Francis X. Diebold*

In finance recently, there has been extensive inquiry into issues such as long-horizon mean reversion in asset returns, persistence in mutual fund performance,
volatility and correlation forecasting with applications to financial risk management, and
selection biases attributable to survival or data snooping.^{(1)} In macroeconomics, we have
seen the development and application of new coincident and leading indicators and
tracking portfolios, diffusion indexes, regime-switching models (with potentially time-varying transition probabilities), and new breeds of macroeconomic models that demand
new tools for estimation and forecasting.

The development and assessment of econometric methods for use in empirical
finance and macroeconomics, with special emphasis on problems of prediction, is very
important. That is the subject of my own research program, as well as of an NBER
working group that Kenneth D. West and I lead.^{(2)} Here I describe some aspects of that
research, ranging from general issues of forecast construction and evaluation to specific
topics such as financial asset return volatility and business cycles.

**Forecast Construction and Evaluation in Finance and Macroeconomics**

Motivated by advances in finance and macroeconomics, recent research has
produced new forecasting methods and refined existing ones.^{(3)} For example, prediction
problems involving asymmetric loss functions arise routinely in many fields, including
finance, as when nonlinear tax schedules have different effects on speculative profits
and losses.^{(4)} In recent work, I have developed methods for optimal prediction under
general loss structures, characterized the optimal predictor, provided workable methods
for computing it, and established tight links to new work on volatility forecastability,
which I discuss later.^{(5)}

In related work motivated by financial considerations, such as "convergence
trades," and macroeconomic considerations, such as long-run stability of the "great
ratios," Peter F. Christoffersen and I have considered the forecasting of co-integrated
variables. We show that at long horizons nothing is lost by ignoring co-integration when
forecasts are evaluated using standard multivariate forecast accuracy measures.^{(6)}
Ultimately, our results suggest not that co-integration is unimportant but that standard
forecast accuracy measures are deficient because they fail to value the maintenance of
co-integrating relationships among variables. We suggest alternative measures that
explicitly do this.

Forecast accuracy is obviously important because forecasts are used to guide decisions. Accuracy is also important to those who produce forecasts, because reputations and fortunes rise and fall with their accuracy. Comparisons of forecast accuracy are also important more generally to economists, as they must discriminate among competing economic hypotheses. Predictive performance and model adequacy are inextricably linked: predictive failure implies model inadequacy.

The evaluation of forecast accuracy is particularly common in finance and
macroeconomics. In finance, one often needs to assess the validity of claims that a
certain model can predict returns relative to a benchmark, such as a martingale. This is
a question of point forecasting, and much has been written about the evaluation and
combination of point forecasts.^{(7)} In particular, Roberto S. Mariano and I have developed
formal methods for testing the null hypothesis: that there is no difference in the accuracy
of two competing forecasts.^{(8)} A wide variety of accuracy measures can be used (in
particular, the loss function need not be quadratic, nor even symmetric), and forecast
errors can be non-Gaussian, non-zero mean, serially correlated, and
contemporaneously correlated. Subsequent research has extended our approach to
account for parameter estimation uncertainty^{(9)} and data snooping bias.^{(10)}

Recent developments in finance and financial risk management encourage the
use of density forecasts: forecasts stated as complete densities rather than as point
forecasts or confidence intervals. However, appraisal of density forecasts has been
hampered by lack of effective tools. In recent work with Todd A. Gunther and Anthony
S. Tay, I have developed a framework for rigorously assessing the adequacy of density
forecasts under minimal assumptions. I have used the new tools to evaluate a variety of
density forecasts involving both simulated and actual equity and exchange rate
returns.^{(11)}

Most recently, Jinyong Hahn, Tay, and I have extended the density forecast
evaluation methods to the multivariate case.^{(12)} Among other things, the multivariate
framework lets us evaluate the adequacy of density forecasts in capturing cross-variable interactions, such as time-varying conditional correlations. We also provide
conditions under which a technique of density forecast "calibration" can be used to
improve density forecasts that are deficient. We show how the calibration method can
be used to generate good density forecasts from econometric models, even when the
conditional density is unknown.

Density forecast evaluation methods are also valuable in macroeconomic
contexts, as my recent work with Tay and Kenneth F. Wallis demonstrates.^{(13)} Since
1968, the Survey of Professional Forecasters has asked respondents to provide a
complete probability distribution of expected U.S. inflation. Evaluation of the adequacy
of those density forecasts reveals several deficiencies. The probability of a large
negative inflation shock is generally overestimated. And, in more recent years, the
probability of a large shock of either sign is overestimated.

