Formulas and Tables
for
Time Series Econometrics / Econometrics II
July, 2022
© Prof. Dr. Kai Carstensen
Institute for Statistics and Econometrics, CAU Kiel
Formulas and Tables for Time Series Econometrics / Econometrics II
Part A. Stationary Time Series Processes
1
1.1
Stationary Time Series Regression
Properties of Stochastic Time-Series Processes
Autocovariance of order k:
Cov(yt , yt−k ) = E{[yt − E(yt )][yt−k − E(yt−k )]}
Autocorrelation of order k:
Corr(yt , yt−k ) =
E{[yt − E(yt )][yt−k − E(yt−k )]}
p
p
.
Var(yt ) Var(yt−k )
Cross covariance of order k:
Cov(xt , yt−k ) = E{[xt − E(xt )][yt−k − E(yt−k )]}.
Cross correlation of order k:
Corr(xt , yt−k ) =
E{[xt − E(xt )][yt−k − E(yt−k )]}
p
p
.
Var(xt ) Var(yt−k )
Weak stationarity: A time series process {yt } is called weakly stationary if the first and
second moments are time-invariant and finite, i.e.,
E(yt ) = µ < ∞ ∀t
Var(yt ) = σ 2 < ∞ ∀t
Cov(yt , yt−k ) = γk < ∞ ∀t
Strong stationarity: A time series process {yt } is called strongly stationary if the joint
probability distribution of any set of k observations in the sequence [yt , yt+1 , . . . , yt+k−1 ] is the
same regardless of the origin, t, in the time scale.
Ergodicity: A strongly stationary time-series process, {yt }, is ergodic if for any two bounded
functions that map vectors in the a and b dimensional real vector spaces to real scalars, f :
Ra → R1 and g : Rb → R1 ,
lim |E[f (yt , . . . , yt+a−1 )g(yt+k , . . . , yt+k+b−1 )]|
k→∞
= |E[f (yt , . . . , yt+a−1 )]| |E[g(yt+k , . . . , yt+k+b−1 )]|
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Ergodic Theorems for scalar processes: If {yt } is a time-series process that is strongly
stationary and ergodic and E[yt ] = µ exists and is a finite constant, then
ȳT = T
−1
T
X
a.s.
yt −→ µ.
t=1
If in addition E[yt2 ] = m exists and is finite then
T −1
T
X
a.s.
yt2 −→ m.
t=1
Ergodic Theorems for vector processes: If {yt } is a K × 1 vector of strongly stationary
and ergodic processes with existing and finite moments E[yt ] = µ and E[yt yt0 ] = M, then
T −1
T
X
a.s.
yt −→ µ
t=1
and
T
−1
T
X
a.s.
yt yt0 −→ M.
t=1
Martingale sequence: A vector sequence zt is a martingale sequence if E[zt |zt−1 , zt−2 , . . .] =
zt−1 .
Martingale difference sequence: A vector sequence zt is a martingale difference sequence
if E[zt |zt−1 , zt−2 , . . .] = 0.
Stationary linear process: A stationary linear process with mean zero is defined as
ut = ψ0 εt + ψ1 εt−1 + ψ2 εt−2 + · · · = ψ(L)εt ,
where εt is iid white noise with mean zero and variance σ 2 > 0, if
(C1)
∞
X
j|ψj | < ∞
j=0
(C2)
ψ(1) = ψ0 + ψ1 + ψ2 + · · · =
6 0
Then ut has mean E(ut ) = 0, variance Var(ut ) = σ 2 (ψ02 + ψ12 + ψ22 + · · · ) and long-run variance
√
λ2 ≡ lim Var( T ūT ) = σ 2 [ψ(1)]2 > 0.
T →∞
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1.2
Asymptotic Properties of Time Series Regressions
Population model and OLS assuptions: The population model with K regressors is
y t = xt β + u t ,
t = 1, . . . , T,
where {[x, u]} is a jointly stationary and ergodic process with finite first and second moments
and the usual OLS assumptions E(x0t ut ) = 0 and rank[E(x0t xt )] = K hold.
Consistency: The OLS estimator of the population model is consistent,
p
β̂ −→ β.
Martingale Difference Central Limit Theorem: If zt is a vector valued stationary and
ergodic martingale difference sequence, with E[zt z0t ] = Σ, where Σ is a finite positive definite
matrix, then
√
T z̄T = T
−1/2
T
X
d
zt −→ Normal(0, Σ).
t=1
Asymptotic distribution of the OLS estimator when ut is white noise: By the martingale difference CLT,
T
−1/2
T
X
d
x0t ut −→ Normal(0, B),
t=1
where B ≡ E(u2t x0t xt ). From this follows
√
d
T (β̂ − β) = −→ Normal(0, V),
where V = A−1 BA−1 and A ≡ E(x0t xt ).
Estimation of the variance of the OLS estimator when ut is white noise: Using the
OLS residuals ût , consistent estimators are
 = T −1
T
X
x0t xt
t=1
and
B̂ = T
−1
T
X
û2t x0t xt
t=1
such that
V̂ = Â−1 B̂Â−1 .
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Formulas and Tables for Time Series Econometrics / Econometrics II
Gordin’s Central Limit Theorem: If {zt } is a stationary and ergodic stochastic process
of dimension K × 1 that satisfies the following conditions:
(1) Asymptotic uncorrelatedness: E[zt |zt−k , zt−k−1 , . . .] converges in mean square to zero as
k → ∞.
(2) Summability of autocovariances: the asymptotic variance Γ∗ is finite, where
√
lim Var( T z̄T ) =
T →∞
∞ X
∞
X
Cov(zt , zs ) =
t=0 s=0
∞
X
Cov(zt , zt−k ) = Γ∗ .
k=−∞
(3) Asymptotic negligibility of innovations: information eventually becomes negligible as it
fades far back in time from the current observation.
Then
√
T z̄T = T
−1/2
T
X
d
zt −→ Normal(0, Γ∗ ).
t=1
Asymptotic distribution of the OLS estimator when ut is temporally dependent:
If x0t ut satisfies the conditions of Gordin’s CLT,
T
−1/2
T
X
d
x0t ut −→ Normal(0, B),
t=1
where B ≡
√
P∞
k=−∞
Cov(x0t ut , x0t−k ut−k ). From this follows
d
T (β̂ − β) = −→ Normal(0, V),
where V = A−1 BA−1 and A ≡ E(x0t xt ).
Estimation of the variance of the OLS estimator when ut is temporally dependent:
A consistent estimate is
V̂ = Â−1 B̂Â−1 ,
where
 = T
−1
T
X
p
x0t xt −→ A.
t=1
and
q
X
0
B̂ = Γ̂0 +
(Γ̂k + Γ̂k ),
k=1
with
Γ̂k = T
−1
T
X
ût ût−k x0t xt−k .
t=k+1
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Formulas and Tables for Time Series Econometrics / Econometrics II
Newey-West estimator: A positive semidefinite estimator of B is
q X
0
k
(Γ̂k + Γ̂k ).
