Long-run Variance Research Articles

Simultaneous inference of the mean of functional time series

For functional time series with physical dependence, we construct confidence bands for its mean function. The physical dependence is a general dependence framework, and it slightly relaxes the conditions of m-approximable dependence. We estimate functional time series mean functions via a spline smoothing technique. Confidence bands have been constructed based on a long-run variance and a strong approximation theorem, which is satisfied with mild regularity conditions. Simulation experiments provide strong evidence that corroborates the asymptotic theories. Additionally, an application to S&P500 index data demonstrates a non-constant volatility mean function at a certain significance level.

Sluggish Private Investment in Japan's Lost Decade: Mixed Frequency Vector Autoregression Approach

It is well known that sluggish private investment plagued the Japanese macroeconomy in the Lost Decade. Past empirical papers have not reached a clear consensus on what caused the investment slowdown. This paper sheds new light on this issue by applying mixed frequency vector autoregressive techniques to monthly stock prices, quarterly bank loans, firm profit, and private investment. First, we reveal a spiral of declining stock prices, profit, and investment. Second, monthly stock prices explain as much as 50.3% of the long-run forecast error variance of investment. Third, the stagnation of bank loans is a consequence (and not a cause) of declining stock prices and investment. Aggregating monthly stock prices into a quarterly level underestimates the influence of stock prices. It also weakens the statistical significance of the spiral of stock prices, profit, and investment. The mixed frequency approach therefore yields richer economic insights into Japan's sluggish investment than the classical single-frequency approach.

Restoring monotonic power in Wald/LM-type tests

Wald/LM-type tests for a shift in mean often exhibit nonmonotonic power, due to incorrect estimation of long-run variance. In this paper, we propose a robust estimator of long-run variance that is built on nonparametric regression residuals and always converges to the true long-run variance under both the null and the alternative hypothesis. Monte Carlo experiments show that the modified tests have monotonic power against the mean with single or multiple breaks in finite samples.

Persistence under Temporal Aggregation and Differencing

Temporal aggregation is known to affect the persistence of time series, where persistence reflects, roughly speaking, the degree of positive serial correlation. It is suggested to measure persistence as percentage of the long-run variance not attributable to the variance (long-run variance difference ratio). Here we study the aggregation of flow variables as well as stock data, and difference-stationarity is allowed for. Moreover, moving averages encountered e.g. when computing annual growth rates (seasonal differences) are investigated. It is clarified when persistence is increased or decreased, and by how much. Our results are exact for finite aggregation level. They are illustrated with numerical examples and real data. Approximate results for growing aggregation level are provided, too.

Persistence under temporal aggregation and differencing

Temporal aggregation is known to affect the persistence of time series. We study the aggregation of flow variables as well as stock data, and difference-stationarity is allowed for. Moreover, moving averages encountered when computing annual growth rates (seasonal differences) are investigated. Using a relative persistence measure (long-run variance ratio), it is clarified when persistence is increased or decreased, and by how much. Our results are exact for a finite aggregation level. They are illustrated with monthly time series. Approximate results for the growing aggregation level are provided, too.

Effects of time delay on stochastic resonance of the stock prices in financial system

Abstract The effect of time delay on stochastic resonance of the stock prices in finance system was investigated. The time delay is introduced into the Heston model driven by the extrinsic and intrinsic periodic information for stock price. The signal power amplification (SPA) was calculated by numerical simulation. The results indicate that an optimal critical value of delay time maximally enhances the reverse-resonance in the behaviors of SPA as a function of long-run variance of volatility or cross correlation coefficient between noises for both cases of intrinsic and extrinsic periodic information. Moreover, in both cases, being a critical value in the delay time, when the delay time takes value below the critical value, reverse-resonance increases with the delay time increasing, however, when the delay time takes value above the critical value, the reverse-resonance decrease with the delay time increasing.

