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

Difference Stationary Processes Suppose that we have a slightly different process 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 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

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

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:

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

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

Estimation of a VAR