B̂ = Γ̂0 +
1−
q+1
k=1
2
2.1
Autoregressive Models
Properties of the Autoregressive Model of Order 1
Autoregressive model of order 1 without constant:
iid
et ∼ (0, σ 2 )
ut = ρut−1 + et ,
Moving average representation:
2
ut = et + ρet−1 + ρ et−2 + · · · =
∞
X
ρi et−i
i=0
Conditional moments:
E(ut |ut−1 ) = ρut−1
Var(ut |ut−1 ) = σ 2
Unconditional moments: The mean is
E(ut ) = 0
and, if |ρ| < 1, the second moments are
1
Var(ut ) = σ 2
1 − ρ2
ρk
Cov(ut , ut−k ) = σ 2
1 − ρ2
Corr(ut , ut−k ) = ρk .
2.2
Estimation of the Autoregressive Model of Order 1
Consistency of OLS when the disturbance is white noise: The OLS estimator of the
AR(1) model without constant,
!−1 T
T
X
X
ρ̂ =
u2t−1
ut−1 ut ,
t=2
t=2
is consistent if et is white noise and thus E(ut−1 et ) = 0.
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Formulas and Tables for Time Series Econometrics / Econometrics II
Asymptotic normality of OLS when the disturbance is white noise: By the martingale difference CLT,
√
d
T (ρ̂ − ρ) −→ Normal(0, V).
If the disturbance et is homoscedastic, the asymptotic covariance can be estimated as
!−1
!−1
T
T
X
X
2
2
[
yt−1
V̂ = σ̂e2 T −1
⇒
Avar(ρ̂)
= σ̂e2
yt−1
.
t=2
t=2
Inconsistency of OLS when the disturbance is autocorrelated: If the disturbance et
is autocorrelated, the OLS estimator of the AR(1) model is inconsistent.
Asymptotic bias of OLS when the disturbance follows an AR(1) process: If et
follows itself an AR(1) process,
et = φet−1 + εt ,
where εt is iid white noise, the asymptotic bias of the OLS estimator of ρ is
ρ+φ
.
plim ρ̂ =
1 + ρφ
2.3
The Autoregressive Model of Order p
Lag operator: The lag operator is defined as Lyt = yt−1 and thus
Lp yt = yt−p ,
p ∈ {. . . , −2, −1, 0, 1, 2, . . .}.
Lag polynomial: A polynomial in L is called a lag polynomial. The lag polynomial
a(L) ≡ a0 + a1 L + . . . + ap Lp
can be applied to a time series variable yt to yield
a(L)yt = a0 yt + a1 yt−1 + . . . + ap yt−p .
Rules for lag polynomials: Lag polynomials can be multiplied. For example, define a(L) =
a0 + a1 L and b(L) = b0 + b1 L + b2 L2 , then
a(L)b(L) = a0 b0 + (a0 b1 + a1 b0 )L + (a0 b2 + a1 b1 )L2 + a1 b2 L3 .
Lag polynomials can be evaluated at 1,
a(1) = a0 + a1 + . . . + ap =
p
X
ai .
i=0
A lag polynomial applied to a time invariant quantity µ yields
a(L)µ = a0 µ + a1 Lµ + . . . + ap Lp µ = a0 µ + a1 µ + . . . + ap µ = a(1)µ.
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The AR(p) model: The autoregressive model of order p is
yt = µ + a1 yt−1 + . . . + ap yt−p + et ,
or
a(L)yt = µ + et ,
where the disturbance process et is iid white noise with mean zero.
OLS Estimation of the AR(p) model: The OLS estimator
 
µ̂
â1 
 
 ..  = (Z0 Z)−1 Z0 Y.
.
âp
is based on the observation matrices



yp+1
1 yp
 .. 
 ..
..
Z = .
Y =  . ,
.
yT
···
1 yT −1 · · ·

y1
..  .
. 
yT −p
If the disturbance is white noise, the OLS estimator is consistent and asymptotically normal.
3
3.1
Dynamic Regression
Autoregressive Distributed Lag Model
Autoregressive distributed lag (ADL) model:
yt = µ + a1 yt−1 + a2 yt−2 + . . . + ap yt−p + b0 xt + b1 xt−1 + . . . + bq xt−q + εt
or, using lag polynomials,
a(L)yt = µ + b(L)xt + εt .
OLS estimation of the ADL model: If E[εt |yt−1 , yt−2 , . . . , xt , xt−1 , xt−2 , . . .] = 0, the joint
process {(yt , xt )} is stationary and ergodic, and there is no perfect multicollinearity among the
regressors 1, yt−1 , . . . , yt−p , xt , . . . , xt−q , then the OLS estimator of the ADL model is consistent
and asymptotically normally distributed.
Interpretation of the ADL coefficients as partial effects:
∂ E(yt |1, yt−1 , . . . , yt−p , xt , . . . , xt−q )
= ai
∂yt−i
∂ E(yt |1, yt−1 , . . . , yt−p , xt , . . . , xt−q )
= bi
∂xt−i
i ∈ [1, . . . , p]
i ∈ [0, . . . , q]
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Formulas and Tables for Time Series Econometrics / Econometrics II
Long-run effect of a permanent shift in x on y:
∂ E(y|x)
b(1)
b0 + b1 + . . . + bq
=
=
∂x
a(1)
1 − a1 − . . . − ap
3.2
Error-Correction Model
Error-correction model (ECM): The ECM is a reparameterization of the ADL model:
∆yt = µ − α(yt−1 − βxt−1 ) + ā1 ∆yt−1 + . . . + āp−1 ∆yt−p+1 + b̄0 ∆xt + . . . + b̄q−1 ∆xt−q+1 + εt
where ∆ = 1 − L and
α = a(1)
β = b(1)/a(1)
p
X
ai ,
āi = −
for i = 1, . . . , p − 1
k=i+1
b̄0 = b0
b̄i = −
q
X
bi ,
for i = 1, . . . , q − 1
k=i+1
Estimation of the ECM by OLS: Writing the ECM as
∆yt = µ + (−α) yt−1 + αβ xt−1 + lagged differences + εt ,
|{z}
| {z }
γ1
γ2
it can be estimated by OLS. The OLS estimator is, under the conditions stated for the ADL
model, consistent and asymptotically normal.
Estimation of long-run parameters of the ECM: Calculate
β̂ = −
γ̂2
.
γ̂1
The asymptotic standard errors of β̂ can be obtained using the delta method.
4
Tests of Autocorrelation and Lag Order Selection
Durbin-Watson test: Test of the null hypothesis that the regression disturbance εt is not
autocorrelated against the alternative that it is AR(1). Test statistic based on an estimation
sample t = 1, . . . , T :
PT
(ε̂t − ε̂t−1 )2
DW = t=2PT 2
.
