In late-2008, short-term nominal interest rates in the U.S. fell to their effective "zero lower bound" (see Bernanke et al., 2004 ). Since standard Gaussian term structure models do not rule out the possibility of negative model-implied yields, they provide a poor approximation to the behavior of nominal yields when the lower bound is binding (Kim and Singleton, 2012; Christensen and Rudebusch, 2013; Bauer and Rudebusch, 2013). Kim and Singleton (2012) find that shadow-rate models in the spirit of Black (1995) successfully capture yield-curve properties observed near the zero lower bound. However, arbitrage-free multi-factor versions of these models tend to be computationally intractable (Christensen and Rudebusch, 2013). Gorovoi and Linetsky (2004) show that bond prices in a one-factor shadow-rate model can be computed analytically by an eigenfunction expansion, but their approach does not generalize to multiple dimensions. Kim and Singleton (2012) and Ichiue and Ueno (2007) successfully estimate shadow-rate models with up to two factors, but they compute bond prices using discretization schemes that are subject to the curse of dimensionality. Christensen and Rudebusch (2013) use a yield formula proposed by Krippner (2012) to estimate shadow-rate Nelson-Siegel models with up to three factors, but 's derivation deviates from the usual no-arbitrage approach. Bauer and Rudebusch (2013) evaluate bond prices by Monte Carlo simulation for given model parameters from an unconstrained Gaussian term structure model, but they do not estimate a shadow-rate version of the model due to the computational burden.
This paper develops and applies a new technique for fast and accurate approximation of arbitrage-free zero-coupon bond yields in multi-factor Gaussian shadow-rate models of the term structure of interest rates. The computational complexity of the method does not increase with the number of yield curve factors, and, empirically, it produces yields that are accurate to within about half a basis point. The method is sufficiently fast to estimate a flexible, arbitrage-free, three-factor term structure model in which the shadow rate follows a Gaussian process. For illustration purposes, I estimate such a model by quasi-maximum likelihood on a sample of U.S. Treasury yields, using the unscented Kalman filter to account for the non-linear mapping between factors and yields.
Consider first the standard, continuous-time N-factor Gaussian term structure model. In particular, let be -dimensional standard Brownian motion on a complete probability space with canonical filtration . Assume there is a pricing measure on that is equivalent to , and denote by Brownian motion under as derived from Girsanov's Theorem (Karatzas and Shreve, 1991). Suppose latent factors (or states) representing uncertainty underlying term-structure securities follow the multivariate Ornstein-Uhlenbeck process
Since is a Gaussian process (Karatzas and Shreve, 1991), it follows from (2) that the short rate takes on strictly negative values with strictly positive probability. To modify the model in a way that accounts for the zero lower bound on nominal yields, Black (1995) proposes to think of as a shadow short rate (and, analogously, of , and as shadow bond price, shadow yield, and shadow instantaneous forward rate, respectively) and define the observed short rate as the shadow rate censored at zero:
The theoretical argument for a lower bound at zero on the nominal short rate (and hence on nominal yields) is based on arbitrage between bonds and currency Black (1995). In practice, the two assets may not be perfect substitutes for reasons such as convenience, default risk, or legal requirements. This may push the empirical lower bound into slightly negative or slightly positive territory. The derivations in Section 2 are easily modified to accommodate a lower bound at . In particular, suppose
The last term is equal to the expression for the yield when the lower bound is zero, except that is subtracted from the shadow rate. Therefore, since , when the lower bound is nonzero we can compute zero-coupon yields as if the bound were zero, with in place of , and then add to the final result. The lower bound can be set to a specific value based on a priori reasoning, or treated as a free parameter in estimation.
A central task in term-structure modeling is the (analytical and/or numerical) computation of arbitrage-free bond prices (and hence yields) based on equation (3). Section 4.1 reviews the standard approach using differential equations formalized by Duffie and Kan (1996), best suited to affine models. While it can be adapted to the shadow-rate framework, it loses much of its analytical tractability, and becomes computationally infeasible as the number of factors increases. Section 4.2 discusses an alternative method proposed by Krippner (2012). It defines a pseudo-forward rate that satisfies the lower bound (though differs from the arbitrage-free forward rate), and uses relationship (5) to approximate bond prices. Finally, Section 4.3 proposes a new approximation technique for yields in the shadow-rate model based on the expansion of a cumulant-generating function.
