current Term_Test_III_solutions

current Term_Test_III_solutions - University of Waterloo...

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Unformatted text preview: University of Waterloo Stat 373 Term Test III (F10) Date: December 1, 2010 Duration: 60 minutes Family Name: First Name: ID. #: Signature: Instructor: P. Balka \ ‘-% v . Instructions: - 0 / [L I Q o This exam has 5 pages including this cover page. The marks for each question are indicath (total 01‘ 25). Show your work. Your grade will be influenced by how clearly you express your ideas, and how well you organize your solutions. Stat 373 Term Test Ill — F10 l) The following is output. from a model selection procedure that. based on the C], criterion, selects the best three models for each subset of k variates for the house prices dataset in the course notes. k Ck *ngareft'n 1 86.169 Baths 1 302.3580 Garage 1 360.5761 "’ Sguare.ft.,_$chool 2 56.7226 Squarefiustyle SquareftuBasement . Square.ft_.,Garage,_Schoo| S_quare.ft.,Basemerfi,School §guare.ft.LBasemenLAge Sguare.ft.,§tfle,§argge,5chool quareftuGarageLBasementJSchool Square.ft_.,Garage,Ag§‘School 32.9169 Sq_uare.ft_.,6arageLBasemenLAggSchool 21.1922 2 60.7708 2 3 3 3 4 4 4 5 Sguare.ft.,§tyfi,Gara_ge,Ag§,School 5 24.3950 5 6 6 6 7 7 7 8 8 62.7850 41.0531 43.1124 40.7244 31.9852 32.2571 Sjuare.ft.,_HeaLGarage,BasementAgg 25.9102 Squaref’guHeatJGarage,BasementAggJSchool 16.3792 Sguare.ft.,§arggeLBasemenLAggFireéchool 16.9584 SquareftuStyleLGaraggBasementAge§chool 18.0239 Sguare.ft.¢teaLGarage,BasementAgeLFirgSchool 11.1061 Sguare.ft.,fieaLStyle,_Gar_agg, BasemenLAgeéchool 14.2888 Sguare.ft. I Bed; HeatLGaragg, Basemen_t,Age,School 14.4709 Sq_uare.ftuBedsLHeat,GaragefiasementlflggFirelyhool 9.4605 SquarettuHeaLfiylgGarage,BasementAgeLFirgSchool 10.7812 Squareft.‘BathsLHeat,Garqge,§asement,Agg,Fire,School .6888 VStuare.ft.LBeds,fleaLStyle,§a@geLBasemenLAgeiireLSchool < 9 9.542 SquarefL,BedsLBathsilefiGarageéasementAggFireéchool 9 . S_quare.ft.,Baths,HeaLStwe,§arqggfiasemenLAggFire§chool 9 12.2402 SguareJLgedsJBaths,_Heat,SLylg,_Gan§1BasememAgeLFirgSchl 10 11.0000 a) [2] Circle the model you consider to be the ‘best’ model, according to the CI, criterion. Briefly explain your choice. (F Is meJ/ "+ / [\v'/ fielfl/é / /"’L fact/mt‘fl“ /' } “WV/f 0’} (f : (ifs/.2 <V/f/ 2‘/U h) [2] From this output. list in order the first three variables that you would select in a forward selection model building procedure. 9 jfmu (gm/kn; (/ CJ’ <./'/ mun/AWAQ/f m;~/{</>/ f2. . u/ ' " A pn/u/ 3‘) jCAcn/ /Jm‘//517L (/4 0% 41/ 1"*—'«”"~’~W’Hé’/L fl J { IN ll/C}% Ii I" ‘fA/L’v: y;- lb’r'L / // y Jclmp/i) ‘. (“M It 8 . j) m.t-r{¢/J %'/»Jll’ (,L’N7L4LA J7. Stat 373 Term Test Ill — F10 2) [3] Consider the set of values {11, 6. 3, 102. 5. 8. 8 . 10, 7. l. 3, 4} of the explanatory variate from a simple linear regression model. Is the 4“ observation an Influential pomt? Explain. ’ 3 I // "‘ ."r\ A111.) A /€l/ / f 1 l/"-/ Na‘f ACCK ‘..fC(/./«~ //\r.2 flat. 1 ’J C 0 cl, F...t /NLL/ 4 ,3 Va: / I ate/e / I y . .» //'l+fll(Z/\7L/¢L// Z/{C/Cr/(;/z/ «A xxx/1.; / ‘1 ,Af‘lug/JNJ a. 7 7 4 t b i 1‘" x i x 3’ i " 1-~¢——ro 'A‘”/* 3) The following output is from a regression analysis ofthe qu Hours x 106) for a large city for 66 successive quarters from the first qu to the second quarter (April—June) of 1996 (Time = I, 2. 