Truncating estimation for the change in stochastic trend with heavy-tailed innovations

A CUSUM estimator is proposed for the change point in stochastic trend with heavy-tailed innovations. In order to avoid the outliers caused by heavy-tailed innovations, we also construct a truncating CUSUM estimator. Results in this paper show that the CUSUM estimators are consistent. Simulations demonstrate that the truncating estimator behaves better for the heavy-tailed innovations.     

System Estimation of Panel Data Models Under Long-Range Dependence

A general dynamic panel data model is considered that incorporates individual and interactive fixed effects allowing for contemporaneous correlation in model innovations. The model accommodates general stationary or nonstationary long-range dependence through interactive fixed effects and innovations, removing the necessity to perform a priori unit-root or stationarity testing. Moreover, persistence in innovations and interactive fixed effects allows for cointegration; innovations can also have vector-autoregressive dynamics; deterministic trends can be featured. Estimations are performed using conditional-sum-of-squares criteria based on projected series by which latent characteristics are proxied. Resulting estimates are consistent and asymptotically normal at standard parametric rates. A simulation study provides reliability on the estimation method. The method is then applied to the long-run relationship between debt and GDP. Supplementary materials for this article are available online.     

True integer value time series

We construct an integer-valued stationary symmetric AR(1) process which can have either a positive or a negative lag-one autocorrelation. Nearly all integer-valued time series models are designed for observations which are non-negative integers or counts. They have innovations which are distributed on the non-negative integers and therefore obviously non-symmetric. We build our model using innovations that come from the difference of two independent identically distributed Poisson random variables. These innovations have a symmetric distribution, which has many advantages; in particular, they will allow us to model negative correlations. For our AR(1) process, we examine its basic properties and consider estimation via conditional least squares.     

Diagnostic Checking of Unobserved-Components Time Series Models

Diagnostic checking of the specification of time series models is normally carried out using the innovations—that is, the one-step-ahead prediction errors. In an unobserved-components model, other sets of residuals are available. These auxiliary residuals are estimators of the disturbances associated with the unobserved components. They can often yield information that is less apparent from the innovations, but they suffer from the disadvantage that they are serially correlated even in a correctly specified model with known parameters. This article shows how the properties of the auxiliary residuals may be obtained, how they are related to each other and to the innovations, and how they can be used to construct test statistics. Applications are presented showing how residuals can be used to detect and distinguish between outliers and structural change.     

Bootstrapping Noncausal Autoregressions: With Applications to Explosive Bubble Modeling

In this article, we develop new bootstrap-based inference for noncausal autoregressions with heavy-tailed innovations. This class of models is widely used for modeling bubbles and explosive dynamics in economic and financial time series. In the noncausal, heavy-tail framework, a major drawback of asymptotic inference is that it is not feasible in practice as the relevant limiting distributions depend crucially on the (unknown) decay rate of the tails of the distribution of the innovations. In addition, even in the unrealistic case where the tail behavior is known, asymptotic inference may suffer from small-sample issues. To overcome these difficulties, we propose bootstrap inference procedures using parameter estimates obtained with the null hypothesis imposed (the so-called restricted bootstrap). We discuss three different choices of bootstrap innovations: wild bootstrap, based on Rademacher errors; permutation bootstrap; a combination of the two (“permutation wild bootstrap”). Crucially, implementation of these bootstraps do not require any a priori knowledge about the distribution of the innovations, such as the tail index or the convergence rates of the estimators. We establish sufficient conditions ensuring that, under the null hypothesis, the bootstrap statistics estimate consistently particular conditionaldistributions of the original statistics. In particular, we show that validity of the permutation bootstrap holds without any restrictions on the distribution of the innovations, while the permutation wild and the standard wild bootstraps require further assumptions such as symmetry of the innovation distribution. Extensive Monte Carlo simulations show that the finite sample performance of the proposed bootstrap tests is exceptionally good, both in terms of size and of empirical rejection probabilities under the alternative hypothesis. We conclude by applying the proposed bootstrap inference to Bitcoin/USD exchange rates and to crude oil price data. We find that indeed noncausal models with heavy-tailed innovations are able to fit the data, also in periods of bubble dynamics. Supplementary materials for this article are available online.     

