STAT 497 LECTURE NOTES 5

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STAT 497LECTURE NOTES 5

• UNIT ROOT TESTS

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UNIT ROOTS IN TIME SERIES MODELS

• Shock is usually used to describe an unexpected
  change in a variable or in the value of the error
  terms at a particular time period.
• When we have a stationary system, effect of a
  shock will die out gradually.
• When we have a non-stationary system, effect of a
  shock is permanent.

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UNIT ROOTS IN TIME SERIES MODELS

• Two types of non-stationarity
• Unit root i.e.,?i 1 homogeneous
  non-stationarity
• ?i gt 1 explosive non-stationarity
• Shock to the system become more influential as
  time goes on.
• Can never be seen in real life

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UNIT ROOTS IN TIME SERIES MODELS
e.g. AR(1)
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UNIT ROOTS IN TIME SERIES MODELS

• A root near 1 of the AR polynomial
• ?differencing
• A root near 1 of the MA polynomial
• ?over-differencing

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UNIT ROOTS IN AUTOREGRESSION

• DICKEY-FULLER (DF) TEST The simplest approach to
  test for a unit root begins with AR(1) model
• DF test actually does not consider ?0 in the
  model, but actually model with ?0 and not ?0
  gives different results.

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DF TEST

• Consider the hypothesis
• The hypothesis is the reverse of KPSS test.

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DF TEST

• To simplify the computation, subtract Yt-1 from
  both sides of the AR(1) model
• If ?0, system has a unit root.

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DF TEST

• DF (1979)

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DF TEST

• Applying OLS method and finding the estimator
  for ?, the test statistic is given by
• The test is a one-sided left tail test. If Yt
  is stationary (i.e.,f lt 1) then it can be shown

• This means that under H0, the limiting
  distribution of t?1 is N(0,1).

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DF TEST

• Under the null hypothesis,
• which does not make any sense. Under the unit
  root null, Yt is not stationary and ergodic,
  and the usual sample moments do not converge to
  fixed constants. Instead, Phillips (1987) showed
  that the sample moments of Yt converge to
  random functions of Brownian motion

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DF TEST

where W(r) denotes a standard Brownian motion
(Wiener process) defined on the unit interval.
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DF TEST

• Using the above results Phillips showed that
  under the unit root null H0 f 1

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DF TEST

• SOME RESULTS
• is super-consistent that is, at
  rate n instead of the usual rate.
• is not asymptotically normally distributed and
  tf1 is not asymptotically standard normal.
• The limiting distribution of tf1 is called the
  Dickey-Fuller (DF) distribution and does not have
  a closed form representation. Consequently,
  quantiles of the distribution must be computed by
  numerical approximation or by simulation.

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DF TEST

• Since the normalized bias (n?1)( - 1) has a
  well defined limiting distribution that does not
  depend on nuisance parameters it can also be used
  as a test statistic for the null hypothesis H0
  f 1.

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DF TEST

• EXAMPLE

?0.05
Critical value ?7.3 for n25 ?7.7 for n50
Not reject H0. There exist a unit root. We need
to take a difference to be able to estimate a
model for the series
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DF TEST

• With a constant term
• The test regression is
• and includes a constant to capture the nonzero
  mean under the alternative. The hypotheses to be
  tested
• This formulation is appropriate for non-trending
  economic and financial series like interest
  rates, exchange rates and spreads.

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DF TEST

• The test statistics tf1 and (n-1)( - 1) are
  computed from the above regression. Under
• H0 f 1, c 0 the asymptotic distributions
  of these test statistics are influenced by the
  presence, but not the coefficient value, of the
  constant in the test regression

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DF TEST

Inclusion of a constant pushes the distributions
of tf1 and (n?1) ( - 1) to the left.
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DF TEST

• With constant and trend term
• The test regression is
• and includes a constant and deterministic time
  trend to capture the deterministic trend under
  the alternative. The hypotheses to be tested

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DF TEST

• This formulation is appropriate for trending time
  series like asset prices or the levels of
  macroeconomic aggregates like real GDP. The test
  statistics tf1 and (n - 1) ( - 1) are
  computed from the above regression.
• Under H0 f 1, d 0 the asymptotic
  distributions of these test statistics are
  influenced by the presence but not the
  coefficient values of the constant and time trend
  in the test regression.

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DF TEST

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DF TEST

• The inclusion of a constant and trend in the test
  regression further shifts the distributions of
  tf1 and (n - 1)( - 1) to the left.

