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Information from a midterm exam for a university course on applied least squares. The exam covers topics such as writing the fitted least squares regression equation, calculating the mean squared error (mse), testing hypotheses about the effect of predictors, and calculating residuals. Matrices and formulas for solving these problems.
Typology: Exams
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Your test should have total of 10 pages including this cover page and two blank pages at the end that you may remove and use for scrap paper.
Note that if you need to use an answer from a previous part and you do not have the answer, use a symbol to represent it.
Note that the matrices are rounded off, so that for example, if you multiply ( X ' X )−^1 by ( X ' X ) as given, you may not get exactly the identity.
30 282 85.11 94. 282 4146 843.54 962. ( ' ) 85.11 843.54 287.92 320. 94.9 962.8 320.63 372.
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(a) Write the fitted least squares regression equation.
(b) The number Y Y ˆ ˆ′^ given above was obtained as a quadratic form Y AY ′^ for some matrix A. Give a formula for this matrix in terms of the matrix X and any other matrices needed. You do not have to numerically compute the matrix A.
iii) One approach to conducting the hypothesis test is to take an
standard error to form a t-statistic and conduct a t-test. note that this is equivalent to using the F-test for the general linear hypothesis. Give a simple unbiased estimate of this difference and give an estimate of its standard error.
Hint : Var A ( − B ) = Var A ( ) + Var B ( ) − 2 Cov A B ( , )
(e) Suppose that I were to remove variable X (^) 2 from the model and then refit. For some data sets, the coefficient estimate for the slope corresponding to variable X 1 would change, but for others it would remain the same. Which would be true for this data set? How can you tell from the given information?
X (^) 1 Y -2 5 0 5 1 10
. 7 . 8
The regression is fit with an intercept and the single predictor. The resulting
projection matrix is given by (^) ( ) 1
where the matrix X is the typical design matrix of a column of 1's along with X (^) 1.
Note that all of the data is not given, hence you will not be able to compute the estimated regression coefficients.
(a) One of the elements of P is missing. What should it be? How do you know this without having to do any matrix multiplication?
(b) Calculate the residual associated with the 1st^ observation?
μ
Let
(a) The quadratic form cY BY ′^ has a known distribution for some constant c. What is the correct constant? What is the distribution? (Be specific, including degrees of freedom, and other parameters if needed).
(b) The vector
μ
can be written in the form
a b
for some a b ,.
Find a and b.
(c) It turns out that the matrix B can be derived as the projection matrix for a simple linear regression with 3 observations, where the design matrix is 1 1 1 2 1 1
Suppose that I were to add an additional predictor to the model to form the
new design matrix
. What would be the rank of this
matrix? How do you know without having to do any additional computation?
(d) Based on the information in this question, explain why B μ = μ (the fact that the matrix multiplication works out that way is not enough of an answer).
