Econometrics II – Section 7.5, Heij et al. (2004)
Regression model with lags
One-equation cointegration
Marius Ooms
Tinbergen Institute, VU University Amsterdam
TI Econometrics II 2006/2007, Chapter 7.5 – p. 1/32
Contents
• Introduction
• Autoregressive Distributed Lag Model in I(0) world
• Equilibrium Correction Model (ECM) in I(0) world
◦ long run equilibrium
◦ equilibrium correction
◦ other economic dynamic regression models
◦ (non)exogeneity, consequences
• Cointegration: ADL/ECM in I(1) world
◦ spurious regression
◦ cointegration
TI Econometrics II 2006/2007, Chapter 7.5 – p. 2/32
Introduction
Section 7.5 considers bivariate dynamic processes with a
dynamic regression interpretation.
One variable is considered dependent and, in addition to lagged
values of the dependent variable, current and lagged values of
the other explanatory variable, that is considered predetermined
in the equation.
Such a relation is called an autoregressive distributed lag
(A(R)DL) relation. MA error terms can sometimes be allowed
for. We first assume joint stationarity of y and x for estimation
purposes: ”I(0)” world.
TI Econometrics II 2006/2007, Chapter 7.5 – p. 3/32
Structural interpretation
Lagged dependent terms in A(R)DL models motivated by
economic theory (partial adjustment, adaptive expectations,
equilibrium correction), rather than just modelling serial
correlation: the ADL model is a structural equation with
interpretable parameters.
Interpretation and estimation parameters depends on
exogeneity assumptions on x.
Example §7.5: yt = πt : US inflation. xt = rn,t : nominal US tbill
rate. See below.
TI Econometrics II 2006/2007, Chapter 7.5 – p. 4/32
AR distributed lag model (ADL(p,r))
We study the ADL(1, 1) model
yt = α + φyt−1 + β0 xt + β1 xt−1 + εt ,
where |φ| < 1: stability condition in this context. The error term,
εt , is White Noise. xt is considered predetermined in the
equation or, in more general terms: (weakly) exogenous with
respect to the estimation of the parameters α, φ, β0 and β1 . This
requires that xt is uncorrelated with εt , εt+1 , . . ..
The model can be re-formulated in DL(∞) form to show the
response of yt , yt+1 , . . . to one-time changes in xt : dynamic
impact (”impulse response”)
TI Econometrics II 2006/2007, Chapter 7.5 – p. 5/32
Dynamic impact, A(R)DL(0,∞) form of A(R)DL(1,1)
(1 − φL)yt = α + (β0 + β1 L)xt + εt ,
φ(L)yt = α + β(L)xt + εt ,
yt = αφ−1 (1) + φ−1 (L)β(L)xt + φ−1 (L)εt .
∂yt
∂xt
∂yt+1
∂xt
= β0 ,
= β1 + φβ0 ,
..
.
∂yt+j
∂xt
= φj−1 (β1 + φβ0 ),
for
j > 0.
Note that the influence on yt+j disappears as j → ∞.
TI Econometrics II 2006/2007, Chapter 7.5 – p. 6/32
A(R)DL example US inflation, Interest rates
US Core inflation (SA), 3-month T-bill, 59.1-99.12
24
20
16
12
8
4
0
-4
-8
60
65
70
75
80
USCOREINFSA
85
90
95
TBAA3M
TI Econometrics II 2006/2007, Chapter 7.5 – p. 7/32
A(R)DL(6,4) estimation example ’reduced form’
Modelling UScoreinfSA by OLS
The estimation sample is: 62 (1) to 99 (12)
Coefficient Std.Error t-value t-prob
UScoreinfSA_1
0.233366
0.04750
4.91
0.000
...
