ECON3350/7350 Cointegration
Eric Eisenstat
The University of Queensland
Lecture 6
Eric Eisenstat
(School of Economics)
ECON3350/7350 Week 6
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Cointegration
Recommended readings
Author
Title
Chapter
Call No
Enders Verbeek
Applied Econometric Time Series, 4e
A Guide to Modern Econometrics
6.1, 6.2 9.2, 9.3
HB139 .E55 2015 HB139 .V465 2012
Eric Eisenstat (School of Economics) ECON3350/7350 Week 6 2 / 18
Spurious Regression
What are the implications of working with I(1) variables?
In general, linear combinations of I(1) variables yields another I(1)
variable; however, in some cases the linear combination can result in I(0). Spurious regression: conclude there is a significant relationship between
I(1) variables when in reality there isn’t one.
Cointegration: I(1) variables are related such that there exists a linear
combination that yields a I(0) process.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 6 3 / 18
Spurious Regression
Let xt = xt−1 + εx,t and yt = yt−1 + εy,t, where εx,t and εy,t are independent white noise.
Since xt and yt are independent random walks, there is no relationship between them.
However, regressing yt = β0 + β1xt + εt typically yields large, significant
β1.
The sampling distribution of β1−β1 under H : β = 0 is not Student-t; it
se(β1) 0 1 β
So, a t-test will tend to reject H0 incorrectly (i.e. α is not the correct significance level).
Eric Eisenstat (School of Economics) ECON3350/7350 Week 6
is more dispersed so that
1
Pr se(β1) > tα,T−k = Pr(reject H0 |H0 is true) > α.
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Spurious Regression
√
T(β1 − β1) −→ ∞. exacerbates the problems with t-tests.
OLS is not consistent, and there is no well-defined sampling distribution.
Significant t-tests occur because even pure random walks tend to resemble trends; when two I(1) variables wander in the same direction for a while, we might mistake this for cointegration.
But in yt = β0 + β1xt + εt, the residual εt is I(1), so would expect low Durbin-Watson statistics.
Practical signs of a spurious regression:
Significant t-statistics; high R2 values; low Durbin-Watson statistics.
In fact, as T −→ ∞, the quantity
OLS estimate of β1 increases with sample size—more data
Eric Eisenstat (School of Economics) ECON3350/7350 Week 6 5 / 18
Cointegration
Consider the following example:
yt = ξt + νy,t, xt = 1ξt +νx,t,
β
ξt = ξt−1 + ηt,
where the errors νy,t, νx,t and ηt are given by
νy,t = −βε1,t, νx,t = − 1 ε2,t, ηt = βε1,t + ε2,t,
β
and ε1,t, ε2,t are both (uncorrelated) white noise processes.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 6 6 / 18
Observe that
∆yt = ηt + νy,t − νy,t−1,
= ε2,t + βε1,t−1 ≡ uy,t,
∆xt = ηt + νx,t − νx,t−1,
= ε1,t + 1 ε2,t−1 ≡ ux,t.
β
When examined individually, yt = yt−1 + uy,t and xt = xt−1 + ux,t are
random walks.
But the linear combination
yt −βxt =νy,t −βνx,t =−βε1,t +ε2,t
is also I(0).
Eric Eisenstat (School of Economics) ECON3350/7350 Week 6 7 / 18
Cointegration and Common Stochastic Trends
In this example, yt and xt are cointegrated.
While each is process is a RW, they in fact wander together.
There exists a long-run stable relationship between yt and xt, such that they never wander too far away from each other.
The key component that ties together yt and xt is the common stochastic trend ξt.
Both yt and xt actually embody stationary movements around the non-stationary ξt.
In economics, many theories suggest that observed economic data are driven by common stochastic trends—variables evolve together over time and there exist long-run equilibria.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 6 8 / 18
Common Stochastic Trend Example
Expectation Theory of the Term Structure of Interest Rates suggests rates are I(1), but spreads are I(0).
16
12
8
4
0
-4
55 60 65 70 75 80 85 90 95 00 05 10 15
GS3 GS5 SPREAD
GS3: ADF statistic is -2.178 (p-value 0.4995); GS5: ADF statistic is -1.977 (p-value 0.6104) SPREAD: ADF statistic is -5.608 (p-value 0.0000)
Eric Eisenstat (School of Economics) ECON3350/7350 Week 6 9 / 18
Definition of Cointegration
Let wt = (w1,t, . . . , wn,t)′ be a n × 1 vector with variables wi,t integrated of order d.
Definition
The components of wt are cointegrated of order d, b, denoted CI(d, b), iff
1 all components of wt are I(d), and
2 there exists a n×1 vector β ̸= 0 such that
zt =β′wt ∼I(d−b), b>0.