3, 66). Note that Quarter2. Quarter3, and Quarter4 are indicator variates (e.g., Quarter2 = I if second quarter. 0 otherwise). Assume that all model assumptions are valid for this model. lm(formula = Usage N Time + QuarterZ + Quarter3 + Quarter4) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 968.3906 16.8756 57.384 <28—16 *** Time 0.9383 0.3377 2.778 0.00725 ** Quarter2 —341.9383 17.9244 —19.077 < 28-16 *** Quarter3 —471.6029 18.1991 -25.914 < 2e—16 *** Quarter4 —230.2287 18.2022 -12.648 < 28-16 *** Residual standard error: 52.25 on **** degrees of freedom Multiple R-Squared: 0.9236, Adjusted R-squared: **** F—statistic: 184.3 on *** and *** DF, p—value: < 2.2e-16 a) [2] Did electricity usage appear to increase over the period of study? Justify your Yes / lt/‘I/r’sux‘ll’ J [/7‘ V?” A: 2007.35) b) [3] Forecast energy usage for the fourth quarter of I996. ,\ a ,4 4 ' - 4: -, s _. = 95833?“ +‘ wows) 4230. MW : 5’01 X10 5 KW LA) 1qu /C,) arterly electricity Usage (Kilowatt— arter (Jan—March) of 1980 illlSWCl‘. J/Aufa e ('a(‘(‘(/ (“hf 5.17///7‘- {a " fad/74'“ Iii/U) Stat 373 Term Test 111 — F10 c) [2] Calculate the adjusted R—squared value for this data. Show your work. 04 e wav/ l; t, “(Jami Q JJ(/€LZ : “kl : [/"9236 l ’ ' MW ” 79w ’ :.o75$/ 55(1’65) _ ‘5‘ch‘v‘ : 8 “wow/(é i) ‘ 6/; 1m) 0 “H ‘ L29 2 ,1- .OJ/w : . W36 % (l) [2] Why would a smoothing method (EWMA or movmg average) not be appropriate to forecast electricity usage for this data? gotch an 7L/\e, J/jnITL/ca/nte o‘/’ %K& %/mc y §C<QUM/ (7M4/7Z6/y/4/ama7L5/J/ {£42} 7/xmc James /1 AV? \ a J/“A'f/M/M/ //0561I/ Jy a Jmo.%,<,7 4t¢/’/pz[ Afif é; W/ynille ‘ e) [2] Consider a reduced model fit without the Time variate. Sketch what you would expect to see in a plot of the fitted residuals vs. index (1.2,3‘....66) from this reduced model. Briefly explain your reasoning. /// A) WI ‘f/I’hC V‘l/K,%C/ f ./ \ y J [CD/(Mk!) Luck/4 fgf/(QIL’ A Y ’ fl ’ . r J MUCH/A} -//e/\r/£ 7‘an /) A0 474/ ALLUIJ/Ivtlél fx/ {.7 Ind. M Stat 373 Term Test III — F10 l‘) [3] Consider a reduced model fit without the Quarter variates. Sketch what you would expect to see in a correlogram (up to lag 10) of the fitted residuals from this reduced model. Briefly ex lain your reasoning. 56ml m flc i ,7 «xx 7, leq// Mcé 7/?{elc Li I- r u a x/jolj Jourrw/ - 1 _ J‘- (juu;c/jl)(m0e/Zae¢// Jf) ci/'/ a J// /J0)/f/Z “id-(7C :fo /KJJ {3/ /°’Z/ lloj (fi 2.91:4 ’(«fa A/Ja JLJJcn’J %4% (me/6t“ h 421%; Wfidc oval/rd /eJ-’4,/I£ /I{ a . . . . ~ ~ . A7‘1i/ 1' 4) Consider the estimatorofthe fitted residuals, r : Y—XB for the linear “IO/(lg? C 40F ‘74 k); ’ R 6 /0 ‘ Y=XB+R‘ R~N(0.OZI) ’ J 2“ i) [2] Show that the F = (I — H)R , where H is the hat matrix. r’tzywfi / / / T’ :xflf/L'XQ(J(/J< y : x/JHZ I XMLJCJJXjér/wtflf = We “5' W“ : 4' /-//{=<I’///Q H ii) [2] Based on the result in i). derive the distribution of? . a?) c riff/H M) : (fl/V) 44/ : 0 (/a/ m : MM am) 0 3 (fa/{flngzflzzm 3 aa‘myyf’g/a) \ F” Wa/fluw/ [/{n (m4. 07L m/mfl/ ’16“ fl ...
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This note was uploaded on 05/03/2011 for the course ECON 202 taught by Professor Na during the Spring '11 term at University of Toronto- Toronto.

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current Term_Test_III_solutions - University of Waterloo...

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