Hello! I could use some help with this problem.

(a) Suppose you're given n outcomes Y = (Y1,...Yn), and each outcome i is associated with p real-valued covariates (X_i1 , ... , X_ip); let X denote the n x p design matrix X, and assume it's full rank with n > p. You run an OLS regression and obtain coefficients:

^= (X^TX)^-1 X^TY

Recall that the p-value p_j associated to the j'th coefficient is hte p-value for a t-test of the null hypothesis that the j'th coefficeint is zero. Suppose the population model takes the form Y = f(X₁, ... , X_p) + for some function f, and error term . Specify the assumptions which would ensure validity of the t-test in this setting.

(b) Suppose we're given n observations of independent and identically distributed pairs (Y₁, X₁), ... , (Y_n, X_n), where the Y_i are binary outcomes, and the X_i are scalar (i.e. real-valued) covariates. The population model is the logistic regression model:

log(1P(Y=1X)P(Y=1X)) =(by definition)= logit(P(Y=1X))=1+ X.

Using the data, you fit an estimated coefficient ^ using maximum likelihood on the preceding population model. Let SE denote the estimated standard error of ^. For a large sample size n, give an approximate 95% CI for P(Y = 1 | X), and explain your reasoning.

Thanks!