Bayesian Forecasting of Many Count-Valued Time Series

Abstract

We develop and exemplify application of new classes of dynamic models for time series of nonnegative counts. Our novel univariate models combine dynamic generalized linear models for binary and conditionally Poisson time series, with dynamic random effects for over-dispersion. These models estimate dynamic regression coefficients in both binary and nonzero count components. Sequential Bayesian analysis allows fast, parallel analysis of sets of decoupled time series. New multivariate models then enable information sharing in contexts when data at a more highly aggregated level provide more incisive inferences on shared patterns such as trends and seasonality. A novel multiscale approach—one new example of the concept of decouple/recouple in time series—enables information sharing across series. This incorporates cross-series linkages while insulating parallel estimation of univariate models, and hence enables scalability in the number of series. The major motivating context is supermarket sales forecasting. Detailed examples drawn from a case study in multistep forecasting of sales of a number of related items showcase forecasting of multiple series, with discussion of forecast accuracy metrics, comparisons with existing methods, and broader questions of probabilistic forecast assessment.     

A note on the estimates of multivariate Gaussian probability

ABSTRACT

Some lower and upper bounds of multivariate Gaussian probability are given based on the univariate Mills’ ratio. These bounds are sharper than known ones on the multivariate Mills’ ratio in many case.     

PITMAN-CLOSENESS AS A MEASURE TO EVALUATE THE QUALITY OF FORECASTS

ABSTRACT

We consider Pitman-closeness to evaluate the performance of univariate and multivariate forecasting methods. Optimal weights for the combination of forecasts are calculated with respect to this criterion. These weights depend on the assumption of the distribution of the individual forecasts errors. In the normal case they are identical with the optimal weights with respect to the MSE-criterion (univariate case) and with the optimal weights with respect to the MMSE-criterion (multivariate case). Further, we present a simple example to show how the different combination techniques perform. There we can see how much the optimal multivariate combination can outperform different other combinations. In practice, we can find multivariate forecasts e.g., in econometrics. There is often the situation that forecast institutes estimate several economic variables.     

New HEAVY Models for Fat-Tailed Realized Covariances and Returns

ABSTRACT

We develop a new score-driven model for the joint dynamics of fat-tailed realized covariance matrix observations and daily returns. The score dynamics for the unobserved true covariance matrix are robust to outliers and incidental large observations in both types of data by assuming a matrix-F distribution for the realized covariance measures and a multivariate Student's t distribution for the daily returns. The filter for the unknown covariance matrix has a computationally efficient matrix formulation, which proves beneficial for estimation and simulation purposes. We formulate parameter restrictions for stationarity and positive definiteness. Our simulation study shows that the new model is able to deal with high-dimensional settings (50 or more) and captures unobserved volatility dynamics even if the model is misspecified. We provide an empirical application to daily equity returns and realized covariance matrices up to 30 dimensions. The model statistically and economically outperforms competing multivariate volatility models out-of-sample. Supplementary materials for this article are available online.     

Model fitting and inference under latent equilibrium processes

This paper presents a methodology for model fitting and inference in the context of Bayesian models of the type f(Y | X,θ)f(X|θ)f(θ), where Y is the (set of) observed data, θ is a set of model parameters and X is an unobserved (latent) stationary stochastic process induced by the first order transition model f(X ^(t+1)|X ^(t),θ), where X ^(t) denotes the state of the process at time (or generation) t. The crucial feature of the above type of model is that, given θ, the transition model f(X ^(t+1)|X ^(t),θ) is known but the distribution of the stochastic process in equilibrium, that is f(X|θ), is, except in very special cases, intractable, hence unknown. A further point to note is that the data Y has been assumed to be observed when the underlying process is in equilibrium. In other words, the data is not collected dynamically over time. We refer to such specification as a latent equilibrium process (LEP) model. It is motivated by problems in population genetics (though other applications are discussed), where it is of interest to learn about parameters such as mutation and migration rates and population sizes, given a sample of allele frequencies at one or more loci. In such problems it is natural to assume that the distribution of the observed allele frequencies depends on the true (unobserved) population allele frequencies, whereas the distribution of the true allele frequencies is only indirectly specified through a transition model. As a hierarchical specification, it is natural to fit the LEP within a Bayesian framework. Fitting such models is usually done via Markov chain Monte Carlo (MCMC). However, we demonstrate that, in the case of LEP models, implementation of MCMC is far from straightforward. The main contribution of this paper is to provide a methodology to implement MCMC for LEP models. We demonstrate our approach in population genetics problems with both simulated and real data sets. The resultant model fitting is computationally intensive and thus, we also discuss parallel implementation of the procedure in special cases.     

