Econometrics Cheat Sheet Page 2
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1
2
3Rejection rule for a two-tail test: If the value of the
Elasticity
test statistic falls in the rejection region, either tail of
percentage change in y
Dy=y
Dy
x
h ¼
¼
¼
Á
the t-distribution, then we reject the null hypothesis
percentage change in x
Dx=x
Dx
y
and accept the alternative.
DEðyÞ=EðyÞ
DEðyÞ
x
x
h ¼
¼
Á
¼ b
Á
Type I error: The null hypothesis is true and we decide
2
Dx=x
Dx
EðyÞ
EðyÞ
to reject it.
Least Squares Expressions Useful for Theory
Type II error: The null hypothesis is false and we decide
not to reject it.
¼ b
þ Sw
b
e
2
i
i
2
p-value rejection rule: When the p-value of a hypoth-
À x
x
i
¼
esis test is smaller than the chosen value of a, then the
w
i
À xÞ
2
Sðx
i
test procedure leads to rejection of the null hypothesis.
¼ 0;
¼ 1;
¼ 1=Sðx
À xÞ
2
2
Sw
Sw
x
Sw
Prediction
i
i
i
i
i
¼ b
þ b
þ e
; ^ y
¼ b
þ b
; f ¼ ^ y
À y
y
x
x
Properties of the Least Squares Estimators
0
0
0
1
2
0
0
1
2
0
0
"
#
"
#
q
ffiffiffiffiffiffiffiffiffiffiffiffiffi
ðx
À xÞ
2
1
2
2
b
b
Sx
s
0
varð f Þ ¼ ^ s
1 þ
þ
; seð f Þ ¼
varð f Þ
2
Þ ¼ s
Þ ¼
2
i
varðb
varðb
À xÞ
2
1
2
N
Sðx
À xÞ
2
À xÞ
2
NSðx
Sðx
i
i
i
"
#
A (1 À a) Â 100% confidence interval, or prediction
Àx
; b
Þ ¼ s
2
interval, for y
covðb
1
2
0
À xÞ
2
^ y
Æ t
seð f Þ
Sðx
i
0
c
Goodness of Fit
Gauss-Markov Theorem: Under the assumptions
À yÞ
2
¼ Sð^ y
À yÞ
2
þ S^ e
2
Sðy
SR1–SR5 of the linear regression model the estimators
i
i
i
b
and b
have the smallest variance of all linear and
1
2
SST ¼ SSR þ SSE
unbiased estimators of b
and b
. They are the Best
1
2
SSR
SSE
¼
¼ 1 À
¼ ðcorrðy; ^ y ÞÞ
2
Linear Unbiased Estimators (BLUE) of b
and b
.
2
R
1
2
SST
SST
If we make the normality assumption, assumption
Log-Linear Model
SR6, about the error term, then the least squares esti-
x þ e; b
lnðyÞ ¼ b
þ b
lnð yÞ ¼ b
þ b
x
mators are normally distributed.
!
!
1
2
1
2
å x
100 Â b
% % change in y given a one-unit change in x:
2
2
2
s
s
$ N b
;
; b
$ N b
;
2
i
b
1
2
1
2
À xÞ
2
À xÞ
2
NSðx
Sðx
^ y
¼ expðb
þ b
i
i
xÞ
n
1
2
^ y
¼ expðb
þ b
=2Þ
2
xÞexpð^ s
Estimated Error Variance
c
1
2
2
S^ e
Prediction interval:
^ s
¼
2
i
h
i
h
i
N À 2
exp b
; exp b
lnðyÞ À t
seð f Þ
lnð yÞ þ t
seð f Þ
c
c
¼ ðcorrðy;^ y
ÞÞ
2
2
Estimator Standard Errors
Generalized goodness-of-fit measure R
n
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
g
b
b
Þ ¼
Þ
;
Þ ¼
Þ
Assumptions of the Multiple Regression Model
seðb
varðb
seðb
varðb
1
1
2
2
¼ b
þ b
þ Á Á Á þ b
þ e
MR1 y
x
x
t-distribution
i
1
2
i2
K
iK
i
) ¼ b
þ b
þ Á Á Á þ b
, E(e
) ¼ 0.
MR2 E(y
x
x
i
1
2
i2
K
iK
i
If assumptions SR1–SR6 of the simple linear regression
) ¼ var(e
) ¼ s
2
MR3 var(y
i
i
model hold, then
) ¼ cov(e
) ¼ 0
MR4 cov(y
, y
, e
À b
i
j
i
j
b
k
t ¼
k
$ t
; k ¼ 1; 2
ðNÀ2Þ
MR5 The values of x
are not random and are not
Þ
ik
seðb
k
exact linear functions of the other explanatory
Interval Estimates
variables.
À t
þ t
)] ¼ 1 À a
b
P[b
se(b
)
b
se(b
$ N½ðb
þ b
þ Á Á Á þ b
Þ; s
Š
2
2
c
2
2
2
c
2
MR6 y
x
x
i
i2
iK
1
2
K
, e
$ Nð0; s
Þ
2
i
Hypothesis Testing
Components of Hypothesis Tests
Least Squares Estimates in MR Model
1. A null hypothesis, H
, . . . , b
0
Least squares estimates b
, b
minimize
1
2
K
2. An alternative hypothesis, H
, . . . , b
Þ ¼ åðy
À b
À b
À Á Á Á À b
Þ
2
1
Sðb
, b
x
x
1
2
K
i
1
2
i2
K
iK
3. A test statistic
4. A rejection region
Estimated Error Variance and Estimator
5. A conclusion
Standard Errors
If the null hypothesis H
¼ c is true, then
: b
0
2
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
å ^ e
À c
2
b
b
2
^ s
¼
Þ ¼
Þ
t ¼
$ t
2
i
seðb
varðb
ðNÀ2Þ
k
k
N À K
Þ
seðb
2
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