A Simple Explanation of the Forecast Combination Puzzle*

This article presents a formal explanation of the forecast combination puzzle, that simple combinations of point forecasts are repeatedly found to outperform sophisticated weighted combinations in empirical applications. The explanation lies in the effect of finite‐sample error in estimating the combining weights. A small Monte Carlo study and a reappraisal of an empirical study by Stock and Watson [Federal Reserve Bank of Richmond Economic Quarterly (2003) Vol. 89/3, pp. 71–90] support this explanation. The Monte Carlo evidence, together with a large‐sample approximation to the variance of the combining weight, also supports the popular recommendation to ignore forecast error covariances in estimating the weight.     

Selecting a Nonlinear Time Series Model using Weighted Tests of Equal Forecast Accuracy*

Nonlinear time series models have become fashionable tools to describe and forecast a variety of economic time series. A closer look at reported empirical studies, however, reveals that these models apparently fit well in‐sample, but rarely show a substantial improvement in out‐of‐sample forecasts, at least over linear models. One of the many possible reasons for this finding is the use of inappropriate model selection criteria and forecast evaluation criteria. In this paper we therefore propose a novel criterion, which we believe does more justice to the very nature of nonlinear models. Simulations show that this criterion outperforms those criteria currently in use, in the sense that the true nonlinear model is more often found to perform better in out‐of‐sample forecasting than a benchmark linear model. An empirical illustration for US GDP emphasizes its relevance.     

Evaluating a Model by Forecast Performance*

Although out‐of‐sample forecast performance is often deemed to be the ‘gold standard’ of evaluation, it is not in fact a good yardstick for evaluating models in general. The arguments are illustrated with reference to a recent paper by Carruth, Hooker and Oswald [Review of Economics and Statistics (1998) , Vol. 80, pp. 621–628], who suggest that the good dynamic forecasts of their model support the efficiency‐wage theory on which it is based.     

Prewhitening Bias in HAC Estimation*

Heteroskedasticity and autocorrelation consistent (HAC) estimation commonly involves the use of prewhitening filters based on simple autoregressive models. In such applications, small sample bias in the estimation of autoregressive coefficients is transmitted to the recolouring filter, leading to HAC variance estimates that can be badly biased. The present paper provides an analysis of these issues using asymptotic expansions and simulations. The approach we recommend involves the use of recursive demeaning procedures that mitigate the effects of small‐sample autoregressive bias. Moreover, a commonly used restriction rule on the prewhitening estimates (that first‐order autoregressive coefficient estimates, or largest eigenvalues, >0.97 be replaced by 0.97) adversely interferes with the power of unit‐root and [ Kwiatkowski, Phillips, Schmidt and Shin (1992) Journal of Econometrics, Vol. 54, pp. 159–178] (KPSS) tests. We provide a new boundary condition rule that improves the size and power properties of these tests. Some illustrations of the effects of these adjustments on the size and power of KPSS testing are given. Using prewhitened HAC estimates and the new boundary condition rule, the KPSS test is consistent, in contrast to KPSS testing that uses conventional prewhitened HAC estimates [ Lee, J. S. (1996) Economics Letters, Vol. 51, pp. 131–137].     

On the Construction of the European Economic Sentiment Indicator*

Economic sentiment surveys are carried out by all European Union member states and are often seen as early indicators for future economic developments. Based on these surveys, the European Commission constructs an aggregate European Economic Sentiment Indicator (ESI). This paper compares the ESI with more sophisticated aggregation schemes based on statistical methods: dynamic factor analysis and partial least squares. The indicator based on partial least squares clearly outperforms the other two indicators in terms of comovement with economic activity. In terms of forecast ability, the ESI, constructed in a rather ad hoc way, can compete with the other indicators.     

Incorporating Asymmetric Preferences into Fan Charts and Path Forecasts*

Ordinary fan charts consist of symmetric marginal forecast intervals, and do not take into consideration the concrete loss function of the user of the forecast. The note shows how to build fan charts that have exact joint coverage even under asymmetric loss, and maintain at the same time the intuition conveyed by ordinary fan charts. The proposed method is computationally simple, and easily implemented with any loss function. The differences between the information conveyed by fan charts with or without asymmetries, and with or without exact joint coverage, are illustrated with a Bayesian forecast exercise of US GDP growth rates.     