Asymptotic Distributions of Innovation Density Estimators in Linear Processes

In this article, we demonstrate that at a fixed point, the asymptotic distribution of the innovation density estimator is normal for stationary linear process. Also, we show that the asymptotic distribution of the global measure of the deviation of the density estimator from the expectation of the kernel innovation density (based on the true innovations) is the same as that in the case when we can observe the true innovations.     

Strong orthogonal decompositions and non‐linear impulse response functions for infinite‐variance processes

The author proves that Wold‐type decompositions with strong orthogonal prediction innovations exist in smooth, reflexive Banach spaces of discrete time processes if and only if the projection operator generating the innovations satisfies the property of iterations. His theory includes as special cases all previous Wold‐type decompositions of discrete time processes, completely characterizes when non‐linear heavy‐tailed processes obtain a strong‐orthogonal moving average representation, and easily promotes a theory of non‐linear impulse response functions for infinite‐variance processes. The author exemplifies his theory by developing a non‐linear impulse response function for smooth transition threshold processes, and discusses how to test decomposition innovations for strong orthogonality and whether the proposed model represents the best predictor. A data set on currency exchange rates allows him to illustrate his methodology.     

A control chart for a general Gaussian process

It has been repeatedly demonstrated that X-bar quality control charts perform poorly when the process subgroups being monitored are correlated. In this paper, we propose and investigate the performance of a control chart that accounts for subgroup correlations in a general Gaussian process. The time-series innovations algorithm is used to construct the desired chart from a set of one-step ahead predictions and prediction variances. The chart is applicable in both stationary and nonstationary settings. A simulation study shows that this ‘innovations’ chart performs as a traditional X-bar chart even when the correlation structure of the process must be estimated from a small number of subgroups. The innovations chart is then used to study a data set of motor shaft diameters which has correlated subgroups. The results here show that erroneous conclusions can be reached if subgroup correlations are ignored.     

Analysis of autoregressive models with symmetric stable innovations

This paper develops algorithms for fitting autoregressive models with symmetric stable innovations using auto-covariation function. A recursive algorithm is proposed for generalized Yule-Walker estimation of autoregressive coefficients and partial auto-covariation function. It also introduces a new information criterion, useful for consistent order selection. Applications of the proposed methods are illustrated using observations simulated from autoregressive models with symmetric stable innovations as well as by analysing a set of real data.     

Fully observed INAR(1) processes

The innovations of an INAR(1) process (integer-valued autoregressive) are usually assumed to be unobservable. There are, however, situations in practice, where also the innovations can be uncovered, i.e. where we are concerned with a fully observed INAR(1) process. We analyze stochastic properties of such a fully observed INAR(1) process and explore the relation between the INAR(1) model and certain metapopulation models. We show how the additional knowledge about the innovations can be used for parameter estimation, for model diagnostics, and for forecasting. Our findings are illustrated with two real-data examples.     

SUBSAMPLING THE JOHANSEN TEST WITH STABLE INNOVATIONS

By taking into account the thick-tail property of the errors, cointegration analysis in vector error-correction models with infinite-variance stable errors is a natural generalization of cointegration analysis in error-correction models with normally distributed errors. We study the Johansen test for cointegrated systems under symmetric stable innovations with discrete spectral measures. The results show that the distributions of the Johansen test statistics under these innovations involve nuisance parameters. To overcome the problem of nuisance parameters, we implement a nonparametric subsampling procedure. We document some subsampling simulation results and demonstrate in an empirical example how the test can be used in practice.     

Periodograms asymptotic distributions in periodically correlated processes and multivariate stationary processes: An alternative approach

Discrete time periodically correlated (PC) processes are viewed as the processes with time-dependent spectra. This, together with an auxiliary operator which is defined here is employed to apply classical results on the asymptotic distribution of the periodogram of the univariate white noise (innovations) to derive the asymptotic distributions of the periodograms for the PC processes and also for the multivariate stationary processes. We assume only the continuity and positive definiteness of the spectral densities together with the independence of the innovations.     

Tail asymptotic of discounted aggregate claims with compound dependence under risky investment

This paper considers the tail asymptotic of discounted aggregate claims with compound dependence under risky investment. The price of risky investment is modeled by a geometric Lévy process, while claims are modeled by a one-sided linear process whose innovations further obeying a so-called upper tail asymptotic independence. When the innovations are heavy tailed, we derive some uniform asymptotic formulas. The results show that the linear dependence has significant impact on the tail asymptotic of discounted aggregate claims but the upper tail asymptotic independence is negligible.     

