Forecasting Latent Volatility through a Markov Chain Approximation Filter

We propose a new methodology for filtering and forecasting the latent variance in a two‐factor diffusion process with jumps from a continuous‐time perspective. For this purpose we use a continuous‐time Markov chain approximation with a finite state space. Essentially, we extend Markov chain filters to processes of higher dimensions. We assess forecastability of the models under consideration by measuring forecast error of model expected realized variance, trading in variance swap contracts, producing value‐at‐risk estimates as well as examining sign forecastability. We provide empirical evidence using two sources, the S&P 500 index values and its corresponding cumulative risk‐neutral expected variance (namely the VIX index). Joint estimation reveals the market prices of equity and variance risk implicit by the two probability measures. A further simulation study shows that the proposed methodology can filter the variance of virtually any type of diffusion process (coupled with a jump process) with a non‐analytical density function. Copyright © 2015 John Wiley & Sons, Ltd.     

A New‐Keynesian DSGE model for forecasting the South African economy

This paper develops a New‐Keynesian Dynamic Stochastic General Equilibrium (NKDSGE) model for forecasting the growth rate of output, inflation, and the nominal short‐term interest rate (91 days Treasury Bill rate) for the South African economy. The model is estimated via maximum likelihood technique for quarterly data over the period of 1970:1–2000:4. Based on a recursive estimation using the Kalman filter algorithm, out‐of‐sample forecasts from the NKDSGE model are compared with forecasts generated from the classical and Bayesian variants of vector autoregression (VAR) models for the period 2001:1–2006:4. The results indicate that in terms of out‐of‐sample forecasting, the NKDSGE model outperforms both the classical and Bayesian VARs for inflation, but not for output growth and nominal short‐term interest rate. However, differences in RMSEs are not significant across the models. Copyright © 2008 John Wiley & Sons, Ltd.     

Discount weighted estimation

The parsimonious method of exponentially weighted regression (EWR) is attractive but limited in application because it depends upon just one discount factor. This paper generalizes the EWR approach to a method called discount weighted estimation (DWE) which allowed distinct model components to have different associated discount factors. The method includes EWR as a special case. The general non-limiting recurrence relationships will be useful in practice, especially when practitioners wish to specify prior information, to intervene with subjective judgement and to derive estimates and forecasts sequentially based upon limited data. Two theorems extend the important EWR limiting results of Dobbie and McKenzie to DWE. The latter permits the derivation of a large class of known processs for which DWE is optimal. The method is illustrated by two applications, one of which uses the famous international airline passenger data. This allows a comparision with the ICI MULDO system which uses a particular two discount factor forecasting method. A companion paper extends the discount methods to Bayesian forecasting, Kalman filtering and state space modelling.     

Estimation and forecasting of long-memory processes with missing values

This paper addresses the issues of maximum likelihood estimation and forecasting of a long-memory time series with missing values. A state-space representation of the underlying long-memory process is proposed. By incorporating this representation with the Kalman filter, the proposed method allows not only for an efficient estimation of an ARFIMA model but also for the estimation of future values under the presence of missing data. This procedure is illustrated through an analysis of a foreign exchange data set. An investment scheme is developed which demonstrates the usefulness of the proposed approach. © 1997 John Wiley & Sons, Ltd.     

Forecasting ability of GARCH vs Kalman filter method: evidence from daily UK time-varying beta

This paper investigates the forecasting ability of four different GARCH models and the Kalman filter method. The four GARCH models applied are the bivariate GARCH, BEKK GARCH, GARCH-GJR and the GARCH-X model. The paper also compares the forecasting ability of the non-GARCH model: the Kalman method. Forecast errors based on 20 UK company daily stock return (based on estimated time-varying beta) forecasts are employed to evaluate out-of-sample forecasting ability of both GARCH models and Kalman method. Measures of forecast errors overwhelmingly support the Kalman filter approach. Among the GARCH models the GJR model appears to provide somewhat more accurate forecasts than the other bivariate GARCH models. Copyright © 2008 John Wiley & Sons, Ltd.     

Reducing uncertainty in short-term projections: Linkage of monthly and quarterly models

This paper shows how monthly data and forecasts can be used in a systematic way to improve the predictive accuracy of a quarterly macroeconometric model. The problem is formulated as a model pooling procedure (equivalent to non-recursive Kalman filtering) where a baseline quarterly model forecast is modified through ‘add-factors’ or ‘constant adjustments’. The procedure ‘automatically’ constructs these adjustments in a covariance-minimizing fashion to reflect the revised expectation of the quarterly model's forecast errors, conditional on the monthly information set. Results obtained using Federal Reserve Board models indicate the potential for significant reduction in forecast error variance through application of these procedures.     

A unified view of statistical forecasting procedures

A large number of statistical forecasting procedures for univariate time series have been proposed in the literature. These range from simple methods, such as the exponentially weighted moving average, to more complex procedures such as Box–Jenkins ARIMA modelling and Harrison–Stevens Bayesian forecasting. This paper sets out to show the relationship between these various procedures by adopting a framework in which a time series model is viewed in terms of trend, seasonal and irregular components. The framework is then extended to cover models with explanatory variables. From the technical point of view the Kalman filter plays an important role in allowing an integrated treatment of these topics.     

