Quantitative Research Methods III
Problem Set 1
Anna Abate Bessomo
2026
Question 1 Let X be a random variable. Let a and b denote two constants. Suppose
we construct a new random variable Y as follows:
Yi = b + aXi
a. Show that the correlation between X and Y is either 1 or −1.
sXY ∈ {−1, 1}
b. If we were to run a simple linear regression of the form:
Yi = β0 + β1 Xi + ui
What would we expect to find for β̂0 , β̂1 , and ûi ? Provide an argument for your
answer.
c. Create a dataset with 100 observations in R. Generate a random variable X ∼
N (0, 1). Choose values for a and b. Check your answer for b. in R.
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Intermediate Steps for a.
• Start with the equation for the correlation and the variance formula:
Cov(X, Y )
sXY = √
Var(X)Var(Y )
&
Var(X) =
n
)2
1 ∑(
Xi − X̄
n − 1 i=1
• Start with the formula for Ȳ and substitute in for Yi . Express Ȳ as a function of
a, b, and X̄.
• Substitute the new expression for Ȳ into Var(Y ) and simplify. Further, substitute
√
into Var(X)Var(Y ) and simplify.
√
– Note: for some x ∈ R, x2 = |x|, i.e. the absolute value.
• Simplify Yi − Ȳ by substituting in. Substitute further into Cov(X, Y ).
• Calculate sXY and argue.
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Question 2
cept.
Consider the “trivial” regression model: we regress Y only on the interYi = β0 + ui
Show that the estimated intercept equals the sample average of Y .
β̂0 = Ȳ
Intermediate Steps:
• Define the fitted values and residuals.
• Write down the SSR.
• Minimise the SSR by finding the first-order condition.
• Solve for β̂0 .
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