ISQS 5349 Final Exam, Spring 2017.

Transcription

1 ISQS 5349 Final Exam, Spring 7. Instructions: Put all answers on paper other than this exam. If you do not have paper, some will be provided to you. The exam is OPEN BOOKS, OPEN NOTES, but NO ELECTRONIC DEVICES OF AN KIND.. The regression model is a specification of the probability distributions p(y x). A. (5) Define heteroscedasticity in terms of the distributions p(y x) Solution: The distributions p( y x) have different variances for different x. For example, suppose = is a low value of and = is a high value of. If p( y ) has different variance than p(y ), then there is heteroscedasticity. B. (5) Draw graphs of two distributions p(y x), on the same set of axes, that illustrates heteroscedasticity. Solution: Figure.. in the book is a good example: p(y x) age=7 age= net worth, y (thousands)

2 . () The matrix form of the regression model is = Show how this form of the model is equivalent to (i.e., exactly the same as) the more common form i = i i, for i =,,, n Solution: The matrix model is unpacked as follows: = = n n n (by substitution) = n n n (by matrix multiplication) = n n n (by matrix addition and algebraic re-arrangement) Now, the classic model states i = i i, for i =,,, n, implying for i =,,, n: =

3 = n = n n Comparing equations one at a time, you can see that the n equations represented by the classic model are exactly the same n equations that are represented by the matrix model. 3. Consider following scenario. The ROA (return on assets) of various firms () is to be predicted from their business sector, either sector A or sector B; and the year, either 4, 5, or 6. Analyze these data as a two-way ANOVA with no interaction. A. (5) Specify the appropriate model in terms of indicator variables. Solution: = I(Sector A) I(4) 3I(5) Here, I(.) is an indicator variable that is = if the condition is true, and = if the condition is false. B. () Carefully interpret (i.e., explain the meaning of) the four parameters of your model in A. Solution: To make your life easy, you really should break down the model in terms of the six categories. True mean values, according to the model Sector A 3 Sector B 3 Interpretation: = mean ROA in Sector B in 6. = difference between mean ROA, sector A minus sector B, within any fixed year. = difference between mean ROA, 4 minus 6, within any sector. 3 = difference between mean ROA, 5 minus 6, within any sector.

4 C. () Why might it be a bad idea to treat ear as an ordinary numeric variable in the model, rather than using as.factor(ear)? Draw a graph to help answer. Solution: Because if you treat ear as ordinary numeric data, it forces the mean ROA to lie on a perfect line for the three years. But who knows, maybe 5 was an off year compared to 4 and 6, in which case the true ROA function should go down, then up? Here is a graph to axplain it: In the graph, the straight line shows the kind of model that you estimate if you do not specify as.factor(ear). The crooked line shows what is allowed when you specify as.factor(ear): the function can have any shape whatsoever.

5 4. () Many models we used are expressed in terms of logarithms as ln{μ(x, )} = x, where μ(x, ) = E( = x, ). Here is some R output for such a model: Coefficients: Estimate Std. Error z value Pr(> z ) (Intercept) e-5 *** age *** gender < e-6 *** The dependent variable is = Number of financial planners used, and gender is for female and for male. Using the fact that exp(-.955) =.4, interpret the effect of gender on mean number of financial planners used. Solution: The mean number of financial planners used is estimated to be 6% less for females than for males, for people of any fixed age. 5. () Explain why the term leverage, as in a lever, a good way to understand the effect of a data point that is an outlier in space. Draw graph(s) to help answer. Solution: When the point is remote, it acts as a level to move the OLS regression line around, depending on the particular value that is observed at that. If the value happens to be very high, it pulls ( leverages the regression line up towards that value. If the value happens to be very low, it pulls ( leverages the regression line up towards that value. The following graph illustrates.

6 The remote is 9.3. When the value is high (the blue dot), the least squares line for the data (not including the red dot) is the blue line. When the value is low (the red dot), the least squares line for the data (not including the blue dot) is the red line. This graph illustrates how the remote value leverages the least squares line towards the value that corresponds to the remote value.

7 6. Suppose a regression model is p(y x, θ ) = (θ x)exp( θ xy). Here is a small data set: Obs, i A. (5) Write down the likelihood function for θ. ou don t need to simplify it algebraically. Solution: L(θ data) = (θ )exp( θ ) (θ 5)exp( θ 58) (θ 5)exp( θ 57) B. (5) Without doing any calculation, use your answer to A. to explain what is the maximum likelihood estimate of θ. Solution: The value of θ giving the maximum value of L(θ data) defined in A. is the maximum likelihood estimate of θ. C. (5) A hypothesis of interest is H : θ =. Explain why the maximized likelihood function of B. is larger than the value of the likelihood function that you get when you plug in θ =. Solution: If you plug in θ = into the likelihood function, you can get a likelihood no larger than the maximized value in B, simply because the maximized value is the maximum over all possible values of θ. D. () Using your answers of A. B., and C., explain how to perform the likelihood ratio test of H : θ =. Solution: Call the maximized value of L(θ data) in B. L, and let LL be its logarithm. Call the value of L( data) that you get when you plug in θ = L, and denote its logarithm by LL. Then the likelihood ratio χ statistic is χ = LL - LL. There is one degree of freedom since there is one free parameter in the unrestricted model and no free parameters in the restricted model. So you reject the hypothesis H : θ = if χ is greater than the.95 quantile of the chi-squared distribution with one degree of freedom. Equivalently, reject H if the p-value is <.5, where the p-value is the probability that a chi squared random variable with one degree of freedom is greater than the observed likelihood ratio chi square statistic.

8 7. An estimated interaction model is ˆ =. x.5x. 5xx. The effect of on is the slope of the regression line relating to, for a given fixed value of. A. (5) Compare (i) the effect of on when = with (ii) the effect of on when =. Solution: When =, we have that ˆ =. x.5().5x () =. x. So the effect of on is. when =. When =, we have that ˆ =. x.5().5x () =. ( 3) x =. x. So the effect of on is -. when =. For smaller, has a positive effect on. For larger has a negative effect on. B. (5) Use your answer to A. to help answer this question: What does it mean to say that the effect of on is moderated by? Solution: Just as the answer to A. shows: If the effect of on variable ( ) on depends on the value of another variable ( ), then the effect of on is said to be moderated by. In the example, dramatically moderates the effect of on rom positive to negative.