Statistical Modeling and Regression

625.661 Statistical Models and Regression

Modules 3-4 Test

Note: You MUST show all work. While math/stat software can be used to check your work, you MUST show how you obtained all answers, steps leading up to your final answer, assumptions needed, etc. The only exception to this is that you MAY use any math/stat software to find the critical value of the normal, t, F, or chi-square distribution. This test must be completed by you alone; help from any other human will be considered cheating.

1. Suppose that we fit the following model to the n observations (y1,x11,x21), ..., (yn,x1n,x2n):

yi = β0 + β1x1i + β2x2i + �i,

for i = 1, ...,n, where all �i are identically and independently distributed as a normal random variable with mean zero and variance σ2 and every xji is fixed.

(a) [25 Points] Suppose the above model is the true model. Show that at any observation yi, the point estimator of the mean response and its residual are two statistically independent normal random variables.

(b) [25 Points] Suppose that the above model is the true model, but we fit the data to the following model (i.e., ignore the variable x2):

yi = β0 + β1x1i + �i,

for i = 1, ...,n. Assume that x̄1 = 0, x̄2 = 0, and ∑n

i=1 x1ix2i = 0. Derive the least-squares estimator of β1 obtained from fitting this new model. Is this least squares estimator biased for β1 in the original model? Why or why not?

2. Ten observations on the response variable y, associated with two re- gressor variables x1 and x2, are given in the following table.

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Observation No. y x1 x2 1 7 9 1 2 8 6 1 3 5 7 1 4 3 8 1 5 2 5 1 6 10 7 -1 7 9 6 -1 8 10 3 -1 9 9 4 -1 10 8 4 -1

The model fitted to these observations is

yi = β0 + β1x1i + β2x2i + �i,

for i = 1, ...,n, where all �i are identically and independently distributed as a normal random variable with mean zero and a known variance of σ2 = 3.

(a) [25 Points] Test the null hypothesis, that there is no difference between the y-intercept for x2 = 1 and the y-intercept for x2 = −1, at a statistical significance level of 0.05.

(b) [25 Points] Now fit the following model to the above ten observa- tions:

yi = β0 + β2x2i + �i.

Calculate the variance of the residual for observation #6. Make sure to state any assumption(s) used!

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