Midterm Examination. Regression and Forecasting Models


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1 IOMS Department Regresson and Forecastng Models Professor Wllam Greene Phone: Offce: KMC 790 Home page: people.stern.nyu.edu/wgreene Emal: Course web page: people.stern.nyu.edu/wgreene/regresson/outlne.htm : Fall 203, Secton Ths examnaton contans 0 questons, each worth 0 ponts. Where questons have more than one part, pont values for the parts are shown. Please answer all questons. All answers are to be gven n ths test booklet. o supplementary materals wll be accepted. Ths s an open book test. You may use any notes, books, or other materals that you lke. You may not use a telephone, Pad, or any other devce that s capable of sendng or recevng a sgnal durng the exam. In cases n whch a computaton nvolvng several values s requested, you may report the values n the approprate formula. For example, f the answer were the result of (+)/2, you may report the exact formula, (+)/2 rather than. The Stern honor code apples to your partcpaton n ths examnaton. YOUR AME
2 . The followng shows the results of regresson of household ncome (HHIC) on years of educaton based on a survey of a large sample of German households. (2)a. How large s the sample? = (2)b. What s the reported value of R 2? (3)c. What s the value of the sum of squared resduals? (3)d. What are the constant term and slope of the estmated regresson equaton? and (0)2.The regresson model states that the relatonshp between a dependent varable y and an ndependent varable x s y = β 0 + β x + ε. If I want to estmate β, I wll compute the regresson of y on x (as we dscussed n class). But, just usng some smple algebra, I can also wrte x = β 0 /β + (/β )y  (/β )ε. Ths means that I could just compute the regresson of x on y, nstead, and take the recprocal of the slope coeffcent that I compute I wll get the dentcal answer. True or false? Justfy your answer. 2
3 If you regress y on x, your b wll be If you regress x on y, your b wll be now take the recprocal and your /b = = = = = ( x x)( y y) ( x x) 2 ( x x)( y y). (ote dfferent denomnators.) 2 ( y y) = = ( y y) ( x x)( y y) Ths s obvously dfferent from the slope n the regresson of y on x. So false. ote ths s not about the theory of the model. The queston asked about what you would get f you dd the computatons wth your data One of the nterestng features of a country s economy, or the world economy, s the rate at whch health expendture ncreases n response to the sze of the economy. In the regresson below, usng the world health organzaton data that we dscussed n class, I have regressed the log of per capta health expendture on the log of the GDP for the 9 countres n ther sample. (5)a. What s the estmate of the elastcty of health expendture wth respect to ncome (GDP)?.8226 (5)b. How do you nterpret the value of the elastcty? The elastcty mples that f gdp ncreases by %, expendtures ncrease by about.8%. 3
4 4. The followng sample of observatons reports the sales n unts for electroncs stores n the UK. To begn my study of how these busnesses work, I computed a regresson of the camera sales on the number of people workng the floor. (Staff). (5)a. The equaton states that f staff equals zero, sales of cameras wll equal Ths s obvously nonsense. If there s no one workng, sales should be zero. There s obvously somethng wrong wth ths equaton. True or false? Explan. There s nothng wrong wth the equaton. The s the constant term that s needed to make the lne pass through the mddle of the data. o store has staff of zero. The lne predcts the outcome n the range of the data. (5) b.. What s the economc nterpretaton of the coeffcent on Staff n ths equaton? Each addtonal person added to the staff s assocated wth an ncrease of cameras of
5 5. Ths queston s based on the regresson n Queston 4. The average store sze s a lttle under 5 people. We wll use 5 for the average value of staff. I want to predct the sales of Cameras for a store that has Staff = 0 employees. (3)a. What s the predcton of Camera sales for a store wth Staff = 0? The predcton s cameras = (0) = (5)b. What are the lower and upper lmts of a predcton nterval for the sales of ths store? (2)c. It s OK f you use 2.0 or.96 for calculatng the wdth of your nterval. But, ths small sample has only observatons. The approprate crtcal value s somewhat larger than 2. The table you need s on slde 37 of otes Part 2. What s the correct value? b. and c. The standard error to use s s + + ( x* x) SE = (0 5) = e ( ) ( ) b The bounds of the nterval are ± t* (58.726). It s typcal to use.96 or 2.00 for t*. Ths s a very small sample. The actual approprate value from the table mentoned n part c s the t wth 2 = 9 degrees of freedom, Beng an economc phlosopher, I decded to use the producton data n Queston 4 to test a couple hypotheses. The followng shows the results of my regresson of the log of vdeo sales on the log of floor space. I nterpret ths as a regresson of the log of output on the log of captal. (5)a. Theory (Marxst) holds that captal s not productve. The null hypothess s that the coeffcent on log captal (logfloor) s zero. Test ths hypothess. t = ( )/.365 = Ths s much larger than 2.262, so the hypothess s rejected. 5
6 (5)b. Theory 2 (bland and noncommttal) holds that captal mght be productve, but there are constant returns to scale. The null hypothess s that the coeffcent s one. Test ths hypothess. t = (.9352 )/.365 = Stll larger than 2.262, so ths s also rejected. (0)7. The P value reported n the Analyss of varance table n the regresson n Queston 6 s Snce ths s a probablty, the program s reportng that the probablty of the model s 0.000,.e., mpossble. Therefore we should dscard the model as worthless and buld a new model. True or false? Explan. The P value reports the probablty of observng a coeffcent as large (far from zero) as what we observed f the actual coeffcent really were zero. Snce P=0.000, we reject that assumpton. The P value says t s (essentally) mpossble to observe ths value b =.9352 f the true value of β really were zero. (0)8. Usng the data from Queston, I decded to regress the log of ncome on age. The results are shown below. otce that Mntab reports that R 2 equals 0.0%. Ths s nonsense. Ths s sloppness on the part of the people who wrote the software. Luckly, you can compute the R 2 for ths regresson usng other numbers that are reported n the results. What s the rght value for R 2? R 2 = / =
7 9. The fgure below shows a scatter plot of the monthly returns on Mcrosoft stock vs. Walmart stock. There are 70 observatons n the sample. (5)a. Based on ths fgure, s the correlaton between these two varables postve or negatve? Justfy your answer. Postve. The slope of the lne s postve. Ths s the same as the sgn of the correlaton. (5)b. Whch s a good guess of the correlaton between these two varables? Explan If you answered negatve n part a, then put a mnus sgn on the guesses above. 0.0 s close to zero, whch would show for an unorganzed blob of ponts. But, the regresson lne slopes up, so ths s not reasonable..95 and.99 are extremely hgh. The ponts are not that organzed around the lne. That leaves (The actual value s.2486.) 7
8 (0)0. Usng the data n the fgure n Queston 9, I computed a lnear regresson of the Mcrosoft prce on the Walmart prce. I then computed the resduals and plotted them n the fgure below. Lookng at these results, I don t see any patterns that would make me concerned about the model they look lke random nose to me. What knds of patterns would make me (an analyst) queston whether my model s an approprate regresson model? Long streaks of postve and/or negatve values would suggest nonrandomness. Large numbers of very large resduals mght also be suggestve. Ths scatter of resduals looks lne an unorganzed blob of ponts that swngs randomly between postve and negatve. 8
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