14 Vector Autoregressions, Unit Roots, and Cointegration

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14 Vector Autoregressions, Unit Roots, and
Cointegration
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What is in this Chapter?

• This chapter discusses work on time-series
  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

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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

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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

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14.2 Vector Autoregressions
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14.2 Vector Autoregressions
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14.2 Vector Autoregressions
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14.2 Vector Autoregressions
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14.2 Vector Autoregressions
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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.

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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.

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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

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14.3 Problems with VAR Models in Practice
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14.4 Unit Roots
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14.4 Unit Roots
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14.5 Unit Root Tests
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14.5 Unit Root Tests
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14.5 Unit Root Tests
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14.5 Unit Root Tests
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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

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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

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14.5 Unit Root Tests
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14.5 Unit Root Tests
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14.5 Unit Root Tests
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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

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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.

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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
• ym1, ..., ymn is another random walk with a
  different drift
• and the combined series is not the DS type

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14.5 Unit Root Tests

• Perron's study was criticized on the argument
  that he "peeked at the data" before analysisthat
  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.

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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.

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14.6 Cointegration
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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

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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.

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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

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14.7 The Cointegrating Regression
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14.7 The Cointegrating Regression
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14.7 The Cointegrating Regression
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14.7 The Cointegrating Regression
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14.7 The Cointegrating Regression
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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 -ßxt 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

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14.9 Cointegration and Error Correction Models
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14.9 Cointegration and Error Correction Models
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14.10 Tests for Cointegration