**Modeling and Forecasting Financial Asset Return Volatility**

Volatility and correlation are central to finance. Recent work has clarified the comparative desirability of alternative estimators of volatility and correlation and has noted the attractive properties of the so-called realized volatility estimator, used prominently in the classic work of Robert Merton, Kenneth French, and others. Realized volatility is trivial to compute. Further, we now know that under standard diffusion assumptions, and when using the high-frequency underlying returns now becoming widely available, realized volatility is effectively an error-free measure. Hence, for many practical purposes, we can treat volatilities and correlations as observed rather than latent.

Observable volatility creates entirely new opportunities: we can analyze it,
optimize it, use it, and forecast it with much simpler techniques than the complex
econometric models required when volatility is latent. My recent work with Torben
Andersen, Tim Bollerslev, and Paul Labys exploits this insight intensively, in
understanding both the unconditional and conditional distributions of realized asset
return volatility, in developing tools for optimizing the construction of realized volatility
measures, in using realized volatility to make sharp inferences about the conditional
distributions of asset returns, and in explicit modeling and forecasting of realized
volatility.^{(14)}

Noteworthy products of the research include a simple normality-inducing volatility
transformation, high contemporaneous correlation across volatilities, high correlation
between correlation and volatilities, pronounced and highly persistent temporal variation
in both volatilities and correlation, evidence of long-memory dynamics in both volatilities
and correlation, and precise scaling laws under temporal aggregation.^{(15)} The results
should be useful in producing improved strategies for asset pricing, asset allocation,
and risk management, which explicitly account for time-varying volatility and correlation.

Any such strategies exploiting time-varying volatility or correlation, however,
require taking a stand on the horizon at which returns are measured. Different horizons
are relevant for different applications (for example, managing a trading desk versus
managing a university's endowment). Hence, related work involving volatility estimation
and forecasting in financial risk management has focused on the return horizon. In a
study with Andrew Hickman, Atsushi Inoue, and Til Schuermann, I examine the
common practice of converting one-day volatility estimates to "h-day" estimates by
scaling by the square root of h. This turns out to be inappropriate except under very
special circumstances routinely violated in practice.^{(16)} Another more broadly focused
study with Christoffersen uses a model-free procedure to assess the forecastability of
volatility at various horizons ranging from a day to a month.^{(17)} Perhaps surprisingly, the
forecastability of volatility turns out to decay rather quickly with the horizon. This
suggests that volatility forecastability, although clearly relevant for risk management at
short horizons, may be much less important at longer horizons. We are currently at an
interesting juncture in regard to long-horizon volatility forecastability: some studies are
indicating long memory in volatility forecastability and others are not. Very much related
is the possibility of structural breaks, which can masquerade as long memory. This is an
important direction for future research, and I have begun to tackle it in recent work with
Inoue.^{(18)}

**Econometric Methods for Business Cycle and Macroeconomic Modeling**

After nearly a decade of strong growth, it is tempting to assert that the business
cycle is dead. It is not. Indeed, a recession is coming -- we just don't know when.
Another strand of my work, much of it with Glenn D. Rudebusch, centers on the
econometrics of business cycles and business cycle modeling. In part, the research is
eclectic and scattered, ranging from early work on business cycle duration dependence
to later work on strategic complementarity and job durations.^{(19)} But much of it is
organized around three general themes, which I discuss briefly in turn.^{(20)}

What are the defining characteristics of the business cycle? Two features are crucial. The first involves the co-movement of economic variables over the cycle, or, roughly speaking, how broadly business cycles are spread throughout the economy. The notion of co-movement -- particularly accelerated or delayed co-movement -- leads naturally to notions of coincident, leading, and lagging business cycle indicators. The second feature involves the timing of the slow switching between expansions and contractions, and the persistence of business cycle regimes.

Central to much of the work is the idea of a dynamic factor model with a Markov
switching factor, which simultaneously captures both co-movement and regime
switching,^{(21)} as recently implemented using Markov chain Monte Carlo methods.^{(22)}

How can business cycle models be evaluated? One way or another, we want to
assess business cycle models empirically, by checking whether the properties of our
model economy match those of the real economy. However, doing so in a rigorous
fashion presents challenges, particularly with the modern breed of dynamic stochastic
general equilibrium models. In recent work with Lee E. Ohanian, I have attempted to
provide a constructive framework for assessing agreement between dynamic
equilibrium models and data, which enables a complete comparison of model and data
means, variances, and serial correlations.^{(23)} The new methods use bootstrap algorithms
to evaluate the significance of deviations between model and data without assuming
that the model under investigation is correctly specified. They also use goodness-of-fit
criteria to produce estimators that optimize economically relevant loss functions.

