Long-run Variance Research Articles

Two Estimators of the Long-Run Variance: Beyond Short Memory

This paper deals with the estimation of the long-run variance of a stationary sequence. We extend the usual Bartlett-kernel heteroskedasticity and autocorrelation consistent (HAC) estimator to deal with long memory and antipersistence. We then derive asymptotic expansions for this estimator and the memory and autocorrelation consistent (MAC) estimator introduced by Robinson (2005). We offer a theoretical explanation for the sensitivity of HAC to the bandwidth choice, a feature which has been observed in the special case of short memory. Using these analytical results, we determine the MSE-optimal bandwidth rates for each estimator. We analyze by simulations the finite-sample performance of HAC and MAC estimators, and the coverage probabilities for the studentized sample mean, giving practical recommendations for the choice of bandwidths.

A Monte Carlo study to compare two recent modifications of the KPSS test for near integration

The objective of this article is to compare the size and power properties of two modifications of the KPSS test of Kwiatkowski et al. (1992) proposed by Sul et al. (2005) and by Harris et al. (2007), using Monte Carlo simulations, in order to decide which version to use in applied research. The two modifications have been proposed to deal with those cases in which the null hypothesis specifies that the time series is near integrated, in the sense that it is very close to the alternative hypothesis of a unit root. It has been shown in the literature that in these cases the KPSS test tends to over-reject the null hypothesis. The modification by Sul et al. (2005) is based on an alternative long-run variance estimator. Harris et al. (2007) propose applying the KPSS test to the filtered series instead of to the original series. We conclude that the test based on the transformation proposed by Sul et al. outperforms the test based on the transformation proposed by Harris et al.

Fixed‐banalysis of LM‐type tests for a shift in mean

Summary We analyse Lagrange Multiplier (LM) tests for a shift in mean of a univariate time series at an unknown date. We consider a class of LM statistics based on non-parametric kernel estimators of the long-run variance and we develop a fixed-b asymptotic theory for the statistics. We provide results for the case of I(0) and I(1) errors. The fixed-b theory suggests that, for a given statistic, kernel and significance level, there exists a bandwidth such that the fixed-b asymptotic critical value is the same for both I(0) and I(1) errors. We compute these ‘robust’ bandwidths for a selection of well-known kernels. In the case of the supremum statistic, the robust bandwidth LM tests have good power that is monotonic and similar for all kernels. In the case of the mean and exponential mean statistics, power is non-monotonic when the robust bandwidths are used. Under the null hypothesis, the Bartlett supremum statistic has little, if any, over-rejection problems in finite samples even when there is a moving average component with a negative coefficient. In contrast, the other supremum statistics tend to over-reject in this case. In practice, we recommend the Bartlett supremum LM statistic implemented with the robust bandwidth.

The Long-Run Risk of Stock Returns

There is a large debate about whether stocks are less risky in the long-run than in the short run. Siegel (2008) in particular has argued that long-run variances are smaller than one-year variances on a per year basis. A recent article by Pastor and Stambaugh (2010) questions this evidence because of various uncertainties in the estimation of a typical data generating process (DGP, henceforth) for returns. As most research in this area however, the model of Pastor and Stambaugh assumes that there is short run predictability in stock returns, which generates mean reversion and thus affects the long-run variance, whereas the variability of stock return shocks is assumed to be constant over time. Our point of departure of the standard literature is simple: the standard DGP, such as the one used by Pastor and Stambaugh, is easily rejected by actual stock return data, where the evidence for time-varying volatility is many times stronger than the evidence for time-varying conditional means. We introduce a new dynamic model that does fit the intricacies of stock return dynamics, while still allowing for potential time-variation in the conditional mean. Our “Bad Environment-Good Environment” (BEGE for short) framework has two types of shocks: ”good environment” positively skewed shocks and ”bad environment” negatively skewed shocks. The relative importance of these shocks varies through time. By a judicious choice of the distribution to model these shocks, we can compute closed-form solutions for all moments of the stock return distribution at all horizons, even though the conditional variance, skewness and kurtosis of stock returns vary through time. The model is also easily estimable from the data and a normally distributed world is a special case of the BEGE framework. Thinking about the recent crisis period provides good motivation for a BEGE framework. It is hard to generate the type of day to day returns, or even month to month returns, we witnessed in 2008 within a Gaussian model. We view the shocks in October 2008 as likely coming from a negatively skewed distribution, which is characteristic for the “bad environment” that dominated the stock market in 2008 and the beginning of 2009 (and perhaps still does). However, even in the midst of the crisis, mini rallies did occur. In our framework, these are driven by a positively skewed shock distribution, which is more typical of a crisis-free environment but is still lurking in the background even in the midst of the crisis. We call the random variable generating such shocks the “good environment” variable. The BEGE framework has two implications for the long-run risk variance question. First, the ratio of long-run to short-term risks varies through time, as a function of the relative importance of the good and bad environment shocks and current volatility. For example, whatever the assumption on the conditional mean, it is difficult to imagine short-term risk not being higher than long-term stock market risk during most of 2008, and again now in August 2010. Second, the non-normalities imply that the variance is not a sufficient statistic to characterize risk, and we compute closed form solutions for higher order moments at all horizons as well.

