This paper shows that yields to maturity of U.S. Treasury bills are cointegrated, and that during periods when the Federal Reserve specifically targeted short-term interest rates, the spreads between yields of different maturity define the cointegrating vectors. This cointegrating relationship implies that a single non-stationary common factor underlies the time series behavior of each yield to maturity and that risk premia are stationary. An error correction model which uses spreads as the error correction terms is unstable over the Federal Reserve's policy regime changes, but a model using post 1982 data is stable and is shown to be useful for forecasting changes in yields.

The modern world has influenced the approach to empirical modeling and consequently the approach to methodology in general. The question of whether to base a model on an economic theory is easier when several models can be constructed, but an empirical evaluation analysis is required. Starting with a widely specified model and using a reduction procedure is currently a popular process; it is unclear if the reduction should be to a single or to a few alternatives. New methodologies are required for conditional predictive distributions, which are the models of the future.

More and more data, greatly increased computing power, a rising number of research enthusiasts, an increased number of finance journals, and sophisticated techniques have been the characteristics of empirical finance in the past 30 years. Topics of current interest relate to conditional means, conditional variances, and conditional distributions. These topics will remain in the forefront for years to come and perhaps be joined by questions that will shake the foundations of finance theory. For example, will we find that market data are characterized by jump diffusions—that is, diffusions with breaks—rather than standard diffusions? In the third of a century since my first investigations into finance theory, empirical finance has changed dramatically—from only a few active workers to hundreds, maybe thousands, of researchers. The number of finance journals has grown from one to dozens, and the techniques have become considerably more advanced. The availability of much more data and greatly increased computer power have produced impressive research publications. Many of these publications have relatively little practical usefulness, however, and in fact, the purpose of much of the work is unclear. I avoid the most popular procedures that have developed (which are covered by several excellent textbooks on financial econometrics) and survey the work that is going on and the work that needs to be done with conditional means, conditional variances, and conditional distributions. For the future, I see, first, continued investigation of topics already under scrutiny. In particular, conditional distributions will continue to be a major subject as finance learns how to generate more of its fundamental theories in distributional forms: arbitrage and portfolio theory, efficient market theory and its consequences, the Black-Scholes formula, and so forth. Structural breaks are also likely, however, in the present framework. In particular, I foresee renewed study of the “facts” researchers found through using basically a linear foundation for studying prices, returns, and volatility. An example of these facts is the conclusion that returns are nearly white noise; that is, they have no serial or autocorrelation. I see problems in the new field of continuous-time finance theory. The mistake that I see is starting with the assumption that a price or a return can be written in terms of a standard diffusion, which is based on a Gaussian distribution. The approach served well in the early days of econometrics but only for mathematical convenience; it was not tested. The assumption in continuous-time theory is dangerous because we have no way to test it. We have no continuous-time data. Finally, I find interesting the recent results indicating that our market data behave in a way that is more consistent with jump diffusions—that is, diffusions with breaks—than with standard diffusions. If this finding holds, the majority of current financial theory will probably have to be rewritten with “jump diffusion” replacing “diffusion” and with some consequent changes in theorems and results. As with all radical new ideas, this change will certainly be opposed by some.

Investor risk is a complicated concept in practice and is not well captured by measures of volatility as is well understood by uncertainty theory. Rather than asking statisticians to attempt to measure risk, it may be better to listen to decision theorists, but their suggestions are not very practical. Diversification is clearly helpful in reducing risk but the risk level of one portfolio cannot be measured without knowing the risks of other major portfolios. A meta-analysis can be used to compare alternative volatility measures in terms of their forecasting utility. Copyright (C) 2002 John Wiley Sons, Ltd.

A spurious regression occurs when a pair of independent series, but with strong temporal properties, are found apparently to be related according to standard inference in an OLS regression. Although this is well known to occur with pairs of independent unit root processes, this paper finds evidence that similar results are found with positively autocorrelated autoregressive series or long moving averages. This occurs regardless of the sample size and for various distributions of the error terms.

This paper argues in favour of a closer link between decision and forecast evaluation problems. Although the idea of using decision theory for forecast evaluation appears early in the dynamic stochastic programming literature, and has continued to be used in meteorological forecasts, it is hardly mentioned in standard academic textbooks on economic forecasting. Some of the main issues involved are illustrated in the context of a two-state, two-action decision problem as well as in a more general setting. Relationships between statistical and economic methods of forecast evaluation are discussed and useful links between Kuipers score, used as a measure of forecast accuracy in the meteorology literature, and the market timing tests used in finance, are established. An empirical application to the problem of stock market predictability is also provided, and the conditions under which such predictability could be exploited in the presence of transaction costs are discussed.

Conventional measures of the risk of a financial asset make use of the unobserved (conditional) variance or standard deviation of its return. In this paper, we treat the observed absolute return as a measure of risk and explore its forecastability. Two simple models are considered. One is a new AR-like model which is applied to the absolute return. The other is an ARCH-like model called Asymmetric Power ARCH. The forecastability is evaluated with the average log-likelihood of absolute return, instead of that of return itself. While the absolute return is interpreted as volatility , some quantities of its entire distribution, such as the 95-th quantiles, can be interpreted as volatility of volatility . We apply both models to three stock indices, namely Hang Seng Index, Nikkei 225 Index and Standard and Poors 500 Index. The new model by and large outperforms the ARCH-like model in both in-sample goodness of fit and post-sample forecastability. It performs exceptionally well in the post-sample period after the outbreak of the Asian financial crisis

With the advent of very large data sets in economics, econometricians, with the help of statisticians, have to re‐evaluate current techniques and develop new procedures and their interpretation. Most tests of significance become irrelevant, for example, and conditional distributions become important, but difficult to report. Time series with high frequency data also present new and interesting questions.