ε̂
t=1 t
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Formulas and Tables for Time Series Econometrics / Econometrics II
LM test for autocorrelation (Breusch-Godfrey test): In the model
y t = xt γ + u t ,
ut = ρ1 ut−1 + . . . + ρr ut−r + εt ,
it tests H0 : ρ1 = . . . = ρr = 0
LM = T
against
εt ∼ iid
H1 : ¬H0 . The test statistic is
ε̃0 PZ ε̃
,
ε̃0 ε̃
where ε̃ = Y − Xγ̂ are the residuals obtained under H0 , Z = [X, ε̃−1 , . . . , ε̃−r ] and PZ =
d
Z(Z0 Z)−1 Z0 . Under H0 , LM −→ χ2r .
Information criteria: Let k be the number of parameters in the ADL model. The following
information criteria can be used:
2k
,
T
2k
2k 2 + 2k
AICc(k) = log(σ̂ 2 ) +
+
,
T
T −k−1
2k log log T
HQ(k) = log(σ̂ 2 ) +
,
T
k log T
BIC(k) = log(σ̂ 2 ) +
.
T
AIC(k) = log(σ̂ 2 ) +
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Formulas and Tables for Time Series Econometrics / Econometrics II
Part B. Econometrics for Integrated I(1)
Time Series Processes
5
5.1
Stochastic Trends
Integrated Processes
A process {yt } is called integrated of order d (I(d)) if it is nonstationary and differencing it at
least d times is necessary to obtain a stationary process. In particular, a nonstationary process
is I(1) if its first difference, ∆yt = (1 − L)yt = yt − yt−1 is I(0).
5.2
Stochastic Trends with iid Increments (Random Walks)
Stochastic trend without drift: Let {εt } be a iid white noise process with mean zero and
finite variance σ 2 . A driftless stochastic trend with iid increments is defined as
s t = s 0 + ε1 + · · · + εt = s 0 +
t
X
εt .
t=1
Assuming s0 = 0, it has mean E(st ) = 0 and variance Var(st ) = tσ 2 .
Stochastic trend with drift: Let {εt } be a iid white noise process with mean zero and
finite variance σ 2 . A stochastic trend with drift and iid increments is defined as
st = s0 + µt + ε1 + · · · + εt = s0 +
t
X
(µ + εt ).
t=1
Assuming s0 = 0, it has mean E(st ) = µt and variance Var(st ) = tσ 2 .
5.3
Stochastic Trends with Autocorrelated Increments
Stochastic trend without drift: Let {ut } be a stationary linear process with mean zero
and long-run variance λ2 . A driftless stochastic trend with autocorrelated increments is defined
as
s̃t = s̃0 + u1 + · · · + ut = s̃0 +
t
X
ut .
t=1
Assuming s̃0 =√0 and normalizing by
limt→∞ Var(s̃t / t) = λ2 .
√
√
t, it has mean E(s̃t / t) = 0 and asymptotic variance
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Stochastic trend with drift: Let {ut } be a stationary linear process with mean zero and
long-run variance λ2 . A stochastic trend with drift and autocorrelated increments is defined as
s̃t = s̃0 + µt + u1 + · · · + ut = s̃0 +
t
X
(µ + ut ).
t=1
Assuming s̃0 = √
0 and normalizing by
limt→∞ Var(s̃t / t) = λ2 .
5.4
√
√
√
t, it has mean E(s̃t / t) = µ t and asymptotic variance
Beveridge-Nelson Decomposition
Let s̃t be I(1) with drift so that ∆s̃t is a linear I(0) process with long-run variance λ2 ,
s̃t = s̃0 + µt +
t
X
⇒
ui
∆s̃t = µ + ut = µ + ψ(L)εt ,
i=1
where Var(εt ) = σ 2 . Then s̃t can be linearly decomposed into (a) a linear deterministic trend,
(b) a random walk, (c) a stationary process η, and (d) an initial condition z0 = s̃0 − η0 :
s̃t =
µt
|{z}
divergence at rate t
6
+
(λ/σ) s
| {z }t
divergence at rate
+
√
t
ηt
|{z}
bounded in probability
+
z0
|{z}
.
bounded in probability
Unit Root Asymptotics
Standard Brownian motion (Wiener process): The standard Brownian motion W (t),
t ∈ [0, 1] is defined as a stochastic process with the following properties.
• Initialization: W (0) = 0.
• Normality: W (t) is distributed as N (0, t).
• Independent increments: For given points in time 0 ≤ t1 < t2 < . . . tk ≤ 1, the increments
(W (t2 ) − W (t1 )), (W (t3 ) − W (t2 )), . . . , (W (tk ) − W (tk−1 )) are stochastically independent.
Any increment W (s) − W (t), s > t, is normally distributed with mean zero and variance
s − t.
• The process is continuous in t.
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Functional central limit theorem: Let ε1 , . . . , εT be iid white noise random variables with
mean zero and and variance σ 2 . Furthermore define [T r], 0 ≤ r ≤ 1, as the largest integer
smaller or equal to T r. Then the step-function
[T r]
1 X
εt , 0 ≤ r ≤ 1
XT (r) = √
σ T t=1
with XT (r) = 0 for [T r] < 1 converges weakly towards a standard Brownian motion, XT (r) ⇒
W (r), as T → ∞.
Continuous mapping theorem: By the continuous mapping theorem, if XT (r) ⇒ W (r)
then f (XT (r)) ⇒ f (W (r)) for continuous functions f .
Convergence of sample moments of random walks without drift: Let ε1 , . . . , εT be
iid white noise random variables with mean zero and and variance σ 2 and define the random
walk without drift yt = yt−1 + εt , y0 = 0. Then the following convergence results hold:
R1
P
⇒ σ 0 W (r)dr
(1) T −3/2 Tt=1 yt−1
(2) T −2
PT
(3) T −1
PT
2
t=1 yt−1
t=1
⇒ σ2
R1
0
(W (r))2 dr
∆yt yt−1 ⇒ 0.5σ 2 (W (1)2 − 1)
Convergence of sample moments of a stochastic trend with correlated increments:
Let {ut } be a linear I(0) process with mean zero, finite variance γ0 and finite long-run variance
λ2 > 0 and define the driftless stochastic trend ỹ t = ỹ t−1 + ut , ut = ψ(L)εt , ỹ 0 = 0. Then the
following convergence results hold:
R1
P
(1) T −3/2 Tt=1 ỹ t−1
⇒ λ 0 W (r)dr
7
7.1
(2) T −2
PT
(3) T −1
PT
t=1
t=1
ỹ 2t−1
⇒ λ2
R1
0
(W (r))2 dr
∆ỹ t ỹ t−1 ⇒ 0.5λ2 W (1)2 − 0.5γ0
Unit Root Tests
Dickey-Fuller (DF) Tests
Dickey-Fuller test without intercept and trend: Estimate by OLS the population model
yt = ρyt−1 + εt ,
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where εt is iid white noise with variance σ 2 and compute the test statistics
!
PT
y
y
t t−1
−1
DF-ρ = T (ρ̂ − 1) = T
Pt=1
T
2
y
t=1 t−1
ρ̂ − 1
ρ̂ − 1
DF-t =
= PT 2
.
