[SOLVED] CS ## – fig.align=center-

## – fig.align=center-
# Some given data
X_1 <- c(1.9,0.8,1.1,0.1,-0.1,4.4,4.6,1.6,5.5,3.4)X_2 <- c(66, 62, 64, 61, 63, 70, 68, 62, 68, 66)Y <- c(0.7,-1.0,-0.2,-1.2,-0.1,3.4,0.0,0.8,3.7,2.0)dataset <-cbind.data.frame(X_1,X_2,Y)## Compute the OLS estimationmy.lm <- lm(Y ~ X_1 + X_2, data = dataset)## Plot sample regression surfacelibrary(“scatterplot3d”) # library for 3d plotsplot3d <- scatterplot3d(x = X_1, y = X_2, z = Y,angle=33, scale.y=0.8, pch=16,color =”red”, main =”OLS Regression Surface”)plot3d$plane3d(my.lm, lty.box = “solid”, col=gray(.5), draw_polygon=TRUE)## ————————————————————————————————————————set.seed(123)n <- 100 # Sample sizeX <- runif(n, 0, 10) # Relevant X variableX_ir<- runif(n, 5, 20) # Irrelevant X variableerror <- rt(n, df = 10)*10# True errorY <- 1 + 5 * X + error# Y variablelm1 <- summary(lm(Y~X)) # Correct OLS regression lm2 <- summary(lm(Y~X+X_ir))# OLS regression with X_ir lm1$r.squared < lm2$r.squared## —- fig.align=”center”————————————————————————————————-set.seed(2)n<- 100K<- 3X<- matrix(runif(n*(K-1), 2, 10), n, K-1)X<- cbind(1,X)beta <- c(1,5,5)# heteroscedastic errors:sigma<- abs(X[,2] + X[,3])^1.5error<- rnorm(n, mean = 0, sd=sigma)Y<- beta[1]*X[,1] + beta[2]*X[,2] + beta[3]*X[,3] + error##lm_fit <- lm(Y~X -1 )## Caution! By default R computes the standard errors ## assuming homoscedastic errors. This can lead to ## false inferences under heteroscedastic errors.summary(lm_fit)$coefficients library(“sandwich”) # HC robust variance estimation library(“lmtest”)## Robust estimation of the variance of hat{beta}:Var_beta_hat_robust <- sandwich::vcovHC(lm_fit, type=”HC3″)Var_beta_hat_robust ## Corresponding regression-output:lmtest::coeftest(lm_fit, vcov = Var_beta_hat_robust)