Like any conditional expectation of an -measurable random variable, (the time price of a zero-coupon bond maturing at time in the unconstrained Gaussian model, discounted to time 0 ) follows a martingale under . This is an immediate consequence of the definition of a martingale, after an application of the Law of Iterated Expectations. Using It's Lemma and the Martingale Representation Theorem, we can therefore represent by the function , where solves the partial differential equation (PDE)
A PDE analogous to (7) can be set up for the observed bond price in the shadow-rate model, . The only required modification is to replace the expression for the shadow short rate, , by that for the observed short rate, . Unfortunately, when this non-linearity is introduced, the separation-of-variables procedure no longer applies, and no solution as straightforward as (8) is available. It is possible to solve the modified version of (7) directly by numerical methods. This is the approach taken by Kim and Singleton (2012). It requires discretizing and on a multidimensional grid, which is computationally intensive and subject to the curse of dimensionality. Kim and Singleton (2012) therefore do not estimate models with more than two factors.
Krippner (2012) proposes an alternative approach to computing yields in shadow-rate models, which is implemented empirically by Christensen and Rudebusch (2013). It is based on an approximation to the forward rate . Substituting for from the shadow-rate version of (3), and differentiating, we obtain
Since the PDE approach to pricing bonds in shadow-rate models becomes computationally intractable as the number of factors increases, and the approach proposed by Krippner (2012) relies on a forward rate that is not equal to the arbitrage-free forward rate, I propose a new cumulant-based technique to approximating yields in Gaussian shadow-rate models.
The first-order approximation (19) is equivalent to the method proposed independently and contemporaneously by Ichiue and Ueno (2013). I present it here both for comparison and to assess its relative performance in Section 5 below. I will, however, mostly focus on the second-order approximation (20), which I argue is particularly promising a priori because it is exact in the Gaussian benchmark case.6 It can therefore be expected to perform well both for short maturities (where the higher-order terms in (18) are relatively small), and for long maturities as long as is small for large (in which case will behave approximately like a Gaussian process over sufficiently long horizons). Indeed, empirically, the second-order approximation turns out to be highly accurate across maturities both during normal times and when rates are low (see Section 5).
Evaluating the first- and second-order approximations (19) and (20) to zero-coupon yields requires computation of the first two cumulants (equivalently, centered moments) of . This subsection will be concerned with the first moment. As an initial step,
Once we know the first moment of , it remains to evaluate
where , , , and denotes the decumulative bivariate normal cdf.
That is, we can compute analytically up to the bivariate normal cdf, and we can then integrate this expression numerically over and to obtain the second cumulant of .
The following steps summarize the approximation procedure for zero-coupon yields for a given set of parameters and state vector :
Note that the complexity of the algorithm does not depend on the number of yield curve factors , so it is not subject to the curse of dimensionality in the same way that some other methods are.9
For illustration purposes, in Appendix B I apply the second-order approximation method to estimate a three-factor shadow-rate model of the U.S. Treasury term structure.
Section 4.3 argued intuitively that the second-order yield approximation (20) should be relatively precise. This section quantifies that claim. To get an initial sense of the relative numerical accuracy, I consider the stylized model used for illustration by Gorovoi and Linetsky (2004), and replicated for the same purpose in Krippner (2012). It is a one-factor model with , , , , and . Gorovoi and Linetsky (2004) derive model-implied yield curves for states corresponding to shadow short rates . The four panels of Figure 1 plot the model-implied yield curves for each initial state. Within each panel, I compare four different yield approximation schemes: Solving PDE (7) numerically (which, in a one-factor setting, is computationally feasible and can be considered the "exact" solution for comparison purposes), Krippner's (2012) approach described in Section 4.2, and the first- and second-order approximations proposed in Section 4.3. As the figure shows, the second-order approximation matches the exact PDE solution most closely, and consistently across states. The yield approximation error is uniformly less than one basis point. The first-order approximation generally overstates yields (an implication of the alternating nature of series expansion (18)), with approximation errors increasing both in yield maturity and the level of the shadow short rate (in both cases, the first-order approximation is off by an increasingly large convexity adjustment arising from Jensen's inequality). Krippner's (2012) method generally undershoots yields, and is relatively more accurate when the shadow short rate is higher. Why this is the case can be seen intuitively by comparing the -measure expressions for the arbitrage-free forward rate (12) and Krippner's (2012) approximate forward rate (16): While both use the same time short rate , Krippner's (2012) formula discounts by the shadow rate rather than the observed short rate. This means the discount factor tends to be larger than it should be when is low, reducing the covariance between discount factor and , and thus lowering the expectation of their product, .