SE(ρ̂)
σ̂( t=1 yt−1 )−1/2
Under the null hypothesis ρ = 1, the test statistics have limiting distribution
0.5(W (1)2 − 1)
DF-ρ ⇒ DFρ ≡ R 1
(W (r))2 dr
0
0.5(W (1)2 − 1)
DF-t ⇒ DFt ≡ qR
.
1
2
(W (r)) dr
0
Dickey-Fuller test with intercept: Estimate by OLS the population model
yt = α + ρyt−1 + εt ,
where εt is iid white noise with variance σ 2 and compute the test statistics
DF-ρµ = T (ρ̂µ − 1)
ρ̂µ − 1
DF-tµ =
.
SE(ρ̂µ )
Under the null hypothesis ρ = 1, the test statistics have limiting distribution
0.5([W µ (1)]2 − W µ (0)2 − 1)
R1
(W µ (r))2 dr
0
0.5([W µ (1)]2 − W µ (0)2 − 1)
qR
DF-tµ ⇒ DFµt ≡
.
1
µ
2
(W (r)) dr
0
DF-ρµ ⇒ DFµρ ≡
Dickey-Fuller test with intercept and trend: Estimate by OLS the population model
yt = α0 + α1 t + ρyt−1 + εt ,
where εt is iid white noise with variance σ 2 and compute the test statistics
DF-ρτ = T (ρ̂τ − 1)
ρ̂τ − 1
DF-tτ =
.
SE(ρ̂τ )
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Under the null hypothesis ρ = 1, the test statistics have limiting distribution
0.5([W τ (1)]2 − W τ (0)2 − 1)
R1
(W τ (r))2 dr
0
0.5([W τ (1)]2 − W τ (0)2 − 1)
qR
DF-tτ ⇒ DFτt ≡
.
1
τ (r))2 dr
(W
0
DF-ρτ ⇒ DFτρ ≡
7.2
Phillips-Perron (PP) Test
Phillips-Perron test without intercept and trend: Estimate the DF-regression
yt = ρyt−1 + ut ,
where ut is potentially autocorrelated. In addition, estimate the autocovariances of ut , γk , up
to order q, from which an estimate of the long-run variance, λ2 , is computed. Then, under the
null of ρ = 1,
PP-ρ ≡ T (ρ̂ − 1) − 0.5 T 2 (SE(ρ̂))2 /σ̂u2 (λ̂2 − σ̂u2 )
converges to DFρ and
s
λ̂2 − σ̂u2
σ̂u2 ρ̂ − 1
− 0.5
PP-t ≡
(T × SE(ρ̂)/σ̂u )
λ̂2 SE(ρ̂)
λ̂
converges to DFt .
7.3
Augmented Dickey-Fuller (ADF) Tests
Augmented Dickey-Fuller test without intercept and trend: Estimate by OLS the
augmented model
yt = ρyt−1 + φ1 ∆ỹ t−1 + · · · + φp ∆ỹ t−p + vt
and compute the t-statistic
ADF-t =
ρ̂ − 1
.
SE(ρ̂)
Under the null hypothesis ρ = 1, the ADF-t statistic has limiting distribution
ADF-t ⇒ DFt .
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Formulas and Tables for Time Series Econometrics / Econometrics II
Augmented Dickey-Fuller test with intercept: Estimate by OLS the augmented model
yt = α + ρyt−1 + φ1 ∆ỹ t−1 + · · · + φp ∆ỹ t−p + vt
and compute the t-statistic
ADF-tµ =
ρ̂µ − 1
.
SE(ρ̂µ )
Under the null hypothesis ρ = 1, the ADF-tµ statistic has limiting distribution
ADF-tµ ⇒ DFµt .
Augmented Dickey-Fuller test with intercept and trend: Estimate by OLS the augmented model
yt = α0 + α1 t + ρyt−1 + φ1 ∆ỹ t−1 + · · · + φp ∆ỹ t−p + vt
and compute the t-statistic
ADF-tτ =
ρ̂τ − 1
.
SE(ρ̂τ )
Under the null hypothesis ρ = 1, the ADF-tτ statistic has limiting distribution
ADF-tτ ⇒ DFτt .
Choice of the lag order: Information criteria such as the AIC or BIC can be used to
determine the lag order p. The maximum lag order pmax in this search can be chosen as the
integer part of 12 × (T /100)1/4 .
8
8.1
Cointegration
Multivariate Beveridge-Nelson decomposition
Linear vector I(0) process: A linear n dimensional zero-mean vector I(0) process is
ut = εt + Ψ1 εt−1 + Ψ2 εt−2 + . . . = Ψ(L)εt ,
Ψ0 = I,
where εt is iid with mean zero and positive definite variance matrix Ω, if
(C1)
(C2)
the sequence I, Ψ1 , Ψ2 , . . . is one-summable so that Ψ(1) = I + Ψ1 + Ψ2 + · · · is finite,
at least one element of Ψ(1) is nonzero.
Long-run variance matrix: The long-run variance matrix of a linear vector I(0) process is
√
Λ ≡ lim Var( T ūT ) = Ψ(1)ΩΨ(1)0 .
T →∞
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Formulas and Tables for Time Series Econometrics / Econometrics II
Vector I(1) process: A vector I(1) process of dimension n is
∆yt = δ + ut = δ + Ψ(L)εt ,
where ut = Ψ(L)εt is a linear zero-mean vector I(0) process.
Multivariate Beveridge-Nelson decomposition: A vector I(1) process of dimension n
can be decomposed into
yt = δ · t + Ψ(1)
t
X
εi + η t + z 0 ,
i=1
where δ · t is a deterministic trend, Ψ(1)[ε1 + · · · + εt ] is a stochastic trend, η t is a linear vector
I(0) process, and z0 is an initial condition.
8.2
Cointegration and common stochastic trends
Cointegration: Let yt be an n-dimensional vector I(1) process with stochastic increment ut
that is a linear vector I(0) process. Then yt is cointegrated with cointegration vector β 1 if
β 1 6= 0 and β 01 yt is stationary. (In a strict sense, stationarity of β 01 yt requires a suitable choice
of initial conditions.)
Cointegration rank: The cointegration rank is the number, r, of linearly independent cointegration vectors β 1 , . . . , β r , and the cointegration space is the space spanned by the cointegration
vectors.
Cointegration matrix: The (n × r) cointegration matrix β = [β 1 , . . . , β r ] is the matrix of
all cointegration vectors. The cointegration matrix has rank r.
Rank of Ψ(1): The matrix Ψ(1) of the multivariate Beveridge-Nelson decomposition has
rank k ≡ n − r.
Common stochastic trends: A cointegrated n-dimensional vector I(1) process with cointegration rank r is driven by k = n − r common stochastic trends.
8.3
Engle-Granger test of the null of no cointegration
Spurious regression: Suppose yt is an n-dimensional vector I(1) process and not cointegrated. Then the regression
y1,t = µ + γ2 y2,t + · · · + γn yn,t + zt∗
is called spurious. The disturbance zt∗ is I(1) and the OLS estimator has a non-standard
asymptotic distribution.
