The unit root can lead to major problems in economic time series analyses. I obtain the asymptotic distributions of the ordinary least squares (OLS) estimator when the true model is trend stationary for the following three cases: 1) the null model is a random walk without drift, and the auxiliary regression model does not contain a constant; 2) the null model is a random walk with drift, and the auxiliary regression model contains a constant; and 3) the null model is a random walk with drift, and the auxiliary regression model contains both a constant and a time trend. In the third case, the asymptotic distribution of the OLS estimator is determined by the first order of the autocorrelation, and we can distinguish between the random walk and trend stationary models, unlike in previous studies. Based on these results, the real US gross domestic product is analyzed. A time trend model with autoregressive error terms is chosen. The results suggest that the impacts of a shock can become larger than the original shock in some periods and then gradually decline. However, the impacts continue for a long period, and policy makers should account for this to design better economic policies.
KeywordsDickey-Fuller TestUnit RootRandom WalkTrend StationaryUS GDP
Dickey, D.A.I. and Fuller, W.A. (1979) Distribution of the Estimators for Autoregressive Time Series with a Unit Root. Journal of the American Statistical Association, 74, 427-431. https://doi.org/10.1080/01621459.1979.10482531
Said, E. and Dickey, D.A.I. (1984) Testing for Unit Roots in Autoregressive-Moving Average Models of Unknown Order. Biometrika, 71, 599-607. https://doi.org/10.1093/biomet/71.3.599
Perron, P. (1989) The Great Crash, the Oil Price Shock, and the Unit Root Hypothesis. Econometrica, 57, 1391-1401. https://doi.org/10.2307/1913712
Dolado, J.J., Gonzalo, J. and Mayoral, L. (2002) A Fractional Dickey-Fuller Test for Unit Roots. Econometrica, 70, 1963-2006. https://doi.org/10.1111/1468-0262.00359
Leybourne, S., Kim, T.H. and Newbold, P. (2005) Examination of Some More Powerful Modifications of the Dickey-Fuller Test. Journal of Time Series Analysis, 26, 355-369. https://doi.org/10.1111/j.1467-9892.2004.00406.x
Hacker, R.S. and Hatemi, A.-J. (2010) The Properties of Procedures Dealing with Uncertainty about Intercept and Deterministic Trend in Unit Root Testing. CESIS Electronic Working Paper Series, Paper No. 214, Centre of Excellence for Science and Innovation Studies (CESIS), Royal Institute of Technology, Stockholm.
Casini, A. and Perron, P. (2018) Structural Breaks in Time Series. https://doi.org/10.1093/acrefore/9780190625979.013.179
Nagai, K., Nishiyama, Y. and Hitomi, K. (2018) Sequential Test for Unit Root in AR(1) Model. KIER Discussion Paper No.1003, Kyoto University, Kyoto.
Paparoditis, E. and Politis, D.N. (2018) The Asymptotic Size and Power of the Augmented Dickey—Fuller Test for a Unit Root. Econometric Reviews, 37, 955-973. https://doi.org/10.1080/00927872.2016.1178887
Muthuramu, P. and Uma Maheswari, T. (2019) Tests for Structural Breaks in Time Series Analysis: A Review of Recent Development. Shanlax International Journal of Economics, 7, 66-79. https://doi.org/10.34293/economics.v7i4.628
Enders, W. (2015) Applied Econometric Time Series. 4th Edition, John Wiley & Sons, Hoboken.
Haldrup, L. and Jansson, N.M. (2006) Improving Size and Power in Unit Root Testing. In: Miller, T.C. and Patterson, K., Eds., Palgrave Handbook of Econometrics, Vol.1, Palgrave Macmillan, London. https://dx.doi.org/10.2139/ssrn.1147594
West, K.D. (1987) A Note on the Power of Least Squares Tests for a Unit Root. Economics Letters, 24, 249-252. https://doi.org/10.1016/0165-1765(87)90125-X
Schmidt, P. and Phillips, P.C.B. (1992) LM Tests for a Unit Root of the Deterministic Trends. Oxford Bulletin of Economics and Statistics, 54, 257-287. https://doi.org/10.1111/j.1468-0084.1992.tb00002.x
Fleissig, A.R. and Strauss, J. (1999) Is OECD Real Per Capita GDP Trend or Difference Stationary? Evidence from Panel Unit Root Tests. Journal of Macroeconomics, 21, 673-690. https://doi.org/10.1016/S0164-0704(99)80001-9
Hegwood, N. and Papell, D.H. (2007) Are Real GDP Levels Trend, Difference, or Regime-Wise Trend Stationary? Evidence from Panel Data Tests Incorporating Structural Change. Southern Economic Journal, 74, 104-113.
