Fractional cointegration in the presence of linear trends

Abstract. We consider bivariate regressions of nonstationary fractionally integrated variables dominated by linear time trends. The asymptotic behaviour of the ordinary least square (OLS) estimators in this case allows limiting normality to arise at a faster rate of convergence than if the individual series were detrended, increasing in this way the power of the tests for fractional cointegration. We also show that the limiting distribution of the t‐ratio of the slope coefficient depends upon the presence or not of a deterministic trend in the conditional regressor. We introduce the concept of local fractional trend to explain the apparently diverging asymptotic theories that apply when a trend is either present or absent in our set‐up.     

Combining non‐cointegration tests

The local power of many popular non‐cointegration tests has recently been shown to depend on a certain nuisance parameter. Depending on the value of that parameter, different tests perform best. This paper suggests combination procedures with the aim of providing meta tests that maintain high power across the range of the nuisance parameter. 1 We demonstrate the local power of the new meta tests to be in general almost as high as that of the most powerful of the underlying tests. When the underlying tests have similar power, the meta tests even appear more powerful than the best underlying test. At the same time, our new meta tests avoid the arbitrary decision which test to use if individual test results conflict. Moreover it avoids the size distortion inherent in separately applying multiple tests for cointegration to the same dataset. We use the new tests to investigate 286 datasets from published cointegration studies. There, in one‐third of all cases individual tests give conflicting results whereas our meta tests provide an unambiguous test decision.     

Effects of outliers on the identification and estimation of GARCH models

Abstract. This paper analyses how outliers affect the identification of conditional heteroscedasticity and the estimation of generalized autoregressive conditionally heteroscedastic (GARCH) models. First, we derive the asymptotic biases of the sample autocorrelations of squared observations generated by stationary processes and show that the properties of some conditional homoscedasticity tests can be distorted. Second, we obtain the asymptotic and finite sample biases of the ordinary least squares (OLS) estimator of ARCH(p) models. The finite sample results are extended to generalized least squares (GLS), maximum likelihood (ML) and quasi‐maximum likelihood (QML) estimators of ARCH(p) and GARCH(1,1) models. Finally, we show that the estimated asymptotic standard deviations are biased estimates of the sample standard deviations.     

Testing for co‐integration and nonlinear adjustment in a smooth transition error correction model

This article introduces a testing procedure for cointegration and nonlinear adjustment in a smooth transition vector error correction model. To overcome the unidentified parameters problem under the null of no‐cointegration, the Wald statistic is optimized over the unidentified parameter space. The asymptotic distribution of the test statistic is shown to be non‐standard but nuisance parameter‐free and hence critical values are obtained by simulations, Simulations show that the proposed test outperforms the alternatives in small sample sizes both in terms of size and power. Application to the exchange rate‐monetary fundamentals relationship show that the proposed test works considerably well. This article also finds that nonlinear adjustment dynamics are symmetric for some currencies and therefore the speed of adjustment depends on the size of the deviations and is asymmetric for others, hence, the adjustment dynamics depend not only on the size but also on the sign of the deviations.     

A refined efficiency rate for ordinary least squares and generalized least squares estimators for a linear trend with autoregressive errors

When a straight line is fitted to time series data, generalized least squares (GLS) estimators of the trend slope and intercept are attractive as they are unbiased and of minimum variance. However, computing GLS estimators is laborious as their form depends on the autocovariances of the regression errors. On the other hand, ordinary least squares (OLS) estimators are easy to compute and do not involve the error autocovariance structure. It has been known for 50 years that OLS and GLS estimators have the same asymptotic variance when the errors are second‐order stationary. Hence, little precision is gained by using GLS estimators in stationary error settings. This article revisits this classical issue, deriving explicit expressions for the GLS estimators and their variances when the regression errors are drawn from an autoregressive process. These expressions are used to show that OLS methods are even more efficient than previously thought. Specifically, we show that the convergence rate of variance differences is one polynomial degree higher than that of least squares estimator variances. We also refine Grenander's (1954) variance ratio. An example is presented where our new rates cannot be improved upon. Simulations show that the results change little when the autoregressive parameters are estimated.     