Confidence intervals for impulse responses under departures from normality

Monte Carlo evidence shows that in structural VAR models with fat-tailed or skewed innovations the coverage accuracy of impulse response confidence intervals may deterorate substantially compared to the same model with Gaussian innovations. Empirical evidance suggests that such departures from normality are quite plausible for economic time series. The simulation results suggest that applied researchers are best off using nonparametric bootstrap intervals for impulse responses, regardless of whether or not there is evidence of fat tails or skewness in the error distribution. Allowing for departures from normality is shown to considerably weaken the evidence of the delayed overshooting puzzle in Eichenbaum and Evans (1995).     

Unit root tests and dramatic shifts with infinite variance processes

A model which explains data that is subject to sudden structural changes of unspecified nature is presented. The structural shifts are generated by a random walk component whose innovations belong to the normal domain of attraction of a symmetric stable law. To test the model against the stationarity case, several non-parametric, and regression-based statistics are studied. The non-parametric tests are a generalization of the variance ratio test to innovations with heavy-tailed distributions. The tests are consistent and shown to have good finite sample size and power properties and are applied to a set of economic variables.     

On first-order integer-valued autoregressive process with Katz family innovations

This paper considers the first-order integer-valued autoregressive (INAR) process with Katz family innovations. This family of INAR processes includes a broad class of INAR(1) processes with Poisson, negative binomial, and binomial innovations, respectively, featuring equi-, over-, and under-dispersion. Its probabilistic properties such as ergodicity and stationarity are investigated and the formula of the marginal mean and variance is provided. Further, a statistical process control procedure based on the cumulative sum control chart is considered to monitor autocorrelated count processes. A simulation and real data analysis are conducted for illustration.     

Large Bayesian VARs: A Flexible Kronecker Error Covariance Structure

ABSTRACT

We introduce a class of large Bayesian vector autoregressions (BVARs) that allows for non-Gaussian, heteroscedastic, and serially dependent innovations. To make estimation computationally tractable, we exploit a certain Kronecker structure of the likelihood implied by this class of models. We propose a unified approach for estimating these models using Markov chain Monte Carlo (MCMC) methods. In an application that involves 20 macroeconomic variables, we find that these BVARs with more flexible covariance structures outperform the standard variant with independent, homoscedastic Gaussian innovations in both in-sample model-fit and out-of-sample forecast performance.     

ESTIMATION OF SPATIAL ARMA MODELS

Spatial ARMA models are considered using the nonsymmetric half plane ordering on a lattice of data. A method is given for the estimation of the orders and the coefficients of such models under an identifiability condition and the condition that the beat linear predictor is the best predictor in the mean square sense. Under these conditions, the strong consistency of the estimators ia established. The usual methods for ARMA modelling in Time Series Analysis require estimation of the innovations. The method of this paper introduces an inveree model complementary to the original model so that the estimation of the innovations is avoided. This leads to a substantial reduction in the computational complexity in the two-dimensional case.     

Prediction Intervals for Time Series: A Modified Sieve Bootstrap Approach

Traditional Box–Jenkins prediction intervals perform poorly when the innovations are not Gaussian. Nonparametric bootstrap procedures overcome this handicap, but most existing methods assume that the AR and MA orders of the process are known. The sieve bootstrap approach requires no such assumption but produces liberal coverage due to the use of residuals that underestimate the actual variance of the innovations and the failure of the methods to capture variations due to sampling error of the mean. A modified approach, that corrects these deficiencies, is implemented. Monte Carlo simulations results show that the modified version achieves nominal or near nominal coverage.     

Semiparametric Time Series Models with Log‐concave Innovations: Maximum Likelihood Estimation and its Consistency

We study semiparametric time series models with innovations following a log‐concave distribution. We propose a general maximum likelihood framework that allows us to estimate simultaneously the parameters of the model and the density of the innovations. This framework can be easily adapted to many well‐known models, including autoregressive moving average (ARMA), generalized autoregressive conditionally heteroscedastic (GARCH), and ARMA‐GARCH models. Furthermore, we show that the estimator under our new framework is consistent in both ARMA and ARMA‐GARCH settings. We demonstrate its finite sample performance via a thorough simulation study and apply it to model the daily log‐return of the FTSE 100 index.