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Problem 2. (Regression without intercept, 50 pts) Suppose you are given the model: Y; = BX;...
Problem 3: Absence of Intercept Consider the regression model Y, = BX,+", where , and X,...
Problem 3: Absence of Intercept Consider the regression model Y, = BX,+", where , and X, satisfy Assumptions SLR1-SLR5. Y (i) Let B denote an estimator of B that is constructed as P where Y and X as are the sample means of Y,and X,, respectively. Show that B is conditionally unbiased. Derive the least squares estimator of B. Show that the estimator is conditionally unbiased. Derive the conditional variance of the estimator. (ii) (iii) (iv) 2
sve v anu i, respectively. 7. Regression without any regressor. Suppose you are given the model:...
sve v anu i, respectively. 7. Regression without any regressor. Suppose you are given the model: Y = pi + uj. Use OLS to find the estimator of Bi. What is its variance and the RSS? Does the estimated By make intuitive sense? Now consider the two-variable model Y = B1 + B2X; +ui. Is it worth adding X, to the model? If not, why bother with regression analysis?
6. Consider the following regression model without an intercept: Y = B,X, +U, One possible estimator...
6. Consider the following regression model without an intercept: Y = B,X, +U, One possible estimator for this model is given by: BE ANXJ Assume that you can make all of the usual ordinary least squares assumptions about the model, including the assumption that the true model does not include an intercept. Is B, an unbiased estimator? Please prove your conclusion, being sure to state the assumptions you use. [5 points]
Exercise 4.11 Consider the regression model Y Po PX+u Suppose that you know Bo 1. Derive...
Exercise 4.11 Consider the regression model Y Po PX+u Suppose that you know Bo 1. Derive the formula for the least squares estimator of p The least squares objective function is OA. n (v2-bo-bx?) i-1 Ов. O B. n (M-bo-bX) /# 1 n Click to select your answer and then click Check Answer. Exercise 4.11 OA n Σ (--B,χ?) O B. E (Y-bo-b,X)2 j= 1 n Σ (Υ-Βo-bΧ) 3. j= 1 D. n Σ (Υ-0-b,) i- 1 Click to select...
mail/u/3/inbox?projector=1 For a multiple regression model Y = B. B.X.+ B.X.-B.X, BX, BX,+ € where is...
mail/u/3/inbox?projector=1 For a multiple regression model Y = B. B.X.+ B.X.-B.X, BX, BX,+ € where is the error term, to represent the relationship between Y and the four X-variables. We got the following results from the data: Source Sum of Squares degrees of freedom mean squares 110.92 Regression Residual Total 215.94 And also given: Variable B. values S(B) Degrees of freedom 0.02 0.056 -0.13 0.021 0.207 -0.05 0.21 0.067 0.001 0.067 Y-intercept is B. = 2.96 d. Find the regression...
Suppose we have a regression model (x, > 0) with n samples of i.i.d. data 0, Varuir] 2, and (a) O...
Suppose we have a regression model (x, > 0) with n samples of i.i.d. data 0, Varuir] 2, and (a) Obtain the OLS estimator β0Ls for β (b) Obtain the optimal WLS estimator ws for B
Suppose we have a regression model (x, > 0) with n samples of i.i.d. data 0, Varuir] 2, and (a) Obtain the OLS estimator β0Ls for β (b) Obtain the optimal WLS estimator ws for B
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...
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....
Suppose that the data (X1, Y), ... (Xn, Yn is generated by the following ("true") model: a+ bX; + сX; +ei, wher...
Suppose that the data (X1, Y), ... (Xn, Yn is generated by the following ("true") model: a+ bX; + сX; +ei, where a, b, c are some parameters and ei are independent errors with zero mean and variance a2. Suppose that we fit the simple linear regression model to the data (i.e. we ignore the quadratic term cX2) using the OLS method. Find the expectation of the residual from the fit.
Suppose that the data (X1, Y), ... (Xn, Yn...
Theoretical questions: Regression without intercept(40 pts) In this question, we consider a two-variable regression model when...
