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.     

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.     

A BVAR model for the connecticut economy

A Bayesian vector autoregressive (BVAR) model is developed for the Connecticut economy to forecast the unemployment rate, nonagricultural employment, real personal income, and housing permits authorized. The model includes both national and state variables. The Bayesian prior is selected on the basis of the accuracy of the out-of-sample forecasts. We find that a loose prior generally produces more accurate forecasts. The out-of-sample accuracy of the BVAR forecasts is also compared with that of forecasts from an unrestricted VAR model and of benchmark forecasts generated from univariate ARIMA models. The BVAR model generally produces the most accurate short- and long-term out-of-sample forecasts for 1988 through 1992. It also correctly predicts the direction of change.     

Forecasting Mixed‐Frequency Time Series with ECM‐MIDAS Models

This paper proposes a mixed‐frequency error correction model for possibly cointegrated non‐stationary time series sampled at different frequencies. We highlight the impact, in terms of model specification, of the choice of the particular high‐frequency explanatory variable to be included in the cointegrating relationship, which we call a dynamic mixed‐frequency cointegrating relationship. The forecasting performance of aggregated models and several mixed‐frequency regressions are compared in a set of Monte Carlo experiments. In particular, we look at both the unrestricted mixed‐frequency model and at a more parsimonious MIDAS regression. Whereas the existing literature has only investigated the potential improvements of the MIDAS framework for stationary time series, our study emphasizes the need to include the relevant cointegrating vectors in the non‐stationary case. Furthermore, it is illustrated that the choice of dynamic mixed‐frequency cointegrating relationship does not matter as long as the short‐run dynamics are adapted accordingly. Finally, the unrestricted model is shown to suffer from parameter proliferation for samples of relatively small size, whereas MIDAS forecasts are robust to over‐parameterization. We illustrate our results for the US inflation rate. Copyright © 2014 John Wiley & Sons, Ltd.     

An outlier robust GARCH model and forecasting volatility of exchange rate returns

Since volatility is perceived as an explicit measure of risk, financial economists have long been concerned with accurate measures and forecasts of future volatility and, undoubtedly, the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model has been widely used for doing so. It appears, however, from some empirical studies that the GARCH model tends to provide poor volatility forecasts in the presence of additive outliers. To overcome the forecasting limitation, this paper proposes a robust GARCH model (RGARCH) using least absolute deviation estimation and introduces a valuable estimation method from a practical point of view. Extensive Monte Carlo experiments substantiate our conjectures. As the magnitude of the outliers increases, the one‐step‐ahead forecasting performance of the RGARCH model has a more significant improvement in two forecast evaluation criteria over both the standard GARCH and random walk models. Strong evidence in favour of the RGARCH model over other competitive models is based on empirical application. By using a sample of two daily exchange rate series, we find that the out‐of‐sample volatility forecasts of the RGARCH model are apparently superior to those of other competitive models. Copyright © 2002 John Wiley & Sons, Ltd.     

Co-integration,error correction and improved medium-term regional VAR forecasting

This study investigates possible improvements in medium-term VAR forecasting of state retail sales and personal income when the two series are co-integrated and represent an error-correction system. For each of North Carolina and New York, three regional vector autoregression (VAR) models are specified; an unrestricted two-equation model consisting of the two state variables, a five-equation unrestricted model with three national variables added and a Bayesian (BVAR) version of the second model. For each state, the co-integration and error-correction relationship of the two state variables is verified and an error-correction version of each model specified. Twelve successive ex ante five-year forecasts are then generated for each of the state models. The results show that including an error-correction mechanism when statistically significant improves medium-term forecasting accuracy in every case.     

Model uncertainty and forecast accuracy

In time-series analysis, a model is rarely pre-specified but rather is typically formulated in an iterative, interactive way using the given time-series data. Unfortunately the properties of the fitted model, and the forecasts from it, are generally calculated as if the model were known in the first place. This is theoretically incorrect, as least squares theory, for example, does not apply when the same data are used to formulates and fit a model. Ignoring prior model selection leads to biases, not only in estimates of model parameters but also in the subsequent construction of prediction intervals. The latter are typically too narrow, partly because they do not allow for model uncertainty. Empirical results also suggest that more complicated models tend to give a better fit but poorer ex-ante forecasts. The reasons behind these phenomena are reviewed. When comparing different forecasting models, the BIC is preferred to the AIC for identifying a model on the basis of within-sample fit, but out-of-sample forecasting accuracy provides the real test. Alternative approaches to forecasting, which avoid conditioning on a single model, include Bayesian model averaging and using a forecasting method which is not model-based but which is designed to be adaptable and robust.     

