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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 yt t ut • This can be differenced to eliminate the trend yt yt yt 1 ut ut 1 • Differencing once more removes the β and therefore make the series stationary yt ut ut 2ut 1 ut 2 2 2 Difference Stationary Processes • Suppose that we have a slightly different process yt yt 1 t • 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 yt yt 1 t t • 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: yt A( L) yt 1 ut where A( L) A1 A2 L A3 L2 ... 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: Y1t B0 B1 X1t B2 X 2t B3Y2t Y2t B4 B5 X 3t B6 X 4t B7Y1t 1 • 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: Y1t f ( X 1t , X 1t 1 ,..X 2t , X 2t 1...) X 1t f ( X 2t , X 2t 1 ,..X 3t , X 2t 1....,Y1t , Y1t 1..) etc.. • The VAR model does not specify specific causation, nor lag structures. Estimation of a VAR