2. THE DIFFUSION PROCESS

We assume that X = {X_t,t ≥ 0} is a diffusion process, with state space I = (l,r), governed by the stochastic differential equation

where {W_t,t ≥ 0} is a (standard) Wiener process, a and b are unknown coefficients or functions, and x is either a constant value or a random value

-measurable independent of W_t. We assume that a and b have continuous derivatives.

Let

be the scale density function (z₀ is an arbitrary point inside I) and m(u) = (b²(u)s(u))⁻¹ the speed density function. Let

where l < x₁ < x < x₂ < r. We now present a set of five assumptions that are used throughout the paper.

According to Arnold (1974, p. 114), if the infinitesimal coefficients a and b have continuous derivatives with respect to x, then there exists a unique continuous process that is defined up to a random explosion time η in the interval t₀ < η ≤ +∞. The A1 condition assures that P[η = +∞|X₀ = x] = 1 (Ikeda and Watanabe, 1981, pp. 362–363). Furthermore, the boundaries l and r are neither attracting nor attainable (see Karlin and Taylor, 1981, Ch. 15), and the process is recurrent, i.e., P[T_y < ∞|X₀ = x] = 1 for every x,y ∈ I where T_y = inf{t ≥ 0,X_t = y} (Ikeda and Watanabe, 1981, Theorem 3.1, Ch. VI). Roughly speaking, the boundaries l and r are never attained although every finite point can be reached with probability one in finite time. Global Lipschitz and growth conditions, which fail to be satisfied for many interesting models in economics in finance (Aït-Sahalia, 1996), are not needed in the presence of the previous assumptions. The A1 condition is not very strong: e.g., the standard Brownian motion satisfies the A1 condition (Karlin and Taylor, 1981, p. 228). Actually, every process with zero drift a(x) = 0 (and b(x) > 0) satisfies the A1 condition.

The A1 and A2 conditions assure that X is ergodic and the invariant distribution P⁰ has density

with respect to the Lebesgue measure (Skorokhod, 1989, Theorem 16). The expression p(x) is usually denoted as stationary density.

Assumption A3 together with A1 and A2 implies that X is stationary (Arnold, 1974). Assumption A3 can, in some cases, be replaced by the following assumption: X₀ is a random variable with mean μ and variance σ² such that ∫ xp(x) dx = μ < +∞ and ∫(x − μ)²p(x) dx = σ² < +∞. In this case, X is a covariance-stationary process. Even if the A3 condition does not hold, it is known that when t is sufficiently large, the distribution of the ergodic process X_t is well approximated by the distribution with density p(x). For simplicity, in our nonparametric estimation framework, we assume stationarity, although more broad assumptions can be considered (see Bandi and Phillips, 2003). As we will see, our estimators have important advantages over the Florens-Zmirou (1993) and Jiang and Knight (1997) estimator when the process exhibits strong reversion effects, which occurs typically in a stationary framework.

These conditions are discussed in Chen, Hansen, and Carrasco (1998) and are similar to ones proposed by Hansen and Scheinkman (1995). Under Assumption A4 the process is ρ-mixing (see Chen et al., 1998). Technically, for a Markov process, the notion of ρ-mixing requires the conditional expectations operator for any interval of time to be a strong contraction for all functions with zero mean and finite variance. As a consequence, the jth autocovariance of f (X_t) tends to zero at exponential rate as j → +∞, for all functions f such that ∫ f (x)p(x) dx = 0 and ∫ f²(x)p(x) dx < +∞ (see Hansen and Scheinkman, 1995, Proposition 8; Florens-Zmirou, 1989). We notice that even if the drift is zero or converges to zero, Assumption A4 can hold, provided that volatility grows at least linearly.

The kernel function K that we will use to define our estimators satisfies the following assumption.

