تعدیل غیر خطی در بازدهی اوراق قرضه ایالات متحده: یک مدل تجربی با ناهمسانی مشروط

Starting from the work by Campbell and Shiller (Campbell, J.Y. and Shiller, R.J. (1987). Cointegration and tests of present value models. Journal of Political Economy, 95(5):1062–1088.), empirical analysis of interest rates has been conducted in the framework of cointegration. However, parts of this approach have been questioned recently, as the adjustment mechanism may not follow a simple linear rule; another line of criticism points out that stationarity of the spreads is difficult to maintain empirically. In this paper, we analyse data on US bond yields by means of an augmented VAR specification which approximates a generic nonlinear adjustment model. We argue that nonlinearity captures macro information via the shape of the yield curve and thus provides an alternative explanation for some findings that recently appeared in the literature. Moreover, we show how conditional heteroskedasticity can be taken into account via GARCH specifications for the conditional variance, either univariate or multivariate.

Interest rates have been the object of extensive research in the cointegration framework in the past 20 years, stemming from the seminal paper by Campbell and Shiller (1987). A fundamental consequence of the expectation hypothesis is that the most appropriate stochastic process to represent their time-series features is some sort of I(1) process. At the same time, interest rate spreads should be stationary, possibly around a non-zero mean. Of course, this translates into very precise hypotheses on the cointegration properties of interest rates, which should cointegrate in pairs, so the cointegration rank should be n − 1 and the cointegration vectors should be of the form [1,0,…,− 1,0,…]. Both ideas can be incorporated in a classic Vector ECM as: equation(1) Γ(L)Δyt=μt+αβ′yt−1+ɛt,Γ(L)Δyt=μt+αβ′yt−1+ɛt, Turn MathJax on where β′yt−1 is a vector containing the (n − 1) lagged spreads. However, the above model is not guaranteed to fit the data flawlessly; in some cases, the spreads may appear non-stationary and the hypothesis that the cointegration rank is (n − 1) may be rejected by conventional tests. Such findings could be interpreted as an outright rejection of the expectation hypothesis; on the other hand, there is the possibility that the empirical model may have to be refined. Several authors have pointed out the shortcomings of a plain VECM model: on one hand, Ang and Piazzesi (2003) suggest that the shape of the yield curve can be influenced by macro factors and, as a consequence, the typical persistence shown by macro data may result in substantial autocorrelation in the spreads, to the point that there are even doubts on their stationarity (Giese, 2006). On the other hand, there is some evidence that the adjustment mechanism implicit in a cointegration model may follow a nonlinear dynamic in the case of bond yields. In most cases, this effect is modelled via a threshold model à la Balke and Fomby (1997). Hansen and Seo (2002) argue that adjustment follows two regimes, and is noticeable in one but not in the other. A similar argument is put forward in Krishnakumar and Neto (2005), where the authors argue that the adjustment is brought about by the monetary authority's interventions, and therefore occurs sporadically. A serious drawback of this class of models is that inference is rather complex, and the issues arising when modelling more than two series are quite difficult to handle. An additional complication may arise because interest rates, like any other financial variable, show considerable changes in volatility if sampled at a monthly frequency or higher. This empirical regularity is widely acknowledged and has spurred the development of the gigantic literature on conditionally heteroskedastic processes, from Engle (1982) onwards. In this context, highly heteroskedastic innovations may have a dramatic impact on standard inferential procedures: estimator efficiency is an obvious issue, but there may also be robustness concerns. In this article, we propose an empirical analysis that combines nonlinear effects in the conditional mean with conditional heteroskedasticity. The paper is structured as follows: Section 2 describes our dataset and provides some preliminary evidence to motivate our preferred models, which are presented in Section 3, while Section 4 contains the estimates, their economic interpretation and an out-of-sample comparison of the forecasts obtained with our models with some of the alternatives. Section 5 concludes.

The main message of our paper is clear: nonlinear adjustment is an important empirical feature in US bond rates. From an economic point of view, this is an interesting result because it suggests that the notion of long-run equilibrium should be broadened to include the concept that equilibrium could be attained in a region of the state space, rather than a single point (or a collection of isolated points). From a statistical point of view, failure to include nonlinearity into an empirical model leads to mis-specification and may hamper predictive ability. Our models for US bonds approximate a nonlinear adjustment mechanism via a simple variable addition to an otherwise ordinary VAR model. Moreover, incorporating conditional heteroskedasticity can be done via standard methods. Hence, they are much less complex to estimate, from a computational point of view, than multivariate threshold models and can also be used when the number of time series is greater than two. In our empirical application, we provide a description of the data that reconciles the findings of different strands of applied work in this area. The most prominent features of our dataset are encompassed, identifying nonlinear adjustment effects in particular periods such as the Brady Plan introduction, the Mexican Peso crisis, the Russian crisis or the “dot-com” bubble burst. In addition, our out-of-sample analysis shows that the 2007 subprime mortgage crisis is handled satisfactorily. Finally, the description of policy shocks transmission to long-term bonds that our model offers prompts some interesting considerations: out of the three interest rates considered, the short-term rate is the one displaying the least compelling evidence for nonlinearity and for which the improvements in forecasting power are least obvious. This may suggest that the adjustment mechanism we have studied is particularly important for longer-term bond rates. This aspect may be investigated in more detail by considering a wider spectrum of maturities. This point will be the object of future research.