LINREG2

In the case(41 σ 2 1 μ y e y j 1 e x j 2 n j 1 e y

Info iconThis preview shows pages 20–23. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: In the case (41), σ 2 ' 1, μ Y ' E ( Y j ) ' 1 % E ( X j ) ' 2, ' n j ' 1 E [( Y j & ¯ Y ) 2 ] ' E ' n j ' 1 ( Y j & μ Y ) & ( ¯ Y & μ Y ) 2 ' E ' n j ' 1 ( Y j & μ Y ) 2 & n ( ¯ Y & μ Y ) 2 ' ( n & 1)var( Y j ) and var( Y j ) ' E [( X j & 1 % U j ) 2 ] ' E [( X j & 1) 2 ] % E [ U 2 j ] ' E [( X j & 1) 2 ] % 1 ' 3, because is distributed and therefore has the same distribution as and it can be shown X j χ 2 1 U 2 j , that for standard normal random variables Thus, the true R 2 in this case U j , E [( U 2 j & 1) 2 ] ' 2. is R 2 ' 1 & n & 2 3( n & 1) ' 2 n & 1 3 n & 3 . 0.7037 for n ' 10 0.6700 for n ' 100 0.6670 for n ' 1000 The estimation results involved are given in Table 2: 21 Table 2 : Artificial regression estimation results ˆ β ˆ α SER ( ' ˆ σ ) R 2 n estimate : 1.11748 0.55912 0.919045 0.8842 10 ( t & value ): (7.817) (1.675) estimate : 1.03309 0.96028 0.992502 0.8284 100 ( t & value ): (21.753) (8.237) estimate : 1.02360 0.98518 0.983608 0.6899 1000 ( t & value ): (47.124) (26.037) Even for a sample size of n = 10 the OLS estimator is already pretty close to its true value 1, ˆ β and the same applies to , but is too far away from the true value α = 1. However, for n = ˆ σ ˆ α 100 the OLS estimators and deviate only about ±4% from their true values α = β = 1, and ˆ β ˆ α ˆ σ deviates about -1% from its true value 1. In the case n = 1000 these deviations reduce to about ±2%. The R 2 's are too high, and only for n = 1000 is the R 2 reasonably close to its true value. 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. 22 APPENDIX Proof of (1): The first-order conditions for a minimum of are: Q (ˆ α , ˆ β ) ' ' n j ' 1 ( Y j & ˆ α & ˆ β X j ) 2 dQ ( $ " , $ $ )/ d $ " ' ] j n j ' 1 2( Y j & $ " & $ $ X j )( & 1) ' ] j n j ' 1 ( Y j & $ " & $ $ X j ) ' ] j n j ' 1 Y j & j n j ' 1 $ " & j n j ' 1 ( $ $ X j ) ' ] j n j ' 1 Y j ' n $ " % $ $ j n j ' 1 X j ' ] ¯ Y ' $ " % $ $ . ¯ X , (42) and dQ ( $ " , $ $ )/ d $ $ ' ] j n j ' 1 2( Y j & $ " & $ $ X j )( & X j ) ' ] j n j ' 1 ( Y j X j & $ " X j & $ $ X 2 j ) ' ] j n j ' 1 X j Y j & $ " j n j ' 1 X j & $ $ j n j ' 1 X 2 j ' ] 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 ' $ " % $ $ ....
View Full Document

{[ snackBarMessage ]}

Page20 / 29

In the case(41 σ 2 1 μ Y E Y j 1 E X j 2 n j 1 E Y j&...

This preview shows document pages 20 - 23. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online