Solution to last brainteaser Covariance of interactive regre

Solution to last brainteaser Covariance of interactive regression Application: I run a regression of y ~ x (e.g., corporate bond spread returns on stock returns). Then I add an independent interaction variable I, and regress y ~ xI, x(1-I). To check the data, I look at the covariance matrix of xI, x(1-I). Sub-question: (Assume for simplicity that x ~ N(0, sigma2) and I is uncorrelated.) If I is a boolean (0 or 1) indicator (e.g. stock exchange), what is the covariance? Question: Now suppose I is uniformly distributed (e.g. rank of any variable, say market cap). What does the covariance matrix look like now, i.e. what is the ratio of the off-diagonal term to the diagonal term? Solve analytically Solve via simulation

Solution

Cov(xI, X(1-l))=E(xI*x(1-l))-E(xI)E(x(1-I))

when I=0, Cov(xI, X(1-l))=0

when I=1, Cov(xI, X(1-l))=0