Unit Roots

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Unit Roots Forecasting

• Methods of Economic Investigation
• Lecture 20

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Last Time

• Descriptive Time Series
• Processes
• Estimating with exogenous serial correlation
• Estimating with endogenous processes

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Todays Class

• Non-stationaryTime Series
• Unit Roots and Spurious Regressions
• Orders of Integration
• Returning to Causal Effects
• Impulse Response Functions
• Forecasting

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Random Walk Processes

• Definition
• Etxt1 xt that is todays value of X is the
  best predictor of tomorrows value.
• This looks very similar to our AR(1) processes,
  where f 1.
• Autocovariances of a random walk are not well
  defined in a technical sense, but imagine AR(1)
  process with f?1 we have nearly perfect
  autocorrelation for any two time periods.
• persistence dies out too slowly so most of
  variance will largely be due to very
  low-frequency shocks.

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Permanence of Shocks in Unit Root

• An innovation (a shock at t ) to a stationary AR
  process dies out eventually (the autocorrelation
  function declines eventually to zero).
• A shock to a random walk is permanent
• Variance is increasing over time
• Var(xt) Var(x0) ts2

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Drifts and Trends

• Deterministic trend
• yt dt xt et
• xt is some stationary process
• yt is trend stationary
• Its easy to add a deterministic trend to a
  random walk

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Orders of Integration

• A series is integrated of order p if a p
  differences render it stationary.
• If a time series is integrated and differencing
  once renders the time series stationary, then it
  is integrated of order 1 or I(1).
• If it is necessary to difference twice before a
  time series is stationary, then it is I(2), and
  so forth.

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Integrated Series

• If a time series has a unit root, it is said to
  be integrated.
• First differencing the time series removes the
  unit root. E.g. in the case of a random walk
• yt yt-1 ut, ut N(0, s2)
• ?yt ut
• the first difference is white noise, which is
  stationary.
• For an AR(p) a unit root implies
• 1 ß1L ß2L2 ... ßpLp (1 L) (1 ?1L
  ?2L2 ... ?pLp-1) 0
• and as a result first differencing also removes
  the unit root.

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Non-stationarity

• Nonstationarity can have important consequences
  for regression modelsand inference.
• Autoregressive coefficients are biased
• t-statistics have non-normal distributions even
  in large samples
• Spurious regression

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Problem Spurious Regression

• imagine we now have two series are generated by
  independent random walks,
• Suppose we run yt on xt using OLS, that is we
  estimate yt a ßxt ?t.
• In this case, you tend to see significant ß
  because the low-frequency changes make it seem as
  if the two series are in some way associated.

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Unit Root Tests

• Standard Dickey-Fuller test appropriate for AR(1)
  processes
• 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
• Called the augmented Dickey-Fuller (ADF) test

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ADF Test 1

• The ADF test tests the null hypothesis that a
  time series yt is I(1) against the alternative
  that it is I(0), assuming that the dynamics in
  the data have an ARMA structure.
• The ADF test is based on estimating the test
  regression

Other serial correlation
Deterministic variables
Potential unit root
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ADF Test - 2

• To see why
• Subtract yt-1 from both sides and define
• F (a1 a2 ap 1) and we get
• Test F 0 against alternative Flt0
• Use special DF upperbound and lowerbound
• Under alternative, test statistic is not normally
  distributed

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Estimating in Time Series

• Non-stationary time series can lead to a lot of
  problems in econometric analysis.
• In order to work with time series, particular in
  regression models, we should therefore transform
  our variables to stationary time series first.
• First differencing removes unit roots or trends.
  Hence, difference a time series until it is I(0).
  
• Differencing too often is less of a problem since
  a differenced stationary series is still
  stationary.
• Regressions of one stationary variable on another
  is less problematic.
• Although observations may not be independent, we
  can expect regression to have similar properties
  as with cross sectional data.

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Impulse Response Function

• One of the most interesting things to do with an
  ARMA model is form predictions of the variable
  given its past.
• we want to know what is Et(xtj )
• Can do inference with Vart(xtj)
• The impulse response function is a simpel way to
  do that
• Follow te path that x follows if it is kicked by
  unit shock
• characterization of the behavior of our models.
• allows us to start thinking about causes and
  effects.

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Impulse Response and MA(8)

• 1. The MA(8) representation is the same thing as
  the impulse response function.
• i.e.
• The easiest way to calculate an MA(8)
  representation is to simulate the
  impulse-response function.
• The impulse response function is the same as
  Et(xtj) - Et-1(xtj).

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Causality and Impulse Response

• Can either forecast or simulate the effect of a
  given shock
• Try to pick a shock time/level to simulate and
  try to replicate observed data
• Issue of whether that shock is what really
  happened
• Know a shock happened in time time t
• See if observed change (more on this next time)
• Granger causality implies a correlation between
  the current value of one variable and the past
  values of others
• it does not necessarily imply that changes in one
  variable causes changes in another.
• Use a F-test to jointly test for the significance
  of the lags on the explanatory variables, this in
  effect tests for Granger causality between
  these variables.
• Can visually see correlation in impulse response
  functions

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Source Cochrane, QJE (1994)
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Next Time

• Estimating Causality in Time Series
• Some additional forecasting stuff
• Testing for breaks
• Regression discontinuity/Event study