Forecasting cointegrated series with BVAR models

In this paper we examine how BVARs can be used for forecasting cointegrated variables. We propose an approach based on a Bayesian ECM model in which, contrary to the previous literature, the factor loadings are given informative priors. This procedure, applied to Italian macroeconomic series, produces more satisfactory forecasts than different prior specifications or parameterizations. Providing an informative prior on the factor loadings is a crucial point: a flat prior on the ECM terms combined with an informative prior on the lagged endogenous variables coefficients gives too much importance to the long‐run properties with respect to the short‐run dynamics. Copyright © 1999 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.     

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.     

The Effect of Nonlinearity between Credit Conditions and Economic Activity on Density Forecasts

This paper examines the effect of nonlinearities on density forecasting. It focuses on the relationship between credit markets and the rest of the economy. The possible nonlinearity of this relationship is captured by a threshold vector autoregressive model estimated on US data using Bayesian methods. Density forecasts thus account for the uncertainty in all model parameters and possible future regime changes. It is shown that considering nonlinearity can improve the probabilistic assessment of the economic outlook. Moreover, three illustrative examples are discussed to shed some light on the possible practical applicability of density forecasts derived from non‐linear models. Copyright © 2015 John Wiley & Sons, Ltd.     

Forecasting an aggregate of cointegrated disaggregates

This study examines the problem of forecasting an aggregate of cointegrated disaggregates. It first establishes conditions under which forecasts of an aggregate variable obtained from a disaggregate VECM will be equal to those from an aggregate, univariate time series model, and develops a simple procedure for testing those conditions. The paper then uses Monte Carlo simulations to show, for a finite sample, that the proposed test has good size and power properties and that whether a model satisfies the aggregation conditions is closely related to out‐of‐sample forecast performance. The paper then shows that ignoring cointegration and specifying the disaggregate model as a VAR in differences can significantly affect analyses of aggregation, with the VAR‐based test for aggregation possibly leading to faulty inference and the differenced VAR forecasts potentially understating the benefits of disaggregate information. Finally, analysis of an empirical problem confirms the basic results. Copyright © 2000 John Wiley & Sons, Ltd.     

Forecasting nonlinear time series with feed‐forward neural networks: a case study of Canadian lynx data

The forecasting capabilities of feed‐forward neural network (FFNN) models are compared to those of other competing time series models by carrying out forecasting experiments. As demonstrated by the detailed forecasting results for the Canadian lynx data set, FFNN models perform very well, especially when the series contains nonlinear and non‐Gaussian characteristics. To compare the forecasting accuracy of a FFNN model with an alternative model, Pitman's test is employed to ascertain if one model forecasts significantly better than another when generating one‐step‐ahead forecasts. Moreover, the residual‐fit spread plot is utilized in a novel fashion in this paper to compare visually out‐of‐sample forecasts of two alternative forecasting models. Finally, forecasting findings on the lynx data are used to explain under what conditions one would expect FFNN models to furnish reliable and accurate forecasts. Copyright © 2005 John Wiley & Sons, Ltd.     

Forecasting the Term Structure of Interest Rates Using Integrated Nested Laplace Approximations

This article discusses the use of Bayesian methods for inference and forecasting in dynamic term structure models through integrated nested Laplace approximations (INLA). This method of analytical approximation allows accurate inferences for latent factors, parameters and forecasts in dynamic models with reduced computational cost. In the estimation of dynamic term structure models it also avoids some simplifications in the inference procedures, such as the inefficient two‐step ordinary least squares (OLS) estimation. The results obtained in the estimation of the dynamic Nelson–Siegel model indicate that this method performs more accurate out‐of‐sample forecasts compared to the methods of two‐stage estimation by OLS and also Bayesian estimation methods using Markov chain Monte Carlo (MCMC). These analytical approaches also allow efficient calculation of measures of model selection such as generalized cross‐validation and marginal likelihood, which may be computationally prohibitive in MCMC estimations. Copyright © 2014 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.     

