Question 1 Consider the following model: y = XB + E, known as the Classical Linear Regression Model (CLRM), where y is the dependent variable, X is the set of independent variables, ß is the vector of parameters to be estimated and & is the error term. a) List and discuss the assumptions you need for the Ordinary Least Squares (OLS) to be a Best Linear Unbiased Estimator (BLUE). b) Derive the OLS estimator, discussing where the assumptions are needed for the derivation. c) Discuss the properties of linearity, unbiasedness and efficiency, discussing what assumption you need for each of these properties to hold.

Question 1 Consider the following model: y = XB + E, known as the Classical Linear Regression Model (CLRM), where y is the dependent variable, X is the set of independent variables, ß is the vector of parameters to be estimated and & is the error term. a) List and discuss the assumptions you need for the Ordinary Least Squares (OLS) to be a Best Linear Unbiased Estimator (BLUE). b) Derive the OLS estimator, discussing where the assumptions are needed for the derivation. c) Discuss the properties of linearity, unbiasedness and efficiency, discussing what assumption you need for each of these properties to hold.

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter1: Equations And Graphs

Section: Chapter Questions

Problem 10T: Olympic Pole Vault The graph in Figure 7 indicates that in recent years the winning Olympic men’s...

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Olympic Pole Vault The graph in Figure 7 indicates that in recent years the winning Olympic men’s pole vault height has fallen below the value predicted by the regression line in Example 2. This might have occurred because when the pole vault was a new event there was much room for improvement in vaulters’ performances, whereas now even the best training can produce only incremental advances. Let’s see whether concentrating on more recent results gives a better predictor of future records. (a) Use the data in Table 2 (page 176) to complete the table of winning pole vault heights shown in the margin. (Note that we are using x=0 to correspond to the year 1972, where this restricted data set begins.) (b) Find the regression line for the data in part ‚(a). (c) Plot the data and the regression line on the same axes. Does the regression line seem to provide a good model for the data? (d) What does the regression line predict as the winning pole vault height for the 2012 Olympics? Compare this predicted value to the actual 2012 winning height of 5.97 m, as described on page 177. Has this new regression line provided a better prediction than the line in Example 2?
Question 22: A least squares regression line : A- may be used to predict a value of y if the corresponding x value is given. B- implies a cause-effect relationship between c and y. C- can only be determined if a good linear relationship exists between x and y . D- All of these answers are correct.
Consider the instrumental variable regression model Yi = β0 +β1Xi + β2Wi + ui,where Xi is correlated with ui and Zi is an instrument. Suppose that the firstthree assumptions in Key Concept 12.4 are satisfied. Which IV assumptionis not satisfied when:a. Zi is independent of (Yi, Xi, Wi)?b. Zi = Wi?c. Wi = 1 for all i?d. Zi = Xi?
Question 1 Provide an algebraic proof that the least squares estimator is not consistent when Cov(x,e)=0 with the regression model y =B1+B2E(x)+e where E(e)=0 So that E(y) = B1 + B2E(x) Full explain this morning and text typing work only thanks
Consider the one-variable regression model Yi = β0 + β1Xi + ui, and suppose it satisfies the least squares assumptions in Key Concept 4.3. Suppose Yi is measured with error, so the data are ?"i = Yi + wi, where wi is the measurement error, which is i.i.d. and independent of Yi. Consider the population regression ?"i = β0 + β1Xi + vi, where vi is the regression error, using the mismeasured dependent variable ?"i.a) Showthatvi =ui +wi.b) Show that the regression ?"i = β0 + β1Xi + vi satisfies the least squares assumptions in KeyConcept 4.3. (Assume that wi is independent of Yj and Xj for all values of i and j and has afinite fourth moment.)c) Are the OLS estimators consistent? part b in another question
QUESTION2 The data set provided by JOHN shows the number of yearly values of flood damage relief items represented by y and the annual rainfall (in centimetres) represented by x over a period of 10 years.y(000s) x (cm)4.0 1101.5 2501.2 2203.0 1503.0 4502.5 2002.0 2102.0 2301.1 2903.0 100 (a) Find the standard deviations of x and y. (b) Find the equation of the least-squares regression line, assuming that the value of the relief items for flood damage depend on the amount of rainfall. (c) Calculate the correlation coefficient using the…
Consider the following model:? = ?? + ?,known as the Classical Linear Regression Model (CLRM), where y is the dependent variable, X is the set of independent variables, ? is the vector of parameters to be estimated and ? is the error term.a) List and discuss the assumptions that make the Ordinary Least Squares (OLS) the Best Linear Unbiased Estimator (BLUE).b) Derive the OLS estimator and variance and discuss where each assumption is needed for the derivation of the two parameters.c) Discuss the properties of linearity, unbiasedness, and efficiency, andwhat assumption you need for each of these properties to hold.d) Present and discuss the R2 and the adjusted R2 Discuss pros and cons of each of the two statistics.
Consider the following model:? = ?? + ?,known as the Classical Linear Regression Model (CLRM), where y is the dependent variable, X is the set of independent variables, ? is the vector of parameters to be estimated and ? is the error term. Present and discuss the R2 and the adjusted R2. Discuss pros and cons of each of the two statistics.
Consider the one-variable regression model Yi = β0 + β1Xi + ui and suppose that it satisfies the least squares assumptions .The regressor Xi is missing, but data on a related variable, Zi, are available, and the value of Xi is estimated usingX ̃i = E(Xi | Zi). Let wi = X ̃ i - Xi.a. Show that X ̃i is the minimum mean square error estimator of Xi using Zi. That is, let X^i = g(Zi) be some other guess of Xi based on Zi, and show that E[(X^i - Xi)2] ≥ E[(X ̃i - Xi)2].b. Show that E(wi | X ̃i) = 0.c. Suppose that E(ui | Zi) = 0 and that X ̃i is used as the regressor in place of Xi. Show that β^1 is consistent. Is β^0 consistent?
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A least squares regression line ______. a. implies a cause-effect relationship between x and y b. may be used to predict a value of y if the corresponding x value is given c. can only be determined if a good linear relationship exists between x and y d. All of these answers are correct.