Given the assumptions of the classical linear regression model, the least-squares estimator Y = B +...
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Given the assumptions of the classical linear regression model, the least-squares estimator Y₁ = B₁ + B₂X₁ + û, for the Population Regression Function Y₁-B₁ + B2Xi+ui is BLUE. In your own words explain what each of the concepts in this acronym mean and provide an (8 marks) example of each concept that supports this explanation. b. Briefly explain the three sources of variation that determine the precision of the estimator for B₂ in the following PRF: Y₁ = B₁ + B₂Xzi B3X3i+u₁. How do the values of these sources each affect the magnitude of the variance of the estimator? (6 marks) c. Explain, by means of a practical example, the following problems and how they are likely to affect the estimated standard error of an OLS estimator in the absence of any correction: (12 marks) i) Multicollinearity; ii) Heteroscedasticity; iii) Autocorrelation. 1. Violating which of the classical linear regression model assumption will lead to an inconsistent OLS estimator? [1] 2. State two methods that can be used to estimate simultaneous equation regression models. [2] 3. In a system of equations given below, Y's are endogenous while X's are exogenous. Which of the equation is over identified? Explain. [2] %₁ = α+₁₂+α₂%₂+α²₂X₁ + α₂X₂ + ¹4₂ - X₂ =B₁ + B₂X₂ + B₂X₂ + 1₂ 73-%0+1/2+1/3 4. State two properties of a good instrument when applying an instrumental variable approach? [2] 5. State one advantage of using a Vector Autoregression (VAR) model. [1] 6. Name the tool used to produce a graph that plots the correlation of each residual with lags of itself and with all other residuals in a VAR model. [1] 7. In the following VAR model, write down the null hypothesis for the Granger causality test of y₂ does not Granger cause y₁? [1] 3₁ = 0 + B₁³-1+B₁₂32-1+₁2+7/12/2-2 +₁₁+₁₂+1₂ 3/2₂ -α+B₂+ B₂2³21-1 +21³₁1-2+722³2-2 +₂³₁ +2³2 +₂ 8. Name the decomposition that identifies the structural shock from the reduced form shock in a VAR model. [1] 9. In a VAR model, name the method of analysing the proportion of the movements in the dependent variables that are due to their "own" shocks, versus shocks to the other variables. [1] Consider a regression model Y₁ = 3+3₁ X₁ + Ui. (1) Write the objective function to minimize in order to obtain the OLS estimator 3₁-(2 marks) (2)Derive a formula for the least squares estimator 3₁ (3 marks) III. Proof (5 points) 1. Let X be the sample mean, i.e., X = ₁, prove that Var (X) Var(X) (2 marks) 73 12 2. Use the fact that û; = 0 and Σû,0; = 0 to prove the following equation (3 marks) i=1 i=1 72 = TSS = RSS + ESS where TSS, RSS, and ESS are the total sum of squares, sum of squared residuals, and the explained sum of squares respectively. Given the assumptions of the classical linear regression model, the least-squares estimator Y₁ = B₁ + B₂X₁ + û, for the Population Regression Function Y₁-B₁ + B2Xi+ui is BLUE. In your own words explain what each of the concepts in this acronym mean and provide an (8 marks) example of each concept that supports this explanation. b. Briefly explain the three sources of variation that determine the precision of the estimator for B₂ in the following PRF: Y₁ = B₁ + B₂Xzi B3X3i+u₁. How do the values of these sources each affect the magnitude of the variance of the estimator? (6 marks) c. Explain, by means of a practical example, the following problems and how they are likely to affect the estimated standard error of an OLS estimator in the absence of any correction: (12 marks) i) Multicollinearity; ii) Heteroscedasticity; iii) Autocorrelation. 1. Violating which of the classical linear regression model assumption will lead to an inconsistent OLS estimator? [1] 2. State two methods that can be used to estimate simultaneous equation regression models. [2] 3. In a system of equations given below, Y's are endogenous while X's are exogenous. Which of the equation is over identified? Explain. [2] %₁ = α+₁₂+α₂%₂+α²₂X₁ + α₂X₂ + ¹4₂ - X₂ =B₁ + B₂X₂ + B₂X₂ + 1₂ 73-%0+1/2+1/3 4. State two properties of a good instrument when applying an instrumental variable approach? [2] 5. State one advantage of using a Vector Autoregression (VAR) model. [1] 6. Name the tool used to produce a graph that plots the correlation of each residual with lags of itself and with all other residuals in a VAR model. [1] 7. In the following VAR model, write down the null hypothesis for the Granger causality test of y₂ does not Granger cause y₁? [1] 3₁ = 0 + B₁³-1+B₁₂32-1+₁2+7/12/2-2 +₁₁+₁₂+1₂ 3/2₂ -α+B₂+ B₂2³21-1 +21³₁1-2+722³2-2 +₂³₁ +2³2 +₂ 8. Name the decomposition that identifies the structural shock from the reduced form shock in a VAR model. [1] 9. In a VAR model, name the method of analysing the proportion of the movements in the dependent variables that are due to their "own" shocks, versus shocks to the other variables. [1] Consider a regression model Y₁ = 3+3₁ X₁ + Ui. (1) Write the objective function to minimize in order to obtain the OLS estimator 3₁-(2 marks) (2)Derive a formula for the least squares estimator 3₁ (3 marks) III. Proof (5 points) 1. Let X be the sample mean, i.e., X = ₁, prove that Var (X) Var(X) (2 marks) 73 12 2. Use the fact that û; = 0 and Σû,0; = 0 to prove the following equation (3 marks) i=1 i=1 72 = TSS = RSS + ESS where TSS, RSS, and ESS are the total sum of squares, sum of squared residuals, and the explained sum of squares respectively.
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1 The classical linear regression model assumption that will lead to an inconsistent OLS estimator if violated is exogeneity of the explanatory variab... View the full answer
Related Book For
College Algebra Graphs and Models
ISBN: 978-0321845405
5th edition
Authors: Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen, Judith A. Penna
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