Sharp Bounds for Generalized Order Statistics via Logarithmic Moments

ABSTRACT

We present sharp bounds for expectations of generalized order statistics with random indices. The bounds are expressed in terms of logarithmic moments E X ^a (log max {1, X}) ^b of the underlying observation X. They are attainable and provide characterizations of some non trivial distributions. No restrictions are imposed on the parameters of the generalized order statistics model.     

Misspecified T2 tests. i. location and scale

Properties of Hotelling's (1931) T ² are studied under model misspecification in the model for a multivariate experiment. Stochastic bounds on T ² and further properties of the T ² test are studied under misspecified location and scale. The bounds are evaluated numerically in selected cases.     

Forecasting multivariate longitudinal binary data with marginal and marginally specified models

Forecasting with longitudinal data has been rarely studied. Most of the available studies are for continuous response and all of them are for univariate response. In this study, we consider forecasting multivariate longitudinal binary data. Five different models including simple ones, univariate and multivariate marginal models, and complex ones, marginally specified models, are studied to forecast such data. Model forecasting abilities are illustrated via a real-life data set and a simulation study. The simulation study includes a model independent data generation to provide a fair environment for model competitions. Independent variables are forecast as well as the dependent ones to mimic the real-life cases best. Several accuracy measures are considered to compare model forecasting abilities. Results show that complex models yield better forecasts.     

Goodness-of-link tests for multivariate regression models

ABSTRACT

This note presents an approximation to multivariate regression models which is obtained from a first-order series expansion of the multivariate link function. The proposed approach yields a variable-addition approximation of regression models that enables a multivariate generalization of the well-known goodness-of-link specification test, available for univariate generalized linear models. Application of this general methodology is illustrated with models of multinomial discrete choice and multivariate fractional data, in which context it is shown to lead to well-established approximation and testing procedures.     

Adjusted Likelihood Inference in an Elliptical Multivariate Errors-in-Variables Model

In this paper, we obtain an adjusted version of the likelihood ratio (LR) test for errors-in-variables multivariate linear regression models. The error terms are allowed to follow a multivariate distribution in the class of the elliptical distributions, which has the multivariate normal distribution as a special case. We derive a modified LR statistic that follows a chi-squared distribution with a high degree of accuracy. Our results generalize those in Melo and Ferrari (Advances in Statistical Analysis, 2010, 94, pp. 75–87) by allowing the parameter of interest to be vector-valued in the multivariate errors-in-variables model. We report a simulation study which shows that the proposed test displays superior finite sample behavior relative to the standard LR test.     

Applying State Space to SPC: Monitoring Multivariate Time Series

Monitoring cross-sectional and serially interdependent processes has become a new issue in statistical process control (SPC). In up-to-date SPC literature, Kalman filtering was reported to monitor univariate autocorrelated processes. This paper applies a Kalman filter or state-space method for SPC to monitoring multivariate time series. We use Aoki's approach to estimate the parameter matrices of a state-space model. Multivariate Hotelling T ² control charts are employed to monitor the residuals of the state-space. Examples of this approach are illustrated.     

Detection of outliers in growth curve models

The growth curve model introduced by Potthoff and Roy (1964) is a general statistical model which includes as special cases regression models and both univariate and multivariate analysis of variance models. In this paper, we discuss procedures for detection of outliers in growth curve models for mean-slippage and dispersion-slippage outlier model. The distributions of the test statistics are discussed and the values of significant probabilities are given using Bonferronl's bounds. Some simulation results are also presented.     

A transitional Markov switching autoregressive model

ABSTRACT

This paper is concerned with properties of a transitional Markov switching autoregressive (TMSAR) model, together with its maximum-likelihood estimation and inference. We extend existing MSAR models by allowing dependence of AR parameters on hidden states at time points prior to the current time t. A stationary solution is given and expressions for the theoretical autocovariance function are derived. Two time series are analyzed and the new model outperforms two existing MSAR models in terms of maximized log-likelihood, residual correlations, and one-step-ahead forecasting performance. The new model also gives more regime changes in agreement with real events.     