UScoreinfSA_6
0.102451
0.04612
2.22
0.027
Constant
0.239006
0.3136
0.762
0.446
tbaa3m
0.867091
0.2383
3.64
0.000
tbaa3m_1
-0.646922
0.3798
-1.70
0.089
...
tbaa3m_4
-0.340460
0.2405
-1.42
0.158
sigma
2.54947 RSS
2885.89892
Rˆ2
0.542968 F(11,444) =
47.95 [0.000]**
log-likelihood
-1067.72 DW
2.04
no. of observations
456 no. of parameters
12
mean(UScoreinfSA)
4.54782 var(UScoreinfSA)
13.8475
TI Econometrics II 2006/2007, Chapter 7.5 – p. 8/32
A(R)DL dynamic impact of x, example
Dynamic impact interest rate on inflation (scaled, sum=1)
2
tbaa3m
Impact tbaa3m (normalized) on UScoreinfSA
1
0
−1
0
5
2.5
10
15
20
25
30
tbaa3m(cum)
Cumulative impact tbaa3m (normalized) on UScoreinfSA
2.0
1.5
1.0
0
5
10
15
20
25
30
35
40
TI Econometrics II 2006/2007, Chapter 7.5 – p. 9/32
Long run “equilibrium relationships” from ADL eq.
Thought experiment: keep xt = x̄ constant, put all εt = 0 and
compute the value for yt after convergence (assuming this is the
only dynamic relationship between xt and yt ), that is ȳ .
The “equilibrium” is
ȳ = α + φȳ + β0 x̄ + β1 x̄,
or, in general,
α
β(1)
ȳ =
+
x̄.
φ(1) φ(1)
TI Econometrics II 2006/2007, Chapter 7.5 – p. 10/32
Long-run elasticity
When y, x is in logs, then
λ=
β(1)
=
φ(1)
∞
X
∂yt+j
j=0
∂xt
can be interpreted as a “long-run elasticity”.
TI Econometrics II 2006/2007, Chapter 7.5 – p. 11/32
Long run relation Inflation Interest Rate?
US interest rate vs. inflation 62-99
(Fisher equation: rn = rr + π or π = rn − rr )
24
20
USCOREINFSA
16
12
8
4
0
-4
-8
2
4
6
8
10
12
14
16
18
TBAA3M
TI Econometrics II 2006/2007, Chapter 7.5 – p. 12/32
Long run equation implied by A(R)DL
By simple calculations one can derive a long run relation
between x and y from the A(R)DL and test its significance:
Solved static long run equation for UScoreinfSA
Coefficient Std.Error t-value
Constant
1.07147
1.445
0.742
tbaa3m
0.558555
0.2175
2.57
t-prob
0.459
0.011
ECM = UScoreinfSA - 1.07147 - 0.558555*tbaa3m; (Equilibrium correction
mechanism)
Inverse Roots of UScoreinfSA lag polynomial (is AR part stable?):
real
imag
modulus
0.92014
0.00000
0.92014
0.32600
0.61182
0.69326
...
-0.34068
-0.48614
0.59363
How to derive the Equilibrium correction term? See next!
TI Econometrics II 2006/2007, Chapter 7.5 – p. 13/32
Equilibrium (Error) Correction Model (E(q)CM)
The ECM explains the change in y using one lagged level of y
and x and one or more lagged differences of y and x. The
ECM representation of the ADL model is easier to interpret and
often easier to estimate. In the univariate case, β(L) = 0, it
reduces to an (A)D-F regression.
yt = α + φyt−1 + β0 xt + β1 xt−1 + εt ,
∆yt = α − (1 − φ)yt−1 + β0 ∆xt + (β0 + β1 )xt−1 + εt ,
= β0 ∆xt − (1 − φ)[yt−1 − δ − λxt−1 ] + εt ,
α
δ=
,
(1 − φ)
(1)
(2)
(3)
(β0 + β1 )
λ=
.
(1 − φ)
δ and λ are ‘the ‘equilibrium” coefficients.
TI Econometrics II 2006/2007, Chapter 7.5 – p. 14/32
Example: ECM unrestricted: eq. (2)
b from
Exercise (1): Compute the long run semi- elasticity λ
Eviews OLS output. Hint: see §7.5.1, (7.34) and Exercise 7.9.