Robust trend inference with series variance estimator and testing-optimal smoothing parameter

The paper develops a novel testing procedure for hypotheses on deterministic trends in a multivariate trend stationary model. The trends are estimated by the OLS estimator and the long run variance (LRV) matrix is estimated by a series type estimator with carefully selected basis functions. Regardless of whether the number of basis functions K is fixed or grows with the sample size, the Wald statistic converges to a standard distribution. It is shown that critical values from the fixed- K asymptotics are second-order correct under the large- K asymptotics. A new practical approach is proposed to select K that addresses the central concern of hypothesis testing: the selected smoothing parameter is testing-optimal in that it minimizes the type II error while controlling for the type I error. Simulations indicate that the new test is as accurate in size as the nonstandard test of Vogelsang and Franses (2005) and as powerful as the corresponding Wald test based on the large- K asymptotics. The new test therefore combines the advantages of the nonstandard test and the standard Wald test while avoiding their main disadvantages (power loss and size distortion, respectively).

Robust Trend Inference with Series Variance Estimator and Testing-Optimal Smoothing Parameter

The paper develops a novel testing procedure for hypotheses on deterministic trends in a multivariate trend stationary model. The trends are estimated by the OLS estimator and the long run variance (LRV) matrix is estimated by a series type estimator with carefully selected basis functions. Regardless of whether the number of basis functions K is fixed or grows with the sample size, the Wald statistic converges to a standard distribution. It is shown that critical values from the fixed-K asymptotics are second order correct under the large-K asymptotics. A new practical approach is proposed to select K that addresses the central concern of hypothesis testing: the selected smoothing parameter is testing-optimal in that it minimizes the type II error while controlling for the type I error. Simulations indicate that the new test is as accurate in size as the nonstandard test of Vogelsang and Franses (2005) and as powerful as the corresponding Wald test based on the large-K asymptotics. The new test therefore combines the advantages of the nonstandard test and the standard Wald test while avoiding their main disadvantages (power loss and size distortion, respectively).

Asymptotically unbiased estimation of autocovariances and autocorrelations for panel data with incidental trends

We consider the estimation of autocovariances using panel data with incidental trends under double asymptotics. The conventional autocovariance estimator suffers from a bias whose value is approximated by twice the long-run variance. We propose a bias-corrected estimator.

Kernel estimators of asymptotic variance for adaptive Markov chain Monte Carlo

We study the asymptotic behavior of kernel estimators of asymptotic variances (or long-run variances) for a class of adaptive Markov chains. The convergence is studied both in $L^p$ and almost surely. The results also apply to Markov chains and improve on the existing literature by imposing weaker conditions. We illustrate the results with applications to the $\operatorname {GARCH}(1,1)$ Markov model and to an adaptive MCMC algorithm for Bayesian logistic regression.

Ratio test to detect change in the variance of linear process

An adaptation of the method of Horváth et al. [Ratio tests for change point detection, Inst. Math. Stat. 1 (2008), pp. 293–304] is proposed to detect change in the variance of a linear process. In comparison with the existing tests, the new one does not require the estimation of the long-run variance. The test is robust against misspecification. Its asymptotic property is investigated under the no change null hypothesis. A Monte Carlo study shows that our test has reasonably good size and power properties in moderate samples. We also discuss how to estimate the change point which has not been discussed by the existing literatures. The consistency and rate of convergence for the change point estimator are also established. We demonstrate the applicability of our method to estimate the time of change of the variance of an ARMA(1,1) process. The effectiveness of the method is examined by analysing a real example.

