It is obvious that forecasts are of great importance and widely used in economics and finance. Quite simply, good forecasts lead to good decisions. The importance of forecast evaluation and combination techniques follows immediately-- forecast users naturally have a keen interest in monitoring and improving forecast performance. More generally, forecast evaluation figures prominently in many questions in empirical economics and finance, such as: Are expectations rational? (e.g., Keane and Runkle, 1990; Bonham and Cohen, 1995) Are financial markets efficient? (e.g., Fama, 1970, 1991) Do macroeconomic shocks cause agents to revise their forecasts at all horizons, or just at short- and medium-term horizons? (e.g., Campbell and Mankiw, 1987; Cochrane, 1988) Are observed asset returns "too volatile"? (e.g., Shiller, 1979; LeRoy and Porter, 1981) Are asset returns forecastable over long horizons? (e.g., Fama and French, 1988; Mark, 1995)

1 The decomposition of real GDP into trend and cycle remains a problem of considerable practical importance, but two widely used methods yield starkly different results. The unobserved component approach, introduced by Harvey (1985) and Clark (1987), implies a very smooth trend with a cycle that is large in amplitude and highly persistent. In contrast, the approach of Beveridge and Nelson (1981) implies that much of the variation in the series is attributable to variation in the trend while the cycle component is small and noisy. This conflict is apparent in Figures 1 and 2 in this paper where the two cycle components are plotted, and has been widely noted; see Watson (1986) Stock and Watson (1988) among others. It should surprise us that the unobserved component (UC) and Beveridge-Nelson (BN) methods produce very different trend-cycle decompositions since both are modelbased. Each implies an ARIMA representation. Neither imposes smoothness in the trend component a priori as does the smoother of Hodrick and Prescott (1997) or as in the polar case of a linear trend that forces all variation, save constant growth, into the cycle. The UC and BN both "let the data speak for itself " in this regard. While it is often stated

: We consider the forecasting of cointegrated variables, and we show that at long horizons nothing is lost by ignoring cointegration when forecasts are evaluated using standard multivariate forecast accuracy measures. In fact, simple univariate Box-Jenkins forecasts are just as accurate. Our results highlight a potentially important deficiency of standard forecast accuracy measures---they fail to value the maintenance of cointegrating relationships among variables--- and we suggest alternatives that explicitly do so. KEY WORDS: Prediction, Loss Function, Integration, Unit Root - 2 - 1. INTRODUCTION Cointegration implies restrictions on the low-frequency dynamic behavior of multivariate time series. Thus, imposition of cointegrating restrictions has immediate implications for the behavior of long-horizon forecasts, and it is widely believed that imposition of cointegrating restrictions, when they are in fact true, will produce superior long-horizon forecasts. Stock (1995, p. 1), for ...

Evaluation of forecast optimality in economics and finance has almost exclusively been conducted under the assumption of mean squared error loss. Under this loss function optimal forecasts should be unbiased and forecast errors serially uncorrelated at the single period horizon with increasing variance as the forecast horizon grows. Using analytical results we show that standard properties of optimal forecasts can be invalid under asymmetric loss and nonlinear data generating processes and thus may be very misleading as a benchmark for an optimal forecast. We establish instead that a suitable transformation of the forecast error- known as the generalized forecast error- possesses an equivalent set of properties. The paper also provides empirical examples to illustrate the significance in practice of asymmetric loss and nonlinearities and discusses the effect of parameter estimation error on optimal forecasts.

In the vehicle routing problem with time windows a number of identical vehicles must be routed to and from a depot to cover a given set of customers, each of whom has a specified time interval indicating when they are available for service. Each customer also has a known demand, and a vehicle may only serve the customers on a route if the total demand does not exceed the capacity of the vehicle. The most effective solution method proposed to date for this problem is due to Kohl, Desrosiers, Madsen, Solomon, and Soumis. Their algorithm uses a cutting-plane approach followed by a branchand -bound search with column generation, where the columns of the LP relaxation represent routes of individual vehicles. We describe a new implementation of their method, using Karger's randomized minimum-cut algorithm to generate cutting planes. The standard benchmark in this area is a set of 87 problem instances generated in 1984 by M. Solomon; making using of parallel processing in both the cutting-pla...

