Assume the following simple regression model, Y = β0 + β1X + ϵ ϵ ∼ N(0, σ^2 ) Now run the following R-code to generate values of σ^2 = sig2, β1 = beta1 and β0 = beta0. Simulate the parameters using the following codes: Code: # Simulation ## set.seed("12345") beta0 <- rnorm(1, mean = 0, sd = 1) ## The true beta0 beta1 <- runif(n = 1, min = 1, max = 3) ## The true beta1 sig2 <- rchisq(n = 1, df = 25) ## The true value of the error variance sigmaˆ2 ## Multiple simulation will require loops ## nsample <- 10 ## Sample size n.sim <- 100 ## The number of simulations sigX <- 0.2 ## The variances of X # # Simulate the predictor variable ## X <- rnorm(nsample, mean = 0, sd = sqrt(sigX)) Q1 Fix the sample size nsample = 10 . Here, the values of X are fixed. You just need to generate ϵ and Y . Execute 100 simulations (i.e., n.sim = 100). For each simulation, estimate the regression coefficients (β0, β1) and the error variance (σ 2 ). Calculate the mean of the estimates from the different simulations. What did you expect the mean to be? Plot the histogram of each of the regression parameter estimates from (b). Explain the pattern of the distributions. Obtain the variance of the regression parameter estimator (i.e., βˆ 0 and βˆ 1) from the simulations. That is, calculate the sample variances of the regression parameter estimates from the 100 simulations. Is this variance approximately equal to the true variances of the regression parameter estimates? Construct the 95% t and z confidence intervals for β0 and β1 during every simulation. What is the proportion of the intervals for each method containing the true value of the parameters? Is this consistent with the definition of confidence interval? Next, what differences do you observe in the t and z confidence intervals? What effect does increasing the number of simulations from 100 have on the confidence intervals? For steps (a)-(d) the sample size was fixed at 10. Start increasing the sample size (e.g., 20, 50, 100) and run steps (a)-(d). Explain what happens to the mean, variance and distribution of the estimators as the sample size increases. Choose the largest sample size you have used in step (f). Fix the sample size to that and start changing the error variance (sig2). You can increase and decrease the value of the error variance. For each value of error variance execute steps (a) - (d). Explain what happens to the mean, variance and distribution of the estimates as the error variance changes.

Assume the following simple regression model, Y = β0 + β1X + ϵ ϵ ∼ N(0, σ^2 ) Now run the following R-code to generate values of σ^2 = sig2, β1 = beta1 and β0 = beta0. Simulate the parameters using the following codes: Code: # Simulation ## set.seed("12345") beta0 <- rnorm(1, mean = 0, sd = 1) ## The true beta0 beta1 <- runif(n = 1, min = 1, max = 3) ## The true beta1 sig2 <- rchisq(n = 1, df = 25) ## The true value of the error variance sigmaˆ2 ## Multiple simulation will require loops ## nsample <- 10 ## Sample size n.sim <- 100 ## The number of simulations sigX <- 0.2 ## The variances of X # # Simulate the predictor variable ## X <- rnorm(nsample, mean = 0, sd = sqrt(sigX)) Q1 Fix the sample size nsample = 10 . Here, the values of X are fixed. You just need to generate ϵ and Y . Execute 100 simulations (i.e., n.sim = 100). For each simulation, estimate the regression coefficients (β0, β1) and the error variance (σ 2 ). Calculate the mean of the estimates from the different simulations. What did you expect the mean to be? Plot the histogram of each of the regression parameter estimates from (b). Explain the pattern of the distributions. Obtain the variance of the regression parameter estimator (i.e., βˆ 0 and βˆ 1) from the simulations. That is, calculate the sample variances of the regression parameter estimates from the 100 simulations. Is this variance approximately equal to the true variances of the regression parameter estimates? Construct the 95% t and z confidence intervals for β0 and β1 during every simulation. What is the proportion of the intervals for each method containing the true value of the parameters? Is this consistent with the definition of confidence interval? Next, what differences do you observe in the t and z confidence intervals? What effect does increasing the number of simulations from 100 have on the confidence intervals? For steps (a)-(d) the sample size was fixed at 10. Start increasing the sample size (e.g., 20, 50, 100) and run steps (a)-(d). Explain what happens to the mean, variance and distribution of the estimators as the sample size increases. Choose the largest sample size you have used in step (f). Fix the sample size to that and start changing the error variance (sig2). You can increase and decrease the value of the error variance. For each value of error variance execute steps (a) - (d). Explain what happens to the mean, variance and distribution of the estimates as the error variance changes.

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter12: Review Of Calculus And Probability

Section12.5: Random Variables, Mean, Variance, And Covariance

Problem 5P

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Question

Assume the following simple regression model,

Y = β0 + β1X + ϵ

ϵ ∼ N(0, σ^2 )

Now run the following R-code to generate values of σ^2 = sig2, β1 = beta1 and β0 = beta0. Simulate the parameters using the following codes:

Code:

# Simulation ##

set.seed("12345")

beta0 <- rnorm(1, mean = 0, sd = 1) ## The true beta0

beta1 <- runif(n = 1, min = 1, max = 3) ## The true beta1

sig2 <- rchisq(n = 1, df = 25) ## The true value of the error variance sigmaˆ2

## Multiple simulation will require loops ##

nsample <- 10 ## Sample size n.sim <- 100 ## The number of simulations

sigX <- 0.2 ## The variances of X #

# Simulate the predictor variable ##

X <- rnorm(nsample, mean = 0, sd = sqrt(sigX))

Fix the sample size nsample = 10 . Here, the values of X are fixed. You just need to generate ϵ and Y . Execute 100 simulations (i.e., n.sim = 100). For each simulation, estimate the regression coefficients (β0, β1) and the error variance (σ 2 ). Calculate the mean of the estimates from the different simulations. What did you expect the mean to be?
Plot the histogram of each of the regression parameter estimates from (b). Explain the pattern of the distributions.
Obtain the variance of the regression parameter estimator (i.e., βˆ 0 and βˆ 1) from the simulations. That is, calculate the sample variances of the regression parameter estimates from the 100 simulations. Is this variance approximately equal to the true variances of the regression parameter estimates?
Construct the 95% t and z confidence intervals for β0 and β1 during every simulation. What is the proportion of the intervals for each method containing the true value of the parameters? Is this consistent with the definition of confidence interval? Next, what differences do you observe in the t and z confidence intervals? What effect does increasing the number of simulations from 100 have on the confidence intervals?
For steps (a)-(d) the sample size was fixed at 10. Start increasing the sample size (e.g., 20, 50, 100) and run steps (a)-(d). Explain what happens to the mean, variance and distribution of the estimators as the sample size increases.
Choose the largest sample size you have used in step (f). Fix the sample size to that and start changing the error variance (sig2). You can increase and decrease the value of the error variance. For each value of error variance execute steps (a) - (d). Explain what happens to the mean, variance and distribution of the estimates as the error variance changes.

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