M-procedures for detection of a change under weak dependence

Procedures detecting a change of regression parameters in a linear model are considered when both the regressors and the errors are weakly dependent in the sense of Lp–m-approximability. M-estimators and weighted M-residuals are used to construct test statistics, and their asymptotic distribution is studied under the null hypothesis of no change and under contiguous alternatives. Estimators of a long-run variance are also considered and studied both analytically and numerically.

The roles of extrinsic periodic information on the stability of stock price

The roles of extrinsic periodic information on the stability of stock price for the case of the stock market crashes are investigated with Heston model. From numerical simulation of the mean escape time (MET), the results indicate that: (i) the varying amplitude of extrinsic periodic information induces and enhances a maximum in curve of the MET versus the mean reversion of volatility or initial position; (ii) there is a critical value of amplitude or frequency of extrinsic periodic information in the behavior of MET versus the long-run variance or amplitude of volatility fluctuations, and when amplitude or frequency takes value below the critical value, the increase in amplitude or frequency enhances the stability of stock price, but restrains it when amplitude or frequency takes value above the critical value.

Conditional Correlation Models of Autoregressive Conditional Heteroscedasticity With Nonstationary GARCH Equations

In this article, we investigate the effects of careful modeling the long-run dynamics of the volatilities of stock market returns on the conditional correlation structure. To this end, we allow the individual unconditional variances in conditional correlation generalized autoregressive conditional heteroscedasticity (CC-GARCH) models to change smoothly over time by incorporating a nonstationary component in the variance equations such as the spline-GARCH model and the time-varying (TV)-GARCH model. The variance equations combine the long-run and the short-run dynamic behavior of the volatilities. The structure of the conditional correlation matrix is assumed to be either time independent or to vary over time. We apply our model to pairs of seven daily stock returns belonging to the S&P 500 composite index and traded at the New York Stock Exchange. The results suggest that accounting for deterministic changes in the unconditional variances improves the fit of the multivariate CC-GARCH models to the data. The effect of careful specification of the variance equations on the estimated correlations is variable: in some cases rather small, in others more discernible. We also show empirically that the CC-GARCH models with time-varying unconditional variances using the TV-GARCH model outperform the other models under study in terms of out-of-sample forecasting performance. In addition, we find that portfolio volatility-timing strategies based on time-varying unconditional variances often outperform the unmodeled long-run variances strategy out-of-sample. As a by-product, we generalize news impact surfaces to the situation in which both the GARCH equations and the conditional correlations contain a deterministic component that is a function of time.

Modeling the relationship between climate oscillations and drought by a multivariate GARCH model

Typical multivariate time series models may exhibit comovement in mean but not in variance of hydrologic and climatic variables. This paper introduces multivariate generalized autoregressive conditional heteroscedasticity (GARCH) models to capture the comovement of the variance or the conditional covariance between two hydroclimatic time series. The diagonal vectorized and Baba-Engle-Kroft-Kroner models are developed to evaluate the covariance between drought and two atmospheric circulations, Southern Oscillation Index (SOI) and North Atlantic Oscillation (NAO) time series during 1954-2000. The univariate generalized autoregressive conditional heteroscedasticity model indicates a strong persistency level in conditional variance for NAO and a moderate persistency level for SOI. The conditional variance of short-term drought index indicates low level of persistency, while the long-term index drought indicates high level of persistency in conditional variance. The estimated conditional covariance between drought and atmospheric indices is shown to be weak and negative. It is also observed that the covariance between drought and atmospheric indices is largely dependent on short-run variance of atmospheric indices rather than their long-run variance. The nonlinearity and stationarity tests show that the conditional covariances are nonlinear but stationary. However, the degree of nonlinearity is higher for the covariance between long-term drought and atmospheric indices. It is also observed that the nonlinearity of NAO is higher than that for SOI, in contrast to the stationarity which is stronger for SOI time series. Key Points Multivariate heteroscedastic models are developed for drought analysis Conditional covariance between drought, SOI, and NAO is not strong Time-varying correlations between drought and atmospheric indices are estimated.