In related work, Lutz Kilian and I propose a measure of predictability based on
the ratio of the expected loss of a short-run forecast to the expected loss of a long-run
forecast.^{(24)}> The predictability measure can be tailored to the forecast horizons of interest,
and it allows for general loss functions, univariate or multivariate information sets, and
stationary or nonstationary data. We propose a simple estimator, and we suggest
resampling methods for inference. We then put the new tools to work in macroeconomic
environments. First, based on fitted parametric models, we assess the predictability of a
variety of macroeconomic series. Second, we analyze the internal propagation
mechanism of a standard dynamic macroeconomic model by comparing the
predictability of model inputs and model outputs. Finally, we compare the predictability
in U.S. macroeconomic data with that implied by leading macroeconomic models.

How can secular growth be distinguished from cyclical fluctuations?
Understanding the difference between the economy's trend and its cycle is crucial for
business cycle analysis. A long debate continues on the appropriate separation of trend
and cycle; Abdelhak S. Senhadji and I have summarized recent elements in this debate
and attempted to sift the relevant evidence.^{(25)} In the end, a great deal of uncertainty
remains; however, it appears that some traditional trend/cycle decompositions with quite
steady trend growth are not bad approximations in practice.

If there is still uncertainty in disentangling trend from cycle, there is less in finding
good cyclical forecasting models. In particular, the low power that plagues unit root tests
and related procedures when testing against nearby alternatives, which are typically the
relevant alternatives in macroeconomics and finance, is not necessarily a concern for
forecasting. Ultimately, the question of interest for forecasting is not whether unit root
pretests select the "true" model, but whether they select models that produce superior
forecasts. My recent work with Kilian suggests that unit root tests are effective when
used for that purpose.^{(26)}

^{1. } J. H. Cochrane, "New Facts in Finance," NBER Working Paper No.
7169 , June 1999,
provides a fine survey of many of the recent developments in finance.

^{2. } The "Forecasting and Empirical Methods in Finance and Macroeconomics" group is
supported by the NBER and the National Science Foundation. Its meetings have
produced several associated symposiums, including those whose proceedings
appear in: __Review of Economics and Statistics__, November 1999, F. X. Diebold, J.
H. Stock, and K. D. West, eds.; __International Economic Review__, November 1998,
F. X. Diebold and K. D. West, eds.; and __Journal of Applied Econometrics__,
September-October 1996, F. X. Diebold and M. W. Watson, eds.

^{3. } For overviews, see F. X. Diebold, __Elements of Forecasting__, Cincinnati: South-Western
College Publishing, 1998, and "The Past, Present, and Future of Macroeconomic
Forecasting," NBER Working Paper No. 6290, November 1997, and __Journal of
Economic Perspectives__, 12 (1998), pp. 175-92.

^{4. } See, for example, A. C. Stockman, "Economic Theory and Exchange Rate Forecasts,"
__International Journal of Forecasting__, 3 (1987), pp. 3-15.

^{5. } P. F. Christoffersen and F. X. Diebold, "Optimal Prediction under Asymmetric Loss,"
NBER Technical Working Paper No. 167, October 1994. Published in two parts,
as "Optimal Prediction under Asymmetric Loss," __Econometric Theory__, 13 (1997),
pp. 808-17, and "Further Results on Forecasting and Model Selection under
Asymmetric Loss," __Journal of Applied Econometrics__, 11 (1996), pp. 561-72.

^{6. } P. F. Christoffersen and F. X. Diebold, "Cointegration and Long-Horizon Forecasting,"
NBER Technical Working Paper No. 217, October 1997, and __Journal of Business
and Economic Statistics__, 16 (1998), pp. 450-8.

^{7. } For a survey, see F. X. Diebold and J. A. Lopez, "Forecast Evaluation and
Combination," NBER Technical Working Paper No. 192, March 1996, and
__Handbook of Statistics__, G. S. Maddala and C. R. Rao, eds., pp. 241-68.
Amsterdam: North-Holland, 1996.

^{8. } F. X. Diebold and R. S. Mariano, "Comparing Predictive Accuracy," __Journal of
Business and Economic Statistics__, 13 (1995), pp. 253-65, and in __Economic
Forecasting__, T. C. Mills, ed., Cheltenham, U.K.: Edward Elgar Publishing, 1998.

^{9. } K. D. West, "Asymptotic Inference about Predictive Ability," __Econometrica__, 64 (1996),
pp. 1067-84.

^{10. } H. White, "A Reality Check for Data Snooping," __Econometrica__, forthcoming, and R.
Sullivan, A. Timmermann, and H. White, "Data Snooping, Technical Trading Rule
Performance, and the Bootstrap," __Journal of Finance__, forthcoming.

