Forecasting time series with long memory and level shifts

It is well known that some economic time series can be described by models which allow for either long memory or for occasional level shifts. In this paper we propose to examine the relative merits of these models by introducing a new model, which jointly captures the two features. We discuss representation and estimation. Using simulations, we demonstrate its forecasting ability, relative to the one‐feature models, both in terms of point forecasts and interval forecasts. We illustrate the model for daily S&P500 volatility. Copyright © 2005 John Wiley & Sons, Ltd.     

On seasonal error correction when the processes include different numbers of unit roots

We propose a seasonal cointegration model (SECM) for quarterly data which includes variables with different numbers of unit roots and thus needs to be transformed in different ways in order to yield stationarity. A Monte Carlo simulation is carried out to investigate the consequences of specifying a SECM with all variables in annual diffrerences in this situation. The SECM in annual differences is compared to the correctly specified model. Pre‐testing for unit roots using two different approaches, and where the models are specified according to the unit root test results, is also considered. The forecast mean squared error criterion and certain parameter estimation results indicate that, in practice, a cointegration model where all variables are transformed with the annual difference filter is more robust than one obtained by pre‐testing for a smaller number of unit roots. The second‐best choice when the true model is not known and when the aim is to forecast, is an ordinary VAR model also in annual differences. Copyright © 2003 John Wiley & Sons, Ltd.     

A hybrid model considering cointegration for interval-valued pork price forecasting in China

Compared with point forecasting, interval forecasting is believed to be more effective and helpful in decision making, as it provides more information about the data generation process. Based on the well-established “linear and nonlinear” modeling framework, a hybrid model is proposed by coupling the vector error correction model (VECM) with artificial intelligence models which consider the cointegration relationship between the lower and upper bounds (Coin-AIs). VECM is first employed to fit the original time series with the residual error series modeled by Coin-AIs. Using pork price as a research sample, the empirical results statistically confirm the superiority of the proposed VECM-CoinAIs over other competing models, which include six single models and six hybrid models. This result suggests that considering the cointegration relationship is a workable direction for improving the forecast performance of the interval-valued time series. Moreover, with a reasonable data transformation process, interval forecasting is proven to be more accurate than point forecasting.     

Term premia and the maturity composition of the Federal debt: new evidence from the term structure of interest rates

This paper models bond term premia empirically in terms of the maturity composition of the federal debt and other observable economic variables in a time‐varying framework with potential regime shifts. We present regression and out‐of sample forecasting results demonstrating that information on the age composition of the Federal debt is useful for forecasting term premia. We show that the multiprocess mixture model, a multi‐state time‐varying parameter model, outperforms the commonly used GARCH model in out‐of‐sample forecasts of term premia. The results underscore the importance of modelling term premia, as a function of economic variables rather than just as a function of asset covariances as in the conditional heteroscedasticity models. Copyright © 2001 John Wiley & Sons, Ltd.     

Forecasting exchange rates using cointegration models and intra‐day data

We present a cointegration analysis on the triangle (USD–DEM, USD–JPY, DEM–JPY) of foreign exchange rates using intra‐day data. A vector autoregressive model is estimated and evaluated in terms of out‐of‐sample forecast accuracy measures. Its economic value is measured on the basis of trading strategies that account for transaction costs. We show that the typical seasonal volatility in high‐frequency data can be accounted for by transforming the underlying time scale. Results are presented for the original and the modified time scales. We find that utilizing the cointegration relation among the exchange rates and the time scale transformation improves forecasting results. Copyright © 2002 John Wiley & Sons, Ltd.     

