Long‐memory dynamic Tobit models

We introduce a long‐memory dynamic Tobit model, defining it as a censored version of a fractionally integrated Gaussian ARMA model, which may include seasonal components and/or additional regression variables. Parameter estimation for such a model using standard techniques is typically infeasible, since the model is not Markovian, cannot be expressed in a finite‐dimensional state‐space form, and includes censored observations. Furthermore, the long‐memory property renders a standard Gibbs sampling scheme impractical. Therefore we introduce a new Markov chain Monte Carlo sampling scheme, which is orders of magnitude more efficient than the standard Gibbs sampler. The method is inherently capable of handling missing observations. In case studies, the model is fit to two time series: one consisting of volumes of requests to a hard disk over time, and the other consisting of hourly rainfall measurements in Edinburgh over a 2‐year period. The resulting posterior distributions for the fractional differencing parameter demonstrate, for these two time series, the importance of the long‐memory structure in the models. Copyright © 2006 John Wiley & Sons, Ltd.     

Long‐memory forecasting of US monetary indices

Several studies have tested for long‐range dependence in macroeconomic and financial time series but very few have assessed the usefulness of long‐memory models as forecast‐generating mechanisms. This study tests for fractional differencing in the US monetary indices (simple sum and divisia) and compares the out‐of‐sample fractional forecasts to benchmark forecasts. The long‐memory parameter is estimated using Robinson's Gaussian semi‐parametric and multivariate log‐periodogram methods. The evidence amply suggests that the monetary series possess a fractional order between one and two. Fractional out‐of‐sample forecasts are consistently more accurate (with the exception of the M3 series) than benchmark autoregressive forecasts but the forecasting gains are not generally statistically significant. In terms of forecast encompassing, the fractional model encompasses the autoregressive model for the divisia series but neither model encompasses the other for the simple sum series. Copyright © 2006 John Wiley & Sons, Ltd.     

Multi‐step forecasting for long‐memory processes

In this paper we present results of a simulation study to assess and compare the accuracy of forecasting techniques for long‐memory processes in small sample sizes. We analyse differences between adaptive ARMA(1,1) L‐step forecasts, where the parameters are estimated by minimizing the sum of squares of L‐step forecast errors, and forecasts obtained by using long‐memory models. We compare widths of the forecast intervals for both methods, and discuss some computational issues associated with the ARMA(1,1) method. Our results illustrate the importance and usefulness of long‐memory models for multi‐step forecasting. Copyright © 1999 John Wiley & Sons, Ltd.     

Optimal prediction with nonstationary ARFIMA model

We propose two methods to predict nonstationary long‐memory time series. In the first one we estimate the long‐range dependent parameter d by using tapered data; we then take the nonstationary fractional filter to obtain stationary and short‐memory time series. In the second method, we take successive differences to obtain a stationary but possibly long‐memory time series. For the two methods the forecasts are based on those obtained from the stationary components. Copyright © 2007 John Wiley & Sons, Ltd.     

Model‐based forecast adjustment: With an illustration to inflation

This paper introduces the idea of adjusting forecasts from a linear time series model where the adjustment relies on the assumption that this linear model is an approximation of a nonlinear time series model. This way of creating forecasts could be convenient when inference for a nonlinear model is impossible, complicated or unreliable in small samples. The size of the forecast adjustment can be based on the estimation results for the linear model and on other data properties such as the first few moments or autocorrelations. An illustration is given for a first‐order diagonal bilinear time series model, which in certain properties can be approximated by a linear ARMA(1, 1) model. For this case, the forecast adjustment is easy to derive, which is convenient as the particular bilinear model is indeed cumbersome to analyze in practice. An application to a range of inflation series for low‐income countries shows that such adjustment can lead to some improved forecasts, although the gain is small for this particular bilinear time series model.     

