Midterm Exam 1, section 1 (Solution) Thursday, February hour, 15 minutes
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1 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.. Show all the calculatos. 4. If you eed more space, use the back of the page. 5. Fully label all graphs. Good Luck
2 . pots. Cosder two radom varables, ad Y, wth probablty desty fuctos:.5y y f, f y otherwse otherwse a. Prove that f s deed a pdf. Frst, f s o-egatve for all : Thus, f s o-egatve for, ad t s gve that f for all other. Secod, we check that f tegrates to over the support: d b. Suppose that the jot pdf of ad Y s y y ; y f, y otherwse Are ad Y statstcally depedet? Prove your aswer. We check f the jot pdf equals the product of the margal pdfs. f f y.5y y y f, y Ideed, ad Y are statstcally depedet because the jot pdf s equal to the product of margal pdfs.
3 c. Calculate the epected value of d d d. Suppose that the mea of s. Fd the varace of. 8 6 var
4 . pots. Prove that f radom varables ad Y are depedet, the they 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 epectato of the product Y, assumg that the radom varables are cotuous: Y yf, y ddy yf f y ddy by def. of dep. f d dy yf y dy yf y dy Y yf y Thus, cov, Y Y Y For ad Y that are dscrete, the steps are the same as above, wth summatos stead of tegrals. Y y y yf y yf, y f y y yf f y yf y by def. of y dep. yf y 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 by. 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 oly 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 by 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. Mathematcally, 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 hourly eargs of dvdual dollars, ad s the dvdual's educato years 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 eample, eperece, motvato, hard work, luck, coectos, qualty of educato, etc. 6
8 c. Suppose that Dea 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 hourly eargs of a worker wth years of schoolg? Usg the ftted equato, wth, gves: Y ˆ $4 per hour d. Suppose that the average hourly eargs the sample s $5 per hour. What s the average years of schoolg the sample? We kow that the ftted le passes through the pot of sample averages, Y : Y Pluggg Y 5 ad solvg for, gves
9 6. pots. Chrsta studes the relatoshp betwee campag spedg ad cogressoal electos outcomes. She collected data o 7 electos, each volvg two caddates A ad B, ad ther campag costs. Her key varables are: sharea percetage of total campag spedg by caddate A. votea percetage of total votes for caddate A. The R commad ad output from Chrsta s study are preseted below. lmvotea ~ sharea, data = vote Resduals: M Q Meda Q Ma Coeffcets: stmate Std. rror t value Pr> t Itercept <e-6 *** sharea <e-6 *** --- Sgf. codes: ***. **. *.5.. Resdual stadard error: 6.85 o 7 degrees of freedom Multple R-squared:.856, Adjusted R-squared:.855 F-statstc: 8 o ad 7 DF, p-value: <.e-6 a. What s the depedet varable the above regresso model? crcle the correct aswer.. sharea. votea b. What s the depedet varable regressor the above regresso model? crcle the correct aswer.. sharea. votea 8
10 c. Iterpret the estmated regresso coeffcets. The estmated slope coeffcet, b. 46, meas that % crease spedg share by caddate A, s predcted to crease the caddate s share total votes by.46%. The estmated tercept or costat, b 6. 8, s the predcted percetage of votes that a caddate wll receve, f that caddate does ot sped ay moey o campag. I other words, caddates who do t campag are predcted to receve less tha 7% of the votes. d. pla the meag the reported R, ad commet o ts magtude. SS The value of R. 86, meas that about 86% of the varato vote TSS precetage ca be eplaed by the ftted model wth spedg share beg the sole regressor. Ths meas that campag budget s etremely mportat factor determg the electos outcomes. 9
Midterm Exam 1, section 2 (Solution) Thursday, February hour, 15 minutes
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..
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