Advanced Time Series PS 791C. Advanced Time Series Techniques A number of topics come under the...
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Transcript of Advanced Time Series PS 791C. Advanced Time Series Techniques A number of topics come under the...
Advanced Time Series
PS 791C
Advanced Time Series Techniques
• A number of topics come under the general heading of “state-of-the-art” time 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
• This can be differenced to eliminate the trend
• Differencing once more removes the β and therefore make the series stationary
tt uty
11 ttttt uuyyy
2122 2 ttttt uuuuy
Difference Stationary Processes
• Suppose that we have a slightly different process
• Also known as a random walk
ttt yy 1
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
• The series is a DSP if =1 and β=0, and a TSP if ||<1
• Cannot use least squares, so they employ a LR test, and provide tables
tttt yy 1
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:
...)(
)(2
321
1
LALAALAwhere
uyLAy ttt
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:
• 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
117463542
23221101
tttt
tttt
YBXBXBBY
YBXBXBBY
A VAR
• The equivalent VAR would look like this:
• The VAR model does not specify specific causation, nor lag structures.
..
..),....,,,..,(
...),,..,(
1111231221
1221111
etc
YYXXXXfX
XXXXfY
ttttttt
ttttt
Estimation of a VAR