PSY544 – Introduction to Factor Analysis Homework assignment 1, Fall 2019 Due midnight, October 21, 2019 Suppose we have the following matrices: 𝑨 = [ 8 4 3 12 ] 𝑩 = [ 6 3 9 11 0 1 ] π‘ͺ = [ 0 βˆ’8 0 0 0 3 5 βˆ’7 0 ] 𝑫 = [ 13 5 1 0 7 3 ] 𝑿 = [ 8 1 2 5 4] Compute the following, if possible. Show your work. 1. X’X 2. DA 3. CB 4. (DB)C 5. BD + A 6. C + D 7. Is B’A equal to AB? 8. Compute |A|. We haven’t covered this in class, so look up online how to do it by hand. Now, suppose we have a simple, ordinary linear regression model based on N = 5 observations. Using scalar notation, we would write the model as follows: 𝑦𝑖 = 𝛽0 + 𝛽1 π‘₯𝑖 + πœ€π‘– for i = 1, 2, ..., N. The π‘₯𝑖s are the predictor (independent variable) scores, and the πœ€π‘–s are error terms (residuals) that have zero means (by definition). Therefore, the expected value of 𝑦𝑖 is a linear function of the regression coefficients and the predictor scores: 𝐸(𝑦𝑖) = πœ‡π‘– = 𝛽0 + 𝛽1 π‘₯𝑖 Let the N x 1 vector of means be denoted as 𝝁. Write down each element of this vector in terms of the regression coefficients and predictor scores: 𝝁 = [ πœ‡1 πœ‡2 πœ‡3 πœ‡4 πœ‡5] = [ 𝛽0 + 𝛽1 π‘₯1 . . . . ] Given what you’ve just written down, re-express the mean vector 𝝁 as a matrix product between a N x 2 matrix X and a 2 x 1 vector 𝜷: 𝝁 = [ πœ‡1 πœ‡2 πœ‡3 πœ‡4 πœ‡5] = π‘Ώπœ· = [ 1 π‘₯1 . . . . . . . . ] [ . .] Now, let’s call the N x 1 vector resulting from the previous equation y. Also, let’s aggregate all the πœ€π‘–s into an N x 1 vector of error terms. How would you formulate the original regression equation using matrix notation? π’š = [ 1 π‘₯1 . . . . . . . . ] [ . .] + [ . . . . .] = _____ + 𝜺