Resampling a time-series process: A method of estimating the probabilities associated with alternative plans for protecting pensions against inflation

Any pension protection formula that falls short of complete compensation for inflation has associated with it a time-series of probability distributions of future purchasing power losses. A method of estimating those distributions is proposed and applied. The method is based on the idea of representing inflation as a multivariate time-series process and using a model fitted to historical data to generate a large artificial sample of ‘realized’ inflation sequences by means of a bootstrapping procedure. The purchasing power losses under a given protection plan can then be simulated for each inflation sequence and the sample distributions calculated.     

Detecting shocks: Outliers and breaks in time series

A single outlier in a regression model can be detected by the effect of its deletion on the residual sum of squares. An equivalent procedure is the simple intervention in which an extra parameter is added for the mean of the observation in question. Similarly, for unobserved components or structural time-series models, the effect of elaborations of the model on inferences can be investigated by the use of interventions involving a single parameter, such as trend or level changes. Because such time-series models contain more than one variance, the effect of the intervention is measured by the change in individual variances.We examine the effect on the estimated parameters of moving various kinds of intervention along the series. The horrendous computational problems involved are overcome by the use of score statistics combined with recent developments in filtering and smoothing. Interpretation of the resulting time-series plots of diagnostics is aided by simulation envelopes.Our procedures, illustrated with four example, permit keen insights into the fragility of inferences to specific shocks, such as outliers and level breaks. Although the emphasis is mostly on parameter estimation, forecast are also considered. Possible extensions include seasonal adjustment and detrending of series.     

A versatile and robust metric entropy test of time-reversibility,and other hypotheses

We examine the performance of a metric entropy statistic as a robust test for time-reversibility (TR), symmetry, and serial dependence. It also serves as a measure of goodness-of-fit. The statistic provides a consistent and unified basis in model search, and is a powerful diagnostic measure with surprising ability to pinpoint areas of model failure. We provide empirical evidence comparing the performance of the proposed procedure with some of the modern competitors in nonlinear time-series analysis, such as robust implementations of the BDS and characteristic function-based tests of TR, along with correlation-based competitors such as the Ljung–Box Q-statistic. Unlike our procedure, each of its competitors is motivated for a different, specific, context and hypothesis. Our evidence is based on Monte Carlo simulations along with an application to several stock indices for the US equity market.     

The common and specific components of dynamic volatility

This paper develops a dynamic approximate factor model in which returns are time-series heteroskedastic. The heteroskedasticity has three components: a factor-related component, a common asset-specific component, and a purely asset-specific component. We develop a new multivariate GARCH model for the factor-related component. We develop a univariate stochastic volatility model linked to a cross-sectional series of individual GARCH models for the common asset-specific component and the purely asset-specific component. We apply the analysis to monthly US equity returns for the period January 1926 to December 2000. We find that all three components contribute to the heteroskedasticity of individual equity returns. Factor volatility and the common component in asset-specific volatility have long-term secular trends as well as short-term autocorrelation. Factor volatility has correlation with interest rates and the business cycle.     

On the evaluation of poly-t density functions

Poly-t densities are defined by the property that their kernel is a product, or a ratio of products, of multivariate t-density kernels. As discussed in Drèze (1977), these densities arise as Bayesian posterior densities for regression coefficients under a variety of specifications for the prior density and the data generating process. We have therefore developed methods and computer algorithms to evaluate integrating constants and other characteristics of poly-t densities with no more than a single quadratic form in the numerator (section 2). As a by-product of our analysis we have also derived an algorithm for the computation of moments of positive definite quadratic forms in Normal variables (section 3). In section 4 we discuss inference on the sampling variances associated with the models discussed in Drèze (1977).     

Forecasting correlated time series with exponential smoothing models

This paper presents the Bayesian analysis of a general multivariate exponential smoothing model that allows us to forecast time series jointly, subject to correlated random disturbances. The general multivariate model, which can be formulated as a seemingly unrelated regression model, includes the previously studied homogeneous multivariate Holt-Winters’ model as a special case when all of the univariate series share a common structure. MCMC simulation techniques are required in order to approach the non-analytically tractable posterior distribution of the model parameters. The predictive distribution is then estimated using Monte Carlo integration. A Bayesian model selection criterion is introduced into the forecasting scheme for selecting the most adequate multivariate model for describing the behaviour of the time series under study. The forecasting performance of this procedure is tested using some real examples.     

