Abstract
Many financial time series are nonstationary and are modeled as ARIMA processes; they are integrated processes (I(n)) which can be made stationary (I(0)) via differencing n times. I(1) processes have a unit root in the autoregressive polynomial. Using OLS with unit root processes often leads to spurious results; a cointegration analysis should be used instead. Unit root tests (URT) decrease spurious cointegration. The Augmented Dickey Fuller (ADF) URT fails to reject a false null hypothesis of a unit root under the presence of structural changes in intercept and/or linear trend. The Zivot and Andrews (ZA) (1992) URT was designed for unknown breaks, but not under the null hypothesis. Lee and Strazicich (2003) argued the ZA URT was biased towards stationarity with breaks and proposed a new URT with breaks in the null. When an ARMA(p,q) process with trend and/or drift that is to be tested for unit roots and has changepoints in trend and/or intercept two approaches that can be taken: One approach is to use a unit root test that is robust to changepoints. In this paper we consider two of these URT's, the Lee-Strazicich URT and the Hybrid Bai-Perron ZA URT(Herranz, 2016.) The other approach we consider is to remove the deterministic components with changepoints using the Bai-Perron breakpoint detection method (1998, 2003), and then use a standard unit root test such as ADF in each segment. This approach does not assume that the entire time series being tested is all I(1) or I(0), as is the case with standard unit root tests. Performances of the tests were compared under various scenarios involving changepoints via simulation studies. Another type of model for breaks, the Self-Exciting-Threshold-Autoregressive (SETAR) model is also discussed.
Highlights
1.1 Time Series and ARIMA ModelsA time series (TS) is an ordered sequence of values of a variable at spaced time intervals,{xt} = {x1, x2, ..., xn}. (1)An auto-regressive integrated moving average (ARIMA(p,d,q)) model is a model of a time series xt where we first difference the series d-times, resulting in ∆d(xt), and we build an ARMA(p,q) model from the differenced series
When an ARMA(p,q) process with trend and/or drift that is to be tested for unit roots and has changepoints in trend and/or intercept two approaches that can be taken: One approach is to use a unit root test that is robust to changepoints
Unit roots are nonstationary ARMA(p,q) processes which have one of more roots of 1 of the auto-regressive polynomial. If deterministic parameters such as the intercept or linear breaks have structural breaks standard unit root tests such as Augmented Dickey Fuller (ADF) will rarely reject the null of a unit root even when φ1 < 1
Summary
A time series (TS) is an ordered sequence of values of a variable at spaced time intervals,. An auto-regressive integrated moving average (ARIMA(p,d,q)) model is a model of a time series xt where we first difference the series d-times, resulting in ∆d(xt) , and we build an ARMA(p,q) model from the differenced series. An ARMA(p,q) model is defined in terms of its lagged values xt and its current and past innovations εt as :.
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