Testing for Stationarity and Cointegration in an Unobserved Components Framework
While tests for unit roots and cointegration have important econometric and economic implications, they do not always offer conclusive results. For example, Rudebusch (1992; 1993) demonstrates that standard unit root tests have low power against estimated trend stationary alternatives. In addition, Perron (1989) shows that standard unit root tests cannot always distinguish unit root from stationary processes that contain segmented or shifted trends. Recent research (Harvey 1993; Engel and Morley 2001; Morley, Nelson et al. 2003; Morley 2004; Sinclair 2004) suggests that unobserved components models can provide a useful framework for representing economic time series which contain unit roots, including those that are cointegrated. These series can be modeled as containing an unobserved permanent component, representing the stochastic trend, and an unobserved transitory component, representing the stationary component of the series. These unobserved components are then estimated using the Kalman filter. The unobserved components framework can also provide a more powerful way to test for unit roots and cointegration than what is currently available (Nyblom and Harvey 2000). This paper develops a new test that nests a partial unobserved components model within a more general unobserved components model. This nesting allows the general and the restricted models to be compared using a likelihood ratio test. The likelihood ratio test statistic has a nonstandard distribution, but Monte Carlo simulation can provide its proper distribution. The simulation uses data generated with the results from the partial unobserved components model as the values for the null hypothesis. Consequently, the null hypothesis for this test is stationarity, which is useful in many cases. In this sense our test is like the well-known KPSS test (Kwiatkowski, Phillips et al. 1992), but our test is a parametric version which provides more power by considering the unobserved components structure in calculation of the test statistic. This more powerful test can be used to evaluate important macroeconomic theories such as the permanent income hypothesis, real business cycle theories, and purchasing power parity for exchange rates
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