The vector β is called the cointegrating vector.
Eric Eisenstat (School of Economics) ECON3350/7350
Week 6 10 / 18
Cointegration Rank
For any given n × 1 vector wt, there may exist 0 ≤ r ≤ n − 1 linearly independent cointegrating vectors β1, . . . , βr. We say that wt has cointegration rank r.
Supposewi,t ∼I(1)fori=1,…,nandletB=(β1,…,βr)bean×r matrix of cointegrating vectors. Then
zt = B′wt is a r × 1 vector with elements zj,t ∼ I(0).
If wt has cointegration rank r, then there are exactly n − r common stochastic trends driving the variables in wt.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 6
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Real Business Cycles Example
King, R. G., C. I. Plosser, J. H. Stock, and M. W. Watson (1991). “Stochastic trends and economic fluctuations.” The American Economic Review, 81, 819-840.
General class of RBC models can be characterized by
Yt = λtKtθN1−θ, t
ln λt = μλ + ln λt−1 + ξt,
where Yt is output, Kt is capital stock, Nt is labour input, and λt is total
factor productivity, which is a RW with drift.
The resource constraint implies Yt = Ct + It, where Ct is consumption and
It is investment.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 6 12 / 18
Real Business Cycles Example
ln λt/θ is a common stochastic trend with growth rate (μλ + ξt)/θ. Slight extension of the Neoclassical Growth Model yields that the great
ratios Ct and It are stationary stochastic processes. Yt Yt
Therefore, theory implies that ln Ct, ln It, and ln Yt are all I(1) but the quantities ln Ct − ln Yt and ln It − ln Yt are I(0).
That is, there exist two cointegrating vectors such that
′ 1 0 −1lnCt lnCt −lnYt
Bwt = 0 1 −1 lnIt= lnI −lnY ∼I(0).
lnYt t t
KPSW use this to compute impulse responses of economic variables to productivity shocks, identified as innovations in the common stochastic trend. Such IRFs are features of the business cycle.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 6 13 / 18
Testing for Cointegration
When I(1) variables cointegrate, a linear combination will be I(0); otherwise all linear combinations will be I(1).
Basic idea to test for cointegration: given a vector wt, estimate the regression
wn,t =β0 +β1w1,t +···+βn−1wn−1,t +εt, and test the residual
ε=wn,t −β0 −β1w1,t −···−βn−1wn−1,t
for a unit root.
Can use the Cointegrating ADF or Cointegrating Regression DW test for
this: rejecting H implies ε ∼ I(0), and therefore, evidence of 0t
cointegration.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 6 14 / 18
Properties of OLS for Cointegrated Variables
If yt and xt are cointegrated, then OLS estimator β0,β1 in yt = β0 + β1xt + εt
will be superconsistent. Normally,
√
T (β − β) −→ N (0, V).
√
That is, β converges at rate
For cointegrated time-series,
T.
T (β − β) −→ N (0, V). That is, β converges at rate T .
Implies that when cointegrating relationships exist, OLS estimates reliably even in small samples; when they do not exist, OLS leads to spurios
relationships.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 6 15 / 18
Income Consumption Example
Define:
ct: (log) consumption; yt: (log) income;
at: (log) wealth;
and arrange them into a 3 × 1 vector wt = (ct, yt, at)′. Extension of the Permanent Income Hypothesis states that for
β=(1,−1,β)′,zt =β′wt =ct−yt−βat isstationary. To test this, regress
ct − yt = β0 + β1at + εt,
and test for stationarity of ε = ct − yt − β0 − β1at.
Use, e.g., Cointegrating ADF with (in this case) two variables–Table C of
Enders.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 6 16 / 18
Cointegration Space
All examples so far assumed cointegrating vectors are either fully known or partially known.
In the permanent income hypothesis example,
1 B = −1.
β
In the RBC example, we can generalize slightly to obtain
1 0 B=0 1.
β1 β2
In all these cases, the cointegrating vectors are unique and we can
estimate unknown components such as β1 and β2 from data.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 6 17 / 18
Cointegration Space
In general, if zt = B′wt is stationary, then for any invertible K, B = BK is another n × r matrix that yields zt = B′wt = K′zt stationary.
Multiplying B by K yields linear combinations of the vectors in B. The r vectors formed by these linear combinations are also valid cointegrating vectors.
Implication: unrestricted B cannot be estimated from data because only the space spanned by (true) cointegrating vectors β1, . . . , βr (termed cointegration space) is identified.
For identification, restrictions on B are necessary:
Impose 0 and 1 restrictions based on theoretical considerations (as in
examples so far), or
Impose the restriction B′B = Ir, i.e., identify the basis of the
cointegration space.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 6 18 / 18