Despite powerful advances in yield curve modeling in the last twenty years, comparatively little attention has been paid to the key practical problem of forecasting the yield curve. In this paper we do so. We use neither the no-arbitrage approach, which focuses on accurately fitting the cross section of interest rates at any given time but neglects time-series dynamics, nor the equilibrium approach, which focuses on time-series dynamics (primarily those of the instantaneous rate) but pays comparatively little attention to fitting the entire cross section at any given time and has been shown to forecast poorly. Instead, we use variations on the Nelson-Siegel exponential components framework to model the entire yield curve, period-by-period, as a three-dimensional parameter evolving dynamically. We show that the three time-varying parameters may be interpreted as factors corresponding to level, slope and curvature, and that they may be estimated with high efficiency. We propose and estimate autoregressive models for the factors, and we show that our models are consistent with a variety of stylized facts regarding the yield curve. We use our models to produce term-structure forecasts at both short and long horizons, with encouraging results. In particular, our forecasts appear much more accurate at long horizons than various standard benchmark forecasts. Finally, we discuss a number of extensions, including generalized duration measures, applications to active bond portfolio management, and arbitrage-free specifications. Acknowledgments: The National Science Foundation and the Wharton Financial Institutions Center provided research support. For helpful comments we are grateful to Dave Backus, Rob Bliss, Michael Brandt, Todd Clark, Qiang Dai, Ron Gallant, Mike Gibbons, Da...

We examine the extent to which fluctuations in the money stock anticipate (or Granger cause) fluctuations in real output using a variety of rolling window and increasing window estimation techniques. Various models are considered using simple sum as well as Divisia measures of M1 and M2, income, prices, and both the Tbill rate and the commercial paper rate. Findings indicate that the relation between income, money, prices, and interest rates is stable, as long as sufficient data are used, and that there is cointegration among the variables considered, although cointegration spaces become very difficult to estimate precisely when smaller windows of data are used. Further, both M1 and M2 are shown to be important predictors of income for the entire period from 1960:2-1996:3, based on modified versions of what we term the "most damaging" specifications from Friedman and Kuttner (1993) and Thoma (1994). Our new evidence is based in part on a rather novel model selection approach to examini...

In this paper we outline conditions under which the Diebold and Mariano (DM: 1995) test for predictive ability can be extended to the case of two forecasting models, each of which may include cointegrating relations, when allowing for parameter estimation error. We show that in the cases where either the loss function is quadratic or the length of the prediction period, P, grows at a slower rate than the length of the regression period, R, the standard DM test can be used. On the other hand, in the case of a generic loss function, if P R ! as T ! 1, 0 < < 1, then the asymptotic normality result of West (1996) no longer holds. We also extend the "data snooping" technique of White (2000) for comparing the predictive ability of multiple forecasting models to the case of cointegrated variables. In a series of Monte Carlo experiments, we examine the impact of both short run and cointegrating vector parameter estimation error on DM, data snooping, and related tests. Our results sugge...

In this paper, we draw on both the consistent specification testing and the predictive ability testing literatures and propose a test for predictive accuracy which is consistent against generic nonlinear alternatives. Broadly speaking, given a particular reference model, assume that the objective is to test whether there exists any alternative model, among an infinite number of alternatives, that has better predictive accuracy than the reference model, for a given loss function. A typical example is the case in which the reference model is a simple autoregressive model and the objective is to check whether a more accurate forecasting model can be constructed by including possibly unknown (non) linear functions of the past of the process or of the past of some other process(es). We propose a statistic which is similar in spirit to that of White (2000), although our approach diers from his as we allow for an innite number of competing models that may be nested. In addition, we allow for non ...

In this paper we examine semiparametric nonlinear autoregressive models with exogenous variables (NLARX) via three classes of artificial neural networks: the first one uses smooth sigmoid activation functions; the second one uses radial basis activation functions; and the third one uses ridgelet activation functions. We provide root mean squared error convergence rates for these ANN estimators of the conditional mean and median functions with stationary beta-mixing data. As an empirical application, we compare the forecasting performance of linear and semiparametric NLARX models of U.S. inflation. We find that all of our semiparametric models outperform a benchmark linear model based on various forecast performance measures. In addition, a semiparametric ridgelet NLARX model which includes various lags of historical inflation and the GDP gap is best in terms of both forecast mean squared error and forecast mean absolute deviation error.

