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

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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

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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

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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

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23221101

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A VAR

• The equivalent VAR would look like this:

• The VAR model does not specify specific causation, nor lag structures.

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..),....,,,..,(

...),,..,(

1111231221

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etc

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Estimation of a VAR