 ``` ## 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()

Koul2StageMde function