To compare the relative performance of the different yield approximations in a more realistic empirical setting, I use the estimated model parameters and smoothed states from Appendix B.3 to compute model-implied yield curves for all dates in the sample.10 Since this is a three-factor model, solving PDE (7) numerically is no longer practicable. I therefore replace this benchmark by a simulated yield that consistently estimates the true yield based on randomly drawn
short-rate paths per sample date.11,12 Table 1 shows the mean simulation error of , and the root-mean-square errors (RMSE) against of the yield computed by Krippner's (2012) method, the first-order approximation defined in (19), and the second-order approximation defined in (20). The table is divided into two panels. The top panel shows errors for the sub-sample Jan 1990-Nov 2008 (when interest rates were at normal levels), and the bottom panel shows errors for the sub-sample Dec 2008-Dec 2012 (when the lower bound on nominal yields was binding at the short end of the yield curve). All methods are generally more precise at shorter maturities. As the first column in both panels shows, the simulated yields are accurate to within approximately one fifth of a basis point at the ten-year maturity point. As shown in the second column of the tables, Krippner's (2012) method produces ten-year yields that are accurate to about one basis point during normal times, and to within four basis points when the lower bound is binding. While the first-order method is more accurate when rates are low (the bottom panel), its errors remain substantial at the long end. The second-order method, the final column in the tables, produces ten-year yields that are accurate to approximately half a basis point, both during normal times and when the lower bound is binding.
To further illustrate the time-varying relative performance of the three approximation schemes, Figure 2 plots the difference over time between the simulated ten-year yield and the three approximated yields. Krippner's (2012) method and the second-order approximation appear to be equally precise in the first few years of the sample,
|6m Sub-sample Kam 1990-Nov 2008||0.04||0.04||0.05||0.04|
|1y Sub-sample Kam 1990-Nov 2008||0.06||0.06||0.18||0.06|
|2y Sub-sample Kam 1990-Nov 2008||0.09||0.10||0.88||0.10|
|3y Sub-sample Kam 1990-Nov 2008||0.12||0.14||2.26||0.13|
|4y Sub-sample Kam 1990-Nov 2008||0.14||0.18||4.22||0.15|
|5y Sub-sample Kam 1990-Nov 2008||0.16||0.27||6.60||0.17|
|7y Sub-sample Kam 1990-Nov 2008||0.19||0.47||12.22||0.21|
|10y Sub-sample Kam 1990-Nov 2008||0.21||0.93||21.81||0.35|
|6m Sub-sample Dec 2008-Dec 2012||0.01||0.01||0.01||0.01|
|1y Sub-sample Dec 2008-Dec 2012||0.02||0.04||0.04||0.02|
|2y Sub-sample Dec 2008-Dec 2012||0.05||0.19||0.33||0.05|
|3y Sub-sample Dec 2008-Dec 2012||0.07||0.51||1.07||0.07|
|4y Sub-sample Dec 2008-Dec 2012||0.10||0.94||2.31||0.09|
|5y Sub-sample Dec 2008-Dec 2012||0.12||1.42||3.99||0.12|
|7y Sub-sample Dec 2008-Dec 2012||0.15||2.43||8.38||0.23|
|10y Sub-sample Dec 2008-Dec 2012||0.17||3.87||16.63||0.52|
with fluctuations presumably largely due to simulation error.13 The discrepancy between simulated yield and Krippner's (2012) method increases over time as the level of yields declines, and exceeds five basis points by December 2012. The discrepancy between simulated yield and second-order approximation remains small and appears to show little systematic variation over time, perhaps trending up modestly towards the end of the sample. The first-order approximation has a large negative discrepancy initially, which shrinks over time but remains at a high absolute level even at the end of the sample.
Figure 2 also confirms that, just like in the simple one-factor model in Figure 1, the approximation errors under Krippner's (2012) method and the first-order scheme are largely systematic (rather than mere noise), in that the first-order approximation overstates arbitrage-free yields while Krippner's (2012) method tends to understate them.