Powerful Unit Root Tests Free of Nuisance Parameters

We propose a variance ratio‐type unit root test where the nuisance parameter cancels asymptotically under both the null of a unit root and a local‐to‐unity alternative. Critical values and asymptotic power curves can be computed using standard numerical techniques. Our test exhibits higher power compared with tests that share the virtue of being free of tuning parameters. In fact, the local asymptotic power curves of our procedure get close to the power functions of the point optimal test, where the latter suffers from the drawback of having to correct for a nuisance parameter consistently.     

The power of unit root tests against nonlinear local alternatives

This article extends the analysis of local power of unit root tests in a nonlinear direction by considering local nonlinear alternatives and tests built specifically against stationary nonlinear models. In particular, we focus on the popular test proposed by Kapetanios et al. (2003, Journal of Econometrics 112, 359–379) in comparison to the linear Dickey–Fuller test. To this end, we consider different adjustment schemes for deterministic terms. We provide asymptotic results which imply that the error variance has a severe impact on the behaviour of the tests in the nonlinear case; the reason for such behaviour is the interplay of non‐stationarity and nonlinearity. In particular, we show that nonlinearity of the data generating process can be asymptotically negligible when the error variance is moderate or large (compared to the ‘amount of nonlinearity’), rendering the linear test more powerful than the nonlinear one. Should however the error variance be small, the nonlinear test has better power against local alternatives. We illustrate this in an asymptotic framework of what we call persistent nonlinearity. The theoretical findings of this article explain previous results in the literature obtained by simulation. Furthermore, our own simulation results suggest that the user‐specified adjustment scheme for deterministic components (e.g. OLS, GLS, or recursive adjustment) has a much higher impact on the power of unit root tests than accounting for nonlinearity, at least under local (linear or nonlinear) alternatives.     

SEMI‐PARAMETRIC ESTIMATION OF LINEAR COINTEGRATING MODELS WITH NONLINEAR CONTEMPORANEOUS ENDOGENEITY

This article considers linear cointegrating models with unknown nonlinear short‐run contemporaneous endogeneity. Two estimators are proposed to estimate the linear cointegrating parameter after the nonlinear endogenous component is estimated by local linear regression approach. Both the proposed estimators are shown to have the same mixed normal limiting distribution with zero mean and smaller asymptotic variance than the fully modified ordinary least squares and instrumental variables estimators. Monte Carlo simulations are used to evaluate the finite sample performance of our proposed estimators, and an empirical application is also included.     

Local Power of Fixed‐T Panel Unit Root Tests With Serially Correlated Errors and Incidental Trends

The asymptotic local power properties of various fixed T panel unit root tests with serially correlated errors and incidental trends are studied. Asymptotic (over N) local power functions are analytically derived, and through them, the effects of general forms of serial correlation are examined. We find that a test based on an instrumental variables (IV) estimator dominates the tests based on the within‐groups (WG) estimator. These functions also show that in the presence of incidental trends, an instrumental variables test based on the first differences of the model has non‐trivial local power in an N^?1/2 neighbourhood of unity. Furthermore, for a test based on the within‐groups estimator, although it is found that it has trivial power in the presence of incidental trends, this ceases to be the case if there is serial correlation as well.     

Joint Hypothesis Tests for a Unit Root When There is a Break in the Innovation Variance

Abstract. We develop extensions of the Dickey–Fuller F‐statistics for the joint null hypothesis of a unit root that allows for a break in the innovation variance. Our statistics are based on the modified generalized least squares (GLS) strategy outlined in Kim, Leybourne and Newbold [Journal of Econometrics (2002) Vol. 109, pp. 365–387] that requires estimation of the break‐date and corresponding pre‐break and post‐break variances. We derive the asymptotic distribution of the new F‐statistics, tabulate their finite sample and asymptotic critical values, and present finite sample simulation evidence regarding their size and power.     