Updating the forecast function of ARIMA models and the link with DLMs

This paper shows that the whole forecast function of ARIMA time series models, and not just the eventual forecast function, may be updated each time an observation is received. The paper also shows that the coefficients in the updating equations for the forecast function may be expressed in exactly the same form as the Kalman filter updating equations for canonical time series DLMs. Moreover, the adaptive factors in the updating equations are shown to be a simple function of the ARIMA model parameters. Copyright © 1999 John Wiley & Sons, Ltd.     

Dynamic harmonic regression

This paper describes in detail a flexible approach to nonstationary time series analysis based on a Dynamic Harmonic Regression (DHR) model of the Unobserved Components (UC) type, formulated within a stochastic state space setting. The model is particularly useful for adaptive seasonal adjustment, signal extraction and interpolation over gaps, as well as forecasting or backcasting. The Kalman Filter and Fixed Interval Smoothing algorithms are exploited for estimating the various components, with the Noise Variance Ratio and other hyperparameters in the stochastic state space model estimated by a novel optimization method in the frequency domain. Unlike other approaches of this general type, which normally exploit Maximum Likelihood methods, this optimization procedure is based on a cost function defined in terms of the difference between the logarithmic pseudo‐spectrum of the DHR model and the logarithmic autoregressive spectrum of the time series. The cost function not only seems to yield improved convergence characteristics when compared with the alternative ML cost function, but it also has much reduced numerical requirements. Copyright © 1999 John Wiley & Sons, Ltd.     

Structural Time-series Modelling of Monetary Aggregates: A Case Study for Eleven European Countries

Monetary aggregates for eleven European countries are analysed using the structural time-series methodology, paying special attention to unit root issues. Estimation of the parameters of the models is carried out by applying the asymptotic least squares (ALS) procedure. A comparison with the maximum likelihood estimates obtained via the Kalman filter shows that ALS is an alternative to Kalman filter estimation. The empirical results show that for only a small number of series the four variance parameters of the basic structural model are strictly positive. For the majority of the series the variance of the irregular component is equal to 0.©1997 John Wiley & Sons, Ltd.     

Forecasting the Daily Time‐Varying Beta of European Banks During the Crisis Period: Comparison Between GARCH Models and the Kalman Filter

This intention of this paper is to empirically forecast the daily betas of a few European banks by means of four generalized autoregressive conditional heteroscedasticity (GARCH) models and the Kalman filter method during the pre‐global financial crisis period and the crisis period. The four GARCH models employed are BEKK GARCH, DCC GARCH, DCC‐MIDAS GARCH and Gaussian‐copula GARCH. The data consist of daily stock prices from 2001 to 2013 from two large banks each from Austria, Belgium, Greece, Holland, Ireland, Italy, Portugal and Spain. We apply the rolling forecasting method and the model confidence sets (MCS) to compare the daily forecasting ability of the five models during one month of the pre‐crisis (January 2007) and the crisis (January 2013) periods. Based on the MCS results, the BEKK proves the best model in the January 2007 period, and the Kalman filter overly outperforms the other models during the January 2013 period. Results have implications regarding the choice of model during different periods by practitioners and academics. Copyright © 2016 John Wiley & Sons, Ltd.     

Adaptive learning from model space

Dynamic model averaging (DMA) is used extensively for the purpose of economic forecasting. This study extends the framework of DMA by introducing adaptive learning from model space. In the conventional DMA framework all models are estimated independently and hence the information of the other models is left unexploited. In order to exploit the information in the estimation of the individual time‐varying parameter models, this paper proposes not only to average over the forecasts but, in addition, also to dynamically average over the time‐varying parameters. This is done by approximating the mixture of individual posteriors with a single posterior, which is then used in the upcoming period as the prior for each of the individual models. The relevance of this extension is illustrated in three empirical examples involving forecasting US inflation, US consumption expenditures, and forecasting of five major US exchange rate returns. In all applications adaptive learning from model space delivers improvements in out‐of‐sample forecasting performance.     

State-space models for predicting IBNR reserve in row-wise ordered runoff triangles: Calendar year IBNR reserves & tail effects

The issue of modeling and forecasting IBNR (incurred but not reported) actuarial reserve under Kalman filter techniques and extensions, using data arranged in a runoff triangle, is a frequent theme in the literature. One quite recent approach is to order the runoff triangle under a row-wise fashion and use linear state-space models for the resulting data set. To allow new possibilities for short-term IBNR reserves as well as to mitigate insolvency risk, in this paper we extend such a state-space method by: (i) a calendar year IBNR reserve prediction; and (ii) a tail effect for the row-wise ordered triangle. The extension is implemented with a real runoff triangle and compared with some traditional IBNR predictors. Empirical results indicate that the approach of this paper outperforms the competing methods in terms of out-of-sample comparisons and gives more conservative IBNR reserves than the original state-space method.     

