scholarly journals BOOTSTRAP UNION TESTS FOR UNIT ROOTS IN THE PRESENCE OF NONSTATIONARY VOLATILITY

2011 ◽  
Vol 28 (2) ◽  
pp. 422-456 ◽  
Author(s):  
Stephan Smeekes ◽  
A.M. Robert Taylor
Keyword(s):  
Linear Trend ◽  
Decision Rules ◽  
Unit Roots ◽  
Ordinary Least Squares ◽  
Data Uncertainty ◽  
Bootstrap Resampling ◽  
Initial Condition ◽  
Wild Bootstrap ◽  
Test Statistics ◽  
Deterministic Trend

Three important issues surround testing for a unit root in practice: uncertainty as to whether or not a linear deterministic trend is present in the data; uncertainty as to whether the initial condition of the process is (asymptotically) negligible or not, and the possible presence of nonstationary volatility in the data. Assuming homoskedasticity, Harvey, Leybourne, and Taylor (2011, Journal of Econometrics, forthcoming) propose decision rules based on a four-way union of rejections of quasi-differenced (QD) and ordinary least squares (OLS) detrended tests, both with and without a linear trend, to deal with the first two problems. In this paper we first discuss, again under homoskedasticity, how these union tests may be validly bootstrapped using the sieve bootstrap principle combined with either the independent and identically distributed (i.i.d.) or wild bootstrap resampling schemes. This serves to highlight the complications that arise when attempting to bootstrap the union tests. We then demonstrate that in the presence of nonstationary volatility the union test statistics have limit distributions that depend on the form of the volatility process, making tests based on the standard asymptotic critical values or, indeed, the i.i.d. bootstrap principle invalid. We show that wild bootstrap union tests are, however, asymptotically valid in the presence of nonstationary volatility. The wild bootstrap union tests therefore allow for a joint treatment of all three of the aforementioned issues in practice.

ASYMPTOTIC DISTRIBUTIONS FOR UNIT ROOT TEST STATISTICS IN NEARLY INTEGRATED SEASONAL AUTOREGRESSIVE MODELS

2000 ◽  
Vol 16 (2) ◽  
pp. 200-230 ◽  
Author(s):  
Seiji Nabeya
Keyword(s):  
Least Squares ◽  
Unit Root ◽  
Linear Trend ◽  
Unit Root Test ◽  
Ordinary Least Squares ◽  
Asymptotic Distributions ◽  
Autoregressive Models ◽  
Test Statistics ◽  
Autoregressive Parameter ◽  
Ordinary Least Squares Estimator

Seasonal autoregressive models with an intercept or linear trend are discussed. The main focus of this paper is on the models in which the intercept or trend parameters do not depend on the season. One of the most important results from this study is the asymptotic distribution for the ordinary least squares estimator of the autoregressive parameter obtained under nearly integrated condition, and another is the approximation to the limiting distribution of the t-statistic under the null for testing the unit root hypothesis.

scholarly journals UNIT ROOT TESTING IN PRACTICE: DEALING WITH UNCERTAINTY OVER THE TREND AND INITIAL CONDITION

2009 ◽  
Vol 25 (3) ◽  
pp. 587-636 ◽  
Author(s):  
David I. Harvey ◽  
Stephen J. Leybourne ◽  
A.M. Robert Taylor
Keyword(s):  
Decision Rule ◽  
Unit Root ◽  
Ordinary Least Squares ◽  
Unit Root Tests ◽  
Trend Detection ◽  
Initial Condition ◽  
Finite Sample ◽  
Testing Procedures ◽  
Unit Root Testing ◽  
Deterministic Trend