In sum, the analysis above suggests that, empirically, the second-order yield approximation is accurate to within about one half of a basis point at maturities up to ten years, both during normal times and when the lower bound is binding. The approximation error is one order of magnitude smaller than both the model-implied observation error in yields (see Table 3) and the next best approximation method proposed by Krippner's (2012). In contrast, the first-order approximation is acceptable at most at the very short end of the yield curve.
To add perspective, the approximation error in is no greater than commonly accepted fitting error in the derivation of constant-maturity zero-coupon bond yields from observed coupon-bearing Treasuries (e.g., Gürkaynak et al., 2006). This puts the second-order approximation roughly on par with the numerical accuracy achieved
This paper develops an approximation to arbitrage-free zero coupon bond yields in Gaussian shadow-rate term structure models. The complexity of the scheme does not depend on the number of factors. Further, I demonstrate that the method is computationally feasible by estimating a three-factor shadow-rate model of the U.S. Treasury yield curve. Based on Monte Carlo simulation, I also show that the yield approximation is approximately as precise as conventional approaches that are considered to be "exact."
Consider the continuous time stochastic process defined in (1). The following derivations hold under both the -measure and the -measure, hence for notational simplicity I will suppress dependence of moments and parameters on the measure. Since is a Gaussian process, all its finite-dimensional distributions are Gaussian Karatzas and Shreve, 1991). In particular, for , the vectors are jointly conditionally Gaussian, with
This section derives two useful mathematical results involving the moments of censored Gaussian random variables.
Thus, it only remains to compute where is a standard normal random variable, for arbitrary . By direct computation of the integral defining the expectation,
where is the standard normal cdf. Further,
where , , , denotes the univariate standard normal pdf, denotes the univariate standard normal cdf, denotes the bivariate normal pdf when both variables have zero means, unit variances, and correlation , and and denote the corresponding cumulative and decumulative bivariate Gaussian distribution functions, where in particular .
The second and third terms can be evaluated using Lemma A.1. For the first term, it suffices to be able to compute for random variables and that are bivariate normal with zero means, unit variances, and correlation , and for arbitrary . Using the properties of , this expectation can be expanded as follows:
The first double integral is simply , the bivariate normal cdf. The second and third double integrals correspond to expected values of truncated bivariate normal random variables, and the last integral is the expected cross product of a truncated bivariate normal random vector. These expected values are known up to the univariate standard normal cdf and the bivariate normal cdf, respectively (see Rosenbaum (1961)). Using the formulas in Rosenbaum (1961) to evaluate the integrals in (A.9), and substituting into (A.10), we obtain (A.9) after simplification.
This appendix empirically estimates a three-factor Gaussian shadow-rate term structure model using the yield approximation methodology proposed in Section 4.3. The main purpose is to demonstrate the computational tractability of the method in the context of a realistic application. For more in-depth discussion and empirical analysis, see Kim and Priebsch (2013).
I use end-of-month zero-coupon U.S. Treasury yields from January 1990 through December 2012, for maturities of 6 months, 1-5, 7, and 10 years. I derive the 6-month yield from the corresponding T-bill quote, while longer-maturity zero yields are extracted from the CRSP U.S. Treasury Database using the unsmoothed Fama and Bliss (1987) methodology.15
I augment the yield data with survey forecasts from Blue Chip, interpolated to constant horizons of 1-4 quarters (available monthly), as well as annually out to 5 years and for 5-to-10 years (available every six months).16 Model-implied survey forecasts are subject to the same lower-bound constraint as yields,17 but their computation is substantially simpler: Forecasters report their expectation of the arithmetic mean of future observed short rates, . This is exactly (19) with the data-generating measure in place of pricing measure . Therefore, the first-order method described in Section 4.3 produces exact model-implied survey forecasts. Intuitively, unlike yields, survey forecasts are not subject to compounding, so there are no higher-order Jensen's inequality terms to consider.