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Formulas and Tables for Time Series Econometrics / Econometrics II
Cointegrating regression: Suppose yt is an n-dimensional vector I(1) process and cointegrated with cointegration rank r = 1. Then the regression
y1,t = µ + γ2 y2,t + · · · + γn yn,t + zt∗ ,
is called cointegrating regression. The disturbance zt∗ is I(0). The OLS estimator is superconsistent but has a non-standard asymptotic distribution.
Engle-Granger test: Suppose yt is an n-dimensional vector I(1) and either cointegrated
with cointegration rank r = 1 or not cointegrated. The hypotheses H0 : r = 0 versus H1 : r = 1
can be tested as follows.
Stage 1. Estimate by OLS the regression
y1,t = µ̂ + γ̂2 y2,t + · · · + γ̂n yn,t + ẑt∗ .
Stage 2. Estimate the ADF regression without intercept and trend on the first-stage
residuals
∗
∗
∗
ẑt∗ = ρẑt−1
+ φ1 ∆ẑt−1
+ · · · + φp ∆ẑt−p
+ vt .
Stage 3. Construct the ADF statistic
ADF-t =
ρ̂ − 1
SE(ρ̂)
and compare it with the appropriate critical values.
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Formulas and Tables for Time Series Econometrics / Econometrics II
Part C. The Vectorautoregressive (VAR)
Model
9
The VAR(1) model (reduced form)
Model equation:
yt = B1 yt−1 + ut ,
where
• yt is a K × 1 vector of economic variables
• A1 is a K × K parameter matrix
• ut is a K × 1 vector of reduced form white noise disturbances
• Sample runs from t = 0, 1, . . . , T .
Moving average representation:
yt = ut + B1 ut−1 + B21 ut−2 + B31 ut−3 + . . .
∞
∞
X
X
i
=
B1 ut−i =
Ci ut−i ,
Ci ≡ Bi1
i=0
i=0
Stability condition: Stability requires that Bi1 → 0 at i → ∞. Hence, a sufficient condition
is that the eigenvalues of B1 are all absolutely smaller than 1.
10
The VAR(p) model (reduced form)
Model equation:
yt = B1 yt−1 + · · · + Bp yt−p + ut
or, compactly,
B(L)yt = ut ,
where B(L) = I − B1 L1 − · · · − Bp Lp is the lag polynomial in lag order p.
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Formulas and Tables for Time Series Econometrics / Econometrics II
Moving average representation: The result is qualitatively the same as for VAR(1):
yt =
∞
X
Ci ut−i = C(L)ut−i
with C(L) =
i=0
∞
X
Ci Li
i=0
Computation of the parameter matrices:
i
p = 1: C0 = IK ,
Ci = B
i = 1, 2, . . .
P1i,
p > 1: C0 = IK ,
Ci = j=1 Ci−j Bj ,
i = 1, 2, . . .
Wold’s decomposition theorem: Every stationary process xt can be written as the sum
of two uncorrelated processes zt and yt , where zt is a deterministic process and yt is a process
with MA representation, i.e.,
yt = C(L)ut ,
11
ut ∼ white noise.
Estimation and Inference
11.1
System OLS
A system OLS estimator B̂ yields the same results as equation-wise OLS but can be written
more efficiently:


yt−1


yt = B1 yt−1 + · · · + Bp yt−p + ut = [B1 , . . . , Bp ]  ...  + ut = BXt−1 + ut
yt−p
where
• B = [B1 , . . . , Bp ] is a K × Kp coefficient matrix
0
0
]0 is a Kp × 1 observation vector
• Xt−1 = [yt−1
, . . . , yt−p
Now stack observations horizontally:
• y = [y1 , . . . , yT ] is a K × T observation matrix
• X = [X0 , . . . , XT −1 ] is a Kp × T observation matrix
• u = [u1 , . . . , uT ] is a K × T error matrix
OLS estimator:
B̂ = yX0 (XX0 )−1 =
T
X
t=1
!
yt X0t−1
T
X
!−1
Xt−1 X0t−1
t=1
To estimate Σ, construct the residuals û = y − B̂X and compute
Σ̂ = ûû0 /T,
Note that OLS is consistent because the regressors are lags of yt and the error term ut is white
noise and thus not correlated with these lags.
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Formulas and Tables for Time Series Econometrics / Econometrics II
11.2
Maximum likelihood estimation
Assuming normality of the disturbances,
ut ∼ N (0, Σ) ,
t = 1, . . . , T,
the parameters of a VAR model can also be estimated using a maximum likelihood estimator.
Define the set of past observations as
It−1 = {yt−1 , yt−2 , yt−3 , . . .}.
Conditional on It−1 , yt is a linear function of ut and thus normally distributed with
E(yt |It−1 ) = B1 yt−1 + . . . + Bp yt−p
Var(yt |It−1 ) = Var(ut ) = Σ
Hence, conditional on the past, we have the normal distribution
yt |It−1 ∼ N (B1 yt−1 + . . . + Bp yt−p , Σ) ,
t = 1, . . . , T
The conditional likelihood of observation t given the past is thus
1
1
log |Σ| − (yt − B1 yt−1 − . . . − Bp yt−p )0 ×
2
2
−1
Σ × (yt − B1 yt−1 − . . . − Bp yt−p ) ,
`t (θ) ≡ log f (yt |It−1 , θ) = const −
where θ is a vector of all parameters in B1 , . . . , Bp , Σ.
11.3
Inference
Stationary variables:
• Recall that the OLS estimator of a time series regression with white noise disturbances
is asymptotically normally distributed by a martingale difference CLT.
• Hence, we can use standard inference for the VAR parameters.
• For functions of these parameters, e.g., impulse response coefficients, we can use the delta
method to find the asymptotic normal distribution of their estimators.
Nonstationary variables: If the VAR includes I(1) variables, fortunately not much changes
as can be seen from reparameterizing a VAR(2) to a multivariate ADF-type representation:
yt = B1 yt−1 + B2 yt−2 + ut = (B1 + B2 ) yt−1 + (−B2 ) ∆yt−1 + ut
| {z }
| {z }
C
Γ1
Estimating this equation by OLS yields Ĉ that may be superconsistent (convergence rate of
T ) because the regressors are I(1) and Γ̂1 that is root-T consistent because the regressors are
differenced and thus I(0).
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Formulas and Tables for Time Series Econometrics / Econometrics II
12
Specification
Information Criteria: Akaike criterion (AIC), Hannan-Quinn criterion (HQC), Schwarz
criterion (SC):
2
× (# freely estimated parameters)
T
2mK 2
= ln |Σ̂| +
T
AIC(m) = ln |Σ̂| +
2 ln ln T
× (# freely estimated parameters)
T
2 ln ln T
= ln |Σ̂| +
mK 2
T
HQC(m) = ln |Σ̂| +
ln T
× (# freely estimated parameters)
T
ln T
= ln |Σ̂| +
mK 2
T
SC(m) = ln |Σ̂| +
where you try all m = 1, . . . , pmax .
Asymptotics: HQC and SC are consistent, hence pick the right lag order as T → ∞. AIC is
not: it has a positive probability to choose a lag order that is too high.
