Question 1: Which of the following would generally cause the variance of the OLS estimator of...
Question 1: Which of the following would generally cause the variance of the OLS estimator of the slope in a regression model to be larger?
1) smaller variance of the error term
2) a larger sample size
3) smaller variance of the independent variable
4) larger variance of Xi
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Question 2: Which of the following is the best description of the sampling distribution of the OLS estimator under the least squares assumptions?
1) it is a Student's t distribution
2) in large samples it is approximately a normal distribution
3) it is a normal distribution
4) in large samples it is approximately a Student's t distribution
Homework Answers
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Question 1: Which of the following would generally cause the
variance of the OLS estimator of...
Which of the following is not one of the least squares assumptions used in Stock and...
Which of the following is not one of the least squares assumptions used in Stock and Watson to show that the OLS estimators are unbiased and consistent and have approximately a normal distribution in large samples? 1) large outliers are unlikely 2) the error term is homoskedastic, i.e., Var(ui ∣ X=x) does not depend on x 3) the sample (Xi,Yi),i=1,…,n constitutes an i.i.d. random sample from the population joint distribution of X and Y 4) the conditional mean of the...
7. When we impose a restriction on the OLS estimation that the intercept estimator is zero, we ca...
7. When we impose a restriction on the OLS estimation that the intercept estimator is zero, we call it regression through the origin. Consider a population model Y- Au + βίχ + u and we estimate an OLS regression model through the origin: Y-β¡XHi (note that the true intercept parameter Bo is not necessarily zero). (i) Under assumptions SLR.1-SLR.4, either use the method of moments or minimize the SSR to show that the βί-1-- ie1 (2) Find E(%) in terms...
0/1 pts Question7 To obtain the slope estimator using the least squares principle, you divide the...
0/1 pts Question7 To obtain the slope estimator using the least squares principle, you divide the sample covariance of X and Y by the sample variance of X sample covariance of X and Y by the sample variance of Y sample variance of X by the sample covariance of X and Y sample variance of X by the sample variance of Y 0/1 pts Incorrect Question8 Question 8 The standard error of the regression (SER) is defined as follows 1-R2...
4. The Gauss-Markov Theorem says that when Assumptions 1-5 of the linear regression model are satisfied:...
4. The Gauss-Markov Theorem says that when Assumptions 1-5 of the linear regression model are satisfied: (a) The least squares estimator is unbiased (b) The least squares estimator has the smallest variance of all linear estimators (c) The least squares estimator has an approximately normal sampling distribution (d) The least squares estimator is consistent (e) None of the above
Question 19 3 pts The ordinary least squares estimator of a slope coefficient is unbiased means...
Question 19 3 pts The ordinary least squares estimator of a slope coefficient is unbiased means if repeated samples of the same size are taken, on average the OLS estimates will be equal to the true slope parameter O the mean of the sampling distribution of the slope coefficient is zero. O the estimated slope coefficient will always be equal to the true parameter value. the estimated slope coefficient will get closer to the true parameter value as the size...
Question 3 True/False/Explain 1. The variance of the OLS estimator of the coefficient of a certain...
Question 3 True/False/Explain 1. The variance of the OLS estimator of the coefficient of a certain variable X; in a regression is higher, the higher is the degree of collinearity between that variable and the other regressors included in the model 2. Suppose that we estimated the following hourly wage equation n(wage) .092educ + .0041ехреr + (007) 022tenure 284 (0017) (.104) (003) where the numbers in parentheses are estimated standard errors. Assuming that the classical normal linear regression model holds,...
1. For the general multivariate regression model, the least squares estimator is given by Show that...
1. For the general multivariate regression model, the least squares estimator is given by Show that for the slope estimator in the simple (bivariate) regression case, this is equivalent to ja! įs] 2. In the general multivariate regression model, the variance of the least squares estimator, Va( is σ2(XX)". Show that for the simple regression case, this is equivalent to a. Var(B- b. Var(B)o i, Σ (Xi-X) 2 C. What is the covariance between β° and β,?
Question 2 (10 points) You are given the following model y-put ei. Consider two alternative estimators of β, b2xvix? and b = Zy/X 1. Which estimator would you choose and why if the model satisfies al...
Question 2 (10 points) You are given the following model y-put ei. Consider two alternative estimators of β, b2xvix? and b = Zy/X 1. Which estimator would you choose and why if the model satisfies all the assumptions of classical regression? Prove your results. (4 points) 2. Now suppose that var(y)-hxi, where h is a positive constant (a) Obtain the correct variance of the OLS estimator. (2 points) (b) Show that the BLU estimator is now 6. Derive its variance....
