# Time Series for Spatial Econometricians

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## Abstract

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

## References

- Baltagi BH (2013) Econometric analysis of panel data, 5th edn. Wiley, ChichesterGoogle Scholar
- Baltagi BH, Bresson G, Pirotte A (2007) Panel unit root tests and spatial dependence. J Appl Economet 22(2):339–360CrossRefGoogle Scholar
- Banerjee A, Carrion-I-Silvestre JL (2017) Testing for panel cointegration using common correlated effects estimators. J Time Ser Anal 38:610–636CrossRefGoogle Scholar
- Davidson JEH (1994) Stochastic limit theory: an introduction for econometricians. Oxford University Press, OxfordCrossRefGoogle Scholar
- Davidson R, MacKinnon JG (2009) Econometric theory and methods. Oxford University Press, New YorkGoogle Scholar
- Dickey D, Fuller W (1981) Likelihood ratio tests for autoregressive processes with a unit root. Econometrica 49:1057–1072CrossRefGoogle Scholar
- Elliot G, Rothenberg T, Stock J (1996) Efficient tests for an autoregressive unit root. Econometrica 64:813–836CrossRefGoogle Scholar
- Enders W (2004) Applied time series analysis, 2nd edn. John Wiley, New YorkGoogle Scholar
- Engle R (1982) Autoregressive conditional heteroscedasticity and with estimates of the variance of United Kingdom inflations. Econometrica 50:987–1008CrossRefGoogle Scholar
- Engle R, Granger CWJ (1987) Co-integration and error correction: representation, estimation and testing. Econometrica 35:251–276CrossRefGoogle Scholar
- Engle RF, Yoo BS (1991) Cointegrated economic time series: an overview with new results. In: Engle RF, Granger CWJ (eds) Long run economic relationships: readings in cointegration. Oxford University Press, OxfordGoogle Scholar
- Ericsson NR, MacKinnon JG (2002) Distributions for error correction tests for cointegration. Econ J 5:285–318Google Scholar
- Granger CWJ, Newbold P (1974) Spurious regressions in econometrics. J Econ 2:111–120CrossRefGoogle Scholar
- Groen J, Kleibergen F (2003) Likelihood-based cointegration analysis in panels of vector error-correction models. J Bus Econ Stat 21:295–317CrossRefGoogle Scholar
- Hadri K (2000) Testing for stationarity in heterogeneous panel data. Econ J 3:148–161Google Scholar
- Hamilton J (1994) Time series analysis. Princeton University Press, Princeton, NJGoogle Scholar
- Hendry DF (1995) Dynamic econometrics. Oxford University Press, OxfordCrossRefGoogle Scholar
- Im K, Pesaran MH, Shin Y (2003) Testing for unit roots in heterogeneous panels. J Econ 115:53–74CrossRefGoogle Scholar
- Johansen S (1988) Statistical analysis of cointegration vectors. J Econ Dyn Control 12:231–254CrossRefGoogle Scholar
- Kwiatowski D, Phillips PCB, Schmidt P, Shin Y (1992) Testing the null hypothesis of stationarity against the alternative of a unit root: how sure are we that economic time series have a unit root. J Econ 54:159–178CrossRefGoogle Scholar
- Larsson R, Lyhagen J, Löthgren M (2001) Likelihood-based cointegration tests in heterogeneous panels. Econ J 4:109–142Google Scholar
- Li H, Maddala GS (1997) Bootstrapping cointegrated regressions. J Econ 80:297–318CrossRefGoogle Scholar
- Lucas RE (1976) Econometric policy evaluation: a critique. Carn-Roch Conf Ser Public Policy 1:19–46CrossRefGoogle Scholar
- MacKinnon JG (1996) Numerical distribution functions for unit root and cointegration tests. J Appl Economet 11:601–618CrossRefGoogle Scholar
- Maddala GS, Kim I-M (1999) Unit roots, cointegration and structural change. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Osterwald-Lenum M (1992) A note with quantiles of the asymptotic distribution of the maximum likelihood cointegration rank test statistics. Oxf Bull Econ Stat 54:461–471Google Scholar
- Pedroni P (1999) Critical values for cointegration tests in heterogeneous panels with multiple regressors. Oxf Bull Econ Stat 61:653–670CrossRefGoogle Scholar
- Pedroni P (2004) Panel cointegration: asymptotic and finite sample properties of pooled time series tests with an application to the PPP hypothesis. Economet Theor 20:597–625CrossRefGoogle Scholar
- Pesaran MH (2007) A simple panel unit root test in the presence of cross section dependence. J Appl Economet 22(2):265–310CrossRefGoogle Scholar
- Pesaran MH (2015) Time series and panel data econometrics. Oxford University Press, OxfordCrossRefGoogle Scholar
- Phillips PCB (1986) Understanding spurious regressions in econometrics. J Econ 33(3):311–340CrossRefGoogle Scholar
- Phillips PCB, Moon H (1999) Linear regression limit theory for nonstationary panel data. Econometrica 67:1057–1011CrossRefGoogle Scholar
- Phillips PCB, Perron P (1988) Testing for a unit root in time series regression. Biometrika 75:335–346CrossRefGoogle Scholar
- Shin Y (1994) A residual-based test of the null of cointegration against the alternative of no cointegration. Economet Theor 10:91–115CrossRefGoogle Scholar
- Sims CA (1980) Macroeconomics and reality. Econometrica 58:1–48CrossRefGoogle Scholar
- Stock J (1987) Asymptotic properties of least squares estimates of cointegrating vectors. Econometrica 55:1035–1056CrossRefGoogle Scholar
- Stock JH, Watson MW (1993) A simple estimator of cointegrating vectors in higher order integrated systems. Econometrica 61(4):783–820CrossRefGoogle Scholar
- Westerlund J (2007) Testing for error correction in panel based data. Oxf Bull Econ Stat 69(6):709–748CrossRefGoogle Scholar
- Yule GU (1897) On the theory of correlation. J R Stat Soc 60:812–854CrossRefGoogle Scholar
- Yule GU (1926) Why do we sometimes get nonsense-correlations between time series? A study in sampling and the nature of time series. J R Stat Soc 89:1–64CrossRefGoogle Scholar