We review key aspects of forecasting using nonlinear models. Because economic models are typically misspecified, the resulting forecasts provide only an approximation to the best possible forecast. Although it is in principle possible to obtain superior approximations to the optimal forecast using nonlinear methods, there are some potentially serious practical challenges. Primary among these are computational difficulties, the dangers of overfit, and potential difficulties of interpretation. In this chapter we discuss these issues in detail. Then we propose and illustrate the use of a new family of methods (QuickNet) that achieves the benefits of using a forecasting model that is nonlinear in the predictors while avoiding or mitigating the other challenges to the use of nonlinear forecasting methods. 1.

In canonical vector time series autoregressions, which permit dependence only on past values, the errors generally show contemporaneous correlation. By contrast structural vector autoregressions allow contemporaneous series dependence and assume errors with no contemporaneous correlation. Such models having a recursive structure can be described by a directed acyclic graph. We show, with the use of a real example, how the identification of these models may be assisted by examination of the conditional independence graph of contemporaneous and lagged variables. In this example we identify the causal dependence of monthly Italian bank loan interest rates on government bond and repurchase agreement rates. When the number of series is larger, the structural modelling of the canonical errors alone is a useful initial step, and we first present such an example to demonstrate the general approach to identifying a directed graphical model.

We examine the prevalence of data, specification, and parameter uncertainty in the formation of simple rules which mimic monetary policy-making decisions. Our approach is to build realtime datasets and simulate a real-time policy-setting environment in which we are able to assess the actual performance of rules, had they been followed in real time. This approach allows us not only to track the performance of alternative rules over time (hence facillitating a type of model selection among competing rules), but also allows us more generally to assess the importance of the data revision process in the formation of macroeconomic time series models. From the perspective of real time data, our results suggest that the use of data which are erroneous, in the sense that they were not available at the time decisions based on forecasts from the rules were used, can lead to the selection of quantitatively di®erent models. From the perspective of policy rules, we find that: our version of "calibration" is better than naive estimation, although both are dominated by an approach to rule formation based on adaptive least squares learning using; rules based on seasonally unadjusted data are more reliable than those based on seasonally adjusted data; and rules based soly on preliminary data do not minimize mean square forecast error (MSE) risk. In particular, early releases of data can be noisy, and for this reason it is useful to also use data which have been revised when making decisions using policy rules.

Using a standard 4-variable linear vector error correction model (VECM), we rst show that the null hypothesis of linearity can be strongly rejected against the alternative of smooth transition autoregressive nonlinearity. An important result from this stage of the analysis is that the quarterly growth rate of money is identied as the transition variable, the variable which governs the smooth switching between regimes. This implies there is a nonlinear causal relationship between money and output. A smooth transition VECM (STVECM) is then used to examine whether money nonlinearly Granger causes output in the sense that lagged values of money enter the model's output equation as regressors. We evaluate this type of nonlinear Granger causality with both in-sample and out-of-sample analysis. For the in-sample analysis we compare alternative models using predictive accuracy tests. These results vary strongly across use of the AIC and SIC. Our use of an out-of-sample forecasting exercise to study money-income Granger causality, both linear and nonlinear, we believe is new to the literature. The forecasting results do not suggest that money is nonlinearly Granger causal for output. In fact, they show that by allowing money to nonlinearly Granger cause output, the forecasting performance of the STVECM is signicantly worsened. We thank participants of the Tinbergen Institute/Macroeconomic Dynamics Conference on "Nonlinear Modeling of Multivariate Macroeconomic Relations" for their helpful comments. Rothman's research was supported by an East Carolina University Faculty Senate Summer Research Grant and van Dijk wishes to acknowledge the hospitality of the Department of Economics at the University of Western Australia. y Department of Economics, Brewster Bu...

A number of variants of seven procedures designed to check for the absence of causal ordering are summarized. Five are based on classical hypothesis testing principles, including: Wald F-tests designed for stationary and difference stationary data; sequential Wald tests that account for cointegration; surplus lag regression type tests; and nonparametric fully modied vector autoregressive type tests. The other two are based on model selection techniques, and include: complexity penalized likelihood criteria; and ex-ante model selection based on predictive ability. In addition, various other approaches to checking for the causal order of economic variables are briey discussed. A small set of Monte Carlo experiments is carried out in order to assess empirical size, and it is found that although all tests perform well in the environments where the true lag dynamics and cointegrating ranks are "accurately" estimated, simple surplus lag type tests of the variety discussed by Toda and Yamamoto ...