Long-run Variance Research Articles

INFERENCE IN MILDLY EXPLOSIVE AUTOREGRESSIONS UNDER UNCONDITIONAL HETEROSKEDASTICITY

Mildly explosive autoregressions have been extensively employed in recent theoretical and applied econometric work to model the phenomenon of asset market bubbles. An important issue in this context concerns the construction of confidence intervals for the autoregressive parameter that represents the degree of explosiveness. Existing studies rely on intervals that are justified only under conditional homoskedasticity/heteroskedasticity. This paper studies the problem of constructing asymptotically valid confidence intervals in a mildly explosive autoregression where the innovations are allowed to be unconditionally heteroskedastic. The assumed variance process is general and can accommodate both deterministic and stochastic volatility specifications commonly adopted in the literature. Within this framework, we show that the standard heteroskedasticity- and autocorrelation-consistent estimate of the long-run variance converges in distribution to a nonstandard random variable that depends on nuisance parameters. Notwithstanding this result, the corresponding t-statistic is shown to still possess a standard normal limit distribution. To improve the quality of inference in small samples, we propose a dependent wild bootstrap-t procedure and establish its asymptotic validity under relatively weak conditions. Monte Carlo simulations demonstrate that our recommended approach performs favorably in finite samples relative to existing methods across a wide range of volatility specifications. Applications to international stock price indices and U.S. house prices illustrate the relevance of the advocated method in practice.

Lag order selection for long-run variance estimation in econometrics

. Estimating the long-run variance (LRV) is crucial for several econometric issues. Constructing reliable heteroskedasticity autocorrelation consistent (HAC) variance-covariance matrices and implementing efficient generalized method of moments (GMM) estimation procedures require a consistent LRV estimate. A good VARHAC estimator (HAC matrix with the spectral density at frequency zero constructed using a VAR spectral estimation) requires accurately estimating the sum of autoregressive (AR) coefficients; however, a criterion that minimizes the innovation variance does not necessarily yield the best spectral estimate. This article implements an optimal VARHAC estimator using an alternative information criterion, considering the bias in the sum of the parameters for the AR estimator of the spectral density at frequency zero.

Prewhitened long-run variance estimation robust to nonstationarity

We introduce a nonparametric nonlinear VAR prewhitened long-run variance (LRV) estimator for the construction of standard errors robust to autocorrelation and heteroskedasticity that can be used for hypothesis testing in a variety of contexts including the linear regression model. Existing methods either are theoretically valid only under stationarity and have poor finite-sample properties under nonstationarity (i.e., fixed-b methods), or are theoretically valid under the null hypothesis but lead to tests that are not consistent under nonstationary alternative hypothesis (i.e., both fixed-b and traditional HAC estimators). The proposed estimator accounts explicitly for nonstationarity, unlike previous prewhitened procedures which are known to be unreliable, and leads to tests with accurate null rejection rates and good monotonic power. We also establish MSE bounds for LRV estimation that are sharper than previously established and use them to determine the data-dependent bandwidths.

ASYMPTOTICALLY UNIFORMLY MOST POWERFUL TESTS FOR UNIT ROOTS IN GAUSSIAN PANELS WITH CROSS-SECTIONAL DEPENDENCE GENERATED BY COMMON FACTORS

This paper considers testing for unit roots in Gaussian panels with cross-sectional dependence generated by common factors. Within our setup, we can analyze restricted versions of the two prevalent approaches in the literature, that of Moon and Perron (2004, Journal of Econometrics 122, 81–126), who specify a factor model for the innovations, and the PANIC setup proposed in Bai and Ng (2004, Econometrica 72, 1127–1177), who test common factors and idiosyncratic deviations separately for unit roots. We show that both frameworks lead to locally asymptotically normal experiments with the same central sequence and Fisher information. Using Le Cam’s theory of statistical experiments, we obtain the local asymptotic power envelope for unit-root tests. We show that the popular Moon and Perron (2004, Journal of Econometrics 122, 81–126) and Bai and Ng (2010, Econometric Theory 26, 1088–1114) tests only attain the power envelope in case there is no heterogeneity in the long-run variance of the idiosyncratic components. We develop a new test which is asymptotically uniformly most powerful irrespective of possible heterogeneity in the long-run variance of the idiosyncratic components. Monte Carlo simulations corroborate our asymptotic results and document significant gains in finite-sample power if the variances of the idiosyncratic shocks differ substantially among the cross-sectional units.

