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Unformatted text preview:  b1x )ùúû = 0
(These are called moment restrictions) VER. 9/25/2012. © P. KOLM 14 Moment restrictions: E [y  b0  b1x ] = 0
E éêëx (y  b0  b1x )ùúû = 0
Sample versions of the moment restrictions:
n 1 n å (y
i =1
n i ˆ
ˆ
 b0  b1x i ) = 0 ˆ
ˆ
n 1 å x i (yi  b0  b1x i ) = 0
i =1 The first condition is the same as saying
ˆ
ˆ
y = b0 + b1x
or ˆ
ˆ
b0 = y  b1x
ˆ
So once we calculate the averages y , x (from the sample!) and the slope b1 (see
ˆ
below) we obtain the intercept b
0 VER. 9/25/2012. © P. KOLM 15 ˆ
How do we calculate the slope b1 ?
Recall: n 1 n å x (y
i i =1 i ˆ
ˆ
 b0  b1x i ) = 0 From this we obtain1
n å x (y
i i =1 i ˆ
ˆ
 (y  b1x )  b1x i ) = 0 n n i =1 i =1 ˆ å x i (yi  y ) = b1 å x i (x i  x )
n i =1 n i =1 2
ˆ
x i  x )(yi  y ) = b1 å (x i  x )
å( n ˆ
And therefore, b1 = å (x
i =1 i n å (x
i =1 VER. 9/25/2012. © P. KOLM  x )(yi  y )
x) 2 i provided that n å (x i  x ) > 0
2 i =1 16 Remarks
ˆ
ˆ b0 and b1 are called the ordinary least squares (OLS) estimates of b0 and b1 For any value of x = x i we obtain a fitted value for y,
ˆ
ˆ
ˆ
yi = b0 + b1x i Note: The slope is the sample covariance between x and y divided by the sample variance of x , i.e.
n ˆ
b1 = = å (x
i =1 i  x )(yi  y )
n 1 Cov(x , y )
º rxy sx sy
Var (x ) ⋅ n 1
n å (x i  x ) 2 i =1 o If x and y are positively correlated ( rxy > 0 ), the slope will be positive
o If x and y are negatively correlated ( rxy < 0 ), the slope will be negative Important: We need x to vary in our sample otherwise n å (x
i =1 VER. 9/25/2012. © P. KOLM x) = 0
2 i 17 Alternative Derivation of OLS2 Least squares: Fitting a line through the sample points such that the sum of squared residuals is as small as possible, hence the term “ordinary least
squares”
ˆ The residual, u , is an estimate of the error term, u , and is the difference between the fitted line (sample regression function) and the sample point.
Solving3
n n min å u = min å (yi  b0  b1x i )
b0 , b1 i =1 2
i b0 , b1 2 i =1 ˆ
ˆ
using calculus gives us b0 and b1 as before VER. 9/...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.
 Fall '14

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