Forecast Pooling for European Macroeconomic Variables*

We compare alternative forecast pooling methods and 58 forecasts from linear, time‐varying and non‐linear models, using a very large dataset of about 500 macroeconomic variables for the countries in the European Monetary Union. On average, combination methods work well but single non‐linear models can outperform them for several series. The performance of pooled forecasts, and of non‐linear models, improves when focusing on a subset of unstable series, but the gains are minor. Finally, on average over the EMU countries, the pooled forecasts behave well for industrial production growth, unemployment and inflation, but they are often beaten by non‐linear models for each country and variable.     

Levels,Differences and ECMs – Principles for Improved Econometric Forecasting*

Unit‐root testing can be a preliminary step in model development, an intermediate step, or an end in itself. Some researchers have questioned the value of any unit‐root and cointegration testing, arguing that restrictions based on theory are at least as effective. Such confusion is unsatisfactory. Needed is a set of principles that limit and define the role of the tacit knowledge of the model builders. In a forecasting context, we enumerate the various possible model selection strategies and, based on simulation and empirical evidence, recommend using these tests to improve the specification of an initial general vector autoregression model.     

GMM Estimation with Non‐causal Instruments*

This note provides a warning against careless use of the generalized method of moments (GMM) with time series data. We show that if time series follow non‐causal autoregressive processes, their lags are not valid instruments, and the GMM estimator is inconsistent. Moreover, endogeneity of the instruments may not be revealed by the J‐test of overidentifying restrictions that may be inconsistent and has, in general, low finite‐sample power. Our explicit results pertain to a simple linear regression, but they can easily be generalized. Our empirical results indicate that non‐causality is quite common among economic variables, making these problems highly relevant.     

An Introduction to Best Empirical Models when the Parameter Space is Infinite Dimensional*

Ploberger and Phillips (Econometrica, Vol. 71, pp. 627–673, 2003) proved a result that provides a bound on how close a fitted empirical model can get to the true model when the model is represented by a parameterized probability measure on a finite dimensional parameter space. The present note extends that result to cases where the parameter space is infinite dimensional. The results have implications for model choice in infinite dimensional problems and highlight some of the difficulties, including technical difficulties, presented by models of infinite dimension. Some implications for forecasting are considered and some applications are given, including the empirically relevant case of vector autoregression (VAR) models of infinite order.     

Forecast Bias and MSFE Encompassing

We show that the standard condition for MSFE encompassing is no longer valid when the forecasts to be compared are biased. We propose a simple modification of such a condition and of tests for its validity. The theoretical results are illustrated by an empirical example on inflation and deficit forecasts, key variables for the formulation of monetary and fiscal policy.     

Estimation Bias and Inference in Overlapping Autoregressions: Implications for the Target‐Zone Literature*

Samples with overlapping observations are used for the study of uncovered interest rate parity, the predictability of long‐run stock returns and the credibility of exchange rate target zones. This paper quantifies the biases in parameter estimation and size distortions of hypothesis tests of overlapping linear and polynomial autoregressions, which have been used in target‐zone applications. We show that both estimation bias and size distortions of hypothesis tests are generally larger, if the amount of overlap is larger, the sample size is smaller, and autoregressive root of the data‐generating process is closer to unity. In particular, the estimates are biased in a way that makes it more likely that the predictions of the Bertola–Svensson model will be supported. Size distortions of various tests also turn out to be substantial even when using a heteroskedasticity and autocorrelation‐consistent covariance matrix.     

Forecasting a long memory process subject to structural breaks

We develop an easy-to-implement method for forecasting a stationary autoregressive fractionally integrated moving average (ARFIMA) process subject to structural breaks with unknown break dates. We show that an ARFIMA process subject to a mean shift and a change in the long memory parameter can be well approximated by an autoregressive (AR) model and suggest using an information criterion (AIC or Mallows’ _Cp$C_p$) to choose the order of the approximate AR model. Our method avoids the issue of estimation inaccuracy of the long memory parameter and the issue of spurious breaks in finite sample. Insights from our theoretical analysis are confirmed by Monte Carlo experiments, through which we also find that our method provides a substantial improvement over existing prediction methods. An empirical application to the realized volatility of three exchange rates illustrates the usefulness of our forecasting procedure. The empirical success of the HAR-RV model can be explained, from an econometric perspective, by our theoretical and simulation results.     