Navigating sustainable horizons: exploring the dynamics of financial stability, green growth, renewable energy, technological innovation, financial inclusion, and soft infrastructure in shaping sustainable development.

This paper examines sustainable development, which employs an integrated approach to tackle environmental, social, and economic challenges. It provides a theoretical underpinning by examining sustainable development's inception, fundamental tenets, and conceptual structures. This study highlights the interdependence of social equity, economic prosperity, and environmental conservation, emphasizing the need for a comprehensive approach. Quantitative methodology is utilized in this study, and the dependent variable is sustainable development. Financial risk, green growth, technological innovation, renewable energy, financial inclusion, and soft infrastructure are all independent variables. The analysis is predicated on secondary data from the Organization for Economic Cooperation and Development and World Development Indicators databases spanning 2004 to 2019. An entropy-weighted method used for the green growth index is a metric that enhances the precision of variable indicators. Cointegration, correlation, VIF, cross-sectional dependency, and stationarity tests are among the diagnostic tests that inform the selection of methods for the panel data set. It is determined that fully modified ordinary least squares is the suitable technique. The findings suggest statistically significant positive correlations among greenhouse gases, financial inclusion, and soft infrastructure. Conversely, significant negative correlations exist between financial risk, green growth, renewable energy, and technological innovation. An estimated 55% long-run variance is present. The study's key finding is that financial risk has an adverse effect on sustainable development, while an impactful relationship where increased green growth is linked to decreased GHG emissions. This association is notably significant. Results show that renewable energy has a negative coefficient and significant negative impact on greenhouse gases, showing an active relation to enhancing sustainable development. In contrast, financial inclusion has a significant positive effect on sustainable development. The implications imply that providing incentives to institutions engaged in alternative energy, precisely renewable sources, could positively impact the environment. Government policies and funding regulations oriented toward sustainable development are indispensable for environmental sustainability. Government policies and incentives are pivotal in advancing an environmentally conscious and sustainable future. This study's contribution lies in elucidating the positive correlation between government interventions and promoting renewable energy adoption, thereby paving the way for a greener tomorrow.

Maximum-Subsampling Test of Equal Predictive Ability

In comparing the accuracy of any two competing forecasts, under the assumption that the loss differentials are covariance stationary, Diebold and Mariano (DM) proposed a test statistic that is asymptotically normal. This test is further studied by Giacomini and White (GW). However, under the DM-GW framework, the estimator of the variance of the DM test can be inaccurate in small samples, which yields size distortions; see, for example, Coroneo and Iacone (CI). To alleviate this challenge, we propose a maximum-subsampling (MS) test that does not require estimating the long-run variance induced by a number of autocovariances. Accordingly, the MS test is applicable for comparing the predictive ability between two competing forecasts even when the loss differentials are serially correlated with arbitrary autocovariance structures. We demonstrate that the MS test converges to the type I extreme value distribution under the DM-GW null hypothesis with proper conditions. In addition, under a set of alternative hypotheses, we show the MS test is consistent. We further compare the MS test with the DM test and two CI tests via five simulation settings, which are either modified or adapted from McCracken and Coroneo and Iacone. Simulation results show that the MS test performs satisfactorily.