As in the book DTBAA3M=TBAA3M - TBAA3M(-1).
Dependent Variable: DUSCOREINFSA
Method: Least Squares
Sample: 1962:01 1999:12
Included observations: 456
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
DUSCOREINFSA(-1)
DUSCOREINFSA(-2)
DUSCOREINFSA(-3)
DUSCOREINFSA(-4)
DUSCOREINFSA(-5)
DTBAA3M
DTBAA3M(-1)
DTBAA3M(-2)
DTBAA3M(-3)
USCOREINFSA(-1)
TBAA3M(-1)
0.239006
-0.543571
-0.319701
-0.297513
-0.206045
-0.102451
0.867091
0.095577
1.090760
0.340460
-0.223063
0.124593
0.313641
0.063107
0.064293
0.061498
0.058424
0.046116
0.238304
0.247830
0.243602
0.240463
0.053931
0.063355
0.762036
-8.613422
-4.972542
-4.837784
-3.526735
-2.221614
3.638592
0.385655
4.477637
1.415852
-4.136090
1.966567
0.4464
0.0000
0.0000
0.0000
0.0005
0.0268
0.0003
0.6999
0.0000
0.1575
0.0000
0.0499
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.393532
0.378507
2.549465
2885.899
-1067.719
2.040891
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
0.003614
3.233931
4.735608
4.844094
26.19159
0.000000
TI Econometrics II 2006/2007, Chapter 7.5 – p. 15/32
Decomposition ECM model
The ECM specification (3) decomposes a change in y into two
components,
1. from change in x: ∆xt : direct short run effect
2. from lagged equilibrium error: zt−1 where zt = yt − δ − λxt
When yt is higher than equilibrium value (positive z ), y will adjust
downwards in order to get back to equilibrium: Equilibrium
(error) correction.
Remember: §7.5 assumes there is no feedback (Granger
non-causality) from y to x!
TI Econometrics II 2006/2007, Chapter 7.5 – p. 16/32
Example: ECM term, definition and time series plot
20
ECM term = UScoreinfSA - 1.07 - 0.56*tbaa3m
15
10
5
0
-5
-10
-15
1965 1970 1975 1980 1985 1990 1995
ECMTERM
TI Econometrics II 2006/2007, Chapter 7.5 – p. 17/32
Example: ECM estimation eq. (3) with known λ
d , from
Exercise (2): Compute ”adjustment coefficient” ,−φ(1)
Eviews output.
Dependent Variable: DUSCOREINFSA
Method: Least Squares
Sample(adjusted): 1962:02 1999:12
Included observations: 455 after adjusting endpoints
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
DUSCOREINFSA(-1)
DUSCOREINFSA(-2)
DUSCOREINFSA(-3)
DUSCOREINFSA(-4)
DUSCOREINFSA(-5)
DTBAA3M
DTBAA3M(-1)
DTBAA3M(-2)
DTBAA3M(-3)
ECMTERM(-1)
0.001846
-0.543513
-0.319760
-0.298213
-0.205761
-0.104175
0.868916
0.093375
1.093816
0.339643
-0.223273
0.119532
0.061869
0.063442
0.060994
0.058014
0.046282
0.236007
0.245796
0.242643
0.239119
0.052104
0.015444
-8.784847
-5.040161
-4.889204
-3.546746
-2.250878
3.681734
0.379888
4.507912
1.420392
-4.285112
0.9877
0.0000
0.0000
0.0000
0.0004
0.0249
0.0003
0.7042
0.0000
0.1562
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.393677
0.380021
2.549155
2885.197
-1065.821
2.039039
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
0.003294
3.237484
4.733280
4.832891
28.82824
0.000000
TI Econometrics II 2006/2007, Chapter 7.5 – p. 18/32
Partial Scatterplot ECM effect, (c.f. §3.2.5 Case 3)
DUSCOREINFSAPARTIAL vs. ECMTERMLAGGEDPARTIAL
DUSCOREINFSAPARTIAL
15
10
5
0
-5
-10
-6
-4
-2
0
2
4
6
8
10
ECMTERMLAGGEDPARTIAL
TI Econometrics II 2006/2007, Chapter 7.5 – p. 19/32
Direct estimation ECM λ and s.e. in Eviews
Exercise (3): Explain why the coefficient of rt is estimated using
2SLS and why it is a consistent estimate of λ. Hint: which RHS
variable is endogenous? Which instrument is used? Try first for
ARDL(1,1) (forget terms with lag > 1). Apply the decomposition
α0 + α1 L = α(1) − α1 ∆, cf. also page 598, to φ(L) and β(L).