The notion of separation in cointegrated systems helps identifying possible sub‐system structures that may reduce the complexity of larger systems by yielding a more parsimonious representation of the times series. In this paper we demonstrate that although the subsystem cointegration analysis in such systems can be conducted in case of both completely and partially separated systems, the dual approach, i.e. calculation of the common stochastic trends, may turn out to yield properties of the trends that differ depending upon the type of separation under consideration. In particular, we demonstrate how persistent‐transitory (P‐T) decompositions and long‐ and short‐memory factorizations of a multivariate time series will interact across systems when considering the presence (or absence) of different types of separation. Generalizations to non‐linear error correction models are briefly discussed.

This paper analyze 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 nonlinear 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 bivariante cointegration system. With monthly data from 1971 to 1990, it is found that cointegration could have occurred during some periods and specially during the bubble and post-bubble periodo However, dummy variables for the intercept of the long-ron relationships are needed during the full sample. For the price of gold the nonlinear models perform better than the random walk in-sample and out-of-sample. In-sample nonlinear 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 nonlinear 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 analyzed between the prices of gold and silver, the dependence is less out-of sample, possibly meaning that the two markets are becoming separated.

This paper considers the combination of forecasts using changing weights derived from switching regression models or from smooth transition regression models. The regimes associated with the switches may not be known to the forecaster and thus need to be estimated. Several approaches to this problem are considered. In two empirical examples, these time-varying combining procedures produced smaller, in some cases substantially smaller, out-of-sample squared forecast errors than those obtained using the simple linear combining model.

This paper develops tests for roots in linear time series which have a modulus of one but which correspond to seasonal frequencies. Critical values for the tests are generated by Monte Carlo methods or are shown to be available from Dickey-Fuller or Dickey-Hasza-Fuller critical values. Representations for multivariate processes with combinations of seasonal and zero-frequency unit roots are developed leading to a variety of autoregressive and error-correction representations. The techniques are used to examine cointegration at different frequencies between consumption and income in the U.K.

This paper examines the evaluation of models claimed to be relevant for policy making purposes. A number of tests are proposed to determine the usefulness of such models in the policy making process. These tests are applied to three empirical examples.

The relationship between co-integration and error correction models, first suggested in Granger (1981), is here extended and used to develop estimation procedures, tests, and empirical examples.If each element of a vector of time series xt first achieves stationarity after differencing, but a linear combination α'xt, is already stationary, the time series xt are said to be co-integrated with co-integrating vector α. There may be several such co-integrating vectors so that α becomes a matrix. Interpreting α'xt, = 0 as a long run equilibrium, co-integration implies that deviations from equilibrium are stationary, with finite variance, even though the series themselves are nonstationary and have infinite variance.The paper presents a representation theorem based on Granger (1983), which connects the moving average, autoregressive, and error correction representations for co-integrated systems. A vector autoregression in differenced variables is incompatible with these representations. Estimation of these models is discussed and a simple but asymptotically efficient two-step estimator is proposed. Testing for co-integration combines the problems of unit root tests and tests with parameters unidentified under the null. Seven statistics are formulated and analyzed. The critical values of these statistics are calculated based on a Monte Carlo simulation. Using these critical values, the power properties of the tests are examined and one test procedure is recommended for application.In a series of examples it is found that consumption and income are co-integrated, wages and prices are not, short and long interest rates are, and nominal GNP is co-integrated with M2, but not M1, M3, or aggregate liquid assets.

Oxford Bulletin of Economics and StatisticsVolume 48, Issue 3 p. 213-228 Free Access DEVELOPMENTS IN THE STUDY OF COINTEGRATED ECONOMIC VARIABLES C. W. J Granger, C. W. J Granger * I would like to acknowledge the excellent hospitality that I enjoyed at Nuffield College and the Institute of Economics and Statistics, Oxford whilst this paper was prepared.Search for more papers by this author C. W. J Granger, C. W. J Granger * I would like to acknowledge the excellent hospitality that I enjoyed at Nuffield College and the Institute of Economics and Statistics, Oxford whilst this paper was prepared.Search for more papers by this author First published: August 1986 https://doi.org/10.1111/j.1468-0084.1986.mp48003002.xCitations: 1,146 AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume48, Issue3August 1986Pages 213-228 ReferencesRelatedInformation

A nonlinear relationship between electricity sales and temperature is estimated using a semiparametric regression procedure that easily allows linear transformations of the data. This accommodates introduction of covariates, timing adjustments due to the actual billing schedules, and serial correlation. The procedure is an extension of smoothing splines with the smoothness parameter estimated from minimization of the generalized cross-validation criterion introduced by Craven and Wahba (1979). Estimates are presented for residential sales for four electric utilities and are compared with models that represent the weather using only heating and cooling degree days or with piecewise linear splines.

It is well known that a linear combination of forecasts can outperform individual forecasts. The common practice, however, is to obtain a weighted average of forecasts, with the weights adding up to unity. This paper considers three alternative approaches to obtaining linear combinations. It is shown that the best method is to add a constant term and not to constrain the weights to add to unity. These methods are tested with data on forecasts of quarterly hog prices, both within and out of sample. It is demonstrated that the optimum method proposed here is superior to the common practice of letting the weights add up to one.