^{11. } F. X. Diebold, T. A. Gunther, and A. S. Tay, "Evaluating Density Forecasts, with
Applications to Financial Risk Management," NBER Technical Working Paper
No. 215, October 1997, and __International Economic Review__, 39 (1998), pp.
863-83.

^{12. } F. X. Diebold, J. Hahn, and A. S. Tay, "Multivariate Density Forecast Evaluation and
Calibration in Financial Risk Management: High-Frequency Returns on Foreign
Exchange," revised and re-titled version of NBER Working Paper No.
6845,
December 1998. Forthcoming in __Review of Economics and Statistics__, 81 (1999).

^{13. } F. X. Diebold, A. S. Tay, and K. F. Wallis, "Evaluating Density Forecasts of Inflation:
The Survey of Professional Forecasters," NBER Working Paper No.
6228,
October 1997, and __Cointegration, Causality, and Forecasting: A Festschrift in
Honor of Clive W. J. Granger, R. Engle and H. White__, eds., pp. 76-90. Oxford:
Oxford University Press, 1999.

^{14. } For an overview, see T. Andersen, T. Bollerslev, F. X. Diebold, and P. Labys,
"Understanding, Optimizing, Using, and Forecasting Realized Volatility and
Correlation," __Risk__, 12 (forthcoming).

^{15. } See T. Andersen, T. Bollerslev, F. X. Diebold, and P. Labys, "The Distribution of
Exchange Rate Volatility," NBER Working Paper No. 6961, February 1999, and
the references therein.

^{16. } See F. X. Diebold, A. Hickman, A. Inoue, and T. Schuermann, "Scale Models," __Risk__,
11 (1998), pp. 104-7, and __Hedging with Trees: Advances in Pricing and Risk
Managing Derivatives__, M. Broadie and P. Glasserman, eds., pp. 233-7. London:
Risk Publications, 1998.

^{17. } P. F. Christoffersen and F. X. Diebold, "How Relevant Is Volatility Forecasting for
Financial Risk Management?" NBER Working Paper No. 6844, December 1998,
and __Review of Economics and Statistics__, 82 (forthcoming).

^{18. } F. X. Diebold and A. Inoue, "Long Memory and Structural Change," manuscript,
Department of Finance, Stern School of Business, New York University, 1999.

^{19. } F. X. Diebold and G. D. Rudebusch, "A Nonparametric Investigation of Duration
Dependence in the American Business Cycle," __Journal of Political Economy__, 98
(1990), pp. 596-616; F. X. Diebold, D. Neumark, and D. Polsky, "Job Stability in
the United States," __Journal of Labor Economics__, 15 (1997), pp. 206-33; and A.
N. Bomfim and F. X. Diebold, "Bounded Rationality and Strategic
Complementarity in a Macroeconomic Model: Policy Effects, Persistence, and
Multipliers," __Economic Journal__, 107 (1997), pp. 1358-75.

^{20. } F. X. Diebold and G. D. Rudebusch, __Business Cycles: Durations, Dynamics, and
Forecasting__. Princeton: Princeton University Press, 1999.

^{21. } F. X. Diebold and G. D. Rudebusch, "Measuring Business Cycles: A Modern
Perspective," __Review of Economics and Statistics__, 78 (1996), pp. 67-77; and F.
X. Diebold, J.-H. Lee, and G. Weinbach, "Regime Switching with Time-Varying
Transition Probabilities," in __Nonstationary Time Series Analysis and
Cointegration__, C. Hargreaves, ed., pp. 283-302. Oxford: Oxford University Press,
1994.

^{22. } C.-J. Kim and C. R. Nelson, "Business Cycle Turning Points, a New Coincident Index,
and Tests of Duration Dependence Based on a Dynamic Factor Model with
Regime-Switching," __Review of Economics and Statistics__, 80 (1998), pp. 188-201,
and __State Space Models with Regime Switching__. Cambridge, Mass.: MIT Press,
1999.

^{23. } F. X. Diebold, L. E. Ohanian, and J. Berkowitz, "Dynamic Equilibrium Economies: A
Framework for Comparing Models and Data," NBER Technical Working Paper
No. 174, February 1995, and __Review of Economic Studies__, 65 (1998), pp.
433-52.

^{24. } F. X. Diebold and L. Kilian, "Measuring Predictability: Theory and Macroeconomic
Applications," NBER Technical Working Paper No. 213, August 1997.

^{25. } F. X. Diebold and A. S. Senhadji, "The Uncertain Unit Root in Real GNP: Comment,"
__American Economic Review__, 86 (1996), pp. 1291-8.

^{26. } F. X. Diebold and L. Kilian, "Unit Root Tests Are Useful for Selecting Forecasting
Models," NBER Working Paper No. 6928, February 1999, and forthcoming in
__Journal of Business and Economic Statistics__, 18.

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