In this paper we focus on two major issues that surround testing for a unit root in practice, namely, (i) uncertainty as to whether or not a linear deterministic trend is present in the data and (ii) uncertainty as to whether the initial condition of the process is (asymptotically) negligible or not. In each case simple testing procedures are proposed with the aim of maintaining good power properties across such uncertainties. For the first issue, if the initial condition is negligible, quasi-differenced (QD) detrended (demeaned) Dickey–Fuller-type unit root tests are near asymptotically efficient when a deterministic trend is (is not) present in the data generating process. Consequently, we compare a variety of strategies that aim to select the detrended variant when a trend is present, and the demeaned variant otherwise. Based on asymptotic and finite-sample evidence, we recommend a simple union of rejections-based decision rule whereby the unit root null hypothesis is rejected whenever either of the detrended or demeaned unit root tests yields a rejection. Our results show that this approach generally outperforms more sophisticated strategies based on auxiliary methods of trend detection. For the second issue, we again recommend a union of rejections decision rule, rejecting the unit root null if either of the QD or ordinary least squares (OLS) detrended/demeaned Dickey–Fuller-type tests rejects. This procedure is also shown to perform well in practice, simultaneously exploiting the superior power of the QD (OLS) detrended/demeaned test for small (large) initial conditions.

A Bounds Approach to Inference Using the Long Run Multiplier

2019 ◽  
Vol 27 (3) ◽  
pp. 281-301 ◽  
Author(s):  
Clayton Webb ◽  
Suzanna Linn ◽  
Matthew Lebo
Keyword(s):  
Unit Root ◽  
Unit Roots ◽  
Small Sample ◽  
Stochastic Simulations ◽  
Upper And Lower Bounds ◽  
Test Statistics ◽  
Test Statistic ◽  
Independent Variables ◽  
Long Run ◽  
Presidential Success

Pesaran, Shin, and Smith (2001) (PSS) proposed a bounds procedure for testing for the existence of long run cointegrating relationships between a unit root dependent variable ($y_{t}$) and a set of weakly exogenous regressors $\boldsymbol{x}_{t}$ when the analyst does not know whether the independent variables are stationary, unit root, or mutually cointegrated processes. This procedure recognizes the analyst’s uncertainty over the nature of the regressors but not the dependent variable. When the analyst is uncertain whether $y_{t}$ is a stationary or unit root process, the test statistics proposed by PSS are uninformative for inference on the existence of a long run relationship (LRR) between $y_{t}$ and $\boldsymbol{x}_{t}$. We propose the long run multiplier (LRM) test statistic as a means of testing for LRRs without knowing whether the series are stationary or unit roots. Using stochastic simulations, we demonstrate the behavior of the test statistic given uncertainty about the univariate dynamics of both $y_{t}$ and $\boldsymbol{x}_{t}$, illustrate the bounds of the test statistic, and generate small sample and approximate asymptotic critical values for the upper and lower bounds for a range of sample sizes and model specifications. We demonstrate the utility of the bounds framework for testing for LRRs in models of public policy mood and presidential success.

scholarly journals Testing for unit roots in the presence of uncertainty over both the trend and initial condition

2012 ◽  
Vol 169 (2) ◽  
pp. 188-195 ◽  
Author(s):  
David I. Harvey ◽  
Stephen J. Leybourne ◽  
A.M. Robert Taylor
Keyword(s):  
Unit Roots ◽  
Initial Condition

On the First-Order Autoregressive Process with Infinite Variance

1989 ◽  
Vol 5 (3) ◽  
pp. 354-362 ◽  
Author(s):  
Ngai Hang Chan ◽  
Lanh Tat Tran
Keyword(s):  
Strong Consistency ◽  
Domain Of Attraction ◽  
Unit Roots ◽  
Autoregressive Process ◽  
Ordinary Least Squares ◽  
Seasonal Difference ◽  
Infinite Variance ◽  
Stable Law ◽  
First Order ◽  
Heavy Tailed

For a first-order autoregressive process Yt = βYt−1 + ∈t where the ∈t'S are i.i.d. and belong to the domain of attraction of a stable law, the strong consistency of the ordinary least-squares estimator bn of β is obtained for β = 1, and the limiting distribution of bn is established as a functional of a Lévy process. Generalizations to seasonal difference models are also considered. These results are useful in testing for the presence of unit roots when the ∈t'S are heavy-tailed.

scholarly journals The impact of China’s lockdown policy on the incidence of CoVID-19: An Interrupted time series analysis

