Suppose the outcome in an online experiment is Y_i = beta_0 + beta_1 X + epsilon_i, where i indexes ...

---

Suppose the outcome in an online experiment is Y_i = beta_0 + beta_1 X + epsilon_i, where i indexes the users participating in the experiment and X indicates whether the user saw the improved version of an app (X = 1 if they did, 0 if they didn't). Users from several different cities are involved, and all users in a single city are affected by common sources of randomness (the weather in that city, local events, attitudes and interests specific to that city). For each city c, let Z_c be an independent normal random variable with mean 0 and variance 1 representing this common source of randomness for people in city c. In addition, each individual i is affected by a separate source of randomness W_i, an independent normal random variable with mean 0 and variance sigma^2. Let epsilon_i = Z_c(i) + W_i, where c(i) is the city in which individual i lives. (a) Calculate the mean and variance of epsilon_i in terms of sigma^2. (b) Calculate the covariance between epsilon_i and epsilon_j in terms of sigma^2 for two individuals i and j in different cities. (c) Calculate the covariance between epsilon_i and epsilon_j in terms of sigma^2 for two individuals i and j in the same city. (d) Compute the correlation between epsilon_i and epsilon_j for two individuals in the same city, where the correlation is Cov(epsilon_i, epsilon_j) / sqrt(Var(epsilon_i) Var(epsilon_j)). Parts (c) and (d) are not 0, which means the errors are not independent; explain why this violates the assumptions of linear regression. Hint: Var[A] = Cov(A, A); Var[A + B] = Var[A] + Var[B] + 2 Cov[A, B]; Cov[A + B, C] = Cov[A, C] + Cov[B, C]; Cov[aA, B] = a Cov[A, B] for deterministic a; Cov[A, B] = 0 if A and B are independent.