On the long-run properties of income and stock prices: the stability of the “golden ratios”

PurposeThe challenge of predicting changes in aggregate income and stock prices is one that has occupied the research agendas of economists. This paper aims to use the consumption–income ratio and the dividend–price ratio to predict future income and stock prices.Design/methodology/approachTo examine the stability of the consumption–income ratio and the dividend–price ratio, the authors run a two-variable, two-lag reduced-form VAR in the vein of Cochrane (1994), using a lag of each respective ratio as exogenous to the VAR. Additionally, the authors estimate an AR(4) model for income and prices.FindingsThe consumption–income ratio and the dividend–price ratio remain key to understanding future movements in income and stock prices. The consumption–income ratio significantly predicts future income in the USA, and aggregate income is easier to predict than consumption in the VAR model. The dividend–price ratio does not significantly predict future price growth. Consumption and dividend shocks have lasting impacts on income and prices.Originality/valueThe consumption–income ratio and the dividend–price ratio are still key to understanding future movements in income and stock prices. The consumption–income ratio significantly predicts future income in the USA, and aggregate income is easier to predict than consumption in the VAR model. However, the dividend–price ratio does not significantly predict future price growth, a change from previous research from the 1990s, despite the increasing complexity of stock markets. Consumption and dividend shocks have lasting impacts on income and prices and appear to be significant drivers in both the short- and long-run variance in income and prices.

Local Whittle estimation of high-dimensional long-run variance and precision matrices

Local Whittle estimation of high-dimensional long-run variance and precision matrices

Statistical inference for mean and variance of oscillatory processes

When constructing confidence intervals for the mean and variance of a stationary continuous-time stochastic process, two approaches have been considered in the literature: one based on the so-called long-run variance of the process and its square, and the other based on the so-called self-normalization. These approaches are revisited here in the context of random oscillatory processes such as random (non-)linear oscillators and related models with particular attention to the problem of estimating the non-zero long-run variances of the processes. The case of the zero long-run variance, which has been studied and is quite different, is also considered. The approaches are extended to the situation where multiple independent records of the stochastic process are available, for example, by introducing an estimator of the long-run variance which improves on other natural candidate estimators. Finally, a simulation study is provided to assess the performance of the proposed methods in estimating the long-run variances and constructing confidence intervals, and a data example is considered.

Bootstrap Inference in Cointegrating Regressions: Traditional and Self-Normalized Test Statistics

Traditional tests of hypotheses on the cointegrating vector are well known to suffer from severe size distortions in finite samples, especially when the data are characterized by large levels of endogeneity or error serial correlation. To address this issue, we combine a vector autoregressive (VAR) sieve bootstrap to construct critical values with a self-normalization approach that avoids direct estimation of long-run variance parameters when computing test statistics. To asymptotically justify this method, we prove bootstrap consistency for the self-normalized test statistics under mild conditions. In addition, the underlying bootstrap invariance principle allows us to prove bootstrap consistency also for traditional test statistics based on popular modified OLS estimators. Simulation results show that using bootstrap critical values instead of asymptotic critical values reduces size distortions associated with traditional test statistics considerably, but combining the VAR sieve bootstrap with self-normalization can lead to even less size distorted tests at the cost of only small power losses. We illustrate the usefulness of the VAR sieve bootstrap in empirical applications by analyzing the validity of the Fisher effect in 19 OECD countries.

ON GMM INFERENCE: PARTIAL IDENTIFICATION, IDENTIFICATION STRENGTH, AND NONSTANDARD ASYMPTOTICS

This paper analyses aspects of generalized method of moments (GMM) inference in moment equality models in settings where standard regularity conditions may break down. Explicit analytic formulations for the asymptotic distributions of estimable functions of the GMM estimator and statistics based on the GMM criterion function are derived under relatively mild assumptions. The moment Jacobian is allowed to be rank deficient, so first order identification may fail, the values of the Jacobian singular values are not constrained, thereby allowing for varying levels of identification strength, the long-run variance of the moment conditions can be singular, and the GMM criterion function weighting matrix may also be chosen sub-optimally. The large-sample properties are derived without imposing a specific structure on the functional form of the moment conditions. Closed-form expressions for the distributions are presented that can be evaluated using standard software without recourse to bootstrap or simulation methods. The practical operation of the results is illustrated via examples involving instrumental variables estimation of a structural equation with endogenous regressors and a common CH features model.

