Midterm Exam 1, section 2 (Solution) Thursday, February hour, 15 minutes

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1 coometrcs, CON Sa Fracsco State Uverst Mchael Bar Sprg 5 Mdterm xam, secto Soluto Thursda, Februar 6 hour, 5 mutes Name: Istructos. Ths s closed book, closed otes exam.. No calculators of a kd are allowed.. Show all the calculatos. 4. If ou eed more space, use the back of the page. 5. Full label all graphs. Good Luck

2 . pots. Cosder two radom varables, ad Y, wth probablt dest fuctos: x x.5 f x, f otherwse otherwse a. Prove that f s deed a pdf. Frst, f s o-egatve for all :.5x x Thus, f s o-egatve for, ad t s gve that f for all other x. Secod, we check that f tegrates to over the support:.5d b. Suppose that the jot pdf of ad Y s x x ; f x, otherwse Are ad Y statstcall depedet? Prove our aswer. We check f the jot pdf equals the product of the margal pdfs. f x f x.5 x x f x, Thus ad Y are ot statstcall depedet because the jot pdf s ot equal to the product of margal pdfs. Remark: I the above soluto m teto was that studets use the gve margal pdfs, eve though the are ot cosstet wth the gve jot pdf. Studets, who derved the margal from the gve jot pdf, wll receve full score as well.

3 c. Calculate the expected value of Y x d d Y d. Suppose that the mea of Y s. Fd the varace of Y. 9 var Y Y Y

4 . pots. Prove that f radom varables ad Y are depedet, the the must be ucorrelated. You ca assume that ad Y are ether cotuous or dscrete. We kow that covarace betwee ad Y ca be wrtte as: cov, Y Y Y Calculatg the frst term o the rght expectato of the product Y, assumg that the radom varables are cotuous: Y xf x, dxd xf x f dxd b def. of dep. xf x dx d f d f d Y f Thus, cov, Y Y Y For ad Y that are dscrete, the steps are the same as above, wth summatos stead of tegrals. Y f x x xf x, xf x x xf x f f b def. of dep. f Y

5 4. pots. Let be a radom varable wth mea ad varace, ad let Z. Prove that Z has mea zero ad varace of.e. Z s stadard radom varable. var var var var Z Z

6 5 4. pots. Let,..., be a radom sample from a populato wth mea ad varace. The sample average s gve b. a. Prove that s a ubased estmator of the populato mea. We eed to prove that. b. Prove that s a cosstet estmator of the populato mea. Sce bas, we ol eed to show that var as. lm var var var

7 5. pots. Cosder the smple regresso model Y u. a. Defe the OLS estmators of the ukow parameters,, ad deote these estmators b b OLS, b OLS. Let the ftted model for some estmates b,b be Yˆ b b, ad the resdual for observato or predcto error be e Y b b. The OLS estmators are such values of b,b that mmze the Resdual Sum of Squares: RSS Y b b where s the sample sze. Mathematcall, the defto of OLS estmators ca be wrtte as follows: OLS b, b OLS arg m b, b Y b b b. Suppose that Y s hourl eargs of dvdual dollars, ad s the dvdual's educato ears of schoolg. What s the meag of the error term u? The error term captures all flueces o eargs, other tha dvdual s level of educato. For example, experece, motvato, hard work, luck, coectos, qualt of educato, etc. 6

8 c. Suppose that Sam estmated the above model usg OLS, ad hs ftted equato s Yˆ b b. Suppose that hs estmates are b, b. Based o these estmates, what s the predcted hourl eargs of a worker wth ears of schoolg? Usg the ftted equato, wth, gves: Y ˆ $4 per hour d. Suppose that the average hourl eargs the sample s $6 per hour. What s the average ears of schoolg the sample? We kow that the ftted le passes through the pot of sample averages, Y : Y Pluggg Y 6 ad solvg for, gves 6 6 7

9 6. pots. Jeffer studes the relatoshp betwee the market performace of compaes shares ad CO salares. She collected a sample of 9 compaes data, ad her ke varables are: roe rate of retur % o compa s equt shares of stock salar compa s CO salar thousads of dollars. The R commad ad output from Jefer s stud are preseted below. lmsalar ~ roe, data = ceosal Resduals: M Q Meda Q Max Coeffcets: stmate Std. rror t value Pr> t Itercept e-5 *** roe Sgf. codes: ***. **. *.5.. Resdual stadard error: 67 o 7 degrees of freedom Multple R-squared:.9, Adjusted R-squared:.84 F-statstc:.767 o ad 7 DF, p-value:.9777 a. What s the depedet varable the above regresso model? crcle the correct aswer.. roe. salar b. What s the depedet varable regressor the above regresso model? crcle the correct aswer.. roe. salar c. Iterpret the estmated regresso coeffcets. The estmated slope coeffcet, b 8. 5, meas that % crease the rate of retur o compa s equt s predcted to crease the CO s salar b $8,5. The estmated tercept or costat, b 96. 9, a CO whose compa s equt had rate of retur of %, s predcted to ear a salar of $96,9 close to mllo dollars. Remark: ths data s from earl 9s, ad toda the umbers are much hgher. 8

10 d. xpla the meag the reported R, ad commet o ts magtude. SS The value of R., meas that.% of the varato the CO salares TSS ca be explaed b the ftted model wth rate of retur o equt beg the sole regressor. Ths meas that ma other factors affect the salares of COs, besdes the stock market performace of the compa s shares. 9

Midterm Exam 1, section 1 (Solution) Thursday, February hour, 15 minutes

Midterm Exam 1, section 1 (Solution) Thursday, February hour, 15 minutes coometrcs, CON Sa Fracsco State Uversty Mchael Bar Sprg 5 Mdterm am, secto Soluto Thursday, February 6 hour, 5 mutes Name: Istructos. Ths s closed book, closed otes eam.. No calculators of ay kd are allowed..