The interest rate term structure responds to shocks of all frequencies. For example, shocks to inflation are empirically associated with relatively equal changes in interest rate across all maturities, indicating a long-run impact. By contrast, shocks to real output growth tend to affect the short end of the yield curve moreso than long-term rates, suggesting a more intermediate horizon. Likewise, the monetary policies of central banks have historically most directly impacted short-term rates, but their effects can spread across the term structure through their influence on market expectations of future short rate movements (e.g., Balduzzi, Bertola, and Foresi (1997), Piazzesi (2005), and Heidari and Wu (2009)). At higher frequencies, large transactions of a particular fixed-income instrument can significantly move rates at the associated maturities, followed by quick dissipation along the yield curve through hedging practices.
In the dynamic term structure literature, no-arbitrage conditions impose strong restrictions on the cross-sectional relation between interest rates of different maturities. Despite the rapid progress in this literature over the past decade, the focus of empirical work remains on low-dimensional models, most typically with three factors. Consideration of substantially higher dimensional affine models has not previously been considered practical because of the classic curse of dimensionality that plagues model identification. A generic three-factor model can have over 20 free parameters, many of which cannot be estimated with statistical significance, and the number of parameters grows approximately quadratically with the factor dimension. However, restricting attention to low-dimensional models may inhibit empirical performance in several areas. First, low-dimensional models face limitations in the cross-sectional fitting of observed interest rates across different maturities. Although the fitting errors can appear small relative to the average interest rate level, the errors can become economically significant when one forms interest rate portfolios to neutralize the exposure to low-frequency movements (Bali, Heidari, and Wu (2009)) and similarly impact the pricing of interest rate options (Heidari and Wu (2008)). Second, low-dimensional term structure models often imply high cross-correlations between interest rates changes of different maturities, but the actual cross correlation estimates are often much lower (Dai and Singleton (2002)). Finally, low-dimensional models generate poor forecasting performances, often worse than the performance of a simple random walk assumption (Duffee (2002)).
In this paper, we overcome these limitations by developing a class of dynamic term structure models that are extremely parsimonious, effectively eliminating the curse of dimensionality for models within the class. The model employs a cascade structure, where a high-frequency component mean-reverts to a lower frequency component, which mean-reverts to another component of even lower frequency, and so on. This cascade structure provides a natural separation of different frequency factors in interest rate movements, completely removing the difficult and often tedious specification issue of factor rotation that typifies general affine models. By adding additional functional form assumptions to specify the progression of mean reversion rates, volatilities, and risk premia across frequencies, we achieve a framework in which the number of parameters required for identification is independent of the number of factors included in the model. In the simplest case, which we take as our base model, the progression of frequencies in our model follows a power law and merely five free parameters completely govern the term structure and its dynamics, irrespective the number of factors. This dimension-invariance feature eliminates the curse of dimensionality and allows us to estimate low and high-dimensional models with equal ease and accuracy.
To gauge the empirical performance of the models, we collect 13 years of data on six U.S. dollar LIBOR series with maturities from one to 12 months and nine swap rates with maturities from two to 30 years. We estimate 15 models with the number of factors going from one to 15, respectively, and compare their performances in pricing the 15 interest rate series. All five parameters are estimated with high statistical significance, regardless of the number of factors we use. Furthermore, the 15-factor model significantly outperforms lower-dimensional models, both statistically and economically. The high-dimensional model generates mean absolute pricing errors less than one basis point, an order of magnitude smaller than those generated from its three factor counterpart. The near-perfect pricing makes the model an ideal candidate to provide arbitrage-free yield curve stripping. The high-dimensional model also overcomes other known limitations of low-dimensional specifications, for example capturing the observed low cross correlations between interest rate changes of different maturities. Moreover, although the three-factor model performs worse than the random walk hypothesis as is typical of low-dimensional specifications, the 15-factor model significantly outperforms both the random walk and an autoregressive specification in predicting future interest rate movements from one to 12 month maturities. Finally, specification analysis shows that our functional form assumptions of iid risk and power-law mean reversion in the base model enjoy empirical support.
In related literature, Duffie and Kan (1996), Duffie, Pan, and Singleton (2000), and Duffie, Filipovic, and Schachermayer (2003) progressively generalize the conditions underlying a wide class of models wherein continuously compounded spot interest rates are affine functions of the state variables. Our model belongs to this general affine class. While their efforts are on generalization, our innovation lies in our specialization into a particular class of models that provide extreme parsimony in the form of invariance of the number of parameters to the dimensionality of the underlying factor structure. This dimension-invariance feature allows us to explore affine models with a large number of factors, which has not been practical under prior general specifications.
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A Multifrequency Theory of the Interest Rate Term Structure