The filtering of transitory noise in earnings numbers

This paper applies the Kalman filtering procedure to estimate persistent and transitory noise components of accounting earnings. Designating the transitory noise component separately (under a label such as extraordinary items) in financial reports should help users predict future earnings. If a firm has no foreknowledge of future earnings, managers can apply a filter to a firm's accounting earnings more efficiently than an interested user. If management has foreknowledge of earnings, application of a filtering algorithm can result in smoothed variables that convey information otherwise not available to users. Application of a filtering algorithm to a sample of firms revealed that a substantial number of firms exhibited a significant transitory noise component of earnings. Also, for those firms whose earnings exhibited a significant departure from the random walk process, the paper shows that filtering can be fruitfully applied to improve predictive ability.     

Long Memory of Financial Time Series and Hidden Markov Models with Time‐Varying Parameters

Hidden Markov models are often used to model daily returns and to infer the hidden state of financial markets. Previous studies have found that the estimated models change over time, but the implications of the time‐varying behavior have not been thoroughly examined. This paper presents an adaptive estimation approach that allows for the parameters of the estimated models to be time varying. It is shown that a two‐state Gaussian hidden Markov model with time‐varying parameters is able to reproduce the long memory of squared daily returns that was previously believed to be the most difficult fact to reproduce with a hidden Markov model. Capturing the time‐varying behavior of the parameters also leads to improved one‐step density forecasts. Finally, it is shown that the forecasting performance of the estimated models can be further improved using local smoothing to forecast the parameter variations. Copyright © 2016 John Wiley & Sons, Ltd.     

A large Bayesian VAR with a block-specific shrinkage: A forecasting application for Italian industrial production

This paper proposes a Bayesian vector autoregression (BVAR) model with the Kalman filter to forecast the Italian industrial production index in a pseudo real-time experiment. Minnesota priors are adopted as a general framework, but a different shrinkage pattern is imposed for both the VAR coefficients and the Kalman gain, depending on the informative contribution of each variable investigated at frequency level. Both a time-varying and a constant selection for the shrinkage are proposed. Overall, the new BVAR models significantly improve the forecasting performance in comparison with the more traditional versions based on standard Minnesota priors with a single shrinkage, equal for all the variables, and selected on the basis of some optimal criteria. Very promising results come out in terms of density forecasting as well.     

MIMO-OFDM系统中基于变分Bayes EM算法的联合符号检测与鲁棒Kalman信道跟踪

基于变分Bayes期望最大化VBEM(variational Bsayes expectation maximization)算法和Turbo原理,提出了快时变信道条件下MIMO-OFDM系统中的联合符号检测与信道估计算法.在VBEM框架下,信号检测和信道估计分别由修正的列表球形译码算法和软输入Kalman算法完成,检测器和估计器分别考虑了信道和检测信号的估计误差协方差矩阵.当信道时变剧烈时,存在较大检测误差的数据在软输入Kalman算法中引入异常值(outliers),由于Kalman算法对于异常值的敏感性,系统会在错误传播的影响下出现误码平台.为削弱异常值的影响,利用鲁棒统计理论设计了VBEM框架下改进的鲁棒软输入Kalman算法,该算法能在出现异常值的条件下保持较好的信道跟踪能力.仿真结果表明:在快速时变多径信道条件下,文中设计的鲁棒VBEM算法优于传统的VBEM算法和EM算法.     

Traditional versus novel forecasting techniques: how much do we gain?

This article applies two novel techniques to forecast the value of US manufacturing shipments over the period 1956–2000: wavelets and support vector machines (SVM). Wavelets have become increasingly popular in the fields of economics and finance in recent years, whereas SVM has emerged as a more user‐friendly alternative to artificial neural networks. These two methodologies are compared with two well‐known time series techniques: multiplicative seasonal autoregressive integrated moving average (ARIMA) and unobserved components (UC). Based on forecasting accuracy and encompassing tests, and forecasting combination, we conclude that UC and ARIMA generally outperform wavelets and SVM. However, in some cases the latter provide valuable forecasting information that it is not contained in the former. Copyright © 2008 John Wiley & Sons, Ltd.     

Forecasting inflation using univariate continuous-time stochastic models

In this paper we investigate the applicability of several continuous-time stochastic models to forecasting inflation rates with horizons out to 20 years. While the models are well known, new methods of parameter estimation and forecasts are supplied, leading to rigorous testing of out-of-sample inflation forecasting at short and long time horizons. Using US consumer price index data we find that over longer forecasting horizons—that is, those beyond 5 years—the log-normal index model having Ornstein–Uhlenbeck drift rate provides the best forecasts.     

Evaluating the role judgment plays in forecast accuracy

The majority of model-based forecasting efforts today rely on relatively simple techniques of estimation and the subjective adjustment of the model's results to produce forecasts. Published forecasts reflect to a great extent the judgment of the forecaster rather than what the model by itself has to say about the future. This paper examines the role judgment plays in the process of producing a macroeconometric forecast. The debate over the use of adjustment constants to alter the statistical results of a model is outlined and an empirical analysis of forecasts generated by the Michigan Quarterly Econometric Model of the US economy is presented using a unique data set which isolates the role of judgment in the forecasting process.