Unit Roots & Forecasting
Methods of Economic
Investigation
Lecture 20
Last Time
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Descriptive Time Series
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Processes
Estimating with exogenous serial correlation
Estimating with endogenous processes
Today’s Class
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Non-stationaryTime Series
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Unit Roots and Spurious Regressions
Orders of Integration
Returning to Causal Effects
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Impulse Response Functions
Forecasting
Random Walk Processes
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Definition:
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Et[xt+1] = xt that is today’s value of X is the
best predictor of tomorrow’s value.
This looks very similar to our AR(1) processes,
where φ = 1.
Autocovariances of a random walk are not well
defined in a technical sense, but imagine AR(1)
process with φ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 lowfrequency “shocks.”
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
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xt  xt 1   t
 xt  2   t   t 1
t
 x0    i 1
i 1
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Variance is increasing over time
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Var(xt) = Var(x0) + tσ2
Drifts and Trends
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Deterministic trend
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yt = δt + xt + εt
xt is some stationary process
yt is “trend” stationary
It’s easy to add a deterministic trend to a
random walk
yt  yt 1    ut
t
 y0  t   ui 1
i 1
Orders of Integration
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A series is integrated of order p if a p
differences render it stationary.
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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.
Integrated Series
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If a time series has a unit root, it is said to be
integrated.
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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, σ2)
Δyt = ut
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the first difference is white noise, which is stationary.
For an AR(p) a unit root implies
1 – β1L – β2L2 – ... – βpLp = (1 – L) (1 – λ1L – λ2L2 ... – λpLp-1) = 0
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and as a result first differencing also removes the
unit root.
Non-stationarity
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Nonstationarity can have important
consequences for regression modelsand
inference.
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Autoregressive coefficients are biased
t-statistics have non-normal distributions even
in large samples
Spurious regression
Problem: Spurious Regression
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imagine we now have two series are
generated by independent random walks,
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Suppose we run yt on xt using OLS, that is
we estimate yt = α + βxt + νt.
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In this case, you tend to see ”significant”
β because the low-frequency changes
make it seem as if the two series are in
some way associated.
Unit Root Tests
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Standard Dickey-Fuller test appropriate
for AR(1) processes
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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
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
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p
yt  xt  yt 1   j yt  j   t
j 1
Deterministic
variables
Potential unit root
Other serial
correlation
ADF Test - 2
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To see why:
yt   xt  1 yt 1   2 yt  2  ...   p yt  p   t
 xt  (1   2  ..   p ) yt 1   2 ( yt  2  yt 1 )  ...
  p ( yt  p  ...  yt 1 )   t
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Subtract yt-1 from both sides and define
Φ = (α1+ α2+…+ αp – 1) and we get
p
yt  xt  yt 1    j yt  j   t
j 1
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Test Φ= 0 against alternative Φ<0
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Use special DF upperbound and lowerbound
Under alternative, test statistic is not normally distributed
Estimating in Time Series
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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.
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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.
Impulse Response Function
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One of the most interesting things to do with an
ARMA model is form predictions of the variable
given its past.
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we want to know what is Et(xt+j )
Can do inference with Vart(xt+j)
The impulse response function is a simpel way to
do that
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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”.
Impulse Response and MA(∞)
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1. The MA(∞) representation is the same thing as
the impulse response function.
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i.e.
yt   j  t  j
j 0
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The easiest way to calculate an MA(∞)
representation is to simulate the impulseresponse function.
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The impulse response function is the same as
Et(xt+j) − Et−1(xt+j).
Causality and Impulse Response
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Can either forecast or simulate the effect of a given shock
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Try to pick a shock time/level to simulate and try to replicate
observed data
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Know a shock happened in time time t
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Issue of whether that shock is what really happened
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
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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
Source: Cochrane, QJE (1994)
Next Time
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Estimating Causality in Time Series
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Some additional forecasting stuff
Testing for breaks
Regression discontinuity/Event study