Time series regression with a unit root
TL;DR: In this paper, it is shown that simple least squares regression consistently estimates a unit root under very general conditions in spite of the presence of autocorrelated errors. But, the results of this paper are restricted to the unit root case.
Abstract: This paper studies the random walk, in a general time series setting that allows for weakly dependent and heterogeneously distributed innovations. It is shown that simple least squares regression consistently estimates a unit root under very general conditions in spite of the presence of autocorrelated errors. The limiting distribution of the standardized estimator and the associated regression t statistic are found using functional central limit theory. New tests of the random walk hypothesis are developed which permit a wide class of dependent and heterogeneous innovation sequences. A new limiting distribution theory is constructed based on the concept of continuous data recording. This theory, together with an asymptotic expansion that is developed in the paper for the unit root case, explain many of the interesting experimental results recently reported in Evans and Savin (1981, 1984).
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TL;DR: In this article, the authors proposed new tests for detecting the presence of a unit root in quite general time series models, which accommodate models with a fitted drift and a time trend so that they may be used to discriminate between unit root nonstationarity and stationarity about a deterministic trend.
Abstract: SUMMARY This paper proposes new tests for detecting the presence of a unit root in quite general time series models. Our approach is nonparametric with respect to nuisance parameters and thereby allows for a very wide class of weakly dependent and possibly heterogeneously distributed data. The tests accommodate models with a fitted drift and a time trend so that they may be used to discriminate between unit root nonstationarity and stationarity about a deterministic trend. The limiting distributions of the statistics are obtained under both the unit root null and a sequence of local alternatives. The latter noncentral distribution theory yields local asymptotic power functions for the tests and facilitates comparisons with alternative procedures due to Dickey & Fuller. Simulations are reported on the performance of the new tests in finite samples.
16,874 citations
TL;DR: In this article, the authors consider pooling cross-section time series data for testing the unit root hypothesis, and they show that the power of the panel-based unit root test is dramatically higher, compared to performing a separate unit-root test for each individual time series.
10,792 citations
TL;DR: In this paper, a test of the null hypothesis that an observable series is stationary around a deterministic trend is proposed, where the series is expressed as the sum of deterministic trends, random walks, and stationary error.
10,068 citations
Cites methods from "Time series regression with a unit ..."
...These results are not changed by allowing for error autocorrelation using the augmented tests of Said and Dickey (1984) or the test statistics of Phillips (1987) and Phillips and Perron (1988)....
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...We therefore proceed in the spirit of Phillips (1987) and Phillips and Perron (1988) by deriving the asymptotic distribution of the statistics under general conditions on the stationary error, and we propose a modified version of the LM statistic that is valid asymptotically under these general…...
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...A consistent estimator of g2, say ~~(0, can be constructed from the residuals e,, as in Phillips (1987) or Phillips and Perron (1988); specifically, we will use an estimator of the form 32(Z) = T-’ ; e; + 2T-’ f: w(s, I) i e,e,_,. (IO) t=1 s=l t=....
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TL;DR: In this paper, the authors consider the null hypothesis that a time series has a unit root with possibly nonzero drift against the alternative that the process is "trend-stationary" and show how standard tests of the unit root hypothesis against trend stationary alternatives cannot reject the unit-root hypothesis if the true data generating mechanism is that of stationary fluctuations around a trend function which contains a one-time break.
Abstract: We consider the null hypothesis that a time series has a unit root with possibly nonzero drift against the alternative that the process is «trend-stationary». The interest is that we allow under both the null and alternative hypotheses for the presence for a one-time change in the level or in the slope of the trend function. We show how standard tests of the unit root hypothesis against trend stationary alternatives cannot reject the unit root hypothesis if the true data generating mechanism is that of stationary fluctuations around a trend function which contains a one-time break
7,471 citations
TL;DR: In this paper, a variation of Perron's test is considered in which the breakpoint is estimated rather than fixed, and the asymptotic distribution of the estimated breakpoint test statistic is determined.
Abstract: Recently, Perron has carried out tests of the unit-root hypothesis against the alternative hypothesis of trend stationarity with a break in the trend occurring at the Great Crash of 1929 or at the 1973 oil-price shock. His analysis covers the Nelson–Plosser macroeconomic data series as well as a postwar quarterly real gross national product (GNP) series. His tests reject the unit-root null hypothesis for most of the series. This article takes issue with the assumption used by Perron that the Great Crash and the oil-price shock can be treated as exogenous events. A variation of Perron's test is considered in which the breakpoint is estimated rather than fixed. We argue that this test is more appropriate than Perron's because it circumvents the problem of data-mining. The asymptotic distribution of the estimated breakpoint test statistic is determined. The data series considered by Perron are reanalyzed using this test statistic. The empirical results make use of the asymptotics developed for the test stati...
6,608 citations
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TL;DR: In this article, the limit distributions of the estimator of p and of the regression t test are derived under the assumption that p = ± 1, where p is a fixed constant and t is a sequence of independent normal random variables.
Abstract: Let n observations Y 1, Y 2, ···, Y n be generated by the model Y t = pY t−1 + e t , where Y 0 is a fixed constant and {e t } t-1 n is a sequence of independent normal random variables with mean 0 and variance σ2. Properties of the regression estimator of p are obtained under the assumption that p = ±1. Representations for the limit distributions of the estimator of p and of the regression t test are derived. The estimator of p and the regression t test furnish methods of testing the hypothesis that p = 1.
23,509 citations
Book•
01 Jan 1970
TL;DR: In this article, a complete revision of a classic, seminal, and authoritative book that has been the model for most books on the topic written since 1970 is presented, focusing on practical techniques throughout, rather than a rigorous mathematical treatment of the subject.
Abstract: From the Publisher:
This is a complete revision of a classic, seminal, and authoritative book that has been the model for most books on the topic written since 1970. It focuses on practical techniques throughout, rather than a rigorous mathematical treatment of the subject. It explores the building of stochastic (statistical) models for time series and their use in important areas of application forecasting, model specification, estimation, and checking, transfer function modeling of dynamic relationships, modeling the effects of intervention events, and process control. Features sections on: recently developed methods for model specification, such as canonical correlation analysis and the use of model selection criteria; results on testing for unit root nonstationarity in ARIMA processes; the state space representation of ARMA models and its use for likelihood estimation and forecasting; score test for model checking; and deterministic components and structural components in time series models and their estimation based on regression-time series model methods.
19,748 citations
TL;DR: In this article, a simple method of calculating a heteroskedasticity and autocorrelation consistent covariance matrix that is positive semi-definite by construction is described.
Abstract: This paper describes a simple method of calculating a heteroskedasticity and autocorrelation consistent covariance matrix that is positive semi-definite by construction. It also establishes consistency of the estimated covariance matrix under fairly general conditions.
18,117 citations
13,292 citations
Book•
01 Jan 1968
TL;DR: Weak Convergence in Metric Spaces as discussed by the authors is one of the most common modes of convergence in metric spaces, and it can be seen as a form of weak convergence in metric space.
Abstract: Weak Convergence in Metric Spaces. The Space C. The Space D. Dependent Variables. Other Modes of Convergence. Appendix. Some Notes on the Problems. Bibliographical Notes. Bibliography. Index.
13,153 citations