3. BIAS REDUCTION IN NONPARAMETRIC ESTIMATION

Florens-Zmirou (1993) proposes the estimator

for b²(x) where K is the uniform kernel. Under some conditions, including Δ_n → 0, he proves that S_n(x) converges in mean square error (MSE) to b²(x) and that the asymptotic distribution of S_n(x) (suitably standardized) is normal. Jiang and Knight (1997) extend these results for the Gaussian kernel, and Bandi and Phillips (2003) consider a more general estimator under broad assumptions on the data generating process. Although S_n(x) requires only mild regularity conditions to be consistent—in particular the time span t_n can be fixed—it is necessary that the step of discretization go to zero (see, e.g., Bandi and Phillips, 2003). However, in most applications the step of discretization is generally fixed (i.e., Δ_n = Δ as n increases). For example, the series could be daily, weekly, or monthly. Even when Δ_n decreases as n increases (which is possible if one passes, e.g., from weekly to daily observations), the condition Δ_n → 0 is too strong as it requires continuous sampling observations at the limit. When high frequency data are available (i.e., Δ_n is small), the advantages of using all the available information are not clear. In fact, high frequency data are replete with empirical anomalies, such as heteroskedasticity, heterokurtosis, nonsynchronous trading, and many other market microstructure “frictions,” which can obviously contaminate the estimates (see Sawyer, 1993; Andersen, Bollerslev, Diebold, and Labys, 2001).

3.1. Behavior of the Nonparametric Estimators When the Step of Discretization Is Fixed

In this section we quantify the bias of the Florens-Zmirou (1993) and Jiang and Knight (1997) estimator for the diffusion coefficient when the step of discretization is fixed and then we propose a bias adjustment that partially compensates for the distortion. We assume the A1–A5 conditions in all theorems.

THEOREM 1. If Δ_n = Δ (constant), t_n → +∞, h_n → 0, nh_n → +∞ as n → +∞, and E [(X_Δ − X₀)⁴] < +∞ then

where f (x) = b²(x)a′(x) + a(x)b(x)b′(x) + ½b²(x)(b′(x))² + ½b³(x)b′′(x).

It is clear that the consistency of S_n depends crucially on whether Δ_n goes to zero or not. When Δ_n = Δ is constant the bias is of order O(Δ). This bias can vary considerably in the state space I. In general, it is minimum at point x₀ such that a(x₀) = 0, because in this case several terms associated with the bias in the stochastic limit (3) vanish. On the contrary, the bias increases in those points x such that |a(x)| is high, which in stationary framework corresponds to the interval where the process exhibits stronger reversion effects.

Based on these results we propose a bias correction with the following estimator:

We have the following theorem.

THEOREM 2. If Δ_n = Δ (constant), t_n → +∞, h_n → 0, nh_n → +∞ as n → +∞, and E [(X_Δ − X₀ − a(X₀)Δ)⁴] < +∞ then

where f (x) is given in Theorem 1.

Therefore, with Δ_n = Δ constant, the V_n(x) estimator generally has smaller bias than the S_n estimator. The bias reduction occurs because of the elimination of the a²(x)Δ term in the stochastic limit in equation (3). Actually, when Δ_n is constant, the evaluation of the errors (X_ti+1 − X_ti − a(X_ti)Δ_n)²/Δ_n enables a more accurate estimation of the infinitesimal conditional variance than the errors (X_ti+1 − X_ti)²/Δ_n. However, the V_n estimator is unfeasible because the function a(x) is unknown. As in Aït-Sahalia (1996), we can assume a(x) linear, i.e., of type a(x) = θ₁ − θ₂ x, θ₁,θ₂ > 0 where θ₁ and θ₂ can easily be estimated using semiparametric methods. However, if we do not want to rely on the parametric specification of the drift, we should use a nonparametric estimator for a(x). In this case, we propose

where T_n(x) is a nonparametric estimator of a(x). We consider

This estimator is a simpler version of a more general estimator proposed in Bandi and Phillips (2003). The Jiang and Knight (1997) estimator for a(x) cannot, in principle, be used here because it depends on the diffusion coefficient estimator (see however Section 5). The following theorem characterizes the bias of the T_n estimator.

THEOREM 3. If Δ_n = Δ (constant), t_n → +∞, h_n → 0, nh_n → +∞ as n → +∞, and E [(X_Δ − X₀)²] < +∞ then¹

I thank a referee for pointing this out.

Therefore, when Δ is fixed, the drift estimator T_n has a bias of order O(Δ). Because V_n* depends explicitly on T_n, it is important to identify the impact that the bias of the drift estimator has on the bias of the diffusion coefficient estimator. The following theorem addresses this question.

THEOREM 4. If Δ_n = Δ (constant), t_n → +∞, h_n → 0, nh_n → +∞ as n → +∞, and E [(X_Δ − X₀ − a(X₀)Δ)⁴] < +∞ then

where f (x) is given in Theorem 1.