Dependent Variable: USCOREINFSA
Method: Two-Stage Least Squares
Date: 02/10/03 Time: 10:51
Sample: 1962:01 1999:12
Included observations: 456
Instrument list: C TBAA3M DTBAA3M(0 TO -3) DUSCOREINFSA(-1
TO -5) USCOREINFSA(-1)
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
TBAA3M
DUSCOREINFSA
DUSCOREINFSA(-1)
DUSCOREINFSA(-2)
DUSCOREINFSA(-3)
DUSCOREINFSA(-4)
DUSCOREINFSA(-5)
DTBAA3M
DTBAA3M(-1)
DTBAA3M(-2)
DTBAA3M(-3)
1.071473
0.558555
-3.483041
-2.436850
-1.433233
-1.333765
-0.923709
-0.459293
3.328652
0.428475
4.889923
1.526295
1.444886
0.217458
1.083884
0.808796
0.555871
0.507371
0.398263
0.257246
1.381126
1.124256
1.678523
1.140696
0.741562
2.568569
-3.213482
-3.012937
-2.578358
-2.628777
-2.319342
-1.785425
2.410101
0.381118
2.913231
1.338038
0.4587
0.0105
0.0014
0.0027
0.0102
0.0089
0.0208
0.0749
0.0164
0.7033
0.0038
0.1816
R-squared
Adjusted R-squared
S.E. of regression
F-statistic
Prob(F-statistic)
-8.185267
-8.412830
11.42936
2.386014
0.007054
Mean dependent var
S.D. dependent var
Sum squared resid
Durbin-Watson stat
4.547815
3.725304
57999.81
2.040891
TI Econometrics II 2006/2007, Chapter 7.5 – p. 20/32
Partial Adjustment interpretation ADL(1,0)
NB: in following two examples: λ is an adjustment parameter,
not equilibrium parameter. δ is an equilibrium parameter, not a
constant term.
Partial adjustment
The economic structural model:
yt = yt−1 + λ(yt∗ − yt−1 ) + εt
yt∗ = γ + δxt
with ADL model:
yt = λγ + (1 − λ)yt−1 + λδxt + εt :
ADL(1, 0).
Exercise (4): Derive this ADL model.
TI Econometrics II 2006/2007, Chapter 7.5 – p. 21/32
Adaptive expectations A(R)DL(1,0)-MA(1)
Adaptive Expectations
The economic structural model, where the true explanatory
variable (i.c. expected x) x∗t+1 is unobserved:
yt
= γ + δx∗t+1 + εt
x∗t+1 = x∗t + λ(xt − x∗t )
The corresponding A(R)DL(1,0) form has MA(1) errors
yt = λγ + (1 − λ)yt−1 + λδxt + εt − (1 − λ)εt−1 ,
which can also be written in A(R)DL(0,∞) form:
yt = γ + (1 − (1 − λ)L))−1 λδxt + εt .
Exercise (5): derive this A(R)DL(0,∞) form.
TI Econometrics II 2006/2007, Chapter 7.5 – p. 22/32
(Weak) exogeneity
In the context of ECM models one often extends the old concept
of predeterminedness (“independence” of regressor variable and
present and future structural equation errors) to the concept of
weak exogeneity. (Short: exogeneity in §4.1.3: plim n1 X ′ ε) = 0).