Can cointegration‐based forecasting outperform univariate models? An application to Asian exchange rates

Conventional wisdom holds that restrictions on low‐frequency dynamics among cointegrated variables should provide more accurate short‐ to medium‐term forecasts than univariate techniques that contain no such information; even though, on standard accuracy measures, the information may not improve long‐term forecasting. But inconclusive empirical evidence is complicated by confusion about an appropriate accuracy criterion and the role of integration and cointegration in forecasting accuracy. We evaluate the short‐ and medium‐term forecasting accuracy of univariate Box–Jenkins type ARIMA techniques that imply only integration against multivariate cointegration models that contain both integration and cointegration for a system of five cointegrated Asian exchange rate time series. We use a rolling‐window technique to make multiple out of sample forecasts from one to forty steps ahead. Relative forecasting accuracy for individual exchange rates appears to be sensitive to the behaviour of the exchange rate series and the forecast horizon length. Over short horizons, ARIMA model forecasts are more accurate for series with moving‐average terms of order >1. ECMs perform better over medium‐term time horizons for series with no moving average terms. The results suggest a need to distinguish between ‘sequential’ and ‘synchronous’ forecasting ability in such comparisons. Copyright © 2002 John Wiley & Sons, Ltd.     

Causal implications: The linkage between time series and econometric modelling

Construction of causally and structurally adequate simultaneous equations models can be accomplished by determining causal relations between potential variables and balancing these statistically derived inferences with economic theory to relate behavioural or technological forces among the variables. An appropriate lag structure for each of the equations can be determined by a two step multiple transfer function approach involving reduced form equations. Testing the specification of already existing simultaneous equations models is done by constructing multiple transfer function models of the reduced form equations of the simultaneous equations models which permit incorporation of lead cross correlations.     

Forecasting time series with increasing seasonal variation

Four options for modeling and forecasting time series data containing increasing seasonal variation are discussed, including data transformations, double seasonal difference models and two kinds of transfer function-type ARIMA models employing seasonal dummy variables. An explanation is given for the typical ARIMA model identification analysis failing to identify double seasonal difference models for this kind of data. A logical process of selecting one option for a particular case is outlined, focusing on issues of linear versus non-linear increasing seasonal variation, and the level of stochastic versus deterministic behavior in a time series. Example models for the various options are presented for six time series, with point forecast and interval forecast comparisons. Interval forecasts from data-transformation models are found to generally be too wide and sometimes illogical in the dependence of their width on the point forecast level. Suspicion that maximum likelihood estimation of ARIMA models leads to excessive indications of unit roots in seasonal moving-average operators is reported.     

Forecasting in large cointegrated processes

It is widely recognized that taking cointegration relationships into consideration is useful in forecasting cointegrated processes. However, there are a few practical problems when forecasting large cointegrated processes using the well‐known vector error correction model. First, it is hard to identify the cointegration rank in large models. Second, since the number of parameters to be estimated tends to be large relative to the sample size in large models, estimators will have large standard errors, and so will forecasts. The purpose of the present paper is to propose a new procedure for forecasting large cointegrated processes which is free from the above problems. In our Monte Carlo experiment, we find that our forecast gains accuracy when we work with a larger model as long as the ratio of the cointegration rank to the number of variables in the process is high. Copyright © 2009 John Wiley & Sons, Ltd.     

Do Long‐Run Theory Restrictions Help in Forecasting?

Do long‐run equilibrium relations suggested by economic theory help to improve the forecasting performance of a cointegrated vector error correction model (VECM)? In this paper we try to answer this question in the context of a two‐country model developed for the Canadian and US economies. We compare the forecasting performance of the exactly identified cointegrated VECMs to the performance of the over‐identified VECMs with the long‐run theory restrictions imposed. We allow for model uncertainty and conduct this comparison for every possible combination of the cointegration ranks of the Canadian and US models. We show that the over‐identified structural cointegrated models generally outperform the exactly identified models in forecasting Canadian macroeconomic variables. We also show that the pooled forecasts generated from the over‐identified models beat most of the individual exactly identified and over‐identified models as well as the VARs in levels and in differences. Copyright © 2011 John Wiley & Sons, Ltd.     

