Yield curve estimation by kernel smoothing methods

Abstract

We introduce a new method for the estimation of discount functions, yield curves and forward curves from government issued coupon bonds. Our approach is nonparametric and does not assume a particular functional form for the discount function although we do show how to impose various restrictions in the estimation. Our method is based on kernel smoothing and is defined as the minimum of some localized population moment condition. The solution to the sample problem is not explicit and our estimation procedure is iterative, rather like the backfitting method of estimating additive nonparametric models. We establish the asymptotic normality of our methods using the asymptotic representation of our estimator as an infinite series with declining coefficients. The rate of convergence is standard for one dimensional nonparametric regression. We investigate the finite sample performance of our method, in comparison with other well-established methods, in a small simulation experiment.

Introduction

The term structure of interest rates is a central concept in monetary and financial economics. Prices of fixed income securities like bonds, swaps, and mortgage backed bonds (MBB's) are functions of the yield curve, and pricing of derivatives also depends on the yield curve. The spread between long and short term interest rates carries information about the level of future interest rates, see for example Campbell and Shiller (1991) and Engsted and Tanggaard (1995). The slope of the yield curve has frequently been used in empirical studies as a predictor of future inflation and national incomes, see Frankel and Lown (1994) and Estrella and Mishkin (1998) for example. Therefore, estimation of yield curves has had a long tradition among financial researchers and practitioners. See Campbell et al. (1997) for further discussion.

A fundamental problem is that the yields to maturity on coupon bonds are not directly comparable between bonds with different maturities or coupons. Thus, there is a need for a standardized way of measuring the term structure of interest rates. One such standard is the yield curve of zero-coupon bonds issued by sovereign lenders.

The construction of this yield curve poses several problems for applied research. First, many governments do not issue longer term (i.e., greater than 1–2 years) zero-coupon bonds. Hence the yield curve must be inferred from other instruments. A simple solution can be derived from the law of one price by assuming the absence of arbitrage. Arbitrage in the bond market will cause the price p of any bond (coupon or zero) with payments b(τ_j) at time τ_j to be equal to the discounted value of the future cash flow π=∑_j=1^mb(τ_j)d(τ_j), where the discount factor is d(τ_j) at time τ_j. The future income stream, b(τ₁),…,b(τ_m), is assumed known and non-random. The second problem is that, in practice, small pricing errors perhaps due to non-synchronous trading, taxation, illiquidity, and bid–ask spreads necessitate adding an error term to π. The error term should be sufficiently small to ensure that they do not represent (gross) violations of the law of one price (no-arbitrage condition).¹

The statistical problem we address is to estimate the function d(·) from a sample of coupon paying bonds. Note that, based on a continuous time approximation, we have d(t)=exp(−ty(t)), where y(t) is the yield curve, and $\text{y(t)=}\text{exp}\text{(−}\text{∫}\text{0}\text{t}\text{f(s)}\text{d}\text{s)}$, where f(t) is the forward curve, see Anderson et al. (1996, pp. 12–13). Both of these relationships are invertible, so that knowing d is equivalent to knowing y or f.

Following the seminal work of McCulloch (1971), the standard approach to estimation here is to assume a parametric specification for d(t) or y(t) or f(t) and to use linear or nonlinear least squares to estimate the unknown parameters. For example, McCulloch 1971, McCulloch 1975, Shea (1984) use regression splines for d(t), Chambers et al. (1984) use polynomials for y(t), Vasicek and Fong (1982) use exponential splines for d(t), while Nelson and Siegel (1988) propose powers of exponentials for y(t). An approach based on linear programming methods has been suggested by Schaefer (1981). If the specification is considered parametric, i.e., to be a complete and correct representation of the functions of interest, i.e., the mean, then standard asymptotic theory can be used to derive the limiting distribution of the estimator and to justify confidence intervals obtained from this. However, a number of these authors are arguing against adherence to any fixed model and really are viewing the problem as being nonparametric. Some recent studies by Fisher et al. (1995) and Tanggaard (1997) have taken this line. When we view the estimation problem as nonparametric, there is little existing theory regarding the distribution of the estimators; for example, no one has established the asymptotic distribution of the spline estimates discussed above.