A variable xt is said to be weakly exogenous for estimating a
parameter of interest λ, if inference on λ conditional on xt
involves no loss of information. In practice this means that joint
modelling of yt and xt does not improve inference on λ if xt is
weakly exogenous.
Strict exogeneity : “Independence” of regressors and structural
equation errors at all leads and lags, does not apply to ECM.
TI Econometrics II 2006/2007, Chapter 7.5 – p. 23/32
Strong exogeneity, Granger non-causality
The ADL model can be used as part of a forecasting procedure
for yt . If we want efficient inference on future yt without making a
joint dynamic model for yt and xt we need a strong exogeneity
assumption.
Strong Exogeneity of xt for (forecasting) equation yt combines
two requirements:
1. (Weak) exogeneity of xt for estimating all the parameters of
the equation for yt
2. Granger non-causality of yt for xt , §7.6.2
• In linear forecasting of xt using lags of xt , additional
lags of yt do not decrease forecast MSE: Partial
correlations of xt with yt−1 , yt−2 , . . . are zero.
• Regression F -test for Granger-noncausality in auxiliary
equation for xt .
TI Econometrics II 2006/2007, Chapter 7.5 – p. 24/32
Extending A(R)DL from the I(0) to the I(1) world
The A(R)DL model can also be used for nonstationary series,
but then one has to be careful with statistical inference.
√
Warning: It is a case where the CLT, ( n consistency,
asymptotic normality), does not apply in general).
Extension: In the I(1) world (see §7.3.1 for definition) the notion
of long run equilibrium corresponds to (stochastic)
cointegration and the existence of a common trend (a
common I(1)-component).
Further extension: The concept of cointegration does not
require exogeneity of x for the estimation of the parameters in
the A(R)DL equation(s). It does require I(1)-ness of both x and
y . Cointegration analysis without exogeneity assumptions is
discussed in §7.6.
TI Econometrics II 2006/2007, Chapter 7.5 – p. 25/32
Cointegration and Spurious Regression
Definition cointegration: yt , xt ∼ CI(1, 1):
yt ∼ I(1), xt ∼ I(1), linear combination (yt − λxt ) ∼ I(0).
yt − λxt integrated of order 1-1=0. Cointegration concerns only
the stochastic part of series, i.e. not the deterministic part.
Spurious regression:
yt ∼ I(1), xt ∼ I(1), yt and xt independent, regress yt on
constant and xt (so that residuals of spurious regression are
also I(1)).
Spurious regression problem:
Apply assumption of I(0) (or even WN) to I(1) variables in
assessing statistical significance of correlations and regression
coefficients. Result: incorrect rejection of H0 : “no linear
relationship”.
TI Econometrics II 2006/2007, Chapter 7.5 – p. 26/32
Spurious regression, non-cointegration
Regress I(1) process yt on independent I(1) process xt . This
means the residual process is also I(1)!
OLS based inference (§2.3): Overrejection of H0 of
independence because of extremely strong positive serial
correlation in the error term.
GMM based inference (§5.5.3) (automatic “Newey-West
correction” for serial correlation in error term) does not work for
I(1) either. Many moments (variances and covariances) that are
used in GMM are not constant over time.
The (old-fashioned) correction for a determistic trend does
not work either. The ’omitted trend variable’ is stochastic and
not removed by correction for a deterministic trend.
TI Econometrics II 2006/2007, Chapter 7.5 – p. 27/32
Cointegration in a ADL(1,1) model
Consider the A(R)DL model with xt ∼ I(1):
yt = α + φyt−1 + β0 xt + β1 xt−1 + εt
xt = xt−1 + ηt
with εt , ηt independent WN, |φ| < 1, β0 + β1 6= 0, so xt ∼ I(1)
and yt = (1 − φL)−1 [α + β0 xt + β1 xt−1 + εt ] ∼ I(1).
