LI Econometrics
Technical Appendix
1
Properties of Summation
If a and b are constants:
■
n
X
a = a + a + . . . + a = na
i=1
■
n
X
xi = x1 + x2 + x3 + . . . + xn
i=1
■
n
X
bxi = bx1 + bx2 + . . . + bxn = b (x1 + . . . + xn ) = b
i=1
n
X
xi
i=1
3
3
3
X
X
X
■
(xi − yi ) = x1 − y1 + x2 − y2 + x3 − y3 =
xi −
yi
i=1
2
i=1
i=1
Properties of the Expected Value
If a and b are constants:
■
E(a) = a
■
E(aX) = aE(X)
■
E(aX + b) = E(aX) + E(b) = aE(X) + b
■
E(X + Y ) = E(X) + E(Y )
■
n
X
E(xi ) = E(x1 + x2 + ... + xn ) = E(x1 ) + . . . + E(xn ) = E
i=1
■
E(aX + bY ) = E(aX) + E(bY ) = aE(X) + bE(Y )
■
unless X and Y are independent: E(XY ) ̸= E(X)E(Y )
n
X
!
xi
i=1
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LI Econometrics
3
Properties of the Variance
If a and b are constants:
■
var(a) = 0 since constants have unchanging values
Proof: var(a) = E[(a − E(a))2 ] = E[(a − a)2 ] = E(0) = 0
■
var(X + b) = var(X)
■
var(aX) = a2 var(X)
■
var(aX + b) = a2 var(X)
■
var(X + Y ) = E[(X + Y − E(X) − E(Y ))2 ] = E[(X − µX )2 + (Y − µY )2 + 2(X − µX )(Y − µY )]
= var(X) + var(Y ) + 2 cov(X, Y )
■
■
■
4
var(X − Y ) = var(X) + var(Y ) − 2 cov(X, Y )
var(aX + bY ) = a2 var(X) + b2 var(Y ) + 2 ab cov(X, Y ) If X and Y are independent:
E(XY ) = E(X)E(Y ) = µX µY ⇒ cov(X, Y ) = 0
If X and Y are independent so that cov(X, Y ) = 0, then var(aX+bY ) = a2 var(X)+b2 var(Y )
Properties of the Covariance
If a and b are constants:
■
cov(X, Y ) = E[(X − µX )(Y − µY )] = E[XY − XµY − Y µX + µX µY ]
= E(XY ) − µY E(X) − µX E(Y ) + µX µY
= E(XY ) − µX µY
■
cov(a + bX, c + dY ) = E[(a + bX − E(a + bX))(c + dY − E(c + dY ))]
= E[(a + bX − a − bµX )(c + dY − c − dµY )]
= E[b(X − µX ) d(Y − µY )]
= bd cov(X, Y )
■
cov(X, X) = E[(X − µX )(X − µX )] = var(X)
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