Variance ratio tests of the seasonal unit root hypothesis

Taylor, Robert

doi:10.1016/j.jeconom.2003.12.012

Cited by 23 publications

(29 citation statements)

References 31 publications

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“…The second set of unit root statistics to use in ( ) are the seasonal variance ratio unit root test statistics proposed in Taylor ( ). For the Nyquist frequency, this is given by

V R_{ω_{2}, m} : = T^{−3} {\overset{Γ}{true}}_{2}^{−1} \sum_{t = 4}^{T} {true5.2ex5.2ex(\sum_{j = 4}^{t} z_{2, j} {\overset{u}{true}}_{2, j} true5.2ex5.2ex)}^{2},

while the variance ratio statistic for the annual frequency is given by

V R_{ω_{1}, m} : = T^{−3} normaltrace true5.2ex5.2ex[{\overset{Γ}{true}}_{1}^{−1} \sum_{t = 3}^{T} true5.2ex5.2ex(\sum_{j = 3}^{t} z_{1, j} {\overset{u}{true}}_{1, j} true5.2ex5.2ex) \sum_{t = 3}^{T} {true5.2ex5.2ex(\sum_{j = 3}^{t} z_{1, j} {\overset{u}{true}}_{1, j} true5.2ex5.2ex)}^{'} true5.2ex5.2ex],

{\overset{u}{true}}_{i, t}

i = 1, 2

, the OLS residuals from ( ) and where

{\overset{Γ}{true}}_{i} : = T^{−1} \sum_{t = f_{i} + 1}^{T} false({\overset{u}{true}}_{i, t} z_{i, t} false) false({\overset{u}{true}}_{i, t} z_{i, t<...>}

…”

Section: Tests For Deterministic Seasonality and Seasonal Mean Shiftsmentioning

confidence: 99%

Robust tests for deterministic seasonality and seasonal mean shifts

Astill

Taylor

2018

The Econometrics Journal

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SummaryWe develop tests for the presence of deterministic seasonal behaviour and seasonal mean shifts in a seasonally observed univariate time series. These tests are designed to be asymptotically robust to the order of integration of the series at both the zero and seasonal frequencies. Motivated by the approach of Hylleberg, Engle, Granger and Yoo [1990, Journal of Econometrics vol. 44, pp. 215-238], we base our approach on linear filters of the data which remove any potential unit roots at the frequencies not associated with the deterministic component(s) under test. Test statistics are constructed using the filtered data such that they have well defined limiting null distributions regardless of whether the data are either integrated or stationary at the frequency associated with the deterministic component(s) under test. In the same manner as Vogelsang [1998, Econometrica vol. 66, pp. 123-148] [2011, Econometric Theory vol. 27, pp. 992-1025, we scale these statistics by a function of an auxiliary seasonal unit root statistic. This allows us to construct tests which are asymptotically robust to the order of integration of the data at both the zero and seasonal frequencies. Monte Carlo evidence suggests that our proposed tests have good finite sample size and power properties. An empirical application to U.K. GDP indicates the presence of seasonal mean shifts in the data.

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“…The second set of unit root statistics to use in ( ) are the seasonal variance ratio unit root test statistics proposed in Taylor ( ). For the Nyquist frequency, this is given by

V R_{ω_{2}, m} : = T^{−3} {\overset{Γ}{true}}_{2}^{−1} \sum_{t = 4}^{T} {true5.2ex5.2ex(\sum_{j = 4}^{t} z_{2, j} {\overset{u}{true}}_{2, j} true5.2ex5.2ex)}^{2},

while the variance ratio statistic for the annual frequency is given by

V R_{ω_{1}, m} : = T^{−3} normaltrace true5.2ex5.2ex[{\overset{Γ}{true}}_{1}^{−1} \sum_{t = 3}^{T} true5.2ex5.2ex(\sum_{j = 3}^{t} z_{1, j} {\overset{u}{true}}_{1, j} true5.2ex5.2ex) \sum_{t = 3}^{T} {true5.2ex5.2ex(\sum_{j = 3}^{t} z_{1, j} {\overset{u}{true}}_{1, j} true5.2ex5.2ex)}^{'} true5.2ex5.2ex],

{\overset{u}{true}}_{i, t}

i = 1, 2

, the OLS residuals from ( ) and where

{\overset{Γ}{true}}_{i} : = T^{−1} \sum_{t = f_{i} + 1}^{T} false({\overset{u}{true}}_{i, t} z_{i, t} false) false({\overset{u}{true}}_{i, t} z_{i, t<...>}

…”

Section: Tests For Deterministic Seasonality and Seasonal Mean Shiftsmentioning

confidence: 99%

Robust tests for deterministic seasonality and seasonal mean shifts

Astill

Taylor

2018

The Econometrics Journal

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show abstract

“…The test statistic is constructed as a ratio of variances of the observed series and its fractional partial sum, thereby canceling nuisance parameters from the limiting distribution. Speci…cally, consider the behavior of the observed univariate time series fz t g d = 1, the latter choice would in fact lead to the statistic suggested by Breitung (2002) to test for a unit root against nonlinear alternatives, see also Taylor (2005). However, that test can be improved upon, at least against linear alternatives, even within the usual class of autoregressive models by admitting non-integer values of d 1 < 1.…”

Section: Variance Ratio Testing Approachmentioning

confidence: 99%

Nonparametric cointegration analysis of fractional systems with unknown integration orders

Nielsen

2010

Journal of Econometrics

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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Abstract In this paper a nonparametric variance ratio testing approach is proposed for determining the cointegration rank in fractionally integrated systems. The test statistic is easily calculated without prior knowledge of the integration order of the data, the strength of the cointegrating relations, or the cointegration vector(s). The latter property makes it easier to implement than regression-based approaches, especially when examining relationships between several variables with possibly multiple cointegrating vectors. Since the test is nonparametric, it does not require the speci…cation of a particular model and is invariant to short-run dynamics. Nor does it require the choice of any smoothing parameters that change the test statistic without being re ‡ected in the asymptotic distribution. Furthermore, a consistent estimator of the cointegration space can be obtained from the procedure. The asymptotic distribution theory for the proposed test is non-standard but easily tabulated or simulated. Monte Carlo simulations demonstrate excellent …nite sample properties, even rivaling those of well-speci…ed parametric tests. The proposed methodology is applied to the term structure of interest rates, where, contrary to both fractional and integer-based parametric approaches, evidence in favor of the expectations hypothesis is found using the nonparametric approach. Terms of use: Documents inJEL Classi…cation: C32.

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“…The statistic (10) generalizes the idea of , Breitung (2002), andTaylor (2005) who used the ratio of the sample variance of y t and that of the partial sum of y t to eliminate the nuisance parameter 2 y and avoid estimation of serial correlation parameters in testing for a unit root. Thus, setting d = 1,ỹ t is the partial sum of y t and (1) is then (the inverse of) the statistic proposed by Breitung (2002), which is therefore also a member of the family of tests in (10).…”

Section: The Nonparametric Variance Ratio Testmentioning

confidence: 99%