The Calculation of Some Limiting Distributions Arising in Near‐Integrated Models with GLS Detrending

Many unit root test statistics are based on detrended data, with the method of generalized least squares (GLS) detrending being popular in the setting of a near‐integrated model. This article determines the properties of some associated limiting distributions when the GLS detrending is based on a linear time trend. A fundamental result for the moment generating function of two key functionals of the relevant stochastic process is provided and used to compute probability density functions and cumulative distribution functions, as well as means and variances, of the limiting distributions of some statistics of interest. The exact moments and percentiles of some of these distributions are compared with those obtained by simulations, and it is found that, even with a large number of replications and a large sample size, the errors resulting from the simulation methods are not negligible. Some further applications, including a comparison of limiting power functions of different unit root test statistics and the consideration of a more complicated statistic, are also provided.     

Parametric and Semi‐Parametric Efficient Tests for Parameter Instability

This article examines asymptotically point optimal tests for parameter instability in realistic circumstances when little information about the unstable parameter process and error distribution is available. We first show that, under a correctly specified error distribution, if the unstable parameter processes converge weakly to a Wiener process, then any asymptotic optimal tests for structural breaks and time‐varying parameters are asymptotically equivalent. Our finding is then extended to a semi‐parametric set‐up in which the error distribution is treated as an unknown infinite‐dimensional nuisance parameter. We find that semi‐parametric tests can be adaptive without further restrictive conditions on the error distribution.     

Explosive Random‐Coefficient AR(1) Processes and Related Asymptotics for Least‐Squares Estimation

Abstract. Large sample properties of the least‐squares and weighted least‐squares estimates of the autoregressive parameter of the explosive random‐coefficient AR(1) process are discussed. It is shown that, contrary to the standard AR(1) case, the least‐squares estimator is inconsistent whereas the weighted least‐squares estimator is consistent and asymptotically normal even when the error process is not necessarily Gaussian. Conditional asymptotics on the event that a certain limiting random variable is non‐zero is also discussed.     

Error-correction Mechanism Tests for Cointegration in a Single-equation Framework

A new test is proposed for cointegration in a single-equation framework where the regressors are weakly exogenous for the parameters of interest. The test is denoted as an error-correction mechanism (ECM) test and is based upon the ordinary least squares coefficient of the lagged dependent variable in an autoregressive distributed lag model augmented with leads of the regressors. The limit distributions of the standardized coeffi cient and t -ratio versions of the ECM tests are obtained and critical values are provided. These limit distributions do not depend upon nuisance parameters but they depend on the number of regressors. Finally, we compare their power properties with those of other cointegration tests available in the literature and find the circumstances under which the ECM tests have a better performance.     

Spectral Regression For Cointegrated Time Series With Long-Memory Innovations

Spectral regression is considered for cointegrated time series with long-memory innovations. The estimates we advocate are shown to be consistent when cointegrating relationships among stationary variables are investigated, while ordinary least squares are inconsistent due to correlation between the regressors and the cointegrating residuals; in the presence of unit roots, these estimates share the same asymptotic distribution as ordinary least squares. As a corollary of the main result, we provide a functional central limit theorem for quadratic forms in non-stationary fractionally integrated processes.     

Implementing Residual‐Based KPSS Tests for Cointegration with Data Subject to Temporal Aggregation and Mixed Sampling Frequencies

We show how different data types (stocks and flows) and temporal aggregation affect the size and power of the dynamic ordinary least squares residual‐based Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test of the null of cointegration. Size may be more effectively controlled by setting the minimum number of leads equal to one – as opposed to zero – when selecting the lag/lead order of the dynamic ordinary least squares regression using aggregated data, but at a cost to power. If high‐frequency data for one or more series are available – that is, the model has mixed sampling frequencies – we show how to effectively utilize the high‐frequency data to increase power while controlling size.     