Ozturk, I. and Kalyoncu, H. (2007) Is Per Capita Real GDP Stationary in the OECD Countries? Evidence from a Panel Unit Root Test. Ekonomski Pregled, 58, 680-688.
Cuestas, J.C. and Garratt, D. (2011) Is Real GDP Per Capita a Stationary Process? Smooth Transitions, Nonlinear Trends and Unit Root Testing. Empirical Economics, 41, 555-563. https://doi.org/10.1007/s00181-010-0389-0
Furuoka, F. (2011) Is GDP in ASEAN Countries Stationary? New Evidence from Panel Unit Root Tests. Economics Bulletin, 31, 1391-1400.
Charles, A. and Darné, O. (2012) Trends and Random Walks in Macroeconomic Time Series: A Reappraisal. Journal of Macroeconomics, 34, 167-180. https://doi.org/10.1016/j.jmacro.2011.10.001
Lee, K.C. (2014) Is Per Capita Real GDP Stationary in China? Sequential Panel Selection Method. Economic Modelling, 37, 507-517. https://doi.org/10.1016/j.econmod.2013.11.040
Oskooe, S.A.P. and Akbari, L.T. (2015) Is Per Capita Real GDP Stationary? Evidence from OPEC Countries. International Journal of Humanities and Social Science, 5, 166-168.
Xing, L.Z., Dong, X.L. and Guan, J. (2017) Global Industrial Impact Coefficient Based on Random Walk Process and Inter-Country Input-Output Table. Physica A: Statistical Mechanics and its Applications, 471, 576-591. https://doi.org/10.1016/j.physa.2016.12.070
Zeren, F. and Islek, H. (2019) Is Per Capita Real GDP Stationary in the D-8 Countries? Evidence from a Panel unit Root Test. In: Akay, E.Ç. and Korkmaz, Ö., Eds., Selected Topics in Applied Econometrics, Peter Lang, Pieterlen and Bern, 67-86.
Phillips, P.C.B. and Perron, P. (1988) Testing for a Unit Root in Time Series Regression. Biometrika, 75, 335-346. https://doi.org/10.1093/biomet/75.2.335
Long, Z.M. and Herrera, R. (2020) Spurious OLS Estimators of Detrending Method by Adding a Linear Trend in Difference-Stationary Processes—A Mathematical Proof and Its Verification by Simulation. Mathematics, 8, 1931. https://doi.org/10.3390/math8111931
Hamilton, J.D. (1994) Time Series Analysis. Princeton University Press, Princeton. https://doi.org/10.1515/9780691218632
Harvey, D.I., Leybourne, S.J. and Taylor, A.M.B. (2009) Unit Root Testing in Practice: Dealing with Uncertainty over the Time Trend and Initial Conditions. Econometric Theory, 25, 587-636. https://doi.org/10.1017/S026646660809018X
Elliot, G., Rothenberg, T.J. and Stock, J.H. (1996) Efficient Tests for an Autoregressive Unit Root. Econometrica, 64, 813-836. https://doi.org/10.2307/2171846
Kwiatkowski, D., Phillips, P.C.B., Schmidt, P. and 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? Journal of Econometrics, 54, 159-178. https://doi.org/10.1016/0304-4076(92)90104-Y
Dickey, D.A.I. and Fuller, W.A. (1981) Likelihood Ratio Statistics Four Autoregressive Time Series with a Unit Root. Econometrica, 49, 1057-1072. https://doi.org/10.2307/1912517
Economic Research Division, Federal Reserve Bank of St. Louis Federal Reserve (2020) Economic Data. https://fred.stlouisfed.org