Theorem 4 shows that the impact of the bias of the T_n estimator on the bias of the V_n* estimator is negligible. In fact, equation (6) is basically the same expression obtained with the V_n estimator (see expression (4)). With the V_n* estimator we obtain a bias adjustment gain that is similar to that of the V_n(x) estimator (ignoring the terms higher than O(Δ²)). A simple justification for this result is the following. Although the bias of the a(x) estimation is of order O(Δ) (by Theorem 3), the bias involving the estimation of the a²(x)Δ function, which is the term we correct and eliminate in equation (3), is of order O(Δ³) (see the proof of Theorem 4 in Appendix B). Thus, the bias in estimating the a(x) coefficient only appears in the stochastic limit of V_n* in the terms of order O(Δ³).

The advantages of using the V_n*(x) (and V_n(x)) estimator over S_n(x) are more evident when x belongs to the interval where |a(x)| is high, which in a stationary framework (with a nonzero drift) corresponds to the interval where the process exhibits stronger reversion effects. This is clear from equations (3) and (6). Obviously, if stationarity is volatility-induced with zero drift then there are no advantages of using the proposed estimators.

3.2. Limiting Properties as the Frequency of Observations Increases

We now address the limiting properties of V_n and V_n* estimators as Δ_n → 0.

THEOREM 5.

(a) If Δ_n → 0, h_n → 0, nh_n → +∞ as n → +∞, and E [(X_Δn − X₀ − a(X₀)Δ_n)⁴] < +∞ then

Let

.

(b) If, in addition to the previous conditions, nh_n³ → 0 and

as n → +∞

²

Let us assume t_n = O(n^γ), h_n = O(n^β), and consequently Δ_n = t_n /n = O(n^γ−1). To assure the conditions Δ_n → 0, h_n → 0, nh_n → +∞, nh_n³ → 0, and

as n → +∞ it is necessary and sufficient that β and γ be such that −1 < β < −1/3 and 0 ≤ γ < ½ − β/2.

then

where

.

Theorem 5 shows that our asymptotic results are essentially equal to those of Florens-Zmirou (1993) and Jiang and Knight (1997). In fact, we get

, and, if we use the uniform kernel (K(u) = 1/2 if |u| < 1 and K(u) = 0 if |u| ≥ 1), as in Florens-Zmirou (1993), we conclude that, by the asymptotic equivalence theorem,

and

have the same asymptotic distribution.

³

With uniform kernel, we have K₂ = ½ and

thus,

, as in Florens-Zmirou (1993), Theorem 1.

Thus, in asymptotic framework with Δ_n → 0, V_n and S_n are equivalent estimators. Our claim, however, is that our estimator is preferable when Δ_n = Δ is constant, as we have pointed out.

We observe that in Theorem 5 it is not necessary that the observation period go to infinity, i.e., t_n can be constant

⁴

From note 2 we can conclude that γ can be zero where γ is defined in t_n = O(n^γ). That is, consistency can be obtained with t_n constant. In this case, the condition

is redundant.

as n → +∞ (in this case, the condition

is redundant). In fact, it is known that every finite interval of continuous observations contains all the information needed about the diffusion coefficient b²(x) to estimate it with probability one, provided that x is visited by X (see Brown and Hewitt, 1975; Bandi and Phillips, 2003).

The next theorem analyzes the asymptotic behavior of the V_n* estimator.

THEOREM 6. Let the assumptions of the previous theorem hold. In addition suppose that t_n h_n → +∞ (thereby t_n → +∞).⁵

To assure the conditions Δ_n → 0, h_n → 0, nh_n → +∞, nh_n³ → 0,

, and t_n h_n → +∞ as n → +∞ it is necessary and sufficient that −1 < β < −1/3, −β < γ < ½ − β/2 (see notations in note 2), which defines a nonempty convex open set in

. Obviously, the condition γ > −β means that the time span cannot be fixed, i.e., we must have γ > 0.

Then

In Theorem 6 one has the additional condition t_n h_n → +∞. Because h_n → 0 must hold, we have to impose t_n → +∞. This is an obvious consequence of the fact that we use the T_n drift estimator in the expression of V_n* and direct identification of the drift coefficient based on this kernel estimator is impossible over a fixed sampling period. Several authors (e.g., Bandi and Phillips, 2003; Phillips and Yu, 2000; Jiang and Knight, 1997) have stressed that the estimation of the drift is usually empirically and theoretically more difficult than the estimation of the diffusion coefficient. Therefore, there is no benefit in using the V_n* estimator (or even the V_n estimator) when the step of discretization tend to zero. In fact, S_n and V_n* are asymptotically equivalent estimators. Furthermore, unlike the S_n estimator, V_n* requires that the time span tend to infinity. Nevertheless, when Δ_n = Δ is fixed, the estimator V_n* is preferable because it reduces the bias of the S_n estimator, as we have stressed.

