پیش بینی اعتبار از چگالی احتمال مشترک بازدهی اوراق قرضه: توان مدل های تکراری برخورد گام تصادفی است؟

Most existing empirical studies on affine term structure models (ATSMs) have mainly focused on in-sample goodness-of-fit of historical bond yields and ignored out-of-sample forecast of future bond yields. Using an omnibus nonparametric procedure for density forecast evaluation in a continuous-time framework, we provide probably the first comprehensive empirical analysis of the out-of-sample performance of ATSMs in forecasting the joint conditional probability density of bond yields. We find that although the random walk models tend to have better forecasts for the conditional mean dynamics of bond yields, some ATSMs provide better forecasts for the joint probability density of bond yields. However, all ATSMs considered are still overwhelmingly rejected by our tests and fail to provide satisfactory density forecasts. There exists room for further improving density forecasts for bond yields by extending ATSMs.

The term structure of interest rates, which concerns the relationship among the yields of default-free bonds with different maturities, is one of the most widely studied topics in economics and finance. Following the pioneering works of Vasicek (1977) and Cox et al. (1985), a large number of multifactor dynamic term structure models (DTSMs) have been developed over the last two decades.1 These models, by imposing cross-sectional and time series restrictions on bond yields in an internally consistent manner, provide important insights for our understanding of term structure dynamics. They have been widely used in financial industry for pricing fixed-income securities and managing interest rate risk. Affine term structure models (ATSMs), first introduced in Duffie and Pan (1996), have become the leading DTSMs in the literature due to their rich model specification and tractability. In ATSMs, the short-term interest rate is an affine function of the underlying state variables which follow affine diffusions (the instantaneous drift and variance are affine functions of the state variables) under both the risk-neutral and physical measures. These assumptions allow closed-form solutions for a wide variety of fixed-income securities (see, e.g., Duffie et al., 2000; Chacko and Das, 2002), which greatly simplify empirical implementations of ATSMs. As a result, ATSMs have become probably the most widely studied DTSMs in the academic literature. Despite the numerous empirical studies on DTSMs, the existing literature has mainly focused on in-sample fit of historical bond yields and ignored out-of-sample forecast of future bond yields. In-sample diagnostic analysis is important and can reveal useful information on possible sources of model misspecifications. However, it is the evolution of the yield curve in the future, not the past, that is most relevant in many financial applications, such as pricing and hedging fixed-income securities and managing interest rate risk. As widely recognized in the literature, the current yield curve contains information about the future yield curve and the state of the economy. Therefore, accurate forecasts of bond yields are also important for savings and investments decisions of households and firms, and for macroeconomic policy decisions of monetary authorities. Furthermore, as pointed out by Duffee (2002, p. 465), “a model that is consistent with finance theory and produces accurate forecasts can make a deeper contribution to finance,” especially for explaining time varying expected bond returns and the failure of the expectation hypothesis. Unfortunately, there is no guarantee that a model that fits historical data well will also perform well in out-of-sample forecast, due to at least three important reasons. First, the extensive search for more complicated models using the same (or similar) data set(s) may suffer from the so-called data snooping bias, as pointed out by Lo and MacKinlay (1989) and White (2000). In the present context, most studies on ATSMs have used either U.S. Treasury yields in the past 50 years or U.S. dollar swap rates in the past 15 years. While a more complicated model can always fit a given data set better than simpler models, it may overfit some idiosyncratic features of the data without capturing the true data-generating process. Out-of-sample forecasting evaluation will alleviate, if not eliminate completely, such data snooping bias. Second, an overparameterized model contains a large number of estimated parameters and inevitably exhibits excessive sampling variation in parameter estimation. Such excessive parameter estimation uncertainty may adversely affect the out-of-sample forecast performance. Third, a model that fits in-sample data well may not forecast the future well because of unforeseen structural changes or regime shifts in the data-generating process. Therefore, in-sample analysis is not adequate and it is important to examine the out-of-sample predictive ability of existing term structure models, especially when comparing competing models. A few studies that do consider the out-of-sample performance of ATSMs have shown that they fail miserably in forecasting the conditional mean of future bond yields. For example, Duffee (2002) shows that the completely ATSMs of Dai and Singleton (2000) have worse forecasts of the conditional mean of bond yields than a simple random walk model in which expected future yields are equal to current yields. Consequently, Duffee (2002, p. 434) concludes that “for the purposes of forecasting, completely ATSMs are largely useless”.2 In fact, it has been shown that the simple random walk model outperforms most sophisticated models in forecasting the conditional mean of many other economic and financial time series. One well-known example is the forecasts of foreign exchange rates: the classic paper of Meese and Rogoff (1983) and many important subsequent studies have shown that the random walk model outperforms most structural and time series models in forecasting the conditional mean of major exchange rates. However, the full dynamics of an intertemporal model is completely characterized by the conditional density of the state variables, which includes not only the conditional mean, but also higher-order conditional moments. A model that has better forecasts of the conditional mean does not necessarily have better forecasts of higher-order conditional moments as well. For example, it is widely known that GARCH and stochastic volatility models provide better forecasts of the conditional variance of many financial time series than simple random walk models. There is a vast literature on volatility forecasting for the purpose of option pricing and risk management (see, e.g., Andersen et al., 2004). As shown by Dai and Singleton (2000), the ATSMs that have the best (in-sample) empirical performance are those that are flexible in modeling the time varying volatilities and