[SOLVED] CS ##

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## Function to generate artificial data
myDataGenerator <- function(n, beta){##X <- cbind(rep(1, n),runif(n, -4, 4),runif(n, -5, 5))##eps<- runif(n, -.5 * abs(X[,2]), +.5 * abs(X[,2]))Y<- X %*% beta + epsdata <- data.frame(“Y”=Y,”X_1″=X[,1], “X_2″=X[,2], “X_3″=X[,3])##return(data)}## —- eval=FALSE, echo=FALSE———————————–## library(“fractional”)## moment_fun <- function(m=3, a, b){fractional((b^(m+1)-a^(m+1))/((m+1)*(b-a)))}## moment_fun(m=4, a=-4,b=4)## —- fig.align=”center”, echo=TRUE, fig.width = 8, fig.height = 5, out.width = “1\textwidth”—-set.seed(123)n <- 100# a largish sample sizebeta_true <- c(2,3,4) # true data vector## Mean and variance of the true asymptotic ## normal distribution of beta_hat_2:# true meanbeta_true_2 <- beta_true[2] # true variancevar_true_beta_2 <- (solve(diag(c(1, 16/3, 25/3)))%*% diag(c(4/9, 64/15, 100/27))%*% solve(diag(c(1, 16/3, 25/3))))[2,2]/n## Let’s generate 5000 realizations from beta_hat_2 ## conditionally on X and check whether their ## distribution is close to the true normal distributionrep<- 5000 # MC replicationsbeta_hat_2 <- rep(NA, times=rep)##for(r in 1:rep){MC_data <- myDataGenerator(n= n,beta = beta_true)lm_obj<- lm(Y ~ X_2 + X_3, data = MC_data)beta_hat_2[r] <- coef(lm_obj)[2]}## Compare:## True beta_2 versus average of beta_hat_2 estimatesbeta_true_2round(mean(beta_hat_2), 3)## True variance of beta_hat_2 versus ## empirical variance of beta_hat_2 estimatesround(var_true_beta_2, 5)round(var(beta_hat_2), 5)## True normal distribution of beta_hat_2 versus ## empirical density of beta_hat_2 estimateslibrary(“scales”)curve(expr = dnorm(x, mean = beta_true_2,sd=sqrt(var_true_beta_2)), xlab=””,ylab=””, col=gray(.2), lwd=3, lty=1, xlim=range(beta_hat_2),ylim=c(0,14.1),main=paste0(“n=”,n))lines(density(beta_hat_2, bw = bw.SJ(beta_hat_2)), col=alpha(“blue”,.5), lwd=3)legend(“topleft”, lty=c(1,1), lwd=c(3,3),col=c(gray(.2), alpha(“blue”,.5)), bty=”n”, legend= c(expression(“Theoretical (Asymptotic) Gaussian Density of”~hat(beta)[2]), expression(“Empirical Density Estimation based on MC realizations from”~hat(beta)[2])))## —- fig.align=”center”, echo=TRUE, fig.width = 8, fig.height = 5, out.width = “1\textwidth”—-set.seed(123)n <- 5 # a small sample sizebeta_true <- c(2,3,4) # true data vector## Mean and variance of the true asymptotic ## normal distribution of beta_hat_2:# true meanbeta_true_2 <- beta_true[2] # true variancevar_true_beta_2 <- (solve(diag(c(1, 16/3, 25/3)))%*% diag(c(4/9, 64/15, 100/27))%*% solve(diag(c(1, 16/3, 25/3))))[2,2]/n## Let’s generate 5000 realizations from beta_hat_2 ## conditionally on X and check whether their ## distribution is close to the true normal distributionrep<- 5000 # MC replicationsbeta_hat_2 <- rep(NA, times=rep)##for(r in 1:rep){MC_data <- myDataGenerator(n= n,beta = beta_true)lm_obj<- lm(Y ~ X_2 + X_3, data = MC_data)beta_hat_2[r] <- coef(lm_obj)[2]}## Compare:## True beta_2 versus average of beta_hat_2 estimatesbeta_true_2round(mean(beta_hat_2), 3)## True variance of beta_hat_2 versus ## empirical variance of beta_hat_2 estimatesround(var_true_beta_2, 5)round(var(beta_hat_2), 5)## True normal distribution of beta_hat_2 versus ## empirical density of beta_hat_2 estimateslibrary(“scales”)curve(expr = dnorm(x, mean = beta_true_2,sd=sqrt(var_true_beta_2)), xlab=””,ylab=””, col=gray(.2), lwd=3, lty=1, xlim=c(2,4), ylim=c(0,3),main=paste0(“n=”,n))lines(density(beta_hat_2, bw = bw.SJ(beta_hat_2)), col=alpha(“blue”,.5), lwd=3)legend(“topleft”, lty=c(1,1), lwd=c(3,3),col=c(gray(.2), alpha(“blue”,.5)), bty=”n”, legend= c(expression(“Theoretical (Asymptotic) Gaussian Density of”~hat(beta)[2]), expression(“Empirical Density Estimation based on MC realizations from”~hat(beta)[2])))## ————————————————————–suppressMessages(library(“car”)) # for linearHyothesis()# ?linearHypothesislibrary(“sandwich”)## Generate dataMC_data <- myDataGenerator(n= 100,beta = beta_true)## Estimate the linear regression model parameterslm_obj <- lm(Y ~ X_2 + X_3, data = MC_data)vcovHC3_mat <- sandwich::vcovHC(lm_obj, type=”HC3″)## Option 1:car::linearHypothesis(model = lm_obj, hypothesis.matrix = c(“X_2=3”, “X_3=5”), vcov=vcovHC3_mat)## Option 2:R <- rbind(c(0,1,0), c(0,0,1))car::linearHypothesis(model = lm_obj, hypothesis.matrix = R, rhs = c(3,5),vcov=vcovHC3_mat)## ————————————————————–# Load librarieslibrary(“lmtest”) # for coeftest()library(“sandwich”) # for vcovHC()## Generate datan <- 100MC_data <- myDataGenerator(n= n,beta = beta_true)## Estimate the linear regression model parameterslm_obj <- lm(Y ~ X_2 + X_3, data = MC_data)## Robust t test## Robust standard error for hat{beta}_3:SE_rob <- sqrt(vcovHC(lm_obj, type = “HC3”)[3,3])## hypothetical (H0) value of beta_3:beta_3_H0 <- 5## estimate for beta_3:beta_3_hat <- coef(lm_obj)[3]## robust t-test statistict_test_stat <- (beta_3_hat – beta_3_H0)/SE_rob## p-valueK <- length(coef(lm_obj))##p_value <- 2 * min( pt(q = t_test_stat, df = n – K),1- pt(q = t_test_stat, df = n – K))p_value