Econometrics Cheat Sheet Page 3
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1
2
3Hypothesis Tests and Interval Estimates for Single Parameters
Regression with Stationary Time Series Variables
À b
b
Finite distributed lag model
k
t ¼
k
$ t
Use t-distribution
ðNÀKÞ
Þ
seðb
¼ a þ b
þ b
þ b
þ Á Á Á þ b
þ v
k
y
x
x
x
x
t
t
tÀ1
tÀ2
tÀq
t
0
1
2
q
t-test for More than One Parameter
Correlogram
¼ å ðy
À yÞðy
À yÞ= å ðy
À yÞ
2
: b
þ cb
¼ a
r
H
k
t
tÀk
t
0
p
ffiffiffi ffi
2
3
: r
¼ 0;
z ¼
$ Nð0; 1Þ
þ cb
À a
For H
T
r
b
0
k
k
t ¼
2
3
$ t
When H
is true
ðNÀKÞ
0
þ cb
Þ
seðb
LM test
2
3
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ b
þ b
þ r^ e
þ ^ v
: r ¼ 0 with t-test
y
x
Test H
t
t
tÀ1
t
0
1
2
b
b
Þ þ 2c  b
þ cb
Þ ¼
Þ þ c
; b
Þ
2
seðb
varðb
varðb
covðb
2
3
2
3
2
3
^ e
¼ g
þ g
þ r^ e
þ ^ v
Test using LM ¼ T Â R
2
x
t
t
tÀ1
t
1
2
¼ b
þ b
þ e
¼ re
þ v
Joint F-tests
AR(1) error
y
x
e
t
t
t
t
tÀ1
t
1
2
To test J joint hypotheses,
Nonlinear least squares estimation
ðSSE
À SSE
Þ=J
R
U
F ¼
¼ b
ð1 À rÞ þ b
þ ry
À b
þ v
y
x
rx
=ðN À KÞ
t
t
tÀ1
tÀ1
t
1
2
2
SSE
U
ARDL(p, q) model
To test the overall significance of the model the null and alternative
¼ d þ d
þ d
þ Á Á Á þ d
þ u
y
x
x
x
y
hypotheses and F statistic are
t
0
t
l
tÀ1
q
tÀq
l
tÀ1
þ Á Á Á þ u
þ v
: b
¼ 0; b
¼ 0; : : : ; b
¼ 0
y
H
p
tÀp
t
0
2
3
K
AR(p) forecasting model
H
: at least one of the b
is nonzero
1
k
¼ d þ u
þ u
þ Á Á Á þ u
þ v
y
y
y
y
ðSST À SSEÞ=ðK À 1Þ
t
l
tÀ1
2
tÀ2
p
tÀp
t
F ¼
^ y
¼ ay
þ ð1 À aÞ^ y
Exponential smoothing
SSE=ðN À KÞ
t
tÀ1
tÀ1
Multiplier analysis
RESET: A Specification Test
þ d
L þ d
2
þ Á Á Á þ d
q
¼ ð1 À u
L À u
2
À Á Á Á À u
p
Þ
d
L
L
L
L
0
1
2
q
1
2
p
¼ b
þ b
þ b
þ e
^ y
¼ b
þ b
þ b
y
x
x
x
x
 ðb
þ b
L þ b
þ Á Á ÁÞ
i
i2
i3
i
1
2
i2
3
i3
2
1
2
3
i
L
0
1
2
¼ b
þ b
þ b
þ g
^ y
2
þ e
;
: g
¼ 0
y
x
x
H
Unit Roots and Cointegration
i
i2
i3
i
0
1
2
3
1
1
i
¼ b
þ b
þ b
þ g
^ y
þ g
^ y
þ e
;
: g
¼ g
¼ 0
2
3
y
x
x
H
i
i2
i3
i
0
Unit Root Test for Stationarity: Null hypothesis:
1
2
3
1
2
1
2
i
i
: g ¼ 0
H
Model Selection
0
Dickey-Fuller Test 1 (no constant and no trend):
AIC ¼ ln(SSE=N) þ 2K=N
¼ gy
þ v
Dy
SC ¼ ln(SSE=N) þ K ln(N)=N
t
tÀ1
t
Dickey-Fuller Test 2 (with constant but no trend):
