Fakultet for informasjonsteknologi, Institutt for matematiske fag

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Laurence Wilkerson
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1 Page 1 of 5 NTNU Noregs eknisk-naurviskaplege universie Fakule for informasjonseknologi, maemaikk og elekroeknikk Insiu for maemaiske fag - English Conac during exam: John Tyssedal / Exam in TMA467 Linear saisical models Augus 013 Time Permied aids: A yellow samped A-5 shee wih your own handwrien noes. Tabeller og formler i saisikk (Tapir forlag). K. Roman: Maemaisk formelsamling. Calculaor HP30S or Ciizen SR-70X. Problem 1 X Assume ha he random vecor V Y is rivariae normal disribued wih mean vecor Z and covariance marix a) Find he join disribuion of U X Y and V Y Z. Also deermine for which values of a and b, U and W ay bz are independen. Problem An experimen was conduced in order o invesigae how he addiion of sand and carbon boh measured in % affeced he hardness and srengh of casing. We will now only consider he hardness. The resuls from he experimen performed is given below. Noice ha boh facors have hree levels.

2 Page of 5 Le Yijk Sand\Carbon 0% 0.5% 0.5% 0% 61, 63 69, 69 67, 69 15% 67, 69 69, 74 69, 74 30% 65, 74 74, 7 74, 74 denoe he hardness obained wih he i-h level of sand, he j-h level of carbon in he k-h replicaion, We will assume ha he following model is appropriae for he daa. ijk i j ij ijk, 1,,3, 1,,3, 1, Y i j k where all ijk N 0,. In addiion i j ij ij i1 j1 i1 j1 0 are independen and a) Wha kind of experimen has been performed? Explain wha he parameers in he model mean. Wrie also down he hree hypohesis ha normally are of paricular ineres when daa from such a model is analysed. An oupu from an analysis wih R is given below. > lmcasing=lm(hardness~sand*carbon,casing) > anova(lmcasing) Analysis of Variance Table Response: Hardness Df Sum Sq Mean Sq F value Pr(>F) Sand * Carbon * Sand:Carbon Residuals b) Wrie down he es saisics ha are used in his oupu and give he conclusion on he hree hypohesis of paricular ineres (suggesed in a). Use a 5% level of significance. In which order should he hypohesis ess be performed. The hree levels of Sand are denoed S1, S and S3 in increasing order. Similarly he hree levels of Carbon are denoed C1, C and C3. Below is some oupu from R. Firs we find he average hardness of all observaions > mean(hardness) Then we have calculaed he average hardness for each level of sand > Sandmean=lm(Hardness~Sand -1) > summary(sandmean)

3 Page 3 of 5 Coefficiens: Esimae Sd. Error value Pr(> ) SandS <e-16 *** SandS <e-16 *** SandS <e-16 *** Thereafer we have calculaed he average hardness for each level of Carbon > Carbonmean=lm(Hardness~Carbon -1) > summary(carbonmean) Coefficiens: Esimae Sd. Error value Pr(> ) CarbonC <e-16 *** CarbonC <e-16 *** CarbonC <e-16 *** Finally we find he average of hardness for each level combinaion of Sand and Carbon > apply(hardness,lis(sand,carbon),mean) C1 C C3 S S S c) Find esimaes for, 1 and 33. Perform a es o invesigae if he hardness on he level combinaion (S3, C3) is larger han on he level combinaion (S1, C1). Use a 5% level of significance. We observe ha he disance beween S and S1 is he same as beween S3 and S. We also observe ha he disance beween C and C1 is he same as beween C3 and C. Hence i is possible o define new facor columns by ransforming S1 o -1, S o 0 and S3 o 1 and similarly for C1, C and C3. Le us denoe he new facor columns obained for Sand and Carbon as x 1 and x. Their ransposed values are given below: x 1 = (-1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1, -1, -1, 0, 0, 1, 1) x = (-1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1) From he ineracion plo given below we also observe ha for each level of sand here seems o be some curvaure in hardness when carbon increases.

4 Page 4 of 5 Le us define x x. Two regression analysis was performed where Hardness was regressed on x1 and xand x 1, x and x respecively. The oupu from R is given below > model_1=lm(hardness~x1+x+x1**) > summary(model_1) Coefficiens: Esimae Sd. Error value Pr(> ) (Inercep) < e-16 *** x ** x ** Residual sandard error:.694 on 15 degrees of freedom Muliple R-squared: 0.606, Adjused R-squared: F-saisic: on and 15 DF, p-value: > model_=lm(hardness~x1+x+x) > summary(model_) Coefficiens: Esimae Sd. Error value Pr(> ) (Inercep) < e-16 *** x ** x ** x Residual sandard error:.494 on 14 degrees of freedom Muliple R-squared: , Adjused R-squared: F-saisic: on 3 and 14 DF, p-value:

5 Page 5 of 5 d) Which model will you sugges for he daa? Explain your answer. Explain also why he esimaes for he coefficiens in fron of x 1 and x are he same in hese wo models. Problem 3 Consider he linear model wrien in sandard form Y = X + wih X an n p 1 marix wih rank 1 vecor 0 and covariance marix Le ˆ =Xˆ. p and a vecor of uncorrelaed errors wih mean I. Le ˆ 1 X X X Y be he leas square esimaor for. a) Find he mean vecor and he covariance marix of ˆ. The ha marix H is defined as 1 H = X X X X. b) Wha are he properies of he H marix ha makes i possible o conclude ha Var ˆ i p 1. rank( H ) = race( H )? Show ha Now suppose you are fiing a model assuming expeced response n i1 E Y x while he rue response is given by Y 0 1x1 x. Le H be he ha marix obained assuming 0 1x1 E Y. Define e = Y - HY. c) Show ha e = I - H x I - H Show ha E n e e x I - H x

Cointegration: The Engle and Granger approach

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