Spurious Regression and Simple Cointegration

1
Spurious Regression and Simple Cointegration
Gloria González-Rivera University of California,
Riverside and Jesús Gonzalo U. Carlos III de
Madrid
2
Spurious Regression
Set-up
Regress
What do you expect to get?
3
Spurious Regression (cont)
What does it really happen?
4
Spurious Regression (cont)
Spurious Regression Problem (SPR) Regression of
an integrated series on another unrelated
integrated series produces t-ratios on the slope
parameter which indicates a relationship much
more often than they should at the nominal test
level. This problem does not dissapear as the
sample size is increased.
The Spurious Regression Problem can appear with
I(0) series too (see Granger, Hyung and Jeon
(1998)). This is telling us that the problem is
generated by using WRONG CRITICAL VALUES!!!! In
a Spurious Regression the errors would be
correlated and the standard t-statistic will be
wrongly calculated because the variance of the
errors is not consistently estimated. In the I(0)
case the solution is
Maybe the same thing can be done to solve the SPR
problem with I(1) variables.
5
Spurious Regression (cont)
How do we detect a Spurious Regression (between
I(1) series)? Looking at the correlogram of the
residuals and also by testing for a unit root on
them. How do we convert a Spurious Regression
into a valid regression? By taking
differences. Does this solve the SPR problem? It
solves the statistical problems but not the
economic interpretation of the regression. Think
that by taking differences we are loosing
information and also that it is not the same
information contained in a regression involving
growth rates than in a regression involved the
levels of the variables.
6
Spurious Regression (cont)
Does it make sense a regression between two I(1)
variables? Yes if the regression errors are
I(0). Can this be possible? The same question
asked David Hendry to Clive Granger time
ago. Clive answered NO WAY!!!!! but he also said
that he would think about. In the plane trip back
home to San Diego, Clive thought about it and
concluded that YES IT IS POSSIBLE. It is possible
when both variables share the same source of the
I(1)ness (co-I(1)), when both variables move
together in the long-run (co-move), ... when both
variables are
COINTEGRATED!!!!!!!!!!!!!!!!!!!!!!!
7
Some Cointegration Examples
Example 1 Theory of Purchasing Power Parity
(PPP) Apart from transportation costs, good
should sell for the same effective price in two
countries
An index of the price level in the USA
per
Price Index for Spain
In logs
A weaker version of the PPP
If the three variables are I(1) and zt is I(0)
then the PPP theory is implying cointegrating
between pt, st and pt .
8
Some Cointegrating Examples (cont)
Example 2 Present Value Models (PVM)
Yt Long-term yields yt
short-term yields Stock Prices
dividends
Consumption
labor income If yt has a unit root and the PVM
holds then Yt and yt will be cointegrated (see
Campbell and Shiller (1987)
9
Geometric Interpretation of Cointegration
What is an ATTRACTOR?
Consider the price (over time) of a commodity
that is traded in two different locations i and
j.
. 3
. 4
. 5
. 2
. 1
Suppose that
Demand will go to location j
The adjustment does not have to be instantaneous
but eventually
Long-run equilibrium this is a linear
attractor. Shocks to the economy make us move out
of the attractor.
10
Geometric Interpretation of Cointegration (cont)
The concept of attractor is the concept of
long-run equilibrium between two stochasic
processes. We allow the two variables to diverge
in the short-run but in the long-run they have
to converge to a common region denominated
attractor region. In other words, if from now on
there are not any shocks in the system, the two
stochastic processes will converge together to a
common attractor set. Question 1 Write in
intuition terms two two economic examples where
cointegration can be present. Explain
why? Question 2 A drunk man leaving a bar
follows a random walk. His dog also follows a
random walk on its own. Then they go into a park
where dogs are not allowed to be untied.
Therefore the drunk man puts a strap on his dog
and both enter into the park Will their paths be
cointegrated? Why?
11
Definition of Cointegration

• From an economic point of view we are interested
  on answering
• Can we test for the existence of this attractor?
• If it exists, how can be introduced into our
  econometric modelling?

Some rules on linear combinations of I(0) and
I(1) processes
Definition
12
Why Two Series Are Cointegrated?
Consider the following construction
The following linear combination
Result 1.
If two I(1) series have a common I(1) factor and
idiosincratic I(0) components, then they are
cointegrated.
It can be proved that Result 1 is an IF and ONLY
IF result.
13
A Simple Test for Cointegration

• This test is due to Engle and Granger (1987)
• Estimate the following regression model in
  levels
• Perform an ADF test on the residuals

• The null hypothesis

• This means that the residuals have a unit root
  and therefore yt and xt are not cointegrated.
• If the residuals are I(0) then yt and xt are
  cointegrated

14
Error Correction Model
Vector Error Correction Model(VECM)
For a bivariate VAR, where are I(1)
and cointegrated,
Result 2.
If are cointegrated, then exists an
ECM representation. Cointegration is a necessary
condition for ECM and viceversa (Granger
Representation Theorem).
15
Geometric intuition of the Error Correction Model
Intuition on ECM
Wherever the system goes at time t1 , depends on
the magnitude and sign of the disequilibrium
error of the previous period t, at least.
Short-run dynamics movements in the short run,
modeled in the ECM, that guide the economy
towards the Long-run equilibrium
16
Cointegration and Econometric Modelling
1. Check the integration of
use the Dickey-Fuller tests 2. Testing for
cointegration between . Find the
cointegrating relation. OLS regression (minimize
the variance of residuals).
Warning we will be tempted to use the
Dickey-Fuller tests but the test is based in
residuals . We need a different set of critical
values, as in Engle-Granger (89) or McKinnon
(90).
17
Cointegration and Economic Modelling (cont)
3. Short-run dynamics ECM

• Engle-Granger two-step estimation method
• Estimate
• Plug in the ECM (SURE estimation)
  estimators in the ECM
• are consistent and efficient.

18
Cointegration with more than two variables
Example 1.
Example 2.
19
Cointegration with more than two variables (cont)
Example 3.
Example 4.
20
Cointegration Testing and Estimation with more
than two variables
Johansens method

• Two major advantages with respect to
  Engle-Granger procedure
• Testing for number of cointegrating vectors when
  Ngt2
• Joint procedure testing and maximum likelihood
  estimation of
• the vector error correction model and long run
  equilibrium relations.

Framework
Consider a VAR(p)
We construct the vector error correction model
transforming the VAR
21
Vector error correction model
Example 5 2 variables, 1 cointegrating vector
22
Objective Construct the likelihood function
under the null and under the alternative, and
construct a likelihood ratio-type test.
Johansens algorithm to maximize the constrained
likelihood is based on canonical correlation
analysis.
Likelihood ratio test has a non-standard
distribution due to the non-stationarity of the
variables.