correlations of the state variables. In fact, it is well known that changes of most financial time series have weak or little dependent structure in conditional mean, but persistent dependence in conditional variance and higher-order conditional moments. Therefore, ATSMs might have good forecasts for the higher-order moments, or even the whole conditional density of bond yields, although they have poor forecasts of the conditional mean dynamics. In this paper, we study whether ATSMs can provide accurate forecasts of the joint conditional probability density of bond yields. We focus on forecasting the conditional density because interest rates, like most other financial data, are highly non-Gaussian. One needs to go beyond the conditional mean and variance to get a complete picture of term structure dynamics. The conditional probability density of the state variables characterizes the full dynamics of a term structure model and essentially checks all conditional moments simultaneously (if the moments exist). In fact, all continuous-time models in finance, including ATSMs, are essentially models for the transition density of the underlying economic process. Because of this, density forecast evaluation is a very natural and suitable way to evaluate these financial models. Density forecasts are important not only for constructing statistical tests, but also for many economic and financial applications. For example, the booming industry of financial risk management is essentially dedicated to provide density forecasts for portfolio returns, and then to track certain aspects of the distribution such as Value at Risk (VaR) to quantify the risk exposure of a portfolio (e.g., Duffie and Pan, 1997, Morgan, 1996 and Jorion, 2000).3 Density forecast in ATSMs is especially important for financial risk management in the huge fixed-income markets. In ATSMs, a finite number of state variables drive the evolution of the whole yield curve. Thus, accurate forecasts of the joint density of the state variables would allow us to forecast the distribution of the whole yield curve. If ATSMs can provide accurate density forecasts, they would be very useful for managing the large fixed-income holdings of many banks given their closed-form solutions for most existing fixed-income securities. For other financial applications of ATSMs, such as derivatives pricing and hedging, density forecasts rather than forecasts of a specific feature of the density will be required. Therefore, when evaluating ATSMs, out-of-sample density forecast is an important dimension of model diagnostics that should not be ignored. Evaluating density forecasts is not trivial, given that the probability density function is not observable even ex post. Unlike point forecast evaluation, there are relatively few statistical tools for density forecast evaluation.4 In a pioneering contribution, Diebold et al. (1998) evaluate density forecasts by examining the dynamic probability integral transforms of the data with respect to a model forecast density. Such a transformed series, often referred to as the “generalized residuals” of the density forecast model, should be i.i.d. U[0,1]U[0,1] if the density forecast model correctly captures the full dynamics of the underlying process. Any departure from i.i.d. U[0,1]U[0,1] is evidence of suboptimal forecasts and model misspecification. We extend an omnibus nonparametric in-sample test for i.i.d. U[0,1]U[0,1] developed in Hong and Li (2005) to out-of-sample density forecast for continuous-time models of a multivariate process, some of whose components may be latent variables. The evaluation statistics, which measure the departure of the “generalized residuals” from i.i.d. U[0,1],U[0,1], can be viewed as a metric of the distance between the forecast model and the true data-generating process. While researchers have to choose a lag order when implementing Hong and Li's (2005) test, we introduce a portmanteau statistic that combines Hong and Li's (2005) test statistics at different lag orders.5 As a result, the power of the portmanteau statistic becomes much less dependent on which lag order we use in practice. The portmanteau statistic has the advantage of detecting a wide range of suboptimal density forecasts and is convenient for comparing the performance of different models.6 Using the density forecast evaluation procedure just described, we provide probably the first comprehensive empirical analysis of the out-of-sample performance of ATSMs in forecasting the joint conditional probability density of bond yields. While we consider similar models as Hong and Li (2005), i.e., the three-factor completely and essentially ATSMs, the focus of our analysis is mainly on the out-of-sample forecasting performance of these models. We find that although the random walk models tend to have better forecasts of the conditional mean of bond yields, some ATSMs provide better density forecasts of the joint probability density of bond yields. However, all affine models are still overwhelmingly rejected and none of them provides satisfactory density forecasts. This suggests that time series models with more flexible specifications might be able to provide better density forecasts than the affine models. The rest of this paper is organized as follows. In Section 2, we introduce the nonparametric procedure for density forecast evaluation tailored to a continuous-time framework. In Section 3, we discuss density forecast for multifactor ATSMs. Section 4 investigates the in-sample and out-of-sample performance of ATSMs. In Section 5, we conclude the paper. Appendix provides the asymptotic theory.

The affine term structure models have become one of the most popular term structure models in the literature due to their rich model specification and tractability. In spite of the numerous empirical studies of the ATSMs, little effort has been devoted to examining their out-of-sample performance in forecasting future bond yields. In this paper, we have contributed to the literature by providing probably the first comprehensive empirical analysis of the performance of three-factor completely and essentially ATSMs in forecasting the joint conditional probability density of bond yields. Density forecasts of bond yields are important for many financial applications, such as pricing and hedging fixed-income securities and managing interest rate risk. Using a new nonparametric omnibus procedure for density forecast evaluation tailored for ATSMs, we find that although the random walk models tend to have better forecasts of the conditional mean of the bond yields, some ATSMs provide better forecasts of the joint probability density of the bond yields. However, all ATSMs we consider are still overwhelmingly rejected and none of them can provide satisfactory density forecasts. There exists room for further improving the density forecasts for future bond yields by extending ATSMs. This is left for future investigation.