How close a relationship does a capital market have with other markets? A reexamination based on the equal variance test

The cointegration test cannot discriminate closer relationships from cointegrating relationships. In most applications, we must assess the degrees of cointegrating relationships, for example, to examine the comovement between international stock markets using the cointegration methodology. Lee et al. (2012) introduced a variance test of cointegration equilibrium errors to measure the similarity of these relationships. However, the key assumption of cross-sectional independence between a panel of two country-pair squared cointegrating equilibrium errors in their model is not desirable. The appearance of cross-sectional dependence of individual (stock) markets in a panel is a common existence. The current paper shows that the consideration of cross-sectional dependence and the method of estimating long-run variance are important. Our results, which extend the cross-sectional dependence of some Asian stock markets during the Asian financial crisis (1997–1998) documented by Lee et al. (2012), indicate that the similarity of background and business cooperation (or trading activities) are all crucial factors for determining the price patterns by the “equal variance test” proposed in this paper. The analysis of the 2007–2009 global financial crisis is included to confirm the robustness of the results.

Asymptotically Unbiased Estimation of Autocovariances and Autocorrelations with Panel Data in the Presence of Individual and Time Effects

AbstractThis article proposes asymptotically unbiased estimators of autocovariances and autocorrelations for panel data with both individual and time effects. We show that the conventional autocovariance estimators suffer from the bias caused by the elimination of individual and time effects. The bias related to individual effects is proportional to the long-run variance, and it related to time effects is proportional to the value of the estimated autocovariance. For the conventional autocorrelation estimators, the elimination of time effects does not cause a bias while the elimination of individual effects does. We develop methods to estimate the long-run variance and propose bias-corrected estimators based on the proposed long-run variance estimator. We also consider the half-panel jackknife estimation for bias correction. The theoretical results are given by employing double asymptotics under which both the number of observations and the length of the time series tend to infinity. Monte Carlo simulations show that the asymptotic theory provides a good approximation to the actual bias and that the proposed bias-correction methods work well.

Asymptotically Unbiased Estimation of Autocovariances and Autocorrelations with Panel Data in the Presence of Individual and Time Effects

This paper proposes asymptotically unbiased estimators of autocovariances and autocorrelations for panel data with both individual and time effects. We show that the conventional autocovariance estimators suffers from the bias caused by the elimination of individual and time effects. The bias related to individual effects is proportional to the long-run variance, and that related to time effects is proportional to the value of the estimated autocovariance. For the conventional autocorrelation estimators, the elimination of time effects does not cause a bias while the elimination of individual effects does. We develop methods to estimate the long-run variance and propose bias-corrected estimators based on the proposed long-run variance estimator. The theoretical results are given by employing double asymptotics under which both the number of observations and the length of the time series tend to infinity. Monte Carlo simulations show that the asymptotic theory provides a good approximation to the actual bias and that the proposed bias correction works.

Improving the bandwidth-free inference methods by prewhitening

In this paper we consider inference of parameters in time series regression models. In the traditional inference approach, the heteroskedasticity and autocorrelation consistent (HAC) estimation is often involved to consistently estimate the asymptotic covariance matrix of regression parameter estimator. Since the bandwidth parameter in the HAC estimation is difficult to choose in practice, there has been a recent surge of interest in developing bandwidth-free inference methods. However, existing simulation studies show that these new methods suffer from severe size distortion in the presence of strong temporal dependence for a medium sample size. To remedy the problem, we propose to apply the prewhitening to the inconsistent long-run variance estimator in these methods to reduce the size distortion. The asymptotic distribution of the prewhitened Wald statistic is obtained and the general effectiveness of prewhitening is shown through simulations.