Question 1 Consider the simple regression model (only one covariate): y= BoB1 u Let B1 be the OLS estimator of B1. a) W...
Question 1 Consider the simple regression model (only one covariate): y= BoB1 u Let B1 be the OLS estimator of B1. a) What are the six assumptions needed for B1 to be unbiased, have a simple expression for its variance, and have normal distribution? (3 points) b) Under Assumptions 1-6, derive the distribution of B1 conditional on x\,..., xn. (3 points) In lecture we described how to test the null hypothesis B1 bo against the alternative hypothesis B1 bo, where...
1. If OLS estimators satisfy asymptotic normality, it implies that a. they are approximately normally distributed b. th...
1. If OLS estimators satisfy asymptotic normality, it implies that a. they are approximately normally distributed b. they are approximately normally distributed in samples with less than 10 observations large enough sample sizes c. they have a constant mean equal to zero and variance equal to d. they have a constant mean equal to one and variance equal to o 2 In a multiple regression model, the OLS estimator is consistent if a. there is no correlation between the dependent...
7. When we impose a restriction on the OLS estimation that the intercept estimator is zero, we call it regression through the origin. Consider a population model Y- Au + βίχ + u and we estimate an OLS regression model through the origin: Y-β¡XHi (note that the true intercept parameter Bo is not necessarily zero). (i) Under assumptions SLR.1-SLR.4, either use the method of moments or minimize the SSR to show that the βί-1-- ie1 (2) Find E(%) in terms...
0/1 pts Question7 To obtain the slope estimator using the least squares principle, you divide the sample covariance of X and Y by the sample variance of X sample covariance of X and Y by the sample variance of Y sample variance of X by the sample covariance of X and Y sample variance of X by the sample variance of Y 0/1 pts Incorrect Question8 Question 8 The standard error of the regression (SER) is defined as follows 1-R2...
4. The Gauss-Markov Theorem says that when Assumptions 1-5 of the linear regression model are satisfied: (a) The least squares estimator is unbiased (b) The least squares estimator has the smallest variance of all linear estimators (c) The least squares estimator has an approximately normal sampling distribution (d) The least squares estimator is consistent (e) None of the above
Question 19 3 pts The ordinary least squares estimator of a slope coefficient is unbiased means if repeated samples of the same size are taken, on average the OLS estimates will be equal to the true slope parameter O the mean of the sampling distribution of the slope coefficient is zero. O the estimated slope coefficient will always be equal to the true parameter value. the estimated slope coefficient will get closer to the true parameter value as the size...
Question 3 True/False/Explain 1. The variance of the OLS estimator of the coefficient of a certain variable X; in a regression is higher, the higher is the degree of collinearity between that variable and the other regressors included in the model 2. Suppose that we estimated the following hourly wage equation n(wage) .092educ + .0041ехреr + (007) 022tenure 284 (0017) (.104) (003) where the numbers in parentheses are estimated standard errors. Assuming that the classical normal linear regression model holds,...
1. For the general multivariate regression model, the least squares estimator is given by Show that for the slope estimator in the simple (bivariate) regression case, this is equivalent to ja! įs] 2. In the general multivariate regression model, the variance of the least squares estimator, Va( is σ2(XX)". Show that for the simple regression case, this is equivalent to a. Var(B- b. Var(B)o i, Σ (Xi-X) 2 C. What is the covariance between β° and β,?
Question 2 (10 points) You are given the following model y-put ei. Consider two alternative estimators of β, b2xvix? and b = Zy/X 1. Which estimator would you choose and why if the model satisfies all the assumptions of classical regression? Prove your results. (4 points) 2. Now suppose that var(y)-hxi, where h is a positive constant (a) Obtain the correct variance of the OLS estimator. (2 points) (b) Show that the BLU estimator is now 6. Derive its variance....
Question 1 Consider the simple regression model (only one covariate): y= BoB1 u Let B1 be the OLS estimator of B1. a) What are the six assumptions needed for B1 to be unbiased, have a simple expression for its variance, and have normal distribution? (3 points) b) Under Assumptions 1-6, derive the distribution of B1 conditional on x\,..., xn. (3 points) In lecture we described how to test the null hypothesis B1 bo against the alternative hypothesis B1 bo, where...
1. If OLS estimators satisfy asymptotic normality, it implies that a. they are approximately normally distributed b. they are approximately normally distributed in samples with less than 10 observations large enough sample sizes c. they have a constant mean equal to zero and variance equal to d. they have a constant mean equal to one and variance equal to o 2 In a multiple regression model, the OLS estimator is consistent if a. there is no correlation between the dependent...
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