```Advanced Time Series
PS 791C
• A number of topics come under the
series
– Unit Root tests
– Granger Causality
– Vector Autoregression Models
– Error Correction Models
– Co-Integration Models
– Fractional Integration
Nested Special Cases
• Many of these techniques can be
considered a more general version of
others.
• For instance
– OLS is a special case of ARIMA
– An ARIMA Model is a Special Case of an SEQ
model
– An SEQ model is a special case of a VAR
Trend Stationary Processes
• A Simple Linear trend
yt    t  ut
• This can be differenced to eliminate the
trend
yt  yt  yt 1    ut  ut 1
• Differencing once more removes the β and
therefore make the series stationary
 yt   ut  ut  2ut 1  ut 2
2
2
Difference Stationary Processes
• Suppose that we have a slightly different process
yt  yt 1     t
• Also known as a random walk
Implications
• If we estimate the wrong model there are
severe consequences for regression
– Regression of a random walk on time will
produce an R2 of about .44 regardless of
sample size, even when there is actually no
relationship at all
– T-tests are not valid
– The residuals are autocorrelated
– Subject to spurious regression
Unit Root Tests
• In order to avoid this, we need to know if
the series is a DSP or TSP process
• This means that we are testing whether
=1.0, and hence has become known as a
Unit Root test
– The Dickey-Fuller test
– The Augmented Dickey-Fuller Test
– The Phillips-Perron test
Dickey-Fuller test
• The Dickey-Fuller test requires estimating the
following model
yt    yt 1  t   t
• The series is a DSP if =1 and β=0, and a TSP if
||&lt;1
• Cannot use least squares, so they employ a LR
test, and provide tables
CoIntegration
• A model in which the X and Y variables
have unit root processes is called a
cointegrated process.
• Such models are exceedingly likely to
exhibit spurious correlation and will likely
have non-stationary residuals.
Granger Causality
• Ordinary regression tests correlation
• Causation is implied by the theory not the
statistic
• Yet if some dynamic series of Xs explains
more of the dynamics of a set of Ys, then
we may say that X Granger-causes Y
• The test statistic is a block-F test
Vector Autoregression models
• Structural Equation Models (SEQ) models
impose a priori restrictions on the theoretical
exposition of the theory
• VAR models seek to implement tests of theory
with fewer restriction.
• They represent a tradeoff between accuracy of
causal inference and quantitative precision.
• They better characterize uncertainty and model
dynamics.
The VAR Model
• Vector Autoregression is not a statistical
technique
– It is a design
• The VAR Model is:
yt  A( L) yt 1  ut
where A( L)  A1  A2 L  A3 L2  ...
Vector Autoregression
• Vector Autoregression Models (VARs) are
best seen in contrast to Simultaneous
Equation Models (SEQs)
• SEQ models involve a set of endogenous
variables regressed on a set of exogenous
variables, with appropriate lag structures
supplied for dynamic processes, including
simultaneity.
An SEQ Model
• For Instance:
Y1t  B0  B1 X1t  B2 X 2t  B3Y2t
Y2t  B4  B5 X 3t  B6 X 4t  B7Y1t 1
• Note that endogenous variables of one equation
may be exogenous in another.
• The lag structure is specifically articulated
• The causal nature of the model is explicit – it is a
product of the theoretical specification of the model
A VAR
• The equivalent VAR would look like this:
Y1t  f ( X 1t , X 1t 1 ,..X 2t , X 2t 1...)
X 1t  f ( X 2t , X 2t 1 ,..X 3t , X 2t 1....,Y1t , Y1t 1..)
etc..
• The VAR model does not specify specific
causation, nor lag structures.
Estimation of a VAR
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