13
Forecasting
Granger causality: A variable y1t is Granger causal for a variable y2t if its knowledge helps
to reduce, on average, the prediction error for y2t .
If b121 = b221 = 0, then y1t is not Granger causal for y2t .
h-step forecast:
yT +h|T = E(yT +h |IT ) = B1 E(yT +h−1 |IT ) + · · · + Bp E(yT +h−p |IT )
Forecast error:
eT +h|T = yT +h − yT +h|T = yT +h − E(yT +h |IT )
The forecast error covariance matrix: the forecast MSE converges, as h increases, toward
the unconditional variance matrix of yt from below:
MSE(eT +h|T ) = Var(eT +h|T ) =
h−1
X
Cj ΣC0j → Var(yt ).
j=0
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Formulas and Tables for Time Series Econometrics / Econometrics II
Sources of uncertainty: future shocks and estimation error:
yT +h − ŷT +h|T = (yT +h − yT +h|T ) + (yT +h|T − ŷT +h|T ) .
|
{z
} |
{z
}
eT +h|T
Estimation error
Since the first part depends on future shocks and the second part on the shocks in the sample,
the two are uncorrelated. Thus the forecast MSE is
MSE(yT +h − ŷT +h|T ) = MSE(eT +h|T ) + MSE(yT +h|T − ŷT +h|T ).
14
Structural VAR
Model equation:
A0 yt = A1 yt−1 + · · · + Ap yt−p + Dεt .
Leaving aside the lags which account for the dynamics, the contemporaneous relationship between variables and structural shocks is
A0 yt = Dεt .
Central assumption: the structural shocks are mutually uncorrelated so that a ceteris
paribus interpretation is possible. Since we do not interpret the scale of the shocks, we normalize
the variances to 1: Var(εt ) = I.
Finding the reduced form VAR from the structural one: multiply by A−1
0 from the
left,
−1
yt = A0−1 A1 yt−1 + · · · + A−1
0 Ap yt−p + A0 Dεt .
From there, define the reduced form parameters as Bi = A−1
0 Ai and the reduced form disturbances ut = A−1
Dε
:
t
0
yt = B1 yt−1 + · · · + Bp yt−p + ut .
14.1
Estimating the parameters of the structural form
1) Taking the variance of ut = A−1
0 Dεt we find
0
0
0 0 −1
−1
0
0 −1
0 −1
= A−1
Σ ≡ E(ut u0t ) = E(A−1
0 DD A0
0 Dεt εt D A0 ) = A0 D E(εt εt )D A0
0
We can first estimate Σ from the reduced form and then solve
0
−1 0
Σ̂ = Â−1
0 D̂D̂ (Â0 )
for the structural parameters in Â0 and D̂.
2) From Bi = A−1
0 Ai we find Ai = A0 Bi , i = 1, . . . , p. Hence, once we know Â0 from step 1,
we can compute
Âi = Â0 B̂i ,
i = 1, . . . , p.
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Formulas and Tables for Time Series Econometrics / Econometrics II
14.2
Identification of the structural form
0
0 −1
Due to symmetry, Σ has K(K + 1)/2 distinct parameters, hence Σ = A−1
imposes
0 DD A0
2
K(K + 1)/2 on the 2K parameters in A0 and D.
The contemporaneous relationship between variables and structural shocks,
A0 yt = Dεt ,
needs to impose enough restrictions such that the parameters in A0 and D can be recovered
from Σ.
Two cases commonly used to achieve identification:
• A0 = I and D is lower triangular, or
• D = I and A0 is lower triangular.
• Both imply that A−1
0 D and its inverse are lower triangular.
15
Impulse response analysis
To find the impulse responses, we start from the MA representation
yt =
∞
X
Ci ut−i
i=0
and substitute ut = A−1
0 Dεt to get the structural shocks:
yt =
∞
X
−1
−1
−1
Ci A−1
0 Dεt−i = A0 Dεt + C1 A0 Dεt−1 + C2 A0 Dεt−2 + . . .
i=0
Hence,
∂yt
= Cτ A−1
0 D.
∂ε0t−τ
Therefore, the impulse response function of variable i, yit , w.r.t. structural shock j collects
elements i, j from
−1
−1
−1
A0 D, C1 A−1
0 D, C2 A0 D, C3 A0 D, . . . .
24
Formulas and Tables for Time Series Econometrics / Econometrics II
Part D. Tables
Percentiles of the χ2–distribution
Fχ 2
r=1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
70
80
90
100
0.0100
0.0002
0.0201
0.1148
0.2971
0.5543
0.8721
1.2390
1.6465
2.0879
2.5582
3.0535
3.5706
4.1069
4.6604
5.2293
5.8122
6.4078
7.0149
7.6327
8.2604
8.8972
9.5425
10.1957
10.8564
11.5240
12.1981
12.8785
13.5647
14.2565
14.9535
22.1643
29.7067
37.4849
45.4417
53.5401
61.7541
70.0649
0.0250
0.0010
0.0506
0.2158
0.4844
0.8312
1.2373
1.6899
2.1797
2.7004
3.2470
3.8157
4.4038
5.0088
5.6287
6.2621
6.9077
7.5642
8.2307
8.9065
9.5908
10.2829
10.9823
11.6886
12.4012
13.1197
13.8439
14.5734
15.3079
16.0471
16.7908
24.4330
32.3574
40.4817
48.7576
57.1532
65.6466
74.2219
0.0500
0.0039
0.1026
0.3518
0.7107
1.1455
1.6354
2.1673
2.7326
3.3251
3.9403
4.5748
5.2260
5.8919
6.5706
7.2609
7.9616
8.6718
9.3905
10.1170
10.8508
11.5913
12.3380
13.0905
13.8484
14.6114
15.3792
16.1514
16.9279
17.7084
18.4927
26.5093
34.7643
43.1880
51.7393
60.3915
69.1260
77.9295
0.1000
0.0158
0.2107
0.5844
1.0636
1.6103
2.2041
2.8331
3.4895
