In a simple linear regression model yi = β0 + β1xi + εi with the usual...

In a simple linear regression model y_i = β₀ + β₁x_i + ε_i with the usual assumptions show algebraically that the least squares estimator β̂₀ = b₀ of the intercept has mean β₀ and variance σ²[(1/n) + x̄² / S_xx].

Expert Solution

orchestra answered 2 years ago

Consider the simple linear regression: Yi = β0 + β1Xi + ui whereYi and Xi...

Consider the simple linear regression: Yi = β0 + β1Xi + ui where Yi and Xi are random variables, β0 and β1 are population intercept and slope parameters, respectively, ui is the error term. Suppose the estimated regression equation is given by: Yˆ i = βˆ 0 + βˆ 1Xi where βˆ 0 and βˆ 1 are OLS estimates for β0 and β1. Define residuals ˆui as: uˆi = Yi − Yˆ i Show that: (a) (2 pts.) Pn i=1...

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.

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.

(i) Consider a simple linear regression yi = β0 + β1xi + ui Write down the...

(i) Consider a simple linear regression yi = β0 + β1xi + ui Write down the formula for the estimated standard error of the OLS estimator and the formula for the White heteroskedasticity-robust standard error on the estimated coefficient bβ1. (ii) What is the intuition for the White test for heteroskedasticity? (You do not need to describe how the test is implemented in practice.)

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...

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?

Suppose you estimate the following regression model using OLS: Yi = β0 + β1Xi + β2Xi2...

Suppose you estimate the following regression model using OLS: Yi = β0 + β1Xi + β2Xi2 + β3Xi3+ ui. You estimate that the p-value of the F-test that β2= β3 =0 is 0.01. This implies: options: You can reject the null hypothesis that the regression function is linear. You cannot reject the null hypothesis that the regression function is either quadratic or cubic. The alternate hypothesis is that the regression function is either quadratic or cubic. Both (a) and (c).

Consider a regression model Yi=β0+β1Xi+ui and suppose from a sample of 10 observations you are provided...

Consider a regression model Yi=β0+β1Xi+ui and suppose from a sample of 10 observations you are provided the following information: ∑10i=1Yi=71; ∑10i=1Xi=42; ∑10i=1XiYi=308; ∑10i=1X2i=196 Given this information, what is the predicted value of Y, i.e.,Yˆ for x = 12? 1. 14 2. 11 3. 13 4. 12 5. 15

1. Consider the linear regression model for a random sample of size n: yi = β0...

1. Consider the linear regression model for a random sample of size n: yi = β0 + vi ； i = 1, . . . , n, where v is a random error term. Notice that this model is equivalent to the one seen in the classroom, but without the slope β1. (a) State the minimization problem that leads to the estimation of β0. (b) Construct the first-order condition to compute a minimum from the above objective function and use...

The regression model Yi = β0 + β1X1i + β2X2i + β3X3i + β4X4i + ui...

The regression model Yi = β0 + β1X1i + β2X2i + β3X3i + β4X4i + ui has been estimated using Gretl. The output is below. Model 1: OLS, using observations 1-50 coefficient std. error t-ratio p-value const -0.6789 0.9808 -0.6921 0.4924 X1 0.8482 0.1972 4.3005 0.0001 X2 1.8291 0.4608 3.9696 0.0003 X3 -0.1283 0.7869 -0.1630 0.8712 X4 0.4590 0.5500 0.8345 0.4084 Mean dependent var 4.2211 S.D. dependent var 2.3778 Sum squared resid 152.79 S.E. of regression 1.8426 R-squared 0 Adjusted...

Question