In modeling series with leading or lagging indicators, it is desirable to begin comparing models in terms of time distance. This paper formalizes the concept of time distance in terms of various metrics, and investigates the behaviors of these metrics. It is shown that under some circumstances, time distance metrics indeed perform better in forecasting than standard measures (such as mean squared forecasting errors), and that some time-distance metrics outperform others. Department of Economics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0508, USA, email:cgranger@ucsd.edu. This researchwas partially funded by NSF grant SBR-9708615. y Center for Basic Research in the Social Sciences, Harvard University, 34 Kirkland Street, Cambridge, MA 02138,USA, email:yjeon@latte.harvard.edu. 1 1 Introduction If one has a pair of time series, x t and y t , their "nearness" is usually measured in terms of their vertical difference z t = x t ; y t , using the abso...

this paper we treat the prediction problem under general loss structures, building on the classic work of Granger (1969). In Section 2, we characterize the optimal predictor for nonGaussian processes under asymmetric loss. The results apply, for example, to important classes of conditionally heteroskedastic processes. In Section 3, we provide analytic solutions for the optimal predictor under two popular analytically-tractable asymmetric loss functions. In Section 4, we provide methods for approximating the optimal predictor under more general loss functions. We conclude in Section 5. 2. OPTIMAL PREDICTION FOR NON-GAUSSIAN PROCESSES Granger (1969) studies Gaussian processes and shows that under asymmetric loss the optimal predictor is the conditional mean plus a constant bias term. Granger's fundamental result, however, has two key limitations. First, the Gaussian assumption implies a constant L(y

Theory predicts that life cycle saving mechanisms will cause real exchange rate variations as the age structure varies. We investigate the impact of demography on the Swedish real exchange rate, measured as the real TCW index, during 1960 to 2000. Time series regressions show that the Swedish demographic structure has significant explanatory power on the real exchange rate. A model using age shares alone as regressors is used for medium term out-ofsample forecasts, outperforming both a naive forecast and forecasts based on an autoregressive model. Finally we use the estimated model in order to make forecasts of the Swedish real exchange rate up to 2015. The model predicts that the Swedish age structure will have a depreciating effect on the real exchange rate up to 2007 followed by an appreciating effect in the end of the forecasting period.

We develop an unobserved components approach to study surveys of forecasts containing multiple forecast horizons. Under the assumption that forecasters optimally update their beliefs about past, current and future state variables as new information arrives, we use our model to extract information on the degree of predictability of the state variable and the importance of measurement errors in the observables. Empirical estimates of the model are obtained using survey forecasts of annual GDP growth and in‡ation in the US with forecast horizons ranging from 1 to 24 months, and the model is found to closely match the joint realization of forecast errors at di¤erent horizons. Our empirical results suggest that professional forecasters face severe measurement error problems for GDP growth in real time, while this is much less of a problem for in‡ation. Moreover, in-‡ation exhibits greater persistence, and thus is predictable at longer horizons, than GDP growth and the persistent component of both variables is well-approximated by a low-order autoregressive speci…cation. Keywords: Fixed-event forecasts, multiple forecast horizons, Kalman …ltering, survey data. *We thank the editor, Serena Ng, an associate editor and two anonymous referees for constructive comments.

As a unified discipline, econometrics is still relatively young and has been transforming and expanding very rapidly over the past few decades. Major advances have taken place in the analysis of cross sectional data by means of semi-parametric and non-parametric techniques. Heterogeneity of economic relations across individuals, firms and industries is increasingly acknowledged and attempts have been made to take them into account either by integrating out their effects or by modeling the sources of heterogeneity when suitable panel data exists. The counterfactual considerations that underlie policy analysis and treat-ment evaluation have been given a more satisfactory foundation. New time series econometric techniques have been developed and employed extensively in the areas of macroeconometrics and finance. Non-linear econometric techniques are used increasingly in the analysis of cross section and time series observations. Applications of Bayesian techniques to econometric problems have been given new impetus largely thanks to advances in computer power and computational techniques. The use of Bayesian techniques have in turn provided the investigators with a unifying framework where the tasks of forecasting, decision making, model evaluation and learning can be considered as parts of the same interactive and iterative process; thus paving the way for establishing the foundation of “real time econometrics”. This paper attempts to provide an overview of some of these developments.