Forecasting euro area variables with German pre‐EMU data

It is investigated whether euro area variables can be forecast better based on synthetic time series for the pre‐euro period or by using just data from Germany for the pre‐euro period. Our forecast comparison is based on quarterly data for the period 1970Q1–2003Q4 for 10 macroeconomic variables. The years 2000–2003 are used as forecasting period. A range of different univariate forecasting methods is applied. Some of them are based on linear autoregressive models and we also use some nonlinear or time‐varying coefficient models. It turns out that most variables which have a similar level for Germany and the euro area such as prices can be better predicted based on German data, while aggregated European data are preferable for forecasting variables which need considerable adjustments in their levels when joining German and European Monetary Union (EMU) data. These results suggest that for variables which have a similar level for Germany and the euro area it may be reasonable to consider the German pre‐EMU data for studying economic problems in the euro area. Copyright © 2008 John Wiley & Sons, Ltd.     

Econometric modelling for short‐term inflation forecasting in the euro area

This paper examines the problem of forecasting macro‐variables which are observed monthly (or quarterly) and result from geographical and sectorial aggregation. The aim is to formulate a methodology whereby all relevant information gathered in this context could provide more accurate forecasts, be frequently updated, and include a disaggregated explanation as useful information for decision‐making. The appropriate treatment of the resulting disaggregated data set requires vector modelling, which captures the long‐run restrictions between the different time series and the short‐term correlations existing between their stationary transformations. Frequently, due to a lack of degrees of freedom, the vector model must be restricted to a block‐diagonal vector model. This methodology is applied in this paper to inflation in the euro area, and shows that disaggregated models with cointegration restrictions improve accuracy in forecasting aggregate macro‐variables. Copyright © 2007 John Wiley & Sons, Ltd.     

Forecasting Stock Returns: Do Commodity Prices Help?

This paper examines the relationship between stock prices and commodity prices and whether this can be used to forecast stock returns. As both prices are linked to expected future economic performance they should exhibit a long‐run relationship. Moreover, changes in sentiment towards commodity investing may affect the nature of the response to disequilibrium. Results support cointegration between stock and commodity prices, while Bai–Perron tests identify breaks in the forecast regression. Forecasts are computed using a standard fixed (static) in‐sample/out‐of‐sample approach and by both recursive and rolling regressions, which incorporate the effects of changing forecast parameter values. A range of model specifications and forecast metrics are used. The historical mean model outperforms the forecast models in both the static and recursive approaches. However, in the rolling forecasts, those models that incorporate information from the long‐run stock price/commodity price relationship outperform both the historical mean and other forecast models. Of note, the historical mean still performs relatively well compared to standard forecast models that include the dividend yield and short‐term interest rates but not the stock/commodity price ratio. Copyright © 2014 John Wiley & Sons, Ltd.     

Investigating the relationship between gold and silver prices

This paper analyses the long-run relationship between gold and silver prices. The three main questions addressed are: the influence of a large bubble from 1979:9 to 1980:3 on the cointegration relationship, the extent to which by including error-correction terms in a non-linear way we can beat the random walk model out-of-sample, and the existence of a strong simultaneous relationship between the rates of return of gold and silver. Different efficient single-equation estimation techniques are required for each of the three questions and this is explained within a simple bivariate cointegrating system. With monthly data from 1971 to 1990, it is found that cointegration could have occurred during some periods and especially during the bubble and post-bubble periods. However, dummy variables for the intercept of the long-run relationships are needed during the full sample. For the price of gold the non-linear models perform better than the random walk in-sample and out-of-sample. In-sample non-linear models for the price of silver perform better than the random walk but this predictive capacity is lost out-of-sample, mainly due to the structural change that occurs (reduction) in the variance of the out-of-sample models. The in-sample and out-of-sample predictive capacity of the non-linear models is reduced when the variables are in logs. Clear and strong evidence is found for a simultaneous relationship between the rates of return of gold and silver. In the three type of relationships that we have analysed between the prices of gold and silver, the dependence is less out-of-sample, possibly meaning that the two markets are becoming separated. © 1998 John Wiley & Sons, Ltd.     