Testing the specification of multivariate models in the presence of alternative hypotheses

In this paper we propose a non-nested hypothesis test for testing the specification of a multivariate econometric model in the presence of an alternative model which purports to explain the same phenomenon. We demonstrate that the new test statistic tends to minus the same random variable as the CPD test statistic introduced by Pesaran and Deaton (1978), provided that the truth is ‘close’ to the null hypothesis. Since the new test is simpler to compute than the multivariate CPD test, it would seem to be the procedure of choice.     

Dynamic functional time-series forecasts of foreign exchange implied volatility surfaces

This paper presents static and dynamic versions of univariate, multivariate, and multilevel functional time-series methods to forecast implied volatility surfaces in foreign exchange markets. We find that dynamic functional principal component analysis generally improves out-of-sample forecast accuracy. Specifically, the dynamic univariate functional time-series method shows the greatest improvement. Our models lead to multiple instances of statistically significant improvements in forecast accuracy for daily EUR–USD, EUR–GBP, and EUR–JPY implied volatility surfaces across various maturities, when benchmarked against established methods. A stylised trading strategy is also employed to demonstrate the potential economic benefits of our proposed approach.     

Robust recurrent network model for intermittent time-series forecasting

This paper describes a deep-learning-based time-series forecasting method that was ranked third in the accuracy challenge of the M5 competition. We solved the problem using a deep-learning approach based on DeepAR, which is an auto-regressive recurrent network model conditioned on historical inputs. To address the intermittent and irregular characteristics of sales demand, we modified the training procedure of DeepAR; instead of using actual values for the historical inputs, our model uses values sampled from a trained distribution and feeds them to the network as past values. We obtained the final result using an ensemble of multiple models to make a robust and stable prediction. To appropriately select a model for the ensemble, each model was evaluated using the average weighted root mean squared scaled error, calculated for all levels of a wide range of past periods.     

Studentization and prediction in a multivariate normal setting

In a simple multivariate normal prediction setting, derivation of a predictive distribution can flow from formal Bayes arguments as well as pivoting arguments. We look at two special cases and show that the classical invariant predictive distribution is based on a pivot whose sampling distribution depends on the parameter – that is, the pivot is not an ancillary statistic. In contrast, a predictive distribution derived by a structural argument is based on a pivot with a parameter free distribution (an ancillary statistic). The classical procedure is formal Bayes for the Jeffreys prior. Our results show that this procedure does not have a structural or fiducial interpretation.     

Bayesian modeling of joint and conditional distributions

In this paper, we study a Bayesian approach to flexible modeling of conditional distributions. The approach uses a flexible model for the joint distribution of the dependent and independent variables and then extracts the conditional distributions of interest from the estimated joint distribution. We use a finite mixture of multivariate normals (FMMN) to estimate the joint distribution. The conditional distributions can then be assessed analytically or through simulations. The discrete variables are handled through the use of latent variables. The estimation procedure employs an MCMC algorithm. We provide a characterization of the Kullback–Leibler closure of FMMN and show that the joint and conditional predictive densities implied by the FMMN model are consistent estimators for a large class of data generating processes with continuous and discrete observables. The method can be used as a robust regression model with discrete and continuous dependent and independent variables and as a Bayesian alternative to semi- and non-parametric models such as quantile and kernel regression. In experiments, the method compares favorably with classical nonparametric and alternative Bayesian methods.     

Estimation for multivariate stable distributions with generalized empirical likelihood

This paper considers the generalized empirical likelihood (GEL) method for estimating the parameters of the multivariate stable distribution. The GEL method is considered to be an extension of the generalized method of moments (GMM). The multivariate stable distributions are widely applicable as they can accommodate both skewness and heavy tails. We treat the spectral measure, which summarizes scale and asymmetry, by discretization. In order to estimate all the model parameters simultaneously, we apply the estimating function constructed by equating empirical and theoretical characteristic functions. The efficacy of the proposed GEL method is demonstrated in Monte Carlo studies. An illustrative example involving daily returns of market indexes is also included.     