Selection of estimation window in the presence of breaks

In situations where a regression model is subject to one or more breaks it is shown that it can be optimal to use pre-break data to estimate the parameters of the model used to compute out-of-sample forecasts. The issue of how best to exploit the trade-off that might exist between bias and forecast error variance is explored and illustrated for the multivariate regression model under the assumption of strictly exogenous regressors. In practice when this assumption cannot be maintained and both the time and size of the breaks are unknown, the optimal choice of the observation window will be subject to further uncertainties that make exploiting the bias–variance trade-off difficult. To that end we propose a new set of cross-validation methods for selection of a single estimation window and weighting or pooling methods for combination of forecasts based on estimation windows of different lengths. Monte Carlo simulations are used to show when these procedures work well compared with methods that ignore the presence of breaks.     

Optimal forecasts in the presence of structural breaks

This paper considers the problem of forecasting under continuous and discrete structural breaks and proposes weighting observations to obtain optimal forecasts in the MSFE sense. We derive optimal weights for one step ahead forecasts. Under continuous breaks, our approach largely recovers exponential smoothing weights. Under discrete breaks, we provide analytical expressions for optimal weights in models with a single regressor, and asymptotically valid weights for models with more than one regressor. It is shown that in these cases the optimal weight is the same across observations within a given regime and differs only across regimes. In practice, where information on structural breaks is uncertain, a forecasting procedure based on robust optimal weights is proposed. The relative performance of our proposed approach is investigated using Monte Carlo experiments and an empirical application to forecasting real GDP using the yield curve across nine industrial economies.     

Convergence of Prices and Rates of Inflation*

We consider how unit‐root and stationarity tests can be used to study the convergence of prices and rates of inflation. We show how the joint use of these tests in levels and first differences allows the researcher to distinguish between series that are converging and series that have already converged, and we set out a strategy to establish whether convergence occurs in relative prices or just in rates of inflation. Special attention is paid to the issue of whether a mean should be extracted in carrying out tests in first differences and whether there is an advantage to adopting a (Dickey–Fuller) unit‐root test based on deviations from the last observation. The asymptotic distribution of this last test statistic is given and Monte Carlo simulation experiments show that the test yields considerable power gains for highly persistent autoregressive processes with ‘relatively large’ initial conditions. The tests are applied to the monthly series of the consumer price index in the Italian regional capitals over the period 1970–2003.     

Nonlinear Correlograms and Partial Autocorrelograms*

This paper proposes neural network‐based measures of predictability in conditional mean, and then uses them to construct nonlinear analogues to autocorrelograms and partial autocorrelograms. In contrast to other measures of nonlinear dependence that rely on nonparametric estimation of densities or multivariate integration, our autocorrelograms are simple to calculate and appear to work well in relatively small samples.     

Estimation and testing of Euler equation models with time-varying reduced-form coefficients

This paper studies the estimation and testing of Euler equation models in the framework of the classical two-step minimum-distance method. The time-varying reduced-form model in the first step reflects the adaptation of private agents’ beliefs to the changing economic environment. The presumed ability of Euler conditions to deliver stable parameters indexing tastes and technology is interpreted as a time-invariant second-step model. This paper shows that, complementary to and independent of one another, both standard specification test and stability test are required for the evaluation of an Euler equation. As an empirical application, a widely used investment Euler equation is submitted to examination. The empirical outcomes appear to suggest that the standard investment model has not been a success for aggregate investment.     

Estimation and model selection of semiparametric multivariate survival functions under general censorship

We study estimation and model selection of semiparametric models of multivariate survival functions for censored data, which are characterized by possibly misspecified parametric copulas and nonparametric marginal survivals. We obtain the consistency and root-n$n$ asymptotic normality of a two-step copula estimator to the pseudo-true copula parameter value according to KLIC, and provide a simple consistent estimator of its asymptotic variance, allowing for a first-step nonparametric estimation of the marginal survivals. We establish the asymptotic distribution of the penalized pseudo-likelihood ratio statistic for comparing multiple semiparametric multivariate survival functions subject to copula misspecification and general censorship. An empirical application is provided.     

New Methods for Forecasting Inflation,Applied to the US*

Models for the 12‐month‐ahead US rate of inflation, measured by the chain‐weighted consumer expenditure deflator, are estimated for 1974–98 and subsequent pseudo out‐of‐sample forecasting performance is examined. Alternative forecasting approaches for different information sets are compared with benchmark univariate autoregressive models, and substantial out‐performance is demonstrated including against Stock and Watson's unobserved components‐stochastic volatility model. Three key ingredients to the out‐performance are: including equilibrium correction component terms in relative prices; introducing nonlinearities to proxy state‐dependence in the inflation process and replacing the information criterion, commonly used in VARs to select lag length, with a ‘parsimonious longer lags’ parameterization. Forecast pooling or averaging also improves forecast performance.