$30.00
1. The simple regression through the origin model is like a simple linear regression model, but
without the intercept:
Yi = β1xi + ei
, i = 1, 2, . . . , n
with E(ei) = 0, Var(ei) = σ
2 > 0, and Cov(ei
, ej ) = 0 if i 6= j.
The ordinary least squares estimate minimizes the residual sum of squares
RSS(β1) = Xn
i=1
yi − β1xi
2
(a) [2 pts] Take the derivative of RSS, and set the resulting expression equal to zero.
(This is sometimes called the called the normal equation.)
(b) [2 pts] Solve the equation of the previous part.
(c) [2 pts] To find your solution in part (b), you made an assumption about the values of
x1, x2, . . . , xn. What is that assumption, and why is it needed?
(d) [2 pts] Show that the expression you found in part (b) really is a minimizer
of RSS(β1). (Hint: Take the second derivative.)
2. For a constant matrix A and a random vector Z,
E(AZ) = A E(Z) Var(AZ) = A Var(Z)AT
(assuming expectations and variances all exist).
Consider the linear model Y = Xβ + e under the Gauss-Markov conditions. For each of the
following random vectors, determine the mean vector and the variance-covariance matrix (in
terms of X, β, and σ
2
). Simplify, if possible.
(a) [2 pts] e
(b) [2 pts] Y
(c) [2 pts] βˆ
(d) [2 pts] Yˆ (the random vector for which the realization is the computed vector yˆ of
fitted values)
3. The data set ais (in package alr4) provides data on athletes. Use help(ais) for
information about the variables. Fit a regression model with weight (kg) as the response,
and sex, height, sum of skin folds, and percent body fat as predictors.
(a) [2 pts] Present a summary of your fitted model. (Use the R summary function.)
(b) [2 pts] Give the least squares estimates of all coefficients.