1. Advanced Time Series PS 791C

2. 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

3. 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

4. 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

5. Difference Stationary Processes Suppose that we have a slightly different process Also known as a random walk

6. Implications If we estimate the wrong model there are severe consequences for regression –Regression of a random walk on time will produce an R 2 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

7. 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

8. 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

9. 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.

10. 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

11. 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.

12. The VAR Model Vector Autoregression is not a statistical technique –It is a design The VAR Model is:

13. 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.

14. 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

15. A VAR The equivalent VAR would look like this: The VAR model does not specify specific causation, nor lag structures.

16. Estimation of a VAR