We adopt a nonparametric approach in which we do not a priori specify the functional form of the discount function or forward curve. We shall suppose that the discount function d(·) is a continuous and indeed smooth function of time to maturity. Although this is not guaranteed by purely arbitrage arguments, it does seem plausible. We propose a new class of methods for estimating d(·) based on perhaps the most central of all smoothing methods, the kernel method. The flexibility of our method is very important in practical applications because parametric estimates are often flawed by specification biases. It is not immediately obvious how to estimate the function d(·) by kernel methods, since this function affects the mean function indirectly through a convolution with the payment function. We first interpret the function d as the solution of some population mean squared error criterion. We then smooth the sample version of this to obtain an empirical criterion function, which is regular enough to provide consistent estimates. Several versions of the localization are possible including local constant and local linear [which has some well-known advantages in other contexts, see Tsybakov (1986) and Fan (1992)]. It turns out that our methods do not generally have explicit solutions, i.e., our estimator is defined as the solution of a linear integral equation. In practice, our solutions are defined through the method of successive approximations.² We also give a ‘backfitting’ interpretation to our procedure, as in Opsomer and Ruppert (1997) and Mammen et al. (1999). We establish the convergence of our iterative scheme and establish the asymptotic properties of the estimator. We obtain a representation of our estimator as an infinite series with declining coefficients, which thereby provides its asymptotic distribution—it is asymptotically normal at the standard rate of convergence for one-dimensional kernel regression. The asymptotic distribution of the implied estimators of y(t) and f(t) can be easily obtained by the delta method. Our regularity conditions are ‘high level’, but we show how they are satisfied in some leading cases. We also exploit the relationships with the yield curve and forward curve to suggest alternative methods, thus we write d(·)=ψ(θ(·)) for some known function ψ , making θ now the object of estimation. The purpose of this is to give some added flexibility and/or to enforce consistency with theory. For example, by taking d(t)=exp(−ty(t)) we can directly impose the restrictions that d(0)=1 and d(t)>0 for all t, at the same time we are directly estimating the yield curve itself.

We point out that the estimation problem is similar to that considered in Engle et al. (1986) in which electricity demand over a billing period is modelled as a sum of individual daily demands each determined by temperature on the day concerned. They used splines, which effectively parameterize the function d and make the estimation problem standard nonlinear regression. They did not provide any asymptotic theory to justify their approach, at least not for the pointwise distribution of the nonparametric part. A similar estimation problem occurs quite widely with grouped data. For example, Chesher (1997) estimates the individual nutrient intake–age relationship from household level intake and individual characteristics like age. Again, he used splines but did not provide any justification for the validity of his method. A related problem arises in nonparametric simultaneous equations (Newey and Powell, 1988) and in estimating solutions of integral equations (Wahba, 1990; Nychka et al., 1984; O'Sullivan, 1986). See also Hausman and Newey (1995) for a related problem involving differential equations. With minor modifications we can provide new estimators in all these situations and find their asymptotic distribution.

In Section 2 we discuss (for reasons of completeness) smoothing of pure discount bonds. In Section 3 we present our new methods for smoothing the yield curve. In Section 3.1 we present the local constant version of our estimate, i.e., the object of interest is the discount function, while Section 3.2 gives the local linear extension. In Section 3.3 we describe the local constant exponential version and its one-step approximation. We present the asymptotic properties of our methods in Section 4. In Section 5 we provide a small simulation study that compares our method with some alternatives. Proofs are given in the appendix. In the sequel all integrals are Lebesgue integrals, and run from −∞ to +∞ unless otherwise stated.