2020 ◽  
Author(s):  
Mooketsi Molefi ◽  
John Tlhakanelo ◽  
Thabo Phologolo ◽  
Shimeles G. Hamda ◽  
Tiny Masupe ◽  
...  
Keyword(s):  
Time Series ◽  
Linear Trend ◽  
Effective Means ◽  
Interrupted Time Series ◽  
Ordinary Least Squares ◽  
World Health ◽  
Least Squares Regression ◽  
Interrupted Time Series Analysis ◽  
Post Intervention ◽  
The Impact

Abstract BackgroundPolicy changes are often necessary to contain the detrimental impact of epidemics such as the coronavirus disease (COVID-19). China imposed strict restrictions on movement on January 23rd, 2020.Interrupted time series methods were used to study the impact of the lockdown on the incidence of COVID-19. MethodsThe number of cases of COVID-19 reported daily from January 12thto March 30th, 2020 were extracted from the World Health Organization (WHO) COVID-19 dashboard ArcGIS® and matched to China’s projected population of 1 408 526 449 for 2020 in order to estimate daily incidences. Data were plotted to reflect daily incidences as data points in the series. A deferred interruption point of 6thFebruary was used to allow a 14-day period of diffusion. The magnitude of change and linear trend analyses were evaluated using the itsafunction with ordinary least-squares regression coefficients in Stata® yielding Newey-West standard errors.ResultsSeventy-eight (78) daily incidence points were used for the analysis, with 11(14.10%) before the intervention. There was a daily increase of 163 cases (β=1.16*10-07, p=0.00) in the pre-intervention period. Although there was no statistically significant drop in the number of cases reported daily in the immediate period following 6thFebruary 2020 when compared to the counterfactual (p=0.832), there was a 241 decrease (β=-1.71*10-07, p=0.00) in cases reported daily when comparing the pre-intervention and post-intervention periods. A deceleration of 78(47%) cases reported daily. ConclusionThe lockdown policy managed to significantly decrease the incidence of CoVID-19 in China. Lockdown provides an effective means of curtailing the incidence of COVID-19.

scholarly journals Prognóstico da série histórica do faturamento da indústria alimentícia brasileira utilizando técnicas de previsão

2020 ◽  
Vol 42 ◽  
pp. e47
Author(s):  
Matisa Andresa Maas ◽  
Cleber Bisognin
Keyword(s):  
Least Squares ◽  
Ordinary Least Squares ◽  
Arithmetic Mean ◽  
Simple Arithmetic ◽  
Absolute Minimum ◽  
Minimum Deviation ◽  
Deterministic Trend ◽  
Historical Series ◽  
Accuracy Measures ◽  
Forecasts Combination

This paper’s objective is to verify which is the best forecasting technique, including the use of the forecasts’ combination to evaluate the prognosis of the Brazilian food industry’s revenues. The historical series of revenues has deterministic trend and seasonality. Thereby, the models chosen to work on were: SARIMA (3,0,0)×(0,1,1)12, SARIMA (4,0,0)×(2,0,0)12 and Holt-Winters Multiplicative. Analyzing the accuracy measures, to perform the series’ forecast it was used the combination of the three models, presented by the methods: Simple Arithmetic Mean, Ordinary Least Squares and Regression of Absolute Minimum Deviation. The results obtained by the forecast were satisfactory, showing that the Brazilian food industry’s revenues will have peaks of growth and decay in the next two years. Therefore, a preparation of the sector is necessary for the period in which a possible decrease in this revenue will occur, as well as dismissal of the workers, since it is the sector that most employs in Brazil.

Minimizing the impact of the initial condition on testing for unit roots

2006 ◽  
Vol 135 (1-2) ◽  
pp. 285-310 ◽  
Author(s):  
Graham Elliott ◽  
Ulrich K. Müller
Keyword(s):  
Unit Roots ◽  
Initial Condition ◽  
The Impact

scholarly journals Unit Roots in Time Series with Changepoints

2017 ◽  
Vol 6 (6) ◽  
pp. 127
Author(s):  
Ed Herranz ◽  
James Gentle ◽  
George Wang
Keyword(s):  
Time Series ◽  
Unit Root ◽  
Null Hypothesis ◽  
Structural Changes ◽  
Linear Trend ◽  
Unit Root Test ◽  
Unit Roots ◽  
Cointegration Analysis ◽  
Unit Root Tests ◽  
Standard Unit

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.

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