Theoretical questions: Regression without intercept(40 pts) In this question, we consider a two-variable regression model when there is no intercept in the model: There is no intercept x0 in the model. Suppose we have n different samples. Then answer the following questions: (b) Write the explícit solution of βι and函, in terms of Ση 1 гг, Σί.1 za, Σ-1 zazi2Σ㈡ raVi and Σ-1Equ.(30 pts) (Hint: You can refer to the SLR example in slides, they have similar idea)
Problem 3: Absence of Intercept Consider the regression model Y, = BX,+", where , and X, satisfy Assumptions SLR1-SLR5. Y (i) Let B denote an estimator of B that is constructed as P where Y and X as are the sample means of Y,and X,, respectively. Show that B is conditionally unbiased. Derive the least squares estimator of B. Show that the estimator is conditionally unbiased. Derive the conditional variance of the estimator. (ii) (iii) (iv) 2
sve v anu i, respectively. 7. Regression without any regressor. Suppose you are given the model: Y = pi + uj. Use OLS to find the estimator of Bi. What is its variance and the RSS? Does the estimated By make intuitive sense? Now consider the two-variable model Y = B1 + B2X; +ui. Is it worth adding X, to the model? If not, why bother with regression analysis?
6. Consider the following regression model without an intercept: Y = B,X, +U, One possible estimator for this model is given by: BE ANXJ Assume that you can make all of the usual ordinary least squares assumptions about the model, including the assumption that the true model does not include an intercept. Is B, an unbiased estimator? Please prove your conclusion, being sure to state the assumptions you use. [5 points]
Exercise 4.11 Consider the regression model Y Po PX+u Suppose that you know Bo 1. Derive the formula for the least squares estimator of p The least squares objective function is OA. n (v2-bo-bx?) i-1 Ов. O B. n (M-bo-bX) /# 1 n Click to select your answer and then click Check Answer. Exercise 4.11 OA n Σ (--B,χ?) O B. E (Y-bo-b,X)2 j= 1 n Σ (Υ-Βo-bΧ) 3. j= 1 D. n Σ (Υ-0-b,) i- 1 Click to select...
mail/u/3/inbox?projector=1 For a multiple regression model Y = B. B.X.+ B.X.-B.X, BX, BX,+ € where is the error term, to represent the relationship between Y and the four X-variables. We got the following results from the data: Source Sum of Squares degrees of freedom mean squares 110.92 Regression Residual Total 215.94 And also given: Variable B. values S(B) Degrees of freedom 0.02 0.056 -0.13 0.021 0.207 -0.05 0.21 0.067 0.001 0.067 Y-intercept is B. = 2.96 d. Find the regression...
Suppose we have a regression model (x, > 0) with n samples of i.i.d. data 0, Varuir] 2, and (a) Obtain the OLS estimator β0Ls for β (b) Obtain the optimal WLS estimator ws for B
Suppose we have a regression model (x, > 0) with n samples of i.i.d. data 0, Varuir] 2, and (a) Obtain the OLS estimator β0Ls for β (b) Obtain the optimal WLS estimator ws for B
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...
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....
Suppose that the data (X1, Y), ... (Xn, Yn is generated by the following ("true") model: a+ bX; + сX; +ei, where a, b, c are some parameters and ei are independent errors with zero mean and variance a2. Suppose that we fit the simple linear regression model to the data (i.e. we ignore the quadratic term cX2) using the OLS method. Find the expectation of the residual from the fit.
Suppose that the data (X1, Y), ... (Xn, Yn...
Theoretical questions: Regression without intercept(40 pts) In this question, we consider a two-variable regression model when there is no intercept in the model: There is no intercept x0 in the model. Suppose we have n different samples. Then answer the following questions: (b) Write the explícit solution of βι and函, in terms of Ση 1 гг, Σί.1 za, Σ-1 zazi2Σ㈡ raVi and Σ-1Equ.(30 pts) (Hint: You can refer to the SLR example in slides, they have similar idea)
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