TESTING FOR A SHIFT IN TREND AT AN UNKNOWN DATE: A FIXED-BANALYSIS OF HETEROSKEDASTICITY AUTOCORRELATION ROBUST OLS-BASED TESTS

This paper analyzes tests for a shift in the trend function of a time series at an unknown date based on ordinary least squares (OLS) estimates of the trend function. Inference about the trend parameters depends on the serial correlation structure of the data through the long-run variance (zero frequency spectral density) of the errors. Asymptotically pivotal tests can be obtained by the use of serial correlation robust standard errors that require an estimate of the long-run variance. The focus is on the class of nonparametric kernel estimators of the long-run variance. Tests based on these estimators present two problems for practitioners. The first is the choice of kernel and bandwidth. The second is the well-known overrejection problem caused by strong serial correlation (or a possible unit root) in the errors.We provide solutions to both problems by using the fixed-basymptotic framework of Kiefer and Vogelsang (2005,Econometric Theory, 21, 1130–1164) in conjunction with the scaling factor approach of Vogelsang (1998,Econometrica65, 123–148). Our results provide practitioners with a family of OLS-based trend function structural change tests that are size robust to the presence of strong serial correlation or a unit root. Specific recommendations are provided for the tuning parameters (kernel and bandwidth) in a way that maximizes asymptotic integrated power.

BLOCK BOOTSTRAP HAC ROBUST TESTS: THE SOPHISTICATION OF THE NAIVE BOOTSTRAP

This paper studies the properties of naive block bootstrap tests that are scaled by zero frequency spectral density estimators (long-run variance estimators). The naive bootstrap is a bootstrap where the formula used in the bootstrap world to compute standard errors is the same as the formula used on the original data. Simulation evidence shows that the naive bootstrap can be much more accurate than the standard normal approximation. The larger the HAC bandwidth, the greater the improvement. This improvement holds for a large number of popular kernels, including the Bartlett kernel, and it holds when the independent and identically distributed (i.i.d.) bootstrap is used and yet the data are serially correlated. Using recently developed fixed-b asymptotics for HAC robust tests, we provide theoretical results that can explain the finite sample patterns. We show that the block bootstrap, including the special case of the i.i.d. bootstrap, has the same limiting distribution as the fixed-b asymptotic distribution. For the special case of a location model, we provide theoretical results that suggest the naive bootstrap can be more accurate than the standard normal approximation depending on the choice of the bandwidth and the number of finite moments in the data. Our theoretical results lay the foundation for a bootstrap asymptotic theory that is an alternative to the traditional approach based on Edgeworth expansions.

DYNAMIC TIME SERIES BINARY CHOICE

This paper considers dynamic time series binary choice models. It proves near epoch dependence and strong mixing for the dynamic binary choice model with correlated errors. Using this result, it shows in a time series setting the validity of the dynamic probit likelihood procedure when lags of the dependent binary variable are used as regressors, and it establishes the asymptotic validity of Horowitz’s smoothed maximum score estimation of dynamic binary choice models with lags of the dependent variable as regressors. For the semiparametric model, the latent error is explicitly allowed to be correlated. It turns out that no long-run variance estimator is needed for the validity of the smoothed maximum score procedure in the dynamic time series framework.

Testing Stationarity in Small- and Medium-Sized Samples when Disturbances are Serially Correlated*

In this article, we study the size distortions of the KPSS test for stationarity when serial correlation is present and samples are small- and medium-sized. It is argued that two distinct sources of the size distortions can be identified. The first source is the finite-sample distribution of the long-run variance estimator used in the KPSS test, while the second source of the size distortions is the serial correlation not captured by the long-run variance estimator because of a too narrow choice of truncation lag parameter. When the relative importance of the two sources is studied, it is found that the size of the KPSS test can be reasonably well controlled if the finite-sample distribution of the KPSS test statistic, conditional on the time-series dimension and the truncation lag parameter, is used. Hence, finite-sample critical values, which can be applied to reduce the size distortions of the KPSS test, are supplied. When the power of the test is studied, it is found that the price paid for the increased size control is a lower raw power against a non-stationary alternative hypothesis.

How useful is yet another data-driven bandwidth in long-run variance estimation?: A simulation study on cointegrating regressions

This paper investigates how bandwidth choice rules in long-run variance estimation affect finite-sample performance of efficient estimators for cointegrating regression models. Monte Carlo results indicate that Hirukawa's (2010) bandwidth choice rule contributes bias reduction in the estimators.

On the Irrelevance of Impossibility Theorems: The Case of the Long-run Variance

It has been argued that estimating the spectral density function of a stationary stochastic process at the zero frequency (or the so-called long-run variance) is an ill-posed problem so that any estimate will have an infinite minimax risk (e.g., Pötscher 2002). Most often it is a nuisance parameter that is present in the limit distribution of some statistic and one then needs an estimate of it to obtain test statistics that have a pivotal distribution. In this context, we argue that such an impossibility result is irrelevant. We show that, in the presence of the discontinuities that cause the ill-posedness of the estimation problem for the long-run variance, using the true value of the spectral density function at frequency zero leads to tests that have either 0 or 100% size and, hence, lead to confidence intervals that are completely uninformative. On the other hand, tests based on standard estimates of the long-run variance will have well defined limit distributions and, accordingly, be more informative.