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DF TEST

• What do we conclude if H0 is not rejected? The
  series contains a unit root, but is that it? No!
  What if YtI(2)? We would still not have
  rejected. So we now need to test
• H0 YtI(2) vs. H1 YtI(1)
• We would continue to test for a further unit root
  until we rejected H0.

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DF TEST

• This test is valid only if at is WN. If there is
  a serial correlation, the test should be
  augmented. So, check for possible autoregression
  in at.
• Many economic and financial time series have a
  more complicated dynamic structure than is
  captured by a simple AR(1) model.
• Said and Dickey (1984) augment the basic
  autoregressive unit root test to accommodate
  general ARMA(p, q) models with unknown orders and
  their test is referred to as the augmented
  Dickey- Fuller (ADF) test

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AUGMENTED DICKEY-FULLER (ADF) TEST

• If serial correlation exists in the DF test
  equation (i.e., if the true model is nor AR(1)),
  then use AR(p) to get rid of the serial
  correlation.

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ADF TEST

• To test for a unit root, we assume that

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ADF TEST

• Hence, testing for a unit root is equivalent to
  testing ?1 in the following model

or
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ADF TEST

• Hypothesis

Reject H0 if t?1ltCV
Reject H0 if t?0ltCV

• We can also use the following test statistics

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ADF TEST

• The limiting distribution of the test statistic
• is non-standard distribution (function of
  Brownian motion _ or Wiener process).

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Choosing the Lag Length for the ADF Test

• An important practical issue for the
  implementation of the ADF test is the
  specification of the lag length p. If p is too
  small, then the remaining serial correlation in
  the errors will bias the test. If p is too large,
  then the power of the test will suffer.

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Choosing the Lag Length for the ADF Test

• Ng and Perron (1995) suggest the following data
  dependent lag length selection procedure that
  results in stable size of the test and minimal
  power loss
• First, set an upper bound pmax for p. Next,
  estimate the ADF test regression with p pmax.
  If the absolute value of the t-statistic for
  testing the significance of the last lagged
  difference is greater than 1.6, then set p pmax
  and perform the unit root test. Otherwise, reduce
  the lag length by one and repeat the process.

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Choosing the Lag Length for the ADF Test

• A useful rule of thumb for determining pmax,
  suggested by Schwert (1989), is
• where x denotes the integer part of x. This
  choice allows pmax to grow with the sample so
  that the ADF test regressions are valid if the
  errors follow an ARMA process with unknown order.

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ADF TEST

• EXAMPLE n54
• Examine the original model and the differenced
  one to determine the order of AR parameters. For
  this example, p3.
• Fit the model with t 4, 5,, 54.

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ADF TEST

• EXAMPLE (contd.) Under H0,

n50, CV-13.3

• H0 cannot be rejected. There is a unit root. The
  series should be differenced.

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ADF TEST

• If the test statistics is positive, you can
  automatically decide to not reject the null
  hypothesis of unit root.
• Augmented model can be extended to allow MA terms
  in at. It is generally believed that MA terms are
  present in many macroeconomic time series after
  differencing. Said and Dickey (1984) developed an
  approach in which the orders of the AR and MA
  components in the error terms are unknown, but
  can be approximated by an AR(k) process where k
  is large enough to allow good approximation to
  the unknown ARMA(p,q) process.

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ADF TEST

• Ensuring that at is approximately WN

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PHILLIPS-PERRON (PP) UNIT ROOT TEST

• Phillips and Perron (1988) have developed a more
  comprehensive theory of unit root
  nonstationarity. The tests are similar to ADF
  tests. The Phillips-Perron (PP) unit root tests
  differ from the ADF tests mainly in how they deal
  with serial correlation and heteroskedasticity in
  the errors. In particular, where the ADF tests
  use a parametric autoregression to approximate
  the ARMA structure of the errors in the test
  regression, the PP tests ignore any serial
  correlation in the test regression.
• The tests usually give the same conclusions as
  the ADF tests, and the calculation of the test
  statistics is complex.

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PP TEST

• Consider a model
• DF at iid
• PP at serially correlated
• Add a correction factor to the DF test statistic.
  (ADF is to add lagged ?Yt to whiten the
  serially correlated residuals)

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PP TEST

• The hypothesis to be tested

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PP TEST

• The PP tests correct for any serial correlation
  and heteroskedasticity in the errors at of the
  test regression by directly modifying the test
  statistics t?0 and . These modified
  statistics, denoted Zt and Z?, are given by

The terms and are consistent estimates
of the variance parameters
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PP TEST

• Under the null hypothesis that ? 0, the PP Zt
  and Z? statistics have the same asymptotic
  distributions as the ADF t-statistic and
  normalized bias statistics.
• One advantage of the PP tests over the ADF tests
  is that the PP tests are robust to general forms
  of heteroskedasticity in the error term at.
  Another advantage is that the user does not have
  to specify a lag length for the test regression.