There is no long run equilibrium value for x.
∆yt is a stationary, invertible ARMA(1,1) process and therefore
I(0). This means yt is I(1).
yt fluctuates around [α0 + (β0 + β1 )xt ]/(1 − φ).
Show that yt and xt are cointegrated with one common trend:
(1 − L)−1 ηt .
TI Econometrics II 2006/2007, Chapter 7.5 – p. 28/32
A(R)DL(1,1) in ECM form in I(1) world
The error/equilibrium correction form is:
∆yt = β0 ∆xt − (1 − φ)zt−1 + εt ,
where deviation from equilibrium zt is defined as
α
(β0 + β1 )
zt = yt −
−
xt .
(1 − φ)
(1 − φ)
Exercise (6): Prove zt is I(0), therefore xt and yt are CI(1,1).
Hints: Apply the ”unit root” decompositions
α(L) = α0 + α1 L = α(1) − α1 ∆ and/or α0 + α1 L = α0 ∆ + α(1)L
to φ(L) and/or β(L). See the connection with Exercise (3).
TI Econometrics II 2006/2007, Chapter 7.5 – p. 29/32
Warning: Coefficient tests in ECM in I(1) world
Aymptotic inference on coefficient tests in ECM with possible
I(1) regressors is not standard, even correcting for serial
correlation in the residuals.
• Nonstandard (ADF-type) t-test examples, test for
- (1 − φ) = 0 (coeff of yt−1 if no ECM): cf. Augmented DF,
- β0 + β1 = 0 (coeff for xt−1 in (2) if no ECM).
tests involving (1 − φ) = 0 and β0 + β1 = 0: ’single equation
error correction tests’ for non-cointegration. These tests
involve I(1) regressors under H0 . See e.g. Ericsson and
MacKinnon (2002).
• Standard (asymptotically normal) t-tests apply in test for:
- φ = 0 (coeff of yt−1 in (2)): test immediate full adjustment
of y to disequilibrium.
- β0 = 0 (coeff of ∆xt in (2)): zero direct impact x on y .
Reference: Ericsson, N. R. and MacKinnon, J. G. (2002) ”Distributions of Error
Correction Tests for Cointegration”, The Econometrics Journal, 5, 285-318.
TI Econometrics II 2006/2007, Chapter 7.5 – p. 30/32
Engle-Granger two-step test for non-cointegration
Engle-Granger is a simple single equation regression testing
approach for cointegration (not in book). We dinstinguish a
dependent variable and independent variables. Regression
(GMM) approaches to cointegration testing apply when
multivariate testing (as discussed in §7.6) is infeasible.
A multivariate approach (within Vector AutoRegressive models)
is discussed in §7.6.3, within a joint model for yt and for xt .
• How do we test whether there really is a cointegrating
relation using Dickey-Fuller tests?
Test for non-cointegration
• First test yt ∼ I(1) and xt ∼ I(1) using Dickey-Fuller tests
• If I(2) rejected and I(0) not rejected for both yt and xt , then
• Apply (A)DF-test on OLS residuals of ’static’ regression of yt
on xt . Call these residuals zbt .
TI Econometrics II 2006/2007, Chapter 7.5 – p. 31/32
Single equation test for non-cointegration, ctd.
(A)DF-test on zbt with estimated coefficients
• H0 : zt ∼ I(1): non-cointegration
• H1 : zt ∼ I(0): cointegration
• test-statistic: DF -t-test (without trend)
• Critical values: depend on presence of deterministics and
on number of I(1) x-variables. NB: x-variables should not
cointegrate amongst themselves!
• Reject for very negative values
• Consequence of rejection: proceed with cointegrating
modelling
NB: More efficient regression tests (involving ’augmented’
estimators of the cointegrating relationship) exist. Not discussed
here.
TI Econometrics II 2006/2007, Chapter 7.5 – p. 32/32