Econometrics Cheat Sheet
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1
2
3The Rules of Summation
Expectations, Variances & Covariances
n
å
¼ x
þ x
þ Á Á Á þ x
covðX; YÞ ¼ E½ðXÀE½XŠÞðYÀE½YŠÞŠ
x
i
1
2
n
i¼1
¼ å
å
½
x À EðXÞ
Š y À EðYÞ
½
Š f ðx; yÞ
n
å
a ¼ na
x
y
i¼1
covðX;YÞ
r ¼
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n
n
å
¼ a å
ax
x
varðXÞvarðYÞ
i
i
i¼1
i¼1
n
n
n
X þ c
Y ) ¼ c
E(X ) þ c
E(c
E(Y )
å
ðx
þ y
Þ ¼ å
þ å
1
2
1
2
x
y
i
i
i
i
E(X þ Y ) ¼ E(X ) þ E(Y )
i¼1
i¼1
i¼1
n
n
n
å
ðax
þ by
Þ ¼ a å
þ b å
var(aX þ bY þ cZ ) ¼ a
var(X) þ b
var(Y ) þ c
2
2
2
x
y
var(Z )
i
i
i
i
i¼1
i¼1
i¼1
þ 2abcov(X,Y ) þ 2accov(X,Z ) þ 2bccov(Y,Z )
n
n
å
ða þ bx
Þ ¼ na þ b å
x
i
i
If X, Y, and Z are independent, or uncorrelated, random
i¼1
i¼1
n
variables, then the covariance terms are zero and:
å
x
þ x
þ Á Á Á þ x
x
i
1
2
n
x ¼
¼
i¼1
varðaX þ bY þ cZÞ ¼ a
n
2
n
varðXÞ
n
å
ðx
À xÞ ¼ 0
þ b
varðYÞ þ c
2
2
varðZÞ
i
i¼1
2
3
2
Normal Probabilities
å
å
f ðx
; y
Þ ¼ å
½
f ðx
; y
Þ þ f ðx
; y
Þ þ f ðx
; y
Þ
Š
i
j
i
1
i
2
i
3
i¼1
j¼1
i¼1
X À m
¼ f ðx
; y
Þ þ f ðx
; y
Þ þ f ðx
; y
Þ
If X $ N(m, s
), then Z ¼
$ Nð0; 1Þ
2
1
1
1
2
1
3
s
þ f ðx
; y
Þ þ f ðx
; y
Þ þ f ðx
; y
Þ
2
1
2
2
2
3
If X $ N(m, s
2
) and a is a constant, then
a À m
PðX ! aÞ ¼ P Z !
Expected Values & Variances
s
If X $ Nðm; s
Þ and a and b are constants; then
2
EðXÞ ¼ x
f ðx
Þ þ x
f ðx
Þ þ Á Á Á þ x
f ðx
Þ
1
1
2
2
n
n
b À m
n
aÀm
bÞ ¼ P
¼ å
f ðx
Þ ¼ å
x f ðxÞ
Pða
X
Z
x
i
i
s
s
i¼1
x
½
Š ¼ å
gðxÞ f ðxÞ
E gðXÞ
Assumptions of the Simple Linear Regression
x
Model
½
ðXÞ þ g
ðXÞ
Š ¼ å
½
ðxÞ þ g
ðxÞ
Š f ðxÞ
E g
g
1
2
1
2
x
The value of y, for each value of x, is y ¼ b
þ
SR1
1
¼ å
ðxÞ f ðxÞ þ å
ðxÞ f ðxÞ
g
g
1
2
x þ e
b
x
x
2
¼ E g
½
ðXÞ
Š þ E g
½
ðXÞ
Š
SR2
The average value of the random error e is
1
2
E(e) ¼ 0 since we assume that E(y) ¼ b
þ b
x
E(c) ¼ c
1
2
The variance of the random error e is var(e) ¼
E(cX ) ¼ cE(X )
SR3
2
¼ var(y)
s
E(a þ cX ) ¼ a þ cE(X )
var(X ) ¼ s
¼ E[X À E(X )]
¼ E(X
) À [E(X )]
SR4
The covariance between any pair of random
2
2
2
2
) ¼ cov(y
) ¼ 0
var(a þ cX ) ¼ E[(a þ cX) À E(a þ cX)]
¼ c
errors, e
and e
is cov(e
, e
, y
2
2
var(X )
i
j
i
j
i
j
SR5
The variable x is not random and must take at
least two different values.
Marginal and Conditional Distributions
SR6
(optional) The values of e are normally dis-
tributed about their mean e $ N(0, s
2
f ðxÞ ¼ å
f ðx; yÞ for each value X can take
)
y
f ðyÞ ¼ å
f ðx; yÞ for each value Y can take
Least Squares Estimation
x
f ðx; yÞ
If b
and b
are the least squares estimates, then
1
2
f ðxjyÞ ¼ P X ¼ xjY ¼ y
½
Š ¼
f ðyÞ
^ y
¼ b
þ b
x
1
2
i
i
^ e
¼ y
À ^ y
¼ y
À b
À b
x
If X and Y are independent random variables, then
i
i
i
1
2
i
i
f (x,y) ¼ f (x)f ( y) for each and every pair of values
The Normal Equations
x and y. The converse is also true.
þ Sx
¼ Sy
Nb
b
1
i
2
i
If X and Y are independent random variables, then the
þ Sx
¼ Sx
2
Sx
b
b
y
conditional probability density function of X given that
i
1
2
i
i
i
Least Squares Estimators
f ðx; yÞ
f ðxÞ f ðyÞ
Y ¼ y is f ðxjyÞ ¼
¼
¼ f ðxÞ
f ðyÞ
f ðyÞ
À xÞðy
À yÞ
Sðx
i
i
¼
b
2
S ðx
À xÞ
2
for each and every pair of values x and y. The converse is
i
¼ y À b
also true.
b
x
1
2
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