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PROBLEM OF PP TEST

• On the other hand, the PP tests tend to be more
  powerful but, also subject to more severe size
  distortions
• Size problem actual size is larger than the
  nominal one when autocorrelations of at are
  negative.
• more sensitive to model misspecification (the
  order of autoregressive and moving average
  components).
• Plotting ACFs help us to detect the potential
  size problem
• Economic time series sometimes have negative
  autocorrelations especially at lag one, we can
  use a Monte Carlo analysis to simulate the
  appropriate critical values, which may not be
  attractive to do.

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Criticism of Dickey-Fuller and Phillips-Perron
Type Tests

• Main criticism is that the power of the tests is
  low if the process is stationary but with a root
  close to the non-stationary boundary.
• e.g. the tests are poor at deciding if f1 or
  f0.95, especially with small sample sizes.
• If the true data generating process (dgp) is
• Yt 0.95Yt-1 at
• then the null hypothesis of a unit root should
  be rejected.
• One way to get around this is to use a
  stationarity test (like KPSS test) as well as the
  unit root tests we have looked at (like ADF or
  PP).

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Criticism of Dickey-Fuller and Phillips-Perron
Type Tests

• The ADF and PP unit root tests are known (from
  MC simulations) to suffer potentially severe
  finite sample power and size problems.
• 1. Power The ADF and PP tests are known to
  have low power against the alternative hypothesis
  that the series is stationary (or TS) with a
  large autoregressive root. (See, e.g., DeJong, et
  al, J. of Econometrics, 1992.)
• 2. Size The ADF and PP tests are known to
  have severe size distortion (in the direction of
  over-rejecting the null) when the series has a
  large negative moving average root. (See, e.g.,
  Schwert. JBES, 1989 MA -0.8, size 100!)

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Criticism of Dickey-Fuller and Phillips-Perron
Type Tests

• A variety of alternative procedures have been
  proposed that try to resolve these problems,
  particularly, the power problem, but the ADF and
  PP tests continue to be the most widely used unit
  root tests. That may be changing!

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STRUCTURAL BREAKS

• A stationary time-series may look like
  nonstationary when there are structural breaks in
  the intercept or trend
• The unit root tests lead to false non-rejection
  of the null when we dont consider the structural
  breaks ? low power
• A single breakpoint is introduced in Perron
  (1989) into the regression model Perron (1997)
  extended it to a case of unknown breakpoint
• Perron, P., (1989), The Great Crash, the Oil
  Price Shock and the Unit Root Hypothesis,
  Econometrica, 57, 13611401.

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STRUCTURAL BREAKS

• Consider the null and alternative hypotheses
• H0 Yt a0 Yt-1 µ1DP at
• H1 Yt a0 a2t µ2DL at
• Pulse break DP 1 if t TB 1 and zero
  otherwise,
• Level break DL 0 for t 1, . . . , TB and one
  otherwise.
• Null Yt contains a unit root with a onetime
  jump in the level of the series at time t TB
  1 .
• Alternative Yt is trend stationary with a
  onetime jump in the intercept at time t TB 1
  .

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Simulated unit root and trend stationary
processes with structural break.

• H0 ------
• a0 0.5,
• DP 1 for n 51 zero otherwise,
• µ1 10.

n 100 at i.i.d. N(0,1) y00

• H1
• a2 0.5,
• DL 1 for n gt 50.
• µ2 10

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Power of ADF tests Rejection frequencies of
ADFtests

• ADF tests are biased toward non-rejection of the
  null.
• Rejection frequency is inversely related to the
  magnitude of the shift.
• Perron estimated values of the autoregressive
  parameter in the DickeyFuller regression was
  biased toward unity and that this bias increased
  as the magnitude of the break increased.

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Testing for unit roots when there are structural
changes

• Perron suggests running the following OLS
  regression
• H0 a1 1 tratio, DF unit root test.
• Perron shows that the asymptotic distribution of
  the t-statistic depends on the location of the
  structural break, ? TB/n
• critical values are supplied in Perron (1989) for
  different assumptions about l, see Table IV.B.