Polynomial approximations in cross-sectional models

The time-series distributed lag techniques of econometrics can be usefully applied to cross-sectional, spatial and cross-section time-series situations. The application is perfectly natural in cross-section, time-series models when regression coefficients evolve systematically as the cross-section grouping variable changes. The evolution of such coefficients lends itself to polynomial approximation or more general smoothing restrictions. These ideas are not new, Gersovitz and McKinnon (1978) and Trivedi and Lee (1981) providing two of the earliest applications of cross-equation smoothing techniques. However, their applications were in the context of coefficient variation due to seasonal changes and this may account for the non-diffusion of these techniques. The approach here is illustrated in the context of age-specific household formation equations based on census data, using Almon polynomials when the regression coefficients vary systematically by age group. A second application is provided, using spatial data, explaining the incidence of crime, by region; using polynomial and geometric smoothing to model distance declining regional effects.     

Robust estimation of ARIMA models

The paper discusses methods of estimating univariate ARIMA models with outliers. The approach calls for a state vector representation of a time-series model, on which we can then operate on using the Kalman filter. One of the additional advantages of Kalman filter operating on the state vector representation is that the method and code could easily be adapted to be applicable to the ARIMA model with missing observations. The paper investigates ways to calculate robust initial estimation of the parameters of the ARIMA model. The method proposed is based on the results obtained by R.D. Martin (1980).     

Testing multivariate distributions in GARCH models

In this paper, we consider testing distributional assumptions in multivariate GARCH models based on empirical processes. Using the fact that joint distribution carries the same amount of information as the marginal together with conditional distributions, we first transform the multivariate data into univariate independent data based on the marginal and conditional cumulative distribution functions. We then apply the Khmaladze's martingale transformation (K-transformation) to the empirical process in the presence of estimated parameters. The K-transformation eliminates the effect of parameter estimation, allowing a distribution-free test statistic to be constructed. We show that the K-transformation takes a very simple form for testing multivariate normal and multivariate t-distributions. The procedure is applied to a multivariate financial time series data set.     

Enhancing leader inclusion while preventing social exclusion in the work group

Research on inclusion and exclusion at work has grown in recent years, but for the most part has been treated as separate domains. In this paper, we integrate these literatures to build greater understanding of leader inclusion and leader exclusion. Leaders play a critical role in determining group member experiences of inclusion and exclusion through direct treatment of employees, and by serving as a role model (Bandura, 1977). According to social identity theory, when the leader is rewarded by the organization, this signifies that the leader is a prototypical organizational member who exemplifies the set of norms and behaviors most consistent with the organizational ideal (Hogg & van Knippenberg, 2003). We argue that through both social learning and social identity mechanisms, the leader can encourage inclusionary and exclusionary behavior in their work group. We first examine leader inclusion and present the types of behaviors that will aid in creating inclusive team member experiences. By exhibiting these behaviors, a leader can be a role model, an advocate and an ally for building work group inclusion. Next, we present the negative roles of ostracizer and bystander adopted by leaders that indicate support for behaving in an exclusionary manner, which can lead to exclusion among coworkers. We then describe leader remedies for social exclusion. Finally, we discuss the implications of our model and directions for future research.     

A comparison of the ratio of variances in distance-based and classical multivariate analysis

In multivariate analysis, the measure of variance accounted for plays a central role. In this paper, we show that an alternative approach, distance-based multivariate analysis, also yields solutions that can be summarized by a ratio of variances. For classical multivariate analysis, this ratio is equal to the variance accounted for (VAF) and in distance-based multivariate analysis it equals distance accounted for (DAF). We show that DAF in distance-based multivariate analysis can always be made higher than VAF in classical multivariate analysis. This property is illustrated for principal components analysis, multiple correspondence analysis, multiple regression, and analysis of variance.     

A new approach for detecting shifts in forecast accuracy

Forecasts play a critical role at inflation-targeting central banks, such as the Bank of England. Breaks in the forecast performance of a model can potentially incur important policy costs. However, commonly-used statistical procedures implicitly place a lot of weight on type I errors (or false positives), which results in a relatively low power of the tests to identify forecast breakdowns in small samples. We develop a procedure which aims to capture the policy cost of missing a break. We use data-based rules to find the test size that optimally trades off the costs associated with false positives with those that can result from a break going undetected for too long. In so doing, we also explicitly study forecast errors as a multivariate system. The covariance between forecast errors for different series, although often overlooked in the forecasting literature, not only enables us to consider testing in a multivariate setting, but also increases the test power. As a result, we can tailor our choice of the critical values for each series not only to the in-sample properties of each series, but also to the way in which the series of forecast errors covary.