Since the statistical properties of the term structure model laid out in Section 2 are formulated in terms of the latent state vector , but the data actually observed by the econometrician consist of yields, , and survey expectations, (see Appendix B.1), I set up a joint estimation and filtering problem to obtain estimates of the model's parameters .18 When discretely sampled at intervals , the state vector follows a first-order Gaussian vector autoregression,
Next, denote by the (non-linear) mapping from states and parameters to model-implied yields , and by the analogous mapping from states and parameters to model-implied survey forecasts . For estimation purposes, I compute through the second-order approximation (20), and through the exact first-order method discussed in Appendix B.1. To simplify notation, denote the stacked mapping by . If we assume that all yields and survey expectations are observed with iid additive Gaussian errors, we obtain the observation equation
The simple (linear) Kalman filter--optimal when measurement and observation equation are linear and all shocks are Gaussian--has been modified in a number of ways to accommodate nonlinearity as in (B.2). The unscented Kalman filter, proposed by Julier et al. (1995), aims to deliver improved accuracy and numerical stability relative to the more traditional extended Kalman filter, without substantially increasing the computational burden.19,20 The algorithm is described in detail in Wan and van der Merwe (2001). As a by-product of the filtering procedure, it conveniently produces estimates of the mean and covariance matrix of conditional on the econometrician's information set as of time . I use these to set up a quasi-maximum likelihood function based on (B.2),21 which I maximize numerically to obtain estimates of the parameters as well as their asymptotic standard errors (following Bollerslev and Wo oldridge, 1992).
To achieve econometric identification of the parameters in light of invariant transformations resulting in observationally equivalent models with different parameters (see Dai and Singleton, 2000 ), I follow Joslin et al. (2011) and impose the normalizations , , is diagonal and therefore completely determined by its ordered eigenvalues , and is lower triangular.
I estimate the model on the data set described in Appendix B.1, using the quasi-maximum likelihood (QML) procedure discussed in Appendix B.2. Table 2 displays the estimated model parameters , as well as their asymptotic standard errors.
|vector, row 1||- 0.1038|
|vector, row 1: standard error||(0.0226)|
|vector, row 2||- 0.3566|
|vector, row 2: standard error||(0.1177)|
|vector, row 3||- 0.8574|
|vector, row 3: standard error||(0.2876)|
|vector, row 1||- 0.0193|
|vector, row 1: standard error||(0.0052)|
|vector, row 2||- 0.0099|
|vector, row 2: standard error||(0.0224)|
|vector, row 3||0.0278|
|vector, row 3: standard error||(0.0233)|
|matrix, row 1, column 1||0.0268|
|matrix, row 1, column 1: standard error||(0.0084)|
|matrix, row 2, column 1||- 0.0324|
|matrix, row 2, column 1: standard error||(0.0110)|
|matrix, row 2, column 2||0.0416|
|matrix, row 2, column 2: standard error||(0.0295)|
|matrix, row 3, column 1||0.0068|
|matrix, row 3, column 1: standard error||(0.0100)|
|matrix, row 3, column 2||- 0.0397|
|matrix, row 3, column 2: standard error||(0.0302)|
|matrix, row 3, column 3||- 0.0090|
|matrix, row 3, column 3: standard error||(0.0007)|
|matrix, row 1, column 1||- 0.4679|
|matrix, row 1, column 1: standard error||(0.1574)|
|matrix, row 1, column 2||- 0.3415|
|matrix, row 1, column 2: standard error||(0.1490)|
|matrix, row 1, column 3||0.3785|
|matrix, row 1, column 3: standard error||(0.6798)|
|matrix, row 2, column 1||- 0.5752|
|matrix, row 2, column 1: standard error||(0.6303)|
|matrix, row 2, column 2||- 1.1881|
|matrix, row 2, column 2: standard error||(1.1335)|
|matrix, row 2, column 3||- 1.1875|
|matrix, row 2, column 3: standard error||(0.5740)|
|matrix, row 3, column 1||0.8908|
|matrix, row 3, column 1: standard error||(0.6982)|
|matrix, row 3, column 2||1.3060|
|matrix, row 3, column 2: standard error||(1.0641)|
|matrix, row 3, column 3||0.3990|
|matrix, row 3, column 3: standard error||(1.3561)|
Table 3 shows the QML-estimated standard deviations of the measurement errors in yields and survey variables ( in equation (B.2)). The average yield error is 8 basis points, and the average error in surveys is 21 basis points. For both yields and surveys, errors follow a U-shaped pattern, being largest at the short and long ends.
Figure 3 plots the model-implied shadow short rate over the sample period, based on the states implied by the Kalman smoother (that is, incorporating all information up to December 2012, the end of the sample). The shadow rate turned negative in December 2008, after the FOMC established a target federal funds rate range of 0 to 0.25 percent and the effective lower bound became binding, and has stayed negative through the end of the sample.
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