ARARMA models for time series analysis and forecasting

Methods of time series forecasting are proposed which can be applied automatically. However, they are not rote formulae, since they are based on a flexible philosophy which can provide several models for consideration. In addition it provides diverse diagnostics for qualitatively and quantitatively estimating how well one can forecast a series. The models considered are called ARARMA models (or ARAR models) because the model fitted to a long memory time series (t) is based on sophisticated time series analysis of AR (or ARMA) schemes (short memory models) fitted to residuals Y(t) obtained by parsimonious‘best lag’non-stationary autoregression. Both long range and short range forecasts are provided by an ARARMA model Section 1 explains the philosophy of our approach to time series model identification. Sections 2 and 3 attempt to relate our approach to some standard approaches to forecasting; exponential smoothing methods are developed from the point of view of prediction theory (section 2) and extended (section 3). ARARMA models are introduced (section 4). Methods of ARARMA model fitting are outlined (sections 5,6). Since‘the proof of the pudding is in the eating’, the methods proposed are illustrated (section 7) using the classic example of international airline passengers.     

Forecasting high‐frequency financial data with the ARFIMA–ARCH model

Financial data series are often described as exhibiting two non‐standard time series features. First, variance often changes over time, with alternating phases of high and low volatility. Such behaviour is well captured by ARCH models. Second, long memory may cause a slower decay of the autocorrelation function than would be implied by ARMA models. Fractionally integrated models have been offered as explanations. Recently, the ARFIMA–ARCH model class has been suggested as a way of coping with both phenomena simultaneously. For estimation we implement the bias correction of Cox and Reid ( 1987 ). For daily data on the Swiss 1‐month Euromarket interest rate during the period 1986–1989, the ARFIMA–ARCH (5,d,2/4) model with non‐integer d is selected by AIC. Model‐based out‐of‐sample forecasts for the mean are better than predictions based on conditionally homoscedastic white noise only for longer horizons (τ > 40). Regarding volatility forecasts, however, the selected ARFIMA–ARCH models dominate. Copyright © 2001 John Wiley & Sons, Ltd.     

Predicting Bid–Ask Spreads Using Long‐Memory Autoregressive Conditional Poisson Models

We introduce a long‐memory autoregressive conditional Poisson (LMACP) model to model highly persistent time series of counts. The model is applied to forecast quoted bid–ask spreads, a key parameter in stock trading operations. It is shown that the LMACP nicely captures salient features of bid–ask spreads like the strong autocorrelation and discreteness of observations. We discuss theoretical properties of LMACP models and evaluate rolling‐window forecasts of quoted bid–ask spreads for stocks traded at NYSE and NASDAQ. We show that Poisson time series models significantly outperform forecasts from AR, ARMA, ARFIMA, ACD and FIACD models. The economic significance of our results is supported by the evaluation of a trade schedule. Scheduling trades according to spread forecasts we realize cost savings of up to 14 % of spread transaction costs. Copyright © 2013 John Wiley & Sons, Ltd.     

Long‐Run and Cyclical Dynamics in the US Stock Market

This paper uses fractional integration to examine the long‐run dynamics and cyclical structure of US inflation, real risk‐free rate, real stock returns, equity premium and price/dividend ratio, annually from 1871 to 2000. It implements a procedure which allows consideration of unit roots with possibly fractional orders of integration both at zero (long‐run) and cyclical frequencies. When focusing exclusively on the former, the estimated order of integration varies considerably, and non‐stationarity is found only for the price/dividend ratio. When the cyclical component is also taken into account, the series appear to be stationary but to exhibit long memory with respect to both components in almost all cases. The exception is the price/dividend ratio, whose order of integration is higher than 0.5 but smaller than 1 for the long‐run frequency, and is between 0 and 0.5 for the cyclical component. Also, mean reversion occurs in all cases. Finally, six different criteria are applied to compare the forecasting performance of the fractional (at both zero and cyclical frequencies) models with others based on fractional and integer differentiation only at the zero frequency. The results, based on a 15‐year horizon, show that the former outperforms the others in a number of cases. Copyright © 2013 John Wiley & Sons, Ltd.     

