The document describes a simple linear regression model without an intercept. It asks to: (1) derive the least squares estimator of β by differentiating the residual sum of squares; (2) calculate the bias and variance of the estimator assuming X is fixed; (3) show that the sum of the residuals multiplied by the predictor equals zero; and (4) explain whether the sum of the residuals necessarily equals zero or not.

1. Consider a simple regression model with no intercept Yi=Xi+ui where is a fixed parameter with E(u
i)=0,V(ui)=2 and E(uiuj)=0 when i=j. (a) Derive the least squares estimator of . (Hint: First define
RSS, the sum of the squares of the residuals, and then differentiate.) (b) Calculate the bias and
the variance of ^. (You can assume X is nonrandom for simplicity.) (c) Show that for the residuals
u^i=yi^xi, i=1nxiui=0. (d) Is it necessarily true that i=1nui=0 ? Explain.