Intervening on time derivatives

Interventionism analyses causal influence in terms of correlations of changes under a distribution of interventions. But the correspondence between correlated changes and causal influence is not obvious. I probe its plausibility with a problem-case involving variables related as time derivative (velocity) to integral (position), such that the latter variable must change given an intervention on the former unless dependencies are introduced among the testing and controlling interventions. Under the orthodox criteria such interventions will fail to be appropriate for causal analysis. I consider various alternatives, including permitting control interventions to be chancy, restricting the available models and mitigating variation of off-path variables. None of these work. I then present a fourth suggestion which modifies the interventionist criteria in order to permit interventions which can influence other variables than just their own targets. The correspondence between correlated changes and causal influence can thereby saved when dependencies are introduced among such interventions. This modification and the required dependencies, I argue, are perfectly in line with practice and may also assist in a wider class of cases.     

Single‐season heteroscedasticity in time series

We consider seasonal time series in which one season has variance that is different from all the others. This behaviour is evident in indices of production where variability is highest for the month with the lowest level of production. We show that when one season has different variability from others there are constraints on the seasonal models that can be used; neither dummy and trigonometric models are effective in modelling this type of behaviour. We define a general model that provides an appropriate representation of single‐season heteroscedasticity and suggest a likelihood ratio test for the presence of periodic variance in one season. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

Real‐time inflation forecast combination for time‐varying coefficient models

We use real‐time macroeconomic variables and combination forecasts with both time‐varying weights and equal weights to forecast inflation in the USA. The combination forecasts compare three sets of commonly used time‐varying coefficient autoregressive models: Gaussian distributed errors, errors with stochastic volatility, and errors with moving average stochastic volatility. Both point forecasts and density forecasts suggest that models combined by equal weights do not produce worse forecasts than those with time‐varying weights. We also find that variable selection, the allowance of time‐varying lag length choice, and the stochastic volatility specification significantly improve forecast performance over standard benchmarks. Finally, when compared with the Survey of Professional Forecasters, the results of the best combination model are found to be highly competitive during the 2007/08 financial crisis.     

Forecasting Government Bond Yields with Neural Networks Considering Cointegration

This paper discusses techniques that might be helpful in predicting interest rates and tries to evaluate a new hybrid forecasting approach. Results of examining government bond yields in Germany and France reported in this study indicate that a hybrid forecasting approach which combines techniques of cointegration analysis with neural network (NN) forecasting models can produce superior results to the use of NN forecasting models alone. The findings documented in this paper could be a consequence of the fact that examining differenced data under certain conditions will lead to a loss of information and that the inclusion of the error correction term from the cointegration model can help to cope with this problem. The paper also discusses some possibly interesting directions for further research. Copyright © 2015 John Wiley & Sons, Ltd.     

The Role of Financial Variables in predicting economic activity

Previous research found that the US business cycle leads the European one by a few quarters, and can therefore be useful in predicting euro area gross domestic product (GDP). In this paper we investigate whether additional predictive power can be gained by adding selected financial variables belonging to either the USA or the euro area. We use vector autoregressions (VARs) that include the US and euro area GDPs as well as growth in the Rest of the World and selected combinations of financial variables. Out‐of‐sample root mean square forecast errors (RMSEs) evidence that adding financial variables produces a slightly smaller error in forecasting US economic activity. This weak macro‐financial linkage is even weaker in the euro area, where financial indicators do not improve short‐ and medium‐term GDP forecasts even when their timely availability relative to GDP is exploited. It can be conjectured that neither US nor European financial variables help predict euro area GDP as the US GDP has already embodied this information. However, we show that the finding that financial variables have no predictive power for future activity in the euro area relates to the unconditional nature of the RMSE metric. When forecasting ability is assessed as if in real time (i.e. conditionally on the information available at the time when forecasts are made), we find that models using financial variables would have been preferred in several episodes and in particular between 1999 and 2002. Copyright 2011 John Wiley & Sons, Ltd.