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Consider mltiple linear regression mocel Y _ XB + € with 8 € Rr+1 AId € N,(6.o?). Split X and G into X = (X, X2) Ad 6 (8T.61 respectively: So(lat ...

Question

Consider mltiple linear regression mocel Y _ XB + € with 8 € Rr+1 AId € N,(6.o?). Split X and G into X = (X, X2) Ad 6 (8T.61 respectively: So(lat Wc' Write Y X8, XBz Let 0s,1 (XTX)-'XTY. 85 (851.852)" (XX)-IXTY #dl A = (XTX)-'XTXLet tz (F) 217 614 + 2" B12 . 5z (y) = 2,70s2 A =(xi"x)-X"xz Prove the following (Wo iclentities E(Y(F) - z" 8)? 2f var(85.1)21 + (22 - ATz)" var(852)(22 A Tz1) . E(z(S) - 2" 8)2 2] var(85,1z1 + (22 - TA 21)T 8p28F2

Consider mltiple linear regression mocel Y _ XB + € with 8 € Rr+1 AId € N,(6.o?). Split X and G into X = (X, X2) Ad 6 (8T.61 respectively: So(lat Wc' Write Y X8, XBz Let 0s,1 (XTX)-'XTY. 85 (851.852)" (XX)-IXTY #dl A = (XTX)-'XTX Let tz (F) 217 614 + 2" B12 . 5z (y) = 2,70s2 A =(xi"x)-X"xz Prove the following (Wo iclentities E(Y(F) - z" 8)? 2f var(85.1)21 + (22 - ATz)" var(852)(22 A Tz1) . E(z(S) - 2" 8)2 2] var(85,1z1 + (22 - TA 21)T 8p28F2(22 ATz1).



Answers

Use the data set in WAGE2 for this problem. As usual, be sure all of the following regressions contain
an intercept.
(i) Run a simple regression of $I Q$ on educ to obtain the slope coefficient, say, $\tilde{\delta}_{1}$
(ii) Run the simple regression of log(wage) on educ, and obtain the slope coefficient, $\widetilde{\beta}_{1}$
(iii) Run the multiple regression of log(wage) on educ and $I Q,$ and obtain the slope coefficients,
$\hat{\beta}_{1}$ and $\hat{\beta}_{2},$ respectively.
(iv) Verify that $\tilde{\beta}_{1}=\hat{\beta}_{1}+\hat{\beta}_{2} \tilde{\delta}_{1}$.

In a Given Boston multiply, Chris and one with minus two. So we get the equation, then my next and me. My 28. Oh, okay. You see Quarto minus 23. Wonderful. Look, a great question. Number three. No. Adding a question number two and three make it. Yeah. Can a do you call toe? 2.2. So we get a sequel to, you know, point toe. Now putting a sequel to the rope in total. In the question one, we get five b plus 22. Okay. Sorry to want to is equal to the 1.7. So solving this, they can be ableto 1.9 that exists in line. A question will be What is it for? To Europe Oneto X bless 100.

Not even concerned. We multiply Carson number one with two reminders, too. So we get the person, my understanding, my next 20 a is your Balto minus what you can for him. It'd papers and number three now adding a question two and three Mega Bentley is the call to 9 27 so age call to 0.97 I'm putting 84 to 0.97 and a question one. They got five. B Last 9.7 is a call to 21. To resolving this, they can be able to 2.1. So the question the question in Porto. Why'd you Porto a plus b x 40 x plus b so I can put Europe +197 x plus opened one

If we were looking at the results of, um, a graphing utility finding our line of regression for data. And we saw that there was a linear relationship y equals a X plus b a would be the slope. All right, this is the slope and B would be the intercept. He is the y intercept and our would be the correlation coefficient. That's telling us, uh, how how well the data fits the line or how strong an association there is. So if we look at this and we wanted to analyze it, we would say, Wow, that's really strong, right? There is a strong It's strong. 0.98 is nearly one. It's very strong, positive correlation. It's linear because our best fit equation was your so very strong, positive linear relationship. In other words, we would expect our data toe lie along a straight line, and for most of it to be sitting quite near or on the line itself maybe varies just a tiny bit. The, um what A tells us is that for it, every unit change in X Delta X ray. When the change in X is one unit, the change in why is going to eagle toe 5.175 That's what Slope is. So the correct interpretation of that is the change in why is the 5.175 bay the the 1.257 is the Y intercept? That has nothing to do with how much wide changes with respect to X of the error in the interpretation had been thinking that this told you how much why changed?


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