Transcript 14 Vector Autoregressions, Unit Roots, and Cointegration

14 Vector Autoregressions,
Unit Roots, and Cointegration
What is in this Chapter?
• This chapter discusses work on timeseries analysis starting in the 1980s.
– First there is a discussion of vector
autoregression models,
– Next we talk of the different unit root tests.
– Finally, we discuss cointegration, which is a
method of analyzing long-run relationships
between nonstationary variables. We discuss
tests for cointegration and estimation of
cointegrating relationships
14.2 Vector Autoregressions
• In previous sections we discussed the analysis
of a single time series
• When we have several time series, we need to
take into account the interdependence between
them
• One way of doing this is to estimate a
simultaneous equations model as discussed in
Chapter 9 but with lags in all the variables
• Such a model is called a dynamic simultaneous
equations model
14.2 Vector Autoregressions
• However, this formulation involves two steps:
– first, we have to classify the variables into two categories,
endogenous and exogenous,
– second, we have toimpose some constraints on the parameters
to achieve identification.
• Sims argues that both these steps involve many arbitrary
decisions and suggests as an alternative, the
vectorautoregression (VAR) approach.
• This is just a multiple time-series generalization of the
AR model.
• The VAR model is easy to estimate because we can use
the OLS method
14.2 Vector Autoregressions
14.2 Vector Autoregressions
14.2 Vector Autoregressions
14.2 Vector Autoregressions
14.2 Vector Autoregressions
14.3 Problems with VAR Models in
Practice
• We have considered only a simple model
with two variables and only one lag for
each.
• In practice, since we are not considering
any moving average errors, the
autoregressions would probably have to
have more lags to be useful for prediction
• Otherwise, univariate ARMA models
would do better.
14.3 Problems with VAR Models in
Practice
• Suppose that we consider say six lags for
each variable and we have a small system
with four variables
• Then each equation would have 24
parameters to be estimated and we thus
have 96 parameters to estimate overall
• This overparameterization is one of the
major problems with VAR models.
14.3 Problems with VAR Models in
Practice
• One such model that has been found particularly
useful in prediction is the Bayesian
vectorautoregression (BVAR)
• In BVAR we assign some prior distributions for
the coefficients in the vector autoregressions
• In each equation, the coefficient of the own
lagged variable has a prior mean 1, all others
have prior means 0, with the variance of the
prior decreasing as the lag length increases
• For instance, with two variables y1t and y2t and
four lags for each, the first equation will be
14.3 Problems with VAR Models in
Practice
14.4 Unit Roots
14.4 Unit Roots
14.5 Unit Root Tests
14.5 Unit Root Tests
14.5 Unit Root Tests
14.5 Unit Root Tests
14.5 Unit Root Tests
• The Low Power of Unit Root Tests
– Schwert (1989) first presented Monte Carlo evidence
to point out the size distortion problems of the
commonly used unit root tests: the ADF and PP tests.
– Whereas Schwert complained about size distortions,
DeJong et al. complained about the low power of unit
root tests
– They argued that the unit root tests have low power
against plausible trend-stationary alternatives
14.5 Unit Root Tests
– They argue that the PP tests have very low
power (generally less than 0.10) against
trend-stationary alternatives but the ADF test
has power approaching 0.33 and thus is likely
to be more useful in practice.
– They conclude that tests with higher power
need to be developed
14.5 Unit Root Tests
14.5 Unit Root Tests
14.5 Unit Root Tests
14.5 Unit Root Tests
• Structural Change and Unit Roots
– In all the studies on unit roots, the issue of
whether a time series is of the DS or TS type
was decided by analyzing the series for the
entire time period during which many major
events took place
– The Nelson-Plosser series, for instance,
covered the period 1909-1970,which includes
the two world wars and the Depression of the
1930s
14.5 Unit Root Tests
• If there have been any changes in the trend
because of these events, the results obtained by
assuming a constant parameter structure during
the entire period will be suspect
• Many studies done using the traditional multiple
regression methods have included dummy
variables (see Sections 8.2 and 8.3) to allow for
different intercepts (and slopes)
• Rappoport and Richlin (1989) show that a
segmented trend model is a feasible alternative
to the DS model.
14.5 Unit Root Tests
• Perron (1989) argues that standard tests for the
unit root hypothesis against the trend-stationary
(TS) alternatives cannot reject the unit root
hypothesis if the time series has a structural
break.
• Of course, one can also construct examples
where, for instance
– y1, y2, • • •, ym is a random walk with drift
– ym+1, ..., ym+n is another random walk with a
different drift
– and the combined series is not the DS type
14.5 Unit Root Tests
• Perron's study was criticized on the argument
that he "peeked at the data" before analysis—
that after looking at the graph, he decided that
there was a break
• But Kim (1990), using Bayesian methods, finds
that even allowing for an unknown breakpoint,
the standard tests of the unit root hypothesis
were biased in favor of accepting the unit root
hypothesis if the series had a structural break at
some intermediate date.
14.5 Unit Root Tests
• When using long time series, as many of
these studies have done, it is important to
take account of structural changes.
Parameter constancy tests have frequently
been used in traditional regression
analysis.
14.6 Cointegration
14.6 Cointegration
• In the Box-Jenkins method, if the time series is
nonstationary (as evidenced by the correlogram
not damping), we difference the series to
achieve stationarity and then use elaborate
ARMA models to fit the stationary series.
• When we are considering two time series, yt and
Xt say, we do the same thing.
• This differencing operation eliminates the trend
or long-term movement in the series
14.6 Cointegration
• However, what we may be interested in is
explaining the relationship between the trends in
yt and Xt
• We can do this by running a regression of Yt on
Xt, but this regression will not make sense if a
long-run relationship does not exist.
• By asking the question of whether yt and Xt are
cointegrated, we are asking whether there is any
long-run relationship between the trends in yt
and xt.
14.6 Cointegration
• The case with seasonal adjustment is similar
• Instead of eliminating the seasonal components from y
and x and then analyzing the de-seasonalized data, we
might also be asking whether there is a relationship
between the seasonals in y and x
• This is the idea behind "seasonal cointegration.
• Note that in this case we do not considerfirst differences
or I(1) processes
• For instance, with monthly data we consider
twelfthdifferences yt-yt-12 Similarly, for Xt we consider
Xt-Xt-12
14.7 The Cointegrating Regression
14.7 The Cointegrating Regression
14.7 The Cointegrating Regression
14.7 The Cointegrating Regression
14.7 The Cointegrating Regression
14.9 Cointegration and Error
Correction Models
• If xt and yt are cointegrated, there is a long-run
relationship between them
• Furthermore, the short-run dynamics can be
described by the error correction model (ECM)
• This is known as the Granger representation
theorem
• If Xt ~ I(1), yt ~I(1), and zt = yt -βxt is I(0), then x
and y are said to be cointegrated
• The Granger representation theorem says that
in this case Xt and yt may be considered to be
generated by ECMs:
14.9 Cointegration and Error
Correction Models
14.9 Cointegration and Error
Correction Models
14.10 Tests for Cointegration