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Econ 320 Spring 2024

Problem Set 2 - Simple Linear Regression

The due date for this assignment is February 29 at 10:00 a.m EST.

1. (20 points) Let

Yi = β0 + β1X1i + εi

be the linear regression model we have discussed in class. Let the assumptions about the

linear regression model be satisfied.

Given a random sample {Yi , Xi} ni , we estimate β0 and β1 by minimizing the sample mean squared errors:

where En[(Yi − b0 − b1Xi) 2 ] = n/1P ni=1(Yi − b0 − b1Xi) 2 .

Show that the least squares estimators of β0 and β1 are

You must show all the intermediate steps of the calculations.

2. (15 points) Suppose you have a random sample from (Y, X) with 5 observations (Y1, X1) = (4, 1),(Y2, X2) = (−2, −2),(Y3, X3) = (6, 5),(Y4, X4) = (4, 2) and (Y5, X5) = (−5, 0), and you are interested in the estimating linear regression model

Yi = β0 + β1Xi + εi , i = 1, 2, 3, 4, 5.

(a) (5 points) Using the formulas we have seen in class, compute βˆ 0 and βˆ 1 using the available data.

(b) (10 points) Suppose you have put the data in the computer and got that the standard error of βˆ 1 is 0.35.

i. (5 points) Test the null hypothesis H0 : β1 = 0 against the two-sided alternative. Do you reject the null at 10% significance level?

ii. (5 points) Construct a 95% confidence interval for β1. Interpret the result.

3. (15 points) Suppose we are interested in assessing the relationship between marital status and wages: Do married people have higher wages than those who are not married? In order to answer this question, we will use the linear regression model

W agei = β0 + β1M arriedi + εi

where the variable M arried takes the value 1 if the individual is married and 0 otherwise.

(a) (5 points) How would you interpret β0 in this regression model?

(b) (10 points) How would you interpret β1 in this regression model?

4. (20 points) Suppose you have been given the following information about the data:

Calculate the regression slope and the intercept of the simple regression model of Y on an intercept and X.

5. (30 points) Suppose that your entire population is composed of four units. Furthermore, you had access to an Oracle, and they told you the value of each potential outcome for each unit. You also have access to the observed treatment status of each unit, Di , which takes value 1 if a unit i is treated and 0 otherwise. The information from potential outcomes and treatment status is summarized in the Table 1 below.

Table 1: Potential Outcomes and Treatment Assignments

(a) (6 points) Based on Table 1, compute the unit-specific treatment effect for each unit.

(b) (6 points) Based on Table 1, compute the Average Treatment Effect (ATE) and the Average Treatment Effect among the Treated (ATT).

(c) (6 points) Based on Table 1, write what would be the observed outcome Yi for each unit, if you did not have access to the Oracle.

(d) (6 points) Based on Table 1 and the answer to the parts above, compute the simple comparison of means among treated units and untreated units as if you did not have access to the Oracle.

(e) (6 points) Compute the Selection bias term arising from the simple comparison of means.