1. If a true model of simple linear regression reads: yi −y ̄ = β0 +β1(xi −x ̄)+εi for i = 1, 2, · · · , n, showβ0 =0andβˆ0 =0. (1pt)
(hint: use the formula of estimator βˆ0 = y ̄ − βˆ1x ̄.)
1. If a true model of simple linear regression reads: yi −y ̄ = β0 +β1(xi...
Consider the linear regression model Yi = β0 + β1 Xi + ui Yi is the ______________, the ______________ or simply the ______________. Xi is the ______________, the ______________ or simply the ______________. is the population regression line, or the population regression function. There are two ______________ in the function (β0 & β1 ). β0 is is the ______________ of the population regression line; β1is is the ______________ of the population regression line; and ui is the ______________. A. Coefficients...
Consider the linear probability model Yi = β0 + β1Xi + ui. Assume E(ui|Xi)=0. Which of the following statements are true? Question 5 options: The predicted value of the dependent variable can be greater than 1 or less than 0. Thus, the OLS estimator of β1 is biased. The predicted value of the dependent variable will always be between 0 and 1. Thus, the OLS estimator of β1 is unbiased. The predicted value of the dependent variable will always be...
We run the following linear regression model in Excel (or any other softwares) Yi = β0 + β1Xi + β2Wi + εi , where i = 1, 2, . . . , 100. The results suggest that the slope on Xi is 97.28 with t-statistics 0.91, and the slope on Wi is 15.81 with t-statistics 11.39. What does it tell us?
4. In linear regression modeling, the regression line is sometimes known to go through the origin (0, 0), and x = 0 is within the scope of the model. In such case, we add the assumption that β0 = 0, so the regression model reduces to: Yi = β Xi + εi . (The assumptions about ε remain the same). (Note: Since now there is only one parameter, we simply denote it β instead of β1 ). (a) Derive the...
Consider a simple linear regression model with nonstochastic regressor: Yi = β1 + β2Xi + ui. 1. [3 points] What are the assumptions of this model so that the OLS estimators are BLUE (best linear unbiased estimates)? 2. [4 points] Let βˆ and βˆ be the OLS estimators of β and β . Derive βˆ and βˆ. 12 1212 3. [2 points] Show that βˆ is an unbiased estimator of β .22
Consider the regression model: yi = β0 + β1Xi + εi for…. i = 1 Where the dummy variable (0 = failure and 1 = success). Suppose that the data set contains n1 failure and n2 successes (and that n1+n2 = n) Obtain the X^T(X) matrix Obtain the X^T(Y) matrix Obtain the least square estimate b
1. Consider the following simple regression model: y = β0 + β1x1 + u (1) and the following multiple regression model: y = β0 + β1x1 + β2x2 + u (2), where x1 is the variable of primary interest to explain y. Which of the following statements is correct? a. When drawing ceteris paribus conclusions about how x1 affects y, with model (1), we must assume that x2, and all other factors contained in u, are uncorrelated with x1. b....
(Do this problem without using R) Consider the simple linear regression model y =β0 + β1x + ε, where the errors are independent and normally distributed, with mean zero and constant variance σ2. Suppose we observe 4 observations x = (1, 1, −1, −1) and y = (5, 3, 4, 0). (a) Fit the simple linear regression model to this data and report the fitted regression line. (b) Carry out a test of hypotheses using α = 0.05 to determine...
2. Consider a simple linear regression model for a response variable Yi, a single predictor variable ri, i-1,... , n, and having Gaussian (i.e. normally distributed) errors Ý,-BzitEj, Ejį.i.d. N(0, σ2) This model is often called "regression through the origin" since E(Yi) 0 if xi 0 (a) Write down the likelihood function for the parameters β and σ2 (b) Find the MLEs for β and σ2, explicitly showing that they are unique maximizers of the likelihood function. (Hint: The function...