Formula that you might use: For a simple linear regression model yi= α + βxi+ui, i=1,2,...,N....

Formula that you might use:

For a simple linear regression model

yi= α + βxi+ui, i=1,2,...,N.

Homework Assignment 1

1. Suppose researchers want to know the effect of elementary school class size on students’ math scores(total score is 100), intuitively they think there exists a negative linear relationship between class size and students’ math scores. The researchers want to know the marginal effect of class size on student’s math scores.

1) Based on the background information, design an linear regression model for this research question, and briefly explain your dependent variable, independent variable and the meaning of the parameters.
2) Briefly explain the assumptions of Classical linear regression model (CLRM)

3) Suppose the assumptions of the CLRM are satisfied, what properties of your estimators should be expected to have?

Expert Solution

orchestra answered 1 year ago

Consider a simple linear regression model with nonstochastic regressor: Yi = β1 + β2Xi + ui...

Consider a simple linear regression model with nonstochastic regressor: Yi = β1 + β2Xi + ui . 1. [3 points] What are the assumptions of this model so that the OLS estimators are BLUE (best linear unbiased estimates)? 2. [4 points] Let βˆ 1 and βˆ 2 be the OLS estimators of β1 and β2. Derive βˆ 1 and βˆ 2. 3. [2 points] Show that βˆ 2 is an unbiased estimator of β2.

(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.)

3. Consider the simple linear regression Yi = 2Xi + ui for i = 1, 2,...

3. Consider the simple linear regression Yi = 2Xi + ui for i = 1, 2, . . . ,n. The ui are IID (0; 2 ). a. Derive OLS estimator of 2 and called it b 2 b. Find its variance c. Is b 2 unbiased, show it? d.What is the risk we run when we do not include an intercept in the regression? Do question d.

Consider the following regression model: Yi = αXi + Ui , i = 1, .., n...

Consider the following regression model: Yi = αXi + Ui , i = 1, .., n (2) The error terms Ui are independently and identically distributed with E[Ui |X] = 0 and V[Ui |X] = σ^2 . 1. Write down the objective function of the method of least squares. 2. Write down the first order condition and derive the OLS estimator αˆ. Suppose model (2) is estimated, although the (true) population regression model corresponds to: Yi = β0 + β1Xi...

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 a simple linear model Yi = β1 + β2Xi + ui . Suppose that we...

Consider a simple linear model Yi = β1 + β2Xi + ui . Suppose that we have a sample with 50 observations and run OLS regression to get the estimates for β1 and β2. We get βˆ 2 = 3.5, P N i=1 (Xi − X¯) 2 = 175, T SS = 560 (total sum of squares), and RSS = 340 (residual sum of squares). 1. [2 points] What is the standard error of βˆ 2? 2. [4 points] Test...

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...

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

In a simple linear regression model yi = β0 + β1xi + εi 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 / Sxx].

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.)

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...

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