Constructing Optimal tests on a Lagged dependent variable

Abstract. Via the leading unit‐root case, the problem of testing on a lagged dependent variable is characterized by a nuisance parameter which is present only under the alternative [see Andrews and Ploberger, Econometrica (1994 ) Vol. 62, pp. 1318–1414]. This has proven to be a barrier to the construction of optimal tests. Moreover, in their absence it is impossible to objectively assess the absolute power properties of existing tests. Indeed, feasible tests based upon the optimality criteria used here are found to have numerically superior power properties to both the original Dickey and Fuller [Econometrica (1981 ) Vol. 49, pp. 1057–1072] statistics and the efficient detrended versions suggested by Elliott et al. [Econometrica (1996 ) Vol. 64, pp. 813–836] and analysed in Burridge and Taylor [Oxford Bulletin of Economics and Statistics (2000 ) Vol. 62, pp. 633–645].     

Bootstrap Unit‐Root Tests: Comparison and Extensions

Abstract. In this article, we study and compare the properties of several bootstrap unit‐root tests recently proposed in the literature. The tests are Dickey–Fuller (DF) or Augmented DF, based either on residuals from an autoregression and the use of the block bootstrap or on first‐differenced data and the use of the stationary bootstrap or sieve bootstrap. We extend the analysis by interchanging the data transformations (differences vs. residuals), the types of bootstrap and the presence or absence of a correction for autocorrelation in the tests. We show that two sieve bootstrap tests based on residuals remain asymptotically valid. In contrast to the literature which focuses on a comparison of the bootstrap tests with an asymptotic test, we compare the bootstrap tests among themselves using response surfaces for their size and power in a simulation study. This study leads to the following conclusions: (i) augmented DF tests are always preferred to standard DF tests; (ii) the sieve bootstrap performs better than the block bootstrap; (iii) difference‐based tests appear to have slightly better size properties, but residual‐based tests appear more powerful.     

A JOINT PORTMANTEAU TEST FOR CONDITIONAL MEAN AND VARIANCE TIME‐SERIES MODELS

In this article, we propose a new joint portmanteau test for checking the specification of parametric conditional mean and variance functions of linear and nonlinear time‐series models. The use of a joint test is motivated for complete control of the asymptotic size since marginal tests for the conditional variance may lead to misleading conclusions when the conditional mean is misspecified. The new test is based on an asymptotically distribution‐free transformation on the sample autocorrelations of both normalized residuals and squared normalized residuals. This makes it unnecessary to full detail the asymptotic properties of the estimates used to obtain residuals, which could be inefficient two‐step ones, avoiding also choices of maximum lag parameters increasing with sample length to control asymptotic size. The robust versions of the new test also properly account for higher‐order moment dependence at a reduced cost. The finite‐sample performance of the new test is compared with that of well‐known tests through simulations.     

Minimum α‐divergence estimation for arch models

Abstract. This paper considers a minimum α‐divergence estimation for a class of ARCH(p) models. For these models with unknown volatility parameters, the exact form of the innovation density is supposed to be unknown in detail but is thought to be close to members of some parametric family. To approximate such a density, we first construct an estimator for the unknown volatility parameters using the conditional least squares estimator given by Tjøstheim [Stochastic processes and their applications (1986) Vol. 21, pp. 251–273]. Then, a nonparametric kernel density estimator is constructed for the innovation density based on the estimated residuals. Using techniques of the minimum Hellinger distance estimation for stochastic models and residual empirical process from an ARCH(p) model given by Beran [Annals of Statistics (1977) Vol. 5, pp. 445–463] and Lee and Taniguchi [Statistica Sinica (2005) Vol. 15, pp. 215–234] respectively, it is shown that the proposed estimator is consistent and asymptotically normal. Moreover, a robustness measure for the score of the estimator is introduced. The asymptotic efficiency and robustness of the estimator are illustrated by simulations. The proposed estimator is also applied to daily stock returns of Dell Corporation.