Time Series for Spatial Econometricians

Key developments in the econometric analysis of nonstationary time series are reviewed. We begin by defining nonstationarity, which arises when data generating processes (DGP) contain unit roots. The distinction is made between difference stationarity where time trends are stochastic, and trend stationarity where time trends are deterministic. We recall that hypotheses involving levels of nonstationary time series cannot be tested by using their first differences or their deviations from deterministic time trends. We also recall that in structural vector autoregressions the structural parameters are under-identified. Consequently, SVAR models merely provide ex post narratives for the time series involved. The concepts of “spurious” regression and “nonsense” regression, which arise when time series data are nonstationary, are introduced. The functional central limit theorem is presented, and its role in the asymptotic theory of nonstationary time series is described. Alternative statistical tests for unit roots are reviewed under the null hypotheses of nonstationarity and stationarity. Alternative statistical tests for spurious and nonsense regression (cointegration tests) are compared and contrasted. Parameter estimates for variables that are cointegrated are “super-consistent”. Instead of root—T consistency, as in stationary time series, they may be T—consistent or T ^1½ —consistent depending on whether the data have stochastic time trends. Super-consistency radically changes the properties of estimators and the conditions for identification. In particular, OLS parameter estimates for endogenous variables are super-consistent. We also review panel unit root tests and cointegration tests for independent and strongly dependent panel data. Finally, we introduce ARCH models (autoregressive conditional heteroscedasticity), and distinguish between unconditional and conditional heteroscedasticity.