Forecasting VIX with time-varying risk aversion

In this paper, we investigate the predictive value of time-varying risk aversion (RA) for VIX via the realized EGARCH-mixed-data sampling model incorporating RA (henceforth REGARCH-MIDAS-RA). The REGARCH-MIDAS-RA model builds on the REGARCH model, which takes into account the high-frequency information by including the realized measure of volatility. Moreover, the model provides a convenient framework to model the long-run variance, which responds to changes in RA. We obtain the risk-neutralization of the REGARCH-MIDAS-RA model and derive the model-implied VIX formula. Our empirical results show that realized measure and RA possess predictive value for VIX. The REGARCH-MIDAS-RA model yields more accurate VIX forecasts compared to a range of competing models, including the GARCH, GJR-GARCH, nonlinear GARCH, EGARCH, REGARCH and REGARCH-MIDAS. In summary, our findings highlight the importance of incorporating the realized measure as well as RA in forecasting VIX.

Volatility swaps valuation under a modified risk-neutralized Heston model with a stochastic long-run variance level

We consider the pricing of discretely sampled volatility swaps under a modified Heston model, whose risk-neutralized volatility process contains a stochastic long-run variance level. We derive an analytical forward characteristic function under this model, which has never been presented in the literature before. Based on this, we further obtain an analytical pricing formula for volatility swaps which can guarantee the computational accuracy and efficiency. We also demonstrate the significant impact of the introduced stochastic long-run variance level on volatility swap prices with synthetic as well as calibrated parameters. doi: 10.1017/S144618112200013X

Simultaneous bandwidths determination for DK-HAC estimators and long-run variance estimation in nonparametric settings

We consider the derivation of data-dependent simultaneous bandwidths for double kernel heteroscedasticity and autocorrelation consistent (DK-HAC) estimators. In addition to the usual smoothing over lagged autocovariances for classical HAC estimators, the DK-HAC estimator also applies smoothing over the time direction. We obtain the optimal bandwidths that jointly minimize the global asymptotic MSE criterion and discuss the tradeoff between bias and variance with respect to smoothing over lagged autocovariances and over time. Unlike the MSE results of Andrews, we establish how nonstationarity affects the bias-variance tradeoff. We use the plug-in approach to construct data-dependent bandwidths for the DK-HAC estimators and compare them with the DK-HAC estimators from Casini that use data-dependent bandwidths obtained from a sequential MSE criterion. The former performs better in terms of size control, especially with stationary and close to stationary data. Finally, we consider long-run variance (LRV) estimation under the assumption that the series is a function of a nonparametric estimator rather than of a semiparametric estimator that enjoys the usual rate of convergence. Thus, we also establish the validity of consistent LRV estimation in nonparametric parameter estimation settings.

Asymptotic F test in regressions with observations collected at high frequency over long span

This paper proposes tests of linear hypotheses when the variables may be continuous-time processes with observations collected at a high sampling frequency over a long span. Utilizing series long run variance (LRV) estimation in place of the traditional kernel LRV estimation, we develop easy-to-implement and more accurate F tests in both stationary and nonstationary environments. The nonstationary environment accommodates exogenous regressors that are general semimartingales. Endogenous regressors are allowed in a nonstationary environment similar to cointegration models in the usual discrete-time setting. The F tests can be implemented in exactly the same way as in the discrete-time setting. The F tests are, therefore, robust to the continuous-time or discrete-time nature of the data. Simulations demonstrate the improved size accuracy and competitive power of the F tests relative to existing continuous-time testing procedures and their improved versions. The F tests are of practical interest as recent work by Chang et al. (2021) demonstrates that traditional inference methods can become invalid and produce spurious results when continuous-time processes are observed on finer grids over a long span.

