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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.
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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 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
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.