Nonparametric Tests for Periodic Integration

We propose two nonparametric methods to test the null hypothesis of periodic integration, one based on the variance ratio unit root test of Breitung (2002) and the other on the Modi…ed Sargan and Bhargava test proposed by Stock (1999). The …rst one does not require speci…cation of the short-run dynamics of the process, while nevertheless delivering a pivotal limiting distribution. And the second one requires estimation of the long-run variance in order to obtain a distribution free from nuisance parameters. In advantage with the previously proposed tests for periodic integration our tests do not required non-linear methods of estimation for their implementation and could straightforwardly implemented for seasonal time series with S > 4. Finally we report monte-carlo results that support our …nding about the distribution of the proposed tests and allow us to compare their performance with the periodic integration test by Boswijk and Franses (1996).

Testing for Change Points in Time Series

This article considers the CUSUM-based (cumulative sum) test for a change point in a time series. In the case of testing for a mean shift, the traditional Kolmogorov–Smirnov test statistic involves a consistent long-run variance estimator, which is needed to make the limiting null distribution free of nuisance parameters. The commonly used lag-window type long-run variance estimator requires to choose a bandwidth parameter and its selection is a difficult task in practice. The bandwidth that is a fixed function of the sample size (e.g., n1/3, where n is sample size) is not adaptive to the magnitude of the dependence in the series, whereas the data-dependent bandwidth could lead to nonmonotonic power as shown in previous studies. In this article, we propose a self-normalization (SN) based Kolmogorov–Smirnov test, where the formation of the self-normalizer takes the change point alternative into account. The resulting test statistic is asymptotically distribution free and its power is monotonic. Furthermore, we extend the SN-based test to test for a change in other parameters associated with a time series, such as marginal median, autocorrelation at lag one, and spectrum at certain frequency bands. The use of the SN idea thus allows a unified treatment and offers a new perspective to the large literature of change point detection in the time series setting. Monte Carlo simulations are conducted to compare the finite sample performance of the new SN-based test with the traditional Kolmogorov–Smirnov test. Illustrations using real data examples are presented.

Reducing the size distortion of the KPSS test

This paper proposes a new stationarity test based on the KPSS test with less size distortion. We extend the boundary rule proposed by Sul, Phillips and Choi (2005) to the autoregressive spectral density estimator and parametrically estimate the long-run variance. We also derive the finite sample bias of the numerator of the test statistic up to the 1/T order and propose a correction to the bias term in the numerator. Finite sample simulations show that the correction term effectively reduces the bias in the numerator and that the finite sample size of our test is close to the nominal one as long as the long-run parameter in the model satisfies the boundary condition.

Quantifying the risk in age and block replacement policies

Preventive maintenance policies have been studied in the literature without considering the risk due to the cost variability. In this paper, we consider the two most popular preventive replacement policies, namely, age and block replacement policies under long-run average cost and expected unit time cost criteria. To quantify the risk in the preventive maintenance policies, we use the long-run variance of the accumulated cost over a time interval. We numerically derive the Risk-sensitive preventive replacement policies and study the impact of the Risk-sensitive optimality criterion on the managerial decisions. We also examine the performance of the expected unit time cost criterion as an alternative to the traditional long-run average cost criterion.

A two-sample test for comparison of long memory parameters

We construct a two-sample test for comparison of long memory parameters based on ratios of two rescaled variance (V/S) statistics studied in Giraitis et al. [L. Giraitis, R. Leipus, A. Philippe, A test for stationarity versus trends and unit roots for a wide class of dependent errors, Econometric Theory 21 (2006) 989–1029]. The two samples have the same length and can be mutually independent or dependent. In the latter case, the test statistic is modified to make it asymptotically free of the long-run correlation coefficient between the samples. To diminish the sensitivity of the test on the choice of the bandwidth parameter, an adaptive formula for the bandwidth parameter is derived using the asymptotic expansion in Abadir et al. [K. Abadir, W. Distaso, L. Giraitis, Two estimators of the long-run variance: beyond short memory, Journal of Econometrics 150 (2009) 56–70]. A simulation study shows that the above choice of bandwidth leads to a good size of our comparison test for most values of fractional and ARMA parameters of the simulated series.