LinearRegression2

# 9252012 p kolm 20 references almgren r c thum e

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Unformatted text preview: ariate case VER. 9/25/2012. © P. KOLM 20 References Almgren, R., C. Thum, E. Hauptmann and H. Li (2005). "Direct Estimation of Equity Market Impact." Risk 18: 57–62. Almgren, R., C. Thum, E. Hauptmann and H. Li (2005). "Equity Market Impact." Risk 18(7): 57-62. Wooldridge, J. M. (2009). Introductory Econometrics: A Modern Approach, South-Western Pub. Footnotes 1 In some cases even a nonlinear regression can be mathematically turned into a linear regression. For example, by taking b logs y = b0 + b1x 2 + u becomes log(y ) = log(b0 ) + log(b1 ) + b2 ⋅ log(x ) + log(u ) @ b0 + b1 log(x ) + u . 2 2 Since Var (u | x ) = E (u 2 | x ) - éêëE (u | x )ùúû and E (u | x ) = 0 , we have that s 2 = E (u 2 | x ) = E (u 2 ) = Var (u ) . Therefore s 2 is also the unconditional variance, also called the error variance. An alternative to making this distributional assumption is to consider the asymptotic case (i.e. assuming that the sample size, n, is very large). Then, we would be in the regime where the central limit theorem holds. In this case, the limiting distributions are all normal, with a certain means and variances. We will cover this setting separately in a separate lecture, if there is time. 3 There is a non-zero covariance between the slope and intercept estimators: ˆˆ Cov(b0, b1 ) = s 2 (-x ) n å (x i =1 VER. 9/25/2012. © P. KOLM i - x i2 ) 21...
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