Lasso inference for high-dimensional time series

In this paper we develop valid inference for high-dimensional time series. We extend the desparsified lasso to a time series setting under Near-Epoch Dependence (NED) assumptions allowing for non-Gaussian, serially correlated and heteroskedastic processes, where the number of regressors can possibly grow faster than the time dimension. We first derive an error bound under weak sparsity, which, coupled with the NED assumption, means this inequality can also be applied to the (inherently misspecified) nodewise regressions performed in the desparsified lasso. This allows us to establish the uniform asymptotic normality of the desparsified lasso under general conditions, including for inference on parameters of increasing dimensions. Additionally, we show consistency of a long-run variance estimator, thus providing a complete set of tools for performing inference in high-dimensional linear time series models. Finally, we perform a simulation exercise to demonstrate the small sample properties of the desparsified lasso in common time series settings.

Adjusted-range self-normalized confidence interval construction for censored dependent data

This paper proposes an adjusted-range based self-normalization (SN) method to construct confidence intervals for censored dependent data, which helps to circumvent the long-run variance estimation and tuning parameter selection problems. Simulation studies confirm the validity of this new approach. • A new adjusted-range approach to handle censored dependent data. • Circumvent long-run variance estimation and tuning parameter selection problems. • Generate much narrower confidence intervals than competing methods.

Segmenting Time Series via Self-Normalisation

AbstractWe propose a novel and unified framework for change-point estimation in multivariate time series. The proposed method is fully non-parametric, robust to temporal dependence and avoids the demanding consistent estimation of long-run variance. One salient and distinct feature of the proposed method is its versatility, where it allows change-point detection for a broad class of parameters (such as mean, variance, correlation and quantile) in a unified fashion. At the core of our method, we couple the self-normalisation- (SN) based tests with a novel nested local-window segmentation algorithm, which seems new in the growing literature of change-point analysis. Due to the presence of an inconsistent long-run variance estimator in the SN test, non-standard theoretical arguments are further developed to derive the consistency and convergence rate of the proposed SN-based change-point detection method. Extensive numerical experiments and relevant real data analysis are conducted to illustrate the effectiveness and broad applicability of our proposed method in comparison with state-of-the-art approaches in the literature.

High-Dimensional Granger Causality Tests with an Application to VIX and News

Abstract We study Granger causality testing for high-dimensional time series using regularized regressions. To perform proper inference, we rely on heteroskedasticity and autocorrelation consistent (HAC) estimation of the asymptotic variance and develop the inferential theory in the high-dimensional setting. To recognize the time-series data structures, we focus on the sparse-group LASSO (sg-LASSO) estimator, which includes the LASSO and the group LASSO as special cases. We establish the debiased central limit theorem for low-dimensional groups of regression coefficients and study the HAC estimator of the long-run variance based on the sg-LASSO residuals. This leads to valid time-series inference for individual regression coefficients as well as groups, including Granger causality tests. The treatment relies on a new Fuk–Nagaev inequality for a class of τ-mixing processes with heavier than Gaussian tails, which is of independent interest. In an empirical application, we study the Granger causal relationship between the VIX and financial news.

VOLATILITY SWAPS VALUATION UNDER A MODIFIED RISK-NEUTRALIZED HESTON MODEL WITH A STOCHASTIC LONG-RUN VARIANCE LEVEL

AbstractWe consider the pricing of discretely sampled volatility swaps under a modified Heston model, whose risk-neutralized volatility process contains a stochastic long-run variance level. We derive an analytical forward characteristic function under this model, which has never been presented in the literature before. Based on this, we further obtain an analytical pricing formula for volatility swaps which can guarantee the computational accuracy and efficiency. We also demonstrate the significant impact of the introduced stochastic long-run variance level on volatility swap prices with synthetic as well as calibrated parameters.