However the r 2 is only a descriptive statistic it

This preview shows page 21 - 26 out of 29 pages.

However, the R 2 is only a descriptive statistic; it does not play a role in hypotheses testing, so that the unreliability of the R 2 in small samples is harmless. Notice the quite dramatic increase of the t-values. Recall that these t-values are the test statistics of the null hypotheses that the corresponding parameters are zero. Because the true parameters are equal to 1, what you see in Table 2 is the increase of the power of the t-test with the sample size.
Image of page 21

Subscribe to view the full document.

22 APPENDIX Proof of (1): The first-order conditions for a minimum of are: Q α , ˆ β ) ' ' n j ' 1 ( Y j & ˆ α & ˆ β X j ) 2 dQ ( $ " , $ $ )/ d $ " ' 0 ] j n j ' 1 2( Y j & $ " & $ $ X j )( & 1) ' 0 ] j n j ' 1 ( Y j & $ " & $ $ X j ) ' 0 ] j n j ' 1 Y j & j n j ' 1 $ " & j n j ' 1 ( $ $ X j ) ' 0 ] j n j ' 1 Y j ' n $ " % $ $ j n j ' 1 X j ' 0 ] ¯ Y ' $ " % $ $ . ¯ X , (42) and dQ ( $ " , $ $ )/ d $ $ ' 0 ] j n j ' 1 2( Y j & $ " & $ $ X j )( & X j ) ' 0 ] j n j ' 1 ( Y j X j & $ " X j & $ $ X 2 j ) ' 0 ] j n j ' 1 X j Y j & $ " j n j ' 1 X j & $ $ j n j ' 1 X 2 j ' 0 ] j n j ' 1 X j Y j ' $ " j n j ' 1 X j % $ $ j n j ' 1 X 2 j ] 1 n j n j ' 1 X j Y j ' $ " ¯ X % $ $ 1 n j n j ' 1 X 2 j (43) where are the sample means of the X j 's and Y j 's, ¯ X ' (1/ n ) ' n j ' 1 X j and ¯ Y ' (1/ n ) ' n j ' 1 Y j respectively. The last equations in (42) and (43) are called the normal equations : ¯ Y ' $ " % $ $ . ¯ X , (44) 1 n j n j ' 1 X j Y j ' $ " . ¯ X % $ $ 1 n j n j ' 1 X 2 j . (45) To solve these normal equations, substitute in (45). Then we get ˆ α ' ¯ Y & ˆ β . ¯ X
Image of page 22
23 1 n j n j ' 1 X j Y j ' ( ¯ Y & ˆ β ¯ X ) 1 n j n j ' 1 X j % ˆ β 1 n j n j ' 1 X 2 j ' ¯ Y . ¯ X & ˆ β ¯ X 2 % ˆ β 1 n j n j ' 1 X 2 j ' ¯ X . ¯ Y % ˆ β 1 n j n j ' 1 X 2 j & ¯ X 2 hence 1 n j n j ' 1 X j Y j & ¯ X . ¯ Y ' $ $ 1 n j n j ' 1 X 2 j & ¯ X 2 . (46) Equation (46) can also be written as 1 n j n j ' 1 ( X j & ¯ X )( Y j & ¯ Y ) ' $ $ 1 n j n j ' 1 ( X j & ¯ X ) 2 , (47) because 1 n j n j ' 1 ( X j & ¯ X )( Y j & ¯ Y ) ' 1 n j n j ' 1 X j Y j & ¯ X . Y j & X j . ¯ Y % ¯ X . ¯ Y ' 1 n j n j ' 1 X j Y j & ¯ X . 1 n j n j ' 1 Y j & ¯ Y . 1 n j n j ' 1 X j % ¯ X . ¯ Y ' 1 n j n j ' 1 X j Y j & ¯ X . ¯ Y (48) and similarly 1 n j n j ' 1 ( X j & ¯ X ) 2 ' 1 n j n j ' 1 X 2 j & ¯ X 2 . (49) Moreover, 1 n j n j ' 1 ( X j & ¯ X )( Y j & ¯ Y ) ' 1 n j n j ' 1 ( X j & ¯ X ) Y j & 1 n j n j ' 1 ( X j & ¯ X ) ¯ Y ' 1 n j n j ' 1 ( X j & ¯ X ) Y j & ( ¯ X & ¯ X ) ¯ Y ' 1 n j n j ' 1 ( X j & ¯ X ) Y j (50)
Image of page 23

Subscribe to view the full document.

24 The result (1) now follows from (44) and (46) through (50). Proof of Proposition 1. Recall from (1) that $ $ ' ' n j ' 1 ( X j & ¯ X ) Y j ' n j ' 1 ( X j & ¯ X ) 2 . (51) Substitute model (2) in (51). Then $ $ ' ' n j ' 1 ( X j & ¯ X )( " % $ X j % U j ) ' n j ' 1 ( X j & ¯ X ) 2 ' " ' n j ' 1 ( X j & ¯ X ) % $ ' n j ' 1 ( X j & ¯ X ) X j % ' n j ' 1 ( X j & ¯ X ) U j ' n j ' 1 ( X j & ¯ X ) 2 ' $ . ' n j ' 1 ( X j & ¯ X ) X j ' n j ' 1 ( X j & ¯ X ) 2 % ' n j ' 1 ( X j & ¯ X ) U j ' n j ' 1 ( X j & ¯ X ) 2 ' $ % ' n j ' 1 ( X j & ¯ X ) U j ' n j ' 1 ( X j & ¯ X ) 2 , (52) where the last step follows from the fact that similar to (50), j n j ' 1 ( X j & ¯ X ) 2 ' j n j ' 1 ( X j & ¯ X )( X j & ¯ X ) ' j n j ' 1 ( X j & ¯ X ) X j . (53) Now take the mathematical expectation at both sides of (52). Then, E [ $ $ ] ' $ % E ' n j ' 1 ( X j & ¯ X ) U j ' n j ' 1 ( X j & ¯ X ) 2 ' $ % ' n j ' 1 ( X j & ¯ X ) E ( U j ) ' n j ' 1 ( X j & ¯ X ) 2 ' $ , (54) because taking the mathematical expectation of a constant ( β ) does not effect that constant, and taking the mathematical expectation of a linear function of random variables is equal to taking the linear function of the mathematical expectation of these random variables. The last conclusion in (54) follows from assumption II, and the second step in (54) can be taken because
Image of page 24
25 we have assumed that the X j 's are non-random (assumption IV).
Image of page 25

Subscribe to view the full document.

Image of page 26

{[ snackBarMessage ]}

Get FREE access by uploading your study materials

Upload your study materials now and get free access to over 25 million documents.

Upload now for FREE access Or pay now for instant access
Christopher Reinemann
"Before using Course Hero my grade was at 78%. By the end of the semester my grade was at 90%. I could not have done it without all the class material I found."
— Christopher R., University of Rhode Island '15, Course Hero Intern

Ask a question for free

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern