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012. (a) The ordinary least squares estimate of B in the classical linear regression model Yi...
Consider the least-squares residuals ei-yi-yi, 1, 2, . . . , linear regression model. Find the...
Consider the least-squares residuals ei-yi-yi, 1, 2, . . . , linear regression model. Find the variance of the residuals Var(e). Is the vari- ance of the residuals a constant? Discuss. n,from the simple
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 4. Least squares solution [6 marks] The ordinary least squares estimate for the slope in...
Question 4. Least squares solution [6 marks] The ordinary least squares estimate for the slope in simple linear regression gives the following: B = (2=1 Xiyi) – nzy (2=127) - na Show that this is the same as Bi 2=1(ki – 7)(yi — ) i=1(xi – T)2 in where n n 1 = - n Xi, y= Yi n i=1 i=1
Consider a simple linear regression model with nonstochastic regressor: Yi = β1 + β2Xi + ui....
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
2. Consider a simple linear regression model for a response variable Yi, a single predictor variable...
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...
7. In a simple regression model, suppose all of the assumptions of the classical linear regression...
7. In a simple regression model, suppose all of the assumptions of the classical linear regression morel apply, except that rather than assume E (ui | Xi) = 0, you assume that E (Ui / X;) = ali and E (xi) = 0 where a > 0 is a constant. (a) What is the conditional expectation of the OLS slope coefficient, i.e. E (B1 | 21, ..., XN)? (b) In this case, is ß1 an unbiased estimator of B1 or...
1. Consider the simple linear regression model where Bo is known. a) Find the least squares...
1. Consider the simple linear regression model where Bo is known. a) Find the least squares estimator bi of B (b) Is this estimator unbiased? Prove your result. (c) Find an expression for Var(b1x1, ,xn) in terms of x1, ,xn and σ2.
f Squares and Properties of Estimators o. Let xi yi denote two series ofn numbers xi:...
f Squares and Properties of Estimators o. Let xi yi denote two series ofn numbers xi: i-1,2...), tyi: i 1,2...n) Assume that xi s drawn from a distribution that is NOHm σ) Show that the sample mean i ΣΙ-1 χί has a variance of σ/n carefully stating any required assunmptions at each step. Is the sample mean an unbiased estimator of u,? 1. ii. The following results are useful when working with linear regressions. Show that: 2 iii. Show that:...
Find the estimator beta_hat in multivariate linear regression. Multivariate Linear Regression Parameter Estimation Ordinary Least Squares...
Find the estimator beta_hat in multivariate linear
regression.
Multivariate Linear Regression Parameter Estimation Ordinary Least Squares The ordinary least squares (OLS) problem is n m BER(p+1)×m BERP+1)xm に1 に1 where || . || denotes the Frobenius norm. The OLS solution has the form where bx and yk denote the k-th columns of B and Y, respectively.
Q1 a) Explain what it means that the ordinary least squares regression estimator is a linear...
Q1 a) Explain what it means that the ordinary least squares regression estimator is a linear estimator, paying specific attention to how it implies independent variables interact with each other. b) Give two examples of models where the parameters of interest cannot be directly estimated using OLS regression because of nonlinear relationships between them. c) What is the minimum set of conditions necessary for the OLS estimator to be the most efficient unbiased estimator (BLUE) of a parameter? List each...
Consider the least-squares residuals ei-yi-yi, 1, 2, . . . , linear regression model. Find the variance of the residuals Var(e). Is the vari- ance of the residuals a constant? Discuss. n,from the simple
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 4. Least squares solution [6 marks] The ordinary least squares estimate for the slope in simple linear regression gives the following: B = (2=1 Xiyi) – nzy (2=127) - na Show that this is the same as Bi 2=1(ki – 7)(yi — ) i=1(xi – T)2 in where n n 1 = - n Xi, y= Yi n i=1 i=1
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...
7. In a simple regression model, suppose all of the assumptions of the classical linear regression morel apply, except that rather than assume E (ui | Xi) = 0, you assume that E (Ui / X;) = ali and E (xi) = 0 where a > 0 is a constant. (a) What is the conditional expectation of the OLS slope coefficient, i.e. E (B1 | 21, ..., XN)? (b) In this case, is ß1 an unbiased estimator of B1 or...
1. Consider the simple linear regression model where Bo is known. a) Find the least squares estimator bi of B (b) Is this estimator unbiased? Prove your result. (c) Find an expression for Var(b1x1, ,xn) in terms of x1, ,xn and σ2.
f Squares and Properties of Estimators o. Let xi yi denote two series ofn numbers xi: i-1,2...), tyi: i 1,2...n) Assume that xi s drawn from a distribution that is NOHm σ) Show that the sample mean i ΣΙ-1 χί has a variance of σ/n carefully stating any required assunmptions at each step. Is the sample mean an unbiased estimator of u,? 1. ii. The following results are useful when working with linear regressions. Show that: 2 iii. Show that:...
Find the estimator beta_hat in multivariate linear
regression.
Multivariate Linear Regression Parameter Estimation Ordinary Least Squares The ordinary least squares (OLS) problem is n m BER(p+1)×m BERP+1)xm に1 に1 where || . || denotes the Frobenius norm. The OLS solution has the form where bx and yk denote the k-th columns of B and Y, respectively.
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