Alternative Procedures to Discriminate Non Nested Multivariate Linear Regression Models

ABSTRACT

This article builds classical and Bayesian testing procedures for choosing between non nested multivariate regression models. Although there are several classical tests for discriminating univariate regressions, only the Cox test is able to consistently handle the multivariate case. We then derive the limiting distribution of the Cox statistic in such a context, correcting an earlier derivation in the literature. Further, we show how to build alternative Bayes factors for the testing of nonnested multivariate linear regression models. In particular, we compute expressions for the posterior Bayes factor, the fractional Bayes factor, and the intrinsic Bayes factor.     

Multivariate autoregressive extreme value process and its application for modeling the time series properties of the extreme daily asset prices

Abstract

In this article we suggest a new multivariate autoregressive process for modeling time-dependent extreme value distributed observations. The idea behind the approach is to transform the original observations to latent variables that are univariate normally distributed. Then the vector autoregressive DCC model is fitted to the multivariate latent process. The distributional properties of the suggested model are extensively studied. The process parameters are estimated by applying a two-stage estimation procedure. We derive a prediction interval for future values of the suggested process. The results are applied in an empirically study by modeling the behavior of extreme daily stock prices.     

Forecasting Performance of an Open Economy DSGE Model

This paper analyzes the forecasting performance of an open economy dynamic stochastic general equilibrium (DSGE) model, estimated with Bayesian methods, for the Euro area during 1994Q1–2002Q4. We compare the DSGE model and a few variants of this model to various reduced-form forecasting models such as vector autoregressions (VARs) and vector error correction models (VECM), estimated both by maximum likelihood and two different Bayesian approaches, and traditional benchmark models, e.g., the random walk. The accuracy of point forecasts, interval forecasts and the predictive distribution as a whole are assessed in an out-of-sample rolling event evaluation using several univariate and multivariate measures. The results show that the open economy DSGE model compares well with more empirical models and thus that the tension between rigor and fit in older generations of DSGE models is no longer present. We also critically examine the role of Bayesian model probabilities and other frequently used low-dimensional summaries, e.g., the log determinant statistic, as measures of overall forecasting performance.     

Forecasting Performance of an Open Economy DSGE Model

This paper analyzes the forecasting performance of an open economy dynamic stochastic general equilibrium (DSGE) model, estimated with Bayesian methods, for the Euro area during 1994Q1-2002Q4. We compare the DSGE model and a few variants of this model to various reduced-form forecasting models such as vector autoregressions (VARs) and vector error correction models (VECM), estimated both by maximum likelihood and two different Bayesian approaches, and traditional benchmark models, e.g., the random walk. The accuracy of point forecasts, interval forecasts and the predictive distribution as a whole are assessed in an out-of-sample rolling event evaluation using several univariate and multivariate measures. The results show that the open economy DSGE model compares well with more empirical models and thus that the tension between rigor and fit in older generations of DSGE models is no longer present. We also critically examine the role of Bayesian model probabilities and other frequently used low-dimensional summaries, e.g., the log determinant statistic, as measures of overall forecasting performance.     

On double hysteretic heteroskedastic model

ABSTRACT

This paper proposes a hysteretic autoregressive model with GARCH specification and a skew Student's t-error distribution for financial time series. With an integrated hysteresis zone, this model allows both the conditional mean and conditional volatility switching in a regime to be delayed when the hysteresis variable lies in a hysteresis zone. We perform Bayesian estimation via an adaptive Markov Chain Monte Carlo sampling scheme. The proposed Bayesian method allows simultaneous inferences for all unknown parameters, including threshold values and a delay parameter. To implement model selection, we propose a numerical approximation of the marginal likelihoods to posterior odds. The proposed methodology is illustrated using simulation studies and two major Asia stock basis series. We conduct a model comparison for variant hysteresis and threshold GARCH models based on the posterior odds ratios, finding strong evidence of the hysteretic effect and some asymmetric heavy-tailness. Versus multi-regime threshold GARCH models, this new collection of models is more suitable to describe real data sets. Finally, we employ Bayesian forecasting methods in a Value-at-Risk study of the return series.     

On bivariate transformation of scale distributions

ABSTRACT

Elsewhere, I have promoted (univariate continuous) “transformation of scale” (ToS) distributions having densities of the form 2g(Π^?1(x)) where g is a symmetric distribution and Π is a transformation function with a special property. Here, I develop bivariate (readily multivariate) ToS distributions. Univariate ToS distributions have a transformation of random variable relationship with Azzalini-type skew-symmetric distributions; the bivariate ToS distribution here arises from marginal variable transformation of a particular form of bivariate skew-symmetric distribution. Examples are given, as are basic properties—unimodality, a covariance property, random variate generation—and connections with a bivariate inverse Gaussian distribution are pointed out.