ARFIMA approximation and forecasting of the limiting aggregate structure of long‐memory process

This article studies Man and Tiao's (2006) low‐order autoregressive fractionally integrated moving‐average (ARFIMA) approximation to Tsai and Chan's (2005b) limiting aggregate structure of the long‐memory process. In matching the autocorrelations, we demonstrate that the approximation works well, especially for larger d values. In computing autocorrelations over long lags for larger d value, using the exact formula one might encounter numerical problems. The use of the ARFIMA(0, d, d?₁) model provides a useful alternative to compute the autocorrelations as a really close approximation. In forecasting future aggregates, we demonstrate the close performance of using the ARFIMA(0, d, d?₁) model and the exact aggregate structure. In practice, this provides a justification for the use of a low‐order ARFIMA model in predicting future aggregates of long‐memory process. Copyright © 2008 John Wiley & Sons, Ltd.     

ARMA Models and the Box–Jenkins Methodology

The purpose of this paper is to apply the Box–Jenkins methodology to ARIMA models and determine the reasons why in empirical tests it is found that the post-sample forecasting the accuracy of such models is generally worse than much simpler time series methods. The paper concludes that the major problem is the way of making the series stationary in its mean (i.e. the method of differencing) that has been proposed by Box and Jenkins. If alternative approaches are utilized to remove and extrapolate the trend in the data, ARMA models outperform the models selected through Box–Jenkins methodology. In addition, it is shown that using ARMA models to seasonally adjusted data slightly improves post-sample accuracies while simplifying the use of ARMA models. It is also confirmed that transformations slightly improve post-sample forecasting accuracy, particularly for long forecasting horizons. Finally, it is demonstrated that AR(1), AR(2) and ARMA(1,1) models can produce more accurate post-sample forecasts than those found through the application of Box–Jenkins methodology.© 1997 John Wiley & Sons, Ltd.     

Is it a short‐memory,long‐memory,or permanently Granger‐causation influence?

Exploring the Granger‐causation relationship is an important and interesting topic in the field of econometrics. In the traditional model we usually apply the short‐memory style to exhibit the relationship, but in practice there could be other different influence patterns. Besides the short‐memory relationship, Chen (2006) demonstrates a long‐memory relationship, in which a useful approach is provided for estimation where the time series are not necessarily fractionally co‐integrated. In that paper two different relationships (short‐memory and long‐memory relationship) are regarded whereby the influence flow is decayed by geometric, or cutting off, or harmonic sequences. However, it limits the model to the stationary relationship. This paper extends the influence flow to a non‐stationary relationship where the limitation is on ?0.5 ≤ d ≤ 1.0 and it can be used to detect whether the influence decays off (?0.5 ≤ d < 0.5) or is permanent (0.5 ≤ d ≤ 1.0). Copyright © 2008 John Wiley & Sons, Ltd.     

Augmented Half‐Life Estimation Based on High‐Frequency Data

Half‐life estimation has been widely used to evaluate the speed of mean reversion for various economic and financial variables. However, half‐life estimation for the same variable are often different due to the length of the annual time series data used in alternative studies. To solve this issue, this paper extends the ARMA model and derives the half‐life estimation formula for high‐frequency monthly data. Our results indicate that half‐life estimation using short‐period monthly data is an effective approximation for that using long‐period annual data. Furthermore, by applying high‐frequency data, the required effective sample size can be reduced by at least 40% at the 95% confidence level. Copyright © 2015 John Wiley & Sons, Ltd.     

The Realised–Implied Volatility Relationship: Recent Empirical Evidence from FTSE‐100 Stocks

This paper examines the long‐run relationship between implied and realised volatility for a sample of 16 FTSE‐100 stocks. We find strong evidence of long‐memory, fractional integration in equity volatility and show that this long‐memory characteristic is not an outcome of structural breaks experienced during the sample period. Fractional cointegration between the implied and realised volatility is shown using recently developed rank cointegration tests by Robinson and Yajima (2002). The predictive ability of individual equity options is also examined and composite implied volatility estimates are shown to contain information on future idiosyncratic or stock‐specific risk that is not captured using popular statistical approaches. Implied volatilities on individual UK equities are thus closely related to realised volatility and are an effective forecasting method particularly over medium forecasting horizons. Copyright © 2011 John Wiley & Sons, Ltd.     

Real‐time or current vintage: does the type of data matter for forecasting and model selection?