Collinearity and Omitted Variables
¼ a þ gy
þ v
Dy
t
tÀ1
t
¼ b
þ b
þ b
þ e
y
x
x
i
i2
i3
i
1
2
3
Dickey-Fuller Test 3 (with constant and with trend):
2
s
Þ ¼
¼ a þ gy
þ lt þ v
Dy
varðb
2
t
tÀ1
t
ð1 À r
Þ å ðx
À x
Þ
2
2
i2
2
23
Augmented Dickey-Fuller Tests:
b
; x
Þ
m
covðx
Ã
Ã
¼ a þ gy
þ å
þ v
2
3
Þ ¼ Eðb
Þ À b
¼ b
Dy
Dy
a
When x
is omitted; biasðb
t
tÀ1
s
tÀs
t
3
b
2
3
2
2
Þ
s¼1
varðx
2
Test for cointegration
¼ g^ e
þ v
D^ e
Heteroskedasticity
t
tÀ1
t
¼ y
þ v
) ¼ var(e
) ¼ s
Random walk:
y
2
var(y
t
tÀ1
t
i
i
i
¼ a þ y
þ v
Random walk with drift:
y
General variance function
t
tÀ1
t
Random walk model with drift and time trend:
¼ expða
þ a
þ Á Á Á þ a
Þ
2
s
z
z
¼ a þ dt þ y
þ v
1
2
i2
S
iS
i
y
t
tÀ1
t
¼ a
¼ Á Á Á ¼ a
¼ 0
Breusch-Pagan and White Tests for H
: a
0
2
3
S
Panel Data
is true x
¼ N Â R
$ x
2
2
2
When H
0
ðSÀ1Þ
Pooled least squares regression
¼ s
: s
6 ¼ s
2
2
2
2
Goldfeld-Quandt test for H
: s
versus H
¼ b
þ b
þ b
þ e
y
x
x
0
1
M
R
M
R
it
2it
3it
it
1
2
3
is true F ¼ ^ s
=^ s
$ F
2
2
) ¼ c
When H
Cluster robust standard errors cov(e
, e
ðN
ÀK
;N
ÀK
Þ
0
M
R
it
is
ts
M
M
R
R
Þ ¼ s
¼ s
2
2
Fixed effects model
Transformed model for varðe
x
i
i
p
ffiffiffi ffi
p
ffiffiffi ffi
i
p
ffiffiffi ffi
p
ffiffiffi ffi
¼ b
þ b
þ b
þ e
y
x
x
b
not random
=
¼ b
ð
Þ þ b
ð
=
Þ þ e
=
y
x
1=
x
x
x
x
it
2it
3it
it
1i
2
3
1i
i
i
i
i
i
i
i
1
2
À y
¼ b
ðx
À x
Þ þ b
ðx
À x
Þ þ ðe
À e
Þ
y
it
2it
2i
3it
3i
it
i
i
2
3
Estimating the variance function
Random effects model
2
Þ ¼ lnðs
Þ þ v
¼ a
þ a
þ Á Á Á þ a
þ v
2
lnð^ e
z
z
i
1
2
i2
S
iS
i
i
i
¼ b
þ b
þ b
þ e
¼ b
þ u
y
x
x
b
random
it
2it
3it
it
i
1i
2
3
it
1
Grouped data
(
Ã
À ay
¼ b
ð1 À aÞ þ b
ðx
À ax
Þ þ b
ðx
À ax
Þ þ v
y
it
2it
2i
3it
3i
i
1
2
3
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
it
i ¼ 1; 2; . . . ; N
2
q
s
M
Þ ¼ s
¼
M
2
varðe
i
a ¼ 1 À s
þ s
i
i ¼ 1; 2; . . . ; N
2
2
2
Ts
s
e
R
u
e
R
Transformed model for feasible generalized least squares
Hausman test
h
i
. ffiffiffiffi ffi
. ffiffiffiffi ffi
. ffiffiffiffi ffi
. ffiffiffiffi ffi
p
p
p
p
1=2
b
Þ À b
t ¼ ðb
À b
Þ
Þ
^ s
¼ b
^ s
þ b
^ s
þ e
^ s
varðb
varðb
y
1
x
FE;k
RE;k
FE;k
RE;k
i
i
i
i
i
i
i
1
2
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