Simple Linear Regression Matrix Form

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Rhoda Gordon
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1 Simple Liear Regressio Matrix Form Q.1. A foam beverage isulator (beer hugger) maufacturer produces their product for firms that wat their logo o beer huggers for marketig purposes. The firm s cost aalyst wats to estimate their cost fuctio. She iterprets as the fixed cost of a productio ru, ad as the uit variable cost (or margial cost). Based o = 5 productio rus she observes the followig pairs (Xi, Yi) where Xi is the umber of beer huggers produced i the i th productio ru (i 1000s), ad Yi was the total cost of the ru (i $1000). i X_i Y_i Obtai the followig matrices ad vectors: 1 X, Y, X' X, X' Y, X'X, β Y, e Q.. Give the expeccted values of SS Y'PY ad SSE Y' I - P Y Q.3. For the matrix form of simple liear regressio: p.3.a. Derive the least squares estimator of p.3.b. Give the mea vector ad variace-covariace matrix for the estimator i p.3.a.for Q.4. For the matrix form of simple liear regressio: p.4.a. Write Y ad e as liear fuctios of Y p.4.b. Show that 1 ' P X X'X X is symmetric ad idempotet p.4.c. Use p.4.a. ad p.4.b. to show that i1 Yi ei Y'e 0 Q.5. A egieer is iterested i the relatioship betwee steel thickess (X) ad its breakig stregth (Y). She obtais the followig matrices from a matrix computer package: 1 60 ' ' X X X Y 800 Y' I - P Y Y' P J Y

2 p.5.a. Compute β β ad V p.5.b. Give a 95% Cofidece Iterval for 1 p.5.c. Test H0 : 1 0 H A : 1 0 at = 0.05 sigificace level. SS SSE Q.6. Write out ad as quadratic forms ivolvig Y. Obtai the samplig distributios for each quatity ad show they are idepedet. Q.7. Write e as a liear fuctio of Y, ad use that to derive the mea vector ad covariace matrix of e. Are the residuals ucorrelated? Q.8. For SLR i matrix form, derive the least squares estimate for Q.9. For SLR i matrix form, obtai i i i1 i matrix form (quadratic forms i i1 SSModel Y ad SSE e Y). Obtai the distributios of the two sum of squares divided by (be specific with regard to their family of distributios, degrees of freedom, ad o-cetrality parameters). Q.10. For SLR i matrix form: Derive the Covariace of the vector of sample meas ad the vector of fitted values: 1-1 Y Y1 J Y Y Xβ XX'X X'Y PY Q.11. Cosider the simple liear regressio model: Yi 0 1 Xi X i i 1,..., i ~ NID 0, p.11.a. Give the correspodig X matrix Y vector, ad (symbolically): X'X, X'X X'Y β p.11.b. Give the followig matrices ad vectors (symbolically): -1

3 Q.1. Cosider the MODEL Sum of Squares for Model. SS(Model) Y'Y Y'AY p.1.a.give the defiig matrix A ad show that it is idempotet. p.1.b. Derive the degrees of freedom for SS(Model) p.1.c. Derive the o-cetrality parameter for the o-cetral chi-square distributio of SS(Model)/ p.1.d. Show that SS(Model)/ ad SS(Residual)/ are idepedet (state SS(Residual) as quadratic form) Q.13. A simple liear regressio was fit, relatig the modulus of a tire (Y) to the amout of weeks (X) heated at 15⁰, with results give below: Weeks (X) Modulus(Y) p.13.a. For SLR i matrix form, obtai X, Y, X X, (X X) -1, ad X Y p.13.b. Compute β, Y, e, MSE, V β p.13.c. Compute a 95% Cofidece Iterval for the chage i the mea of modulus as weeks (X) icreases by 1 Q.14. A simple regressio model is fit, with 1 predictor ad a itercept. Defie the projectio matrix as: P = XX'X -1 X'. p.14.a. Show that P is symmetric ad idempotet. (Hit: (X X) -1 is symmetric) p.14.b. What does p.14.c. What does Pij equal? Where Pij is the elemet of P i the i th row ad j th colum j1 Pij equal? j1 ij equal? j1 p.14.d. What does P P i

4 Q.15. The total salaries (X, i millios of pouds) ad the umber of poits eared (Y) for the = 0 Eglish Premier League teams i 1995/6 are used to fit a simple liear regressio model. For this problem, we will treat this as a sample from a populatio of all possible league teams. Team It WageK Poits(Y) X'X X'Y Arseal Asto Villa Blackbur Bolto Chelsea Covetry Everto Leeds Liverpool Machester City Machester Uited Middlesbrough Newcastle Nottigham Forest Quees Park Sheffield Southampto Totteham West Ham Uited Wimbledo p.15.a. Compute X'X -1 ad β p.15.b. Compute the fitted value ad residual for Arseal. 0 p.15.c. i Give Y 5.1 ad Y Y Compute SSR ad SSE i1 p.15.d. Obtai a 95% Cofidece Iterval for 1 Q.16. For the Aalysis of Variace i SLR, with observatios ad 1 predictor, complete the followig parts. p.16.a. Write the Regressio ad Residual sums of squares as quadratic forms. p.16.b. Derive the distributios of SSR/ ad SSE/ p.16.c. Show that SSR/ ad SSE/ are idepedet

5 Q.17. The followig measuremets were obtaied at =5 locatios o the earth s surface, where Y=measured gravity, ad X= latitude. Cosider fittig SLR, cotaiig a itercept. Note this icludes the fitted values ad residuals from results of p.4.a, ad makes use of both scalar model ad matrix model. Obs# X(Lat) Y(Grav) Y-hat e p.17.a. Obtai the followig matrices ad vectors. X'X X'Y INV(X'X) Beta-hat p.17.b. Compute the residual sum of squares, ad give its degrees of freedom (based o scalar model). p.17.c. Compute the regressio sum of squares, ad give its degrees of freedom (based o scalar model). p.17.d. Coduct the F-test, testig H0: 1 = 0 vs HA: 1 0 at = 0.05 sigificace level. p.17.d.i. Test Statistic: p.17.d.ii. Rejectio Regio: Q.18. I the matrix form of the simple liear regressio model, the least squares estimator for is 1 β X'X X'Y where the elemets of X are fixed costats i a cotrolled laboratory experimet. p.18.a. Derive p.18.b. Derive Eβ Show all work V β Show all work

6 Q.19. A experimet is coducted, relatig weekly sales for a food delivery (Y) service to the amout of advertisig (X) durig the week. The results for a sample of = 6 weeks are give below. Fit SLR i matrix form by fillig i the followig matrices. Week Ad Sped (X) Sales (Y) X Y X'X X'Y 1 X'X β Y e SSE e'e Q.0. For the simple regressio model, with a itercept term, complete the followig parts 1 p.0.a. Y Y e Y Xβ Y= J Y prove that Y Y ' Y Y Y Y' Y Y e' e p.0.b. Derive the samplig distributios of Y- Y ad e

Regression, Inference, and Model Building

Regression, Inference, and Model Building Regressio, Iferece, ad Model Buildig Scatter Plots ad Correlatio Correlatio coefficiet, r -1 r 1 If r is positive, the the scatter plot has a positive slope ad variables are said to have a positive relatioship