Question 1 [2 points] Imagine we are regressing a dependent variable y on two in

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Question 1 [2 points] Imagine we are regressing a dependent variable y on two independent variables: x·,1 {0,1} is a dummy variable, and x·,2 is continuous. We estimate the following model: yi = 0 +1xi,1 +2xi,2 +3(xi,1 × xi,2)+²i (1) where (xi,1 × xi,2) is an interaction between the component terms. (If it’s helpful, you can imagine that fitting the model returned the following estimates: ˆ = [1.5, 0.75, 1.5].)

a. When we interact a continuous variable with a dummy variable, it reflects a belief that the effect of the continuous variable is conditional on the level of the dummy. In other words, it means we believe that the effect of x·,2 depends on whether x·,1 = 0 or x·,1 = 1. Explain what we mean by that. Offer an example of when the effect of a continuous variable might depend on the level of another independent variable. 1

b. Imagine a theoretical observation with x·,1 = 0 and x·,2 = 0. What is the expected value of y? (I.e., what is yˆ)? You may answer with the ‘estimates’ given above, or just use Greek notation.

c. Imagine a theoretical observation with x·,1 = 1 and x·,2 = 0. What is the expected value of y? (I.e., what is yˆ)? You may answer with the ‘estimates’ given above, or just use Greek notation.

d. Explain why, based on the above, we think of the dummy variable x·,1 as ‘shifting the intercept’ in the regression model.

Question l 12points] Imagine we are regressing a dependent variable y on two independent variables: x. 1 E 0,1 is a dummy variable, and x 2 is continuous. We estimate the following model: where (x 1 x x,2) is an interaction between the component terms. (If it's helpful, you can imagine that fitting the model returned the following estimates: 6 1.5, 0.75, 1.50.) a. When we interact a continuous variable with a dummy variable, it reflects a belief that the effect of the continuous variable is conditional on the level ofthe dummy. In other words, it means we believe that the effect of x.2 depends on whether x. 0 or x. IE 1. Explain what we mean by that. Offer an example of when the effect of a continuous variable m depend on the level of another independent variable b. Imagine a theoretical observation with 0 and x..2 0. What is the expected value of y? (I what is y)? You may answer with the 'estimates given above, or just use Greek notation c. Imagine a theoretical observation with x.1 1 and x.2 0. What is the expected value of y? (I what is y)? You may answer with the estimates given above, orjust use Greek notation d. Explain why, based on the above, we think ofthe dummy variable x. as shifting the intercept' in the regression model Activate Windows o to Settings to activate Window

Explanation / Answer

Dummy Vatriable : A dummy variable is one that takes the value 0 or 1 to indicate the absence or presence of some categorical effect that may be expected to shift the outcome.

Here xi,1 is the dummy variable and xi, 2 is the continous independent variable

Here we will take example of xi, 2 as year of education and xi,1 as the gender where male = 0 and female = 1

and y = level of income in thousands dollars

a. when we interact a continous variable, here we are talking about the term 3(xi,1 * x i,2) so it is obvious by our taken example that level of education may be different for males and females and can be substantially produce different income set. By the given corrollary in the question, we mean that continous variable is dependent on what is the value of dummy variable (0,1) .

b. Let say xi, 2 = 0 and xi,1 = o also

then the expected value of y = 0

c. If say xi, 2 = 0 and xi,1 = 1 then

then the expected value of y = 0 + 1 because the rest terms are equal to 0

d. This is the most important thing of having a dummy variable

the continous function will vary as per the value of x

but as we saw the answers of queesion (B) and (c), there are nothing but the value of intercepts of linear regression between X and Y.

In our example case of y(education) and x2 (income) and x1 (male, female) for different values of dummy variables

let say male y = 0 + 2 X2 ( 0 intercept)

and for in case of female y = 0 + 1 + 2 X2 + 3X2 (0 + 1 is the intercept here)

so it shifts the intercept

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