4.1682
4.8652
5.5778
6.3038
7.0415
7.7895
8.5468
9.3122
10.0852
10.8649
11.6509
12.4426
13.2396
14.0415
14.8480
15.6587
16.4734
17.2919
18.1139
18.9392
19.7677
20.5992
29.0505
37.6886
46.4589
55.3289
64.2778
73.2911
82.3581
0.9000
2.7055
4.6052
6.2514
7.7794
9.2364
10.6446
12.0170
13.3616
14.6837
15.9872
17.2750
18.5493
19.8119
21.0641
22.3071
23.5418
24.7690
25.9894
27.2036
28.4120
29.6151
30.8133
32.0069
33.1962
34.3816
35.5632
36.7412
37.9159
39.0875
40.2560
51.8051
63.1671
74.3970
85.5270
96.5782
107.5650
118.4980
0.9500
3.8415
5.9915
7.8147
9.4877
11.0705
12.5916
14.0671
15.5073
16.9190
18.3070
19.6751
21.0261
22.3620
23.6848
24.9958
26.2962
27.5871
28.8693
30.1435
31.4104
32.6706
33.9244
35.1725
36.4150
37.6525
38.8851
40.1133
41.3371
42.5570
43.7730
55.7585
67.5048
79.0819
90.5312
101.8795
113.1453
124.3421
0.9750
5.0239
7.3778
9.3484
11.1433
12.8325
14.4494
16.0128
17.5345
19.0228
20.4832
21.9200
23.3367
24.7356
26.1189
27.4884
28.8454
30.1910
31.5264
32.8523
34.1696
35.4789
36.7807
38.0756
39.3641
40.6465
41.9232
43.1945
44.4608
45.7223
46.9792
59.3417
71.4202
83.2977
95.0232
106.6286
118.1359
129.5612
0.9900
6.6349
9.2103
11.3449
13.2767
15.0863
16.8119
18.4753
20.0902
21.6660
23.2093
24.7250
26.2170
27.6882
29.1412
30.5779
31.9999
33.4087
34.8053
36.1909
37.5662
38.9322
40.2894
41.6384
42.9798
44.3141
45.6417
46.9629
48.2782
49.5879
50.8922
63.6907
76.1539
88.3794
100.4252
112.3288
124.1163
135.8067
25
Formulas and Tables for Time Series Econometrics / Econometrics II
Percentiles of the t–distribution
Ft
r=1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
∞
0.9000
3.0777
1.8856
1.6377
1.5332
1.4759
1.4398
1.4149
1.3968
1.3830
1.3722
1.3634
1.3562
1.3502
1.3450
1.3406
1.3368
1.3334
1.3304
1.3277
1.3253
1.3232
1.3212
1.3195
1.3178
1.3163
1.3150
1.3137
1.3125
1.3114
1.3104
1.2816
0.9500
6.3138
2.9200
2.3534
2.1318
2.0150
1.9432
1.8946
1.8595
1.8331
1.8125
1.7959
1.7823
1.7709
1.7613
1.7531
1.7459
1.7396
1.7341
1.7291
1.7247
1.7207
1.7171
1.7139
1.7109
1.7081
1.7056
1.7033
1.7011
1.6991
1.6973
1.6449
0.9750
12.7062
4.3027
3.1824
2.7764
2.5706
2.4469
2.3646
2.3060
2.2622
2.2281
2.2010
2.1788
2.1604
2.1448
2.1314
2.1199
2.1098
2.1009
2.0930
2.0860
2.0796
2.0739
2.0687
2.0639
2.0595
2.0555
2.0518
2.0484
2.0452
2.0423
1.9600
0.9900
31.8205
6.9646
4.5407
3.7469
3.3649
3.1427
2.9980
2.8965
2.8214
2.7638
2.7181
2.6810
2.6503
2.6245
2.6025
2.5835
2.5669
2.5524
2.5395
2.5280
2.5176
2.5083
2.4999
2.4922
2.4851
2.4786
2.4727
2.4671
2.4620
2.4573
2.3263
0.9950
63.6567
9.9248
5.8409
4.6041
4.0321
3.7074
3.4995
3.3554
3.2498
3.1693
3.1058
3.0545
3.0123
2.9768
2.9467
2.9208
2.8982
2.8784
2.8609
2.8453
2.8314
2.8188
2.8073
2.7969
2.7874
2.7787
2.7707
2.7633
2.7564
2.7500
2.5758
26
Percentiles of the F –distribution
The Table shows the values k, for which P (v ≤ k) = F (k) = 0.95 holds. (r1 = degrees of freedom of the nominator,r2 = degrees of
freedom of the denominator)
r1 = 1
161.4476
18.5128
10.1280
7.7086
6.6079
5.9874
5.5914
5.3177
5.1174
4.9646
4.8443
4.7472
4.6672
4.6001
4.5431
4.4940
4.4513
4.4139
4.3807
4.3512
4.3248
4.3009
4.2793
4.2597
4.2417
4.2252
4.2100
4.1960
4.1830
4.1709
4.0847
4.0012
3.9201
3.8416
2
199.5000
19.0000
9.5521
6.9443
5.7861
5.1433
4.7374
4.4590
4.2565
4.1028
3.9823
3.8853
3.8056
3.7389
3.6823
3.6337
3.5915
3.5546
3.5219
3.4928
3.4668
3.4434
3.4221
3.4028
3.3852
3.3690
3.3541
3.3404
3.3277
3.3158
3.2317
3.1504
3.0718
2.9958
3
215.7073
19.1643
9.2766
6.5914
5.4095
4.7571
4.3468
4.0662
3.8625
3.7083
3.5874
3.4903
3.4105
3.3439
3.2874
3.2389
3.1968
3.1599
3.1274
3.0984
3.0725
3.0491
3.0280
3.0088
2.9912
2.9752
2.9604
2.9467
2.9340
2.9223
2.8387
2.7581
2.6802
2.6050
4
224.5832
19.2468
9.1172
6.3882
5.1922
4.5337
4.1203
3.8379
3.6331
3.4780
3.3567
3.2592
3.1791
3.1122
3.0556
3.0069
2.9647
2.9277
2.8951
2.8661
2.8401
2.8167
2.7955
2.7763
2.7587
2.7426
2.7278
2.7141
2.7014
2.6896
2.6060
2.5252
2.4472
2.3720
5
230.1619
19.2964
9.0135
6.2561
5.0503
4.3874
3.9715
3.6875
3.4817
3.3258
3.2039
3.1059
3.0254
2.9582
2.9013
2.8524
2.8100
2.7729
2.7401
2.7109
2.6848
2.6613
2.6400
2.6207
2.6030
2.5868
2.5719
2.5581
2.5454
2.5336
2.4495
2.3683
2.2899
2.2142
6
233.9860
19.3295
8.9406
6.1631
4.9503
4.2839
3.8660
3.5806
3.3738
3.2172
3.0946
2.9961
2.9153
2.8477
2.7905
2.7413
2.6987
2.6613
2.6283
2.5990
2.5727
2.5491
2.5277
2.5082
2.4904
2.4741
2.4591
2.4453
2.4324
2.4205
2.3359
2.2541
2.1750
2.0987
7
236.7684
19.3532
8.8867
6.0942
4.8759
4.2067
3.7870
3.5005
3.2927
3.1355
3.0123
2.9134
2.8321
2.7642
2.7066
2.6572
2.6143
2.5767
2.5435
2.5140
2.4876
2.4638