The Effect of Estimating Parameters on Long‐Term Forecasts for Cointegrated Systems

This paper concerns Long‐term forecasts for cointegrated processes. First, it considers the case where the parameters of the model are known. The paper analytically shows that neither cointegration nor integration constraint matters in Long‐term forecasts. It is an alternative implication of Long‐term forecasts for cointegrated processes, extending the results of previous influential studies. The appropriate Mote Carlo experiment supports our analytical result. Secondly, and more importantly, it considers the case where the parameters of the model are estimated. The paper shows that accuracy of the estimation of the drift term is crucial in Long‐term forecasts. Namely, the relative accuracy of various Long‐term forecasts depends upon the relative magnitude of variances of estimators of the drift term. It further experimentally shows that in finite samples the univariate ARIMA forecast, whose drift term is estimated by the simple time average of differenced data, is better than the cointegrated system forecast, whose parameters are estimated by the well‐known Johansen's ML method. Based upon finite sample experiments, it recommends the univariate ARIMA forecast rather than the conventional cointegrated system forecast in finite samples for its practical usefulness and robustness against model misspecifications. Copyright © 2011 John Wiley & Sons, Ltd.     

Multiple prediction intervals for time series: Comparison of simultaneous and marginal intervals

Simultaneous prediction intervals for forecasts from time series models that contain L (L ≤ 1) unknown future observations with a specified probability are derived. Our simultaneous intervals are based on two types of probability inequalities, i.e. the Bonferroni- and product-types. These differ from the marginal intervals in that they take into account the correlation structure between the forecast errors. For the forecasting methods commonly used with seasonal time series data, we show how to construct forecast error correlations and evaluate, using an example, the simultaneous and marginal prediction intervals. For all the methods, the simultaneous intervals are accurate with the accuracy increasing with the use of higher-order probability inequalities, whereas the marginal intervals are far too short in every case. Also, when L is greater than the seasonal period, the simultaneous intervals based on improved probability inequalities will be most accurate.     

An integrated bayesian vector auto regression and error correction model for forecasting electricity consumption and prices

The analysis and forecasting of electricity consumption and prices has received considerable attention over the past forty years. In the 1950s and 1960s most of these forecasts and analyses were generated by simultaneous equation econometric models. Beginning in the 1970s, there was a shift in the modeling of economic variables from the structural equations approach with strong identifying restrictions towards a joint time-series model with very few restrictions. One such model is the vector auto regression (VAR) model. It was soon discovered that the unrestricted VAR models do not forecast well. The Bayesian vector auto regression (BVAR) approach as well the error correction model (ECM) and models based on the theory of co integration have been offered as alternatives to the simple VAR model. This paper argues that the BVAF., ECM, and co integration models are simply VAR models with various restrictions placed on the coefficients. Based on this notion of a restricted VAR model, a four-step procedure for specifying VAR forecasting models is presented and then applied to monthly data on US electricity consumption and prices.     

The Effects of Disaggregation on Forecasting Nonstationary Time Series

This paper focuses on the effects of disaggregation on forecast accuracy for nonstationary time series using dynamic factor models. We compare the forecasts obtained directly from the aggregated series based on its univariate model with the aggregation of the forecasts obtained for each component of the aggregate. Within this framework (first obtain the forecasts for the component series and then aggregate the forecasts), we try two different approaches: (i) generate forecasts from the multivariate dynamic factor model and (ii) generate the forecasts from univariate models for each component of the aggregate. In this regard, we provide analytical conditions for the equality of forecasts. The results are applied to quarterly gross domestic product (GDP) data of several European countries of the euro area and to their aggregated GDP. This will be compared to the prediction obtained directly from modeling and forecasting the aggregate GDP of these European countries. In particular, we would like to check whether long‐run relationships between the levels of the components are useful for improving the forecasting accuracy of the aggregate growth rate. We will make forecasts at the country level and then pool them to obtain the forecast of the aggregate. The empirical analysis suggests that forecasts built by aggregating the country‐specific models are more accurate than forecasts constructed using the aggregated data. Copyright © 2014 John Wiley & Sons, Ltd.     

Why do regime‐switching models forecast so badly?

Most non‐linear techniques give good in‐sample fits to exchange rate data but are usually outperformed by random walks or random walks with drift when used for out‐of‐sample forecasting. In the case of regime‐switching models it is possible to understand why forecasts based on the true model can have higher mean squared error than those of a random walk or random walk with drift. In this paper we provide some analytical results for the case of a simple switching model, the segmented trend model. It requires only a small misclassification, when forecasting which regime the world will be in, to lose any advantage from knowing the correct model specification. To illustrate this we discuss some results for the DM/dollar exchange rate. We conjecture that the forecasting result is more general and describes limitations to the use of switching models for forecasting. This result has two implications. First, it questions the leading role of the random walk hypothesis for the spot exchange rate. Second, it suggests that the mean square error is not an appropriate way to evaluate forecast performance for non‐linear models. Copyright © 1999 John Wiley & Sons, Ltd.