In this paper we investigate the impact of data revisions on forecasting and model selection procedures. A linear ARMA model and nonlinear SETAR model are considered in this study. Two Canadian macroeconomic time series have been analyzed: the real‐time monetary aggregate M3 (1977–2000) and residential mortgage credit (1975–1998). The forecasting method we use is multi‐step‐ahead non‐adaptive forecasting. Copyright © 2008 John Wiley & Sons, Ltd.     

Modelling and Trading the English and German Stock Markets with Novelty Optimization Techniques

The motivation for this paper was the introduction of novel short‐term models to trade the FTSE 100 and DAX 30 exchange‐traded funds (ETF) indices. There are major contributions in this paper which include the introduction of an input selection criterion when utilizing an expansive universe of inputs, a hybrid combination of partial swarm optimizer (PSO) with radial basis function (RBF) neural networks, the application of a PSO algorithm to a traditional autoregressive moving model (ARMA), the application of a PSO algorithm to a higher‐order neural network and, finally, the introduction of a multi‐objective algorithm to optimize statistical and trading performance when trading an index. All the machine learning‐based methodologies and the conventional models are adapted and optimized to model the index. A PSO algorithm is used to optimize the weights in a traditional RBF neural network, in a higher‐order neural network (HONN) and the AR and MA terms of an ARMA model. In terms of checking the statistical and empirical accuracy of the novel models, we benchmark them with a traditional HONN, with an ARMA, with a moving average convergence/divergence model (MACD) and with a naïve strategy. More specifically, the trading and statistical performance of all models is investigated in a forecast simulation of the FTSE 100 and DAX 30 ETF time series over the period January 2004 to December 2015 using the last 3 years for out‐of‐sample testing. Finally, the empirical and statistical results indicate that the PSO‐RBF model outperforms all other examined models in terms of trading accuracy and profitability, even with mixed inputs and with only autoregressive inputs. Copyright © 2016 John Wiley & Sons, Ltd.     

Identifying the time‐effect factors of multiple time series

The Peña–Box model is considered for finding the time‐effect factors of a multiple time series. This paper first establishes the connection between the Peña–Box model and the vector ARMA model. According to the Peña–Box model, some series can be ignored while modelling the vector ARMA model. A consistent estimator is then proposed to identify the model for nonlinear and nonstationary time series. Finally, the finite‐sample behaviour of the estimator is illustrated via simulations. Copyright © 2005 John Wiley & Sons, Ltd.     

Simultaneous transfer functions versus vector arma models

This work compares two classes of multiple time series models which have been developed in past decades and are usually believed to be equivalent: the vector ARMA model and the system of simultaneous transfer functions (STF). The first part analyzes the mathematical structure of the two schemes; their properties of stability, structural identification and realization. In the second, algorithms of order identification and parameter estimation are derived, following the approach of stochastic approximation. The proposed solutions are easily implementable on standard statistical software and in an extended empirical example their performance is checked. The superiority of the STF model will be well established.     

Estimating the long memory granger causality effect with a spectrum estimator

This paper discusses the Granger causality test by a spectrum estimator which allows the transfer function to have long memory properties. In traditional methodology the relationship among variables is usually assumed to be short memory or contemporaneous. Hence, we have to make sure they are of the same integrated order, else there might be a spurious regression problem. In practice, not all the variables are fractionally co‐integrated in the economic model. They may have the same random resources, but under a different integrated order. This paper focuses on how to capture the long memory Granger causality effect in the transfer function. This does not necessarily assume the variables are of the same fractional integrated order. Moreover, by the transfer function we construct an estimator to test the long memory effect with the Granger causality sense. Copyright © 2006 John Wiley & Sons, Ltd.     

A fractionally integrated exponential model for UK unemployment

Fractionally integrated models with the disturbances following a Bloomfield ( 1973 ) exponential spectral model are proposed in this article for modelling UK unemployment. This gives us a better understanding of the low‐frequency dynamics affecting the series without relying on any particular ARMA specification for its short‐run components which, in general, require many more parameters to estimate. The results indicate that this exponential model, confounded with fractional integration, may be a feasible way of modelling unemployment. It also shows that its order of integration is much higher than one and thus leads to the conclusion that the standard practice of taking first differences may lead to erroneous results. Copyright © 2001 John Wiley & Sons, Ltd.