2.4422
2.4226
2.4047
2.3883
2.3732
2.3593
2.3463
2.3343
2.2490
2.1665
2.0868
2.0097
8
238.8827
19.3710
8.8452
6.0410
4.8183
4.1468
3.7257
3.4381
3.2296
3.0717
2.9480
2.8486
2.7669
2.6987
2.6408
2.5911
2.5480
2.5102
2.4768
2.4471
2.4205
2.3965
2.3748
2.3551
2.3371
2.3205
2.3053
2.2913
2.2783
2.2662
2.1802
2.0970
2.0164
1.9385
9
240.5433
19.3848
8.8123
5.9988
4.7725
4.0990
3.6767
3.3881
3.1789
3.0204
2.8962
2.7964
2.7144
2.6458
2.5876
2.5377
2.4943
2.4563
2.4227
2.3928
2.3660
2.3419
2.3201
2.3002
2.2821
2.2655
2.2501
2.2360
2.2229
2.2107
2.1240
2.0401
1.9588
1.8800
10
241.8817
19.3959
8.7855
5.9644
4.7351
4.0600
3.6365
3.3472
3.1373
2.9782
2.8536
2.7534
2.6710
2.6022
2.5437
2.4935
2.4499
2.4117
2.3779
2.3479
2.3210
2.2967
2.2747
2.2547
2.2365
2.2197
2.2043
2.1900
2.1768
2.1646
2.0772
1.9926
1.9105
1.8308
20
248.0131
19.4458
8.6602
5.8025
4.5581
3.8742
3.4445
3.1503
2.9365
2.7740
2.6464
2.5436
2.4589
2.3879
2.3275
2.2756
2.2304
2.1906
2.1555
2.1242
2.0960
2.0707
2.0476
2.0267
2.0075
1.9898
1.9736
1.9586
1.9446
1.9317
1.8389
1.7480
1.6587
1.5706
120
253.2529
19.4874
8.5494
5.6581
4.3985
3.7047
3.2674
2.9669
2.7475
2.5801
2.4480
2.3410
2.2524
2.1778
2.1141
2.0589
2.0107
1.9681
1.9302
1.8963
1.8657
1.8380
1.8128
1.7896
1.7684
1.7488
1.7306
1.7138
1.6981
1.6835
1.5766
1.4673
1.3519
1.2216
Formulas and Tables for Time Series Econometrics / Econometrics II
F (k) = 0.95
r2 = 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
60
120
∞
27
Formulas and Tables for Time Series Econometrics / Econometrics II
Critical values for the DF tests
Table 1: Empirical cumulative distribution of the DF-ρ statistic under H0 : ρ = 1
Probability that the statistic is less than entry
Sample size (T)
0.01
0.025
0.05
0.10
0.90
0.95
0.975
0.99
(a) No intercept, no trend
25
-11.8
50
-12.8
100
-13.3
250
-13.6
500
-13.7
∞
-13.8
-9.3
-9.9
-10.2
-10.4
-10.4
-10.5
-7.3
-7.7
-7.9
-8.0
-8.0
-8.1
-5.3
-5.5
-5.6
-5.7
-5.7
-5.7
1.01
0.97
0.95
0.94
0.93
0.93
1.41
1.34
1.31
1.29
1.28
1.28
1.78
1.69
1.65
1.62
1.61
1.60
2.28
2.16
2.09
2.05
2.04
2.03
(b) Intercept, no trend
25
-17.2
50
-18.9
100
-19.8
250
-20.3
500
-20.5
∞
-20.7
-14.6
-15.7
-16.3
-16.7
-16.8
-16.9
-12.5
-13.3
-13.7
-13.9
-14.0
-14.1
-10.2
-10.7
-11.0
-11.1
-11.2
-11.3
-0.76
-0.81
-0.83
-0.84
-0.85
-0.85
0.00
-0.07
-0.11
-0.13
-0.14
-0.14
0.65
0.53
0.47
0.44
0.42
0.41
1.39
1.22
1.14
1.08
1.07
1.05
(c) Intercept and trend
25
-22.5
50
-25.8
100
-27.4
250
-28.5
500
-28.9
∞
-29.4
-20.0
-22.4
-23.7
-24.4
-24.7
-25.0
-17.9
-19.7
-20.6
-21.3
-21.5
-21.7
-15.6
-16.8
-17.5
-17.9
-18.1
-18.3
-3.65
-3.71
-3.74
-3.76
-3.76
-3.77
-2.51
-2.60
-2.63
-2.65
-2.66
-2.67
-1.53
-1.67
-1.74
-1.79
-1.80
-1.81
-0.46
-0.67
-0.76
-0.83
-0.86
-0.88
Source: Hayashi (2000), p. 576.
28
Formulas and Tables for Time Series Econometrics / Econometrics II
Table 2: Empirical cumulative distribution of the DF-t statistic under H0 : ρ = 1
Probability that the statistic is less than entry
Sample size (T)
0.01
0.025
0.05
0.10
0.90
0.95
0.975
0.99
(a) No intercept, no trend
25
-2.65
50
-2.62
100
-2.60
250
-2.58
500
-2.58
∞
-2.58
-2.26
-2.25
-2.24
-2.24
-2.23
-2.23
-1.95
-1.95
-1.95
-1.95
-1.95
-1.95
-1.60
-1.61
-1.61
-1.62
-1.62
-1.62
0.92
0.91
0.90
0.89
0.89
0.89
1.33
1.31
1.29
1.28
1.28
1.28
1.70
1.66
1.64
1.63
1.62
1.62
2.15
2.08
2.04
2.02
2.01
2.01
(b) Intercept, no trend
25
-3.75
50
-3.59
100
-3.50
250
-3.45
500
-3.44
∞
-3.42
-3.33
-3.23
-3.17
-3.14
-3.13
-3.12
-2.99
-2.93
-2.90
-2.88
-2.87
-2.86
-2.64
-2.60
-2.59
-2.58
-2.57
-2.57
-0.37
-0.41
-0.42
-0.42
-0.44
-0.44
0.00
-0.04
-0.05
-0.06
-0.07
-0.08
0.34
0.28
0.26
0.24
0.24
0.23
0.72
0.66
0.63
0.62
0.61
0.60
(c) Intercept and trend
25
-4.38
50
-4.15
100
-4.05
250
-3.98
500
-3.97
∞
-3.96
-3.95
-3.80
-3.73
-3.69
-3.67
-3.66
-3.60
-3.50
-3.45
-3.42
-3.42
-3.41
-3.24
-3.18
-3.15
-3.13
-3.13
-3.12
-1.14
-1.19
-1.22
-1.23
-1.24
-1.25
-0.81
-0.87
-0.90
-0.92
-0.93
-0.94
-0.50
-0.58
-0.62
-0.64
-0.65
-0.66
-0.15
-0.24
-0.28
-0.31
-0.32
-0.33
Source: Hayashi (2000), p. 578.
29
Formulas and Tables for Time Series Econometrics / Econometrics II
Critical values for the ADF test for cointegration
(Engle-Granger test)
Number of regressors,
excluding constant
1%
2.5%
5%
10%
A. Regressors have no drift
1
-3.96
2
-4.31
3
-4.73
4
-5.07
5
-5.28
-3.64
-4.02
-4.37
-4.71
-4.98
-3.37
-3.77
-4.11
-4.45
-4.71
-3.07
-3.45
-3.83
-4.16
-4.43
B. Some regressors have drift
1
-3.96
2
-4.36
3
-4.65
4
-5.04
5
-5.36
-3.67
-4.07
-4.39
-4.77
-5.02
-3.41
-3.80
-4.16
-4.49
-4.74
-3.13
-3.52
-3.84
-4.20
-4.46
Source: Hayashi (2000), Table 10.1, p. 646.
30