Assume that the population regression function is Yi = BXi + ei (B is beta, e...

Assume that the population regression function is Y_i = BX_i + e_i (B is beta, e is the error term). This is a regression through the origin (no intercept).

A. Under the homoskedastic normal regression assumptions, the t-statistic will have a Student t distribution with n-1 degrees of freedom (not n-2 degrees of freedom). Explain.

B. Will the residuals sum to zero in this case? Explain and show your derivations

Expert Solution

orchestra answered 2 years ago

Consider the model: yi = βxi + ei, i = 1,...,n where E(ei) = 0 and...

Consider the model: yi = βxi + ei, i = 1,...,n where E(ei) = 0 and Variance(ei) = σ2 and ei(s) are non-correlated errors. a) Obtain the minimum-square estimator for β and propose an unbiased estimator for σ2. b) Specify the approximate distribution of the β estimator. c) Specify an approximate confidence interval for the parameter β with confidence coefficient γ, 0 < γ < 1.

Consider the simple regression model: !Yi = β0 + β1Xi + ei (a) Explain how the...

Consider the simple regression model: !Yi = β0 + β1Xi + ei (a) Explain how the Ordinary Least Squares (OLS) estimator formulas for β̂0 and β̂1 are derived. (b) Under the Classical Linear Regression Model assumptions, the ordinary least squares estimator, OLS estimators are the “Best Linear Unbiased Estimators (B.L.U.E.).” Explain. (c) Other things equal, the standard error of β̂1 will decline as the sample size increases. Explain the importance of this.

Question 1: Consider the simple regression model: !Yi = β0 + β1Xi + ei (a) Explain...

Question 1: Consider the simple regression model: !Yi = β0 + β1Xi + ei (a) Explain how the Ordinary Least Squares (OLS) estimator formulas for !β̂ and !β̂ are derived. (b) Under the Classical Linear Regression Model assumptions, the ordinary least squares estimator, OLS estimators are the “Best Linear Unbiased Estimators (B.L.U.E.).” Explain. (c) Other things equal, the standard error of β! ̂ will decline as the sample size increases. Explain the importance of this. Question 2: Consider the following...

Following is a simple linear regression model: yi = a+ bxi +ei The following results were...

Following is a simple linear regression model: yi = a+ bxi +ei The following results were obtained from some statistical software. R2 = 0.523 syx(regression standard error) = 3.028 n (total observations) = 41 Significance level = 0.05 = 5% Variable Parameter Estimate Std. Err. of Parameter Est. Interecpt 0.519 0.132 Slope of X -0.707 0.239 Note: For all the calculated numbers, keep three decimals. 6. A 95% confidence interval for the slope b in the simple linear regression model...

Consider 2 models: yi = β1 + β2xi + ei (1) Y = X0β + e;...

Consider 2 models: yi = β1 + β2xi + ei (1) Y = X0β + e; (2) where Equation (1) represents a system of n scalar equations for individuals i = 1; ...; n , and Equation (2) is a matrix representation of the same system. The vector Y is n x 1. The matrix X0 is n x 2 with the first column made up entirely of ones and the second column is x1; x2; ...; xn. a. Set...

Suppose you estimate a simple linear regression Yi = β0 + β1Xi + ei. Next suppose...

Suppose you estimate a simple linear regression Yi = β0 + β1Xi + ei. Next suppose you estimate a regression only going through the origin, Yi = β ̃1Xi + ui. Which regression will give a smaller SSR (sum of squared residuals)? Why?

There are two OLS regression specification Yi = aSexi + ui (1) Yi = b Malei...

There are two OLS regression specification Yi = aSexi + ui (1) Yi = b Malei + c Femalei + ui(2) a, b, c, are constants. Sexi = 1 if the person is Female, and 0 otherwise. Malei = 1 if Sexi = 0, Femalei = 1 if Sexi = 1, and both are 0 otherwise. Why are neither regression(1) nor regression(2) directly tell you if the difference in Yi between Males and Females is statistically significant? What would be...

Consider the simple regression model: Yi = β0 + β1Xi + e (a) Explain how the...

Consider the simple regression model: Yi = β0 + β1Xi + e (a) Explain how the Ordinary Least Squares (OLS) estimator formulas for β0 and β1are derived. (b) Under the Classical Linear Regression Model assumptions, the ordinary least squares estimator, OLS estimators are the “Best Linear Unbiased Estimators (B.L.U.E.).” Explain. (c) Other things equal, the standard error of will decline as the sample size increases. Explain the importance of this.

Following is a simple linear regression model: yi = /alpha + /beta xi + /epsilon i...

Following is a simple linear regression model: yi = /alpha + /beta xi + /epsilon i The following results were obtained from some statistical software. R2 = 0.735 syx (regression standard error) = 5.137 n (total observations) = 60 Significance level = 0.05 = 5% Variable Parameter Estimate Std. Err. of Parameter Est. Interecpt 0.325 0.097 Slope of X -1.263 0.309 1. Write the fitted model. (I ALREADY KNOW THE ANSWER TO THIS. I LEFT IT INCASE IT IS NEEDED...

Consider the following (generic) population regression model: Yi = β0 + β1X1,i + β2X2,i + ui,=...

Consider the following (generic) population regression model: Yi = β0 + β1X1,i + β2X2,i + ui,= (∗) transform the regression so that you can use a t-statistic to test a. β1 = β2 b. β1 + 2β2 = 0. c. β1 + β2 = 1. (Hint: You must redefine the dependent variable in the regression.)

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