Econometric Policy Evaluation in a Two Sector Model

This paper estimates a New-Keynesian structural model and evaluates alternative monetary policy regimes. An estimated two sector model and alternative policy regimes are used to calculate the optimal monetary policy frontier. The structural model that consists of two sectors: durable and nondurable, is estimated by the Full Information Maximum Likelihood (FIML) method. Given the structure of the economy and discretionary monetary policy regimes, we find that changes in monetary policy responses to aggregate or disaggregate inflation and output gap can imply substantial differences in long-run inflation and output variances. More importantly, this paper shows that in terms of monetary policy frontier, monetary policy could have been more effective if the central bank had focused on sectoral stabilization as opposed to traditional stabilization.

Inference for modulated stationary processes

We study statistical inferences for a class of modulated stationary processes with time-dependent variances. Due to non-stationarity and the large number of unknown parameters, existing methods for stationary or locally stationary time series are not applicable. Based on a self-normalization technique, we address several inference problems, including self-normalized central limit theorem, self-normalized cumulative sum test for the change-point problem, long-run variance estimation through blockwise self-normalization, and self-normalization-based wild boot-strap. Monte Carlo simulation studies show that the proposed self-normalization-based methods outperform stationarity-based alternatives. We demonstrate the proposed methodology using two real data sets: annual mean precipitation rates in Seoul during 1771-2000, and quarterly U.S. Gross National Product growth rates during 1947-2002.

Powerful tests for structural changes in volatility

Detecting structural changes in volatility is important for understanding volatility dynamics and stylized facts observed for financial returns such as volatility persistence. We propose modified CUSUM and LM tests that are built on a robust estimator of the long-run variance of squared series. We establish conditions under which the new tests have standard null distributions and diverge faster than standard tests under the alternative. The theory allows smooth and abrupt structural changes that can be small. The smoothing parameter is automatically selected such that the proposed test has good finite-sample size and meanwhile achieves decent power gain.

A FIXED-bPERSPECTIVE ON THE PHILLIPS–PERRON UNIT ROOT TESTS

In this paper we extend fixed-basymptotic theory to the nonparametric Phillips–Perron (PP) unit root tests. We show that the fixed-blimits depend on nuisance parameters in a complicated way. These nonpivotal limits provide an alternative theoretical explanation for the well-known finite-sample problems of the PP tests. We also show that the fixed-blimits depend on whether deterministic trends are removed using one-step or two-step detrending approaches. This is in contrast to the asymptotic equivalence of the one- and two-step approaches under a consistency approximation for the long-run variance estimator. Based on these results we introduce modified PP tests that allow for asymptotically pivotal fixed-binference. The theoretical analysis is cast in the framework of near-integrated processes, which allows us to study the asymptotic behavior both under the unit root null hypothesis and for local alternatives. The performance of the original and modified PP tests is compared by means of local asymptotic power and a small finite-sample simulation study.

Tests of Short Memory With Thick-Tailed Errors

In this article, we consider the robustness to fat tails of four stationarity tests. We also consider their sensitivity to the number of lags used in long-run variance estimation, and the power of the tests. Lo's modified rescaled range (MR/S) test is not very robust. Choi's Lagrange multiplier (LM) test has excellent robustness properties but is not generally as powerful as the Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test. As an analytical framework for fat tails, we suggest local-to-finite variance asymptotics, based on a representation of the process as a weighted sum of a finite variance process and an infinite variance process, where the weights depend on the sample size and a constant. The sensitivity of the asymptotic distribution of a test to the weighting constant is a good indicator of its robustness to fat tails. This article has supplementary material online.

A Comparison of the Robustness of Several Tests of Short Memory to Autocorrelated Errors

In this paper we consider the robustness to error autocorrelation of four stationarity tests. The size and power properties of these tests are investigated by simulation. Size is improved by using fixed-b critical values to account for the number of lags used in long-run variance estimation. Lo’s MR/S test is not very robust. Choi’s LM test has excellent robustness properties but this comes at some cost in power; it is not as powerful as the KPSS test or the rescaled variance (V/S) test.