4. EVALUATION OF THE BIAS USING A PARAMETRIC INTEREST RATE MODEL

In this section, we compare the bias of the S_n and V_n* estimators from some specified (parametric) models in the framework n → +∞. Given a(x) and b²(x) it is easy, from equations (3) and (6), to evaluate the bias of the S_n and V_n* estimators as n → +∞ for a fixed step of discretization Δ (thereby t_n = nΔ → +∞). In fact, assuming that O(Δ²) is negligible, the biases are given by the expressions

where f (x) = b²(x)a′(x) + a(x)b(x)b′(x) + ½b²(x)(b′(x))² + ½b³(x)b′′(x) (note that we suppress the subscript n associated with the estimators because we are dealing with the case n → +∞). It follows that S(x) = b²(x) + a²(x)Δ + f (x)Δ and V*(x) = b²(x) + f (x)Δ (assuming n → +∞). We start by selecting the coefficients a(x) and b²(x) based on some estimated models by Aït-Sahalia (1999) using monthly Fed funds data from January 1963 through December 1998. From Table VI of Aït-Sahalia (1999) we select two models:

where the parameters (expressed in annualized form) were estimated by the maximum likelihood method and the asymptotic standard errors are shown below the parameters. From these models we evaluate the functions Bias₁(x), Bias₂(x), S(x) = b²(x) + a²(x)Δ + f (x)Δ, and V*(x) = b²(x) + f (x)Δ running x (interest rate expressed in annualized form) in the interval]0,0.22] (the maximum value observed in the period was 0.2236). The constant Δ is equal to

. Before we present the results, it is worth mentioning that this simulation seemingly does not best serve our purposes because, in the interest rate estimation framework, the drift coefficient is numerically close to zero. As we have pointed out, this is the case where V_n* and S_n present similar results. However, we will see, even in this situation, that the V_n* estimator seems to be better. Then to appreciate how V_n* can improve the S_n estimates, we slightly increase the parameters associated with the reversion effects.

From Figures 1 and 2 one can conclude that in low/moderate values of interest rate both nonparametric estimators produce similar and good estimates whereas in high/very high values, the S_n estimates are generally worse than the V_n* estimates. We give a short explanation for these results. Coherently with most empirical studies of interest rate series, we observe the following pattern. In the interval of moderate/low values not only is the volatility very low but also the process displays martingale behavior, in the sense that a(x) ≈ 0 for x belonging to the interval of moderate/low values. In this interval, the asymptotic bias of S_n (and also of V_n*) is in general very low. By contrast, in the interval of high/very high values of interest rates, the volatility increases significantly and the process exhibits stronger reversion effects. It is in this interval, where |a(x)| is higher, that S_n produces greater bias and, simultaneously, that V_n* shows significant improvement over S_n.

Behavior of the S and V* estimators (as n → +∞) when and the model is dXt = 0.261(0.0717 − Xt) dt + 0.02237dWt.

Behavior of the S and V* estimators (as n → +∞) when and the model is .

It is now interesting to compare the S_n and V_n* estimates for the diffusion coefficient using the Fed fund data referred to previously.

From Figure 3 it seems that the volatility at high values of interest rate, although considerable, is not as great as the S_n estimator suggests. However, no definitive conclusions can be made because in the areas where the drift coefficient is substantially negative (high in absolute value) there are not sufficient observations to draw precise statistical inference (for a discussion on this topic, see Bandi, 2002).

Nonparametric diffusion estimation—Fed fund data.

Let us consider a slight increase in the parameters of the preceding models associated with the reversion effects (everything else remains unchanged):

(in both cases, the parameter β defined in the drift function a(x) = β(μ − x) has changed to 0.9). Although these models certainly no longer reflect the behavior of the interest rate process, they allow us to appreciate the advantages of the proposed estimator when stronger reversion effects characterize the dynamics of the conditional mean. In fact, as can be seen from Figures 4 and 5, greater improvement is achieved by the V_n* estimates over the S_n estimates.

Behavior of the S and V* estimators (as n → +∞) when and the model is dXt = 0.9(0.0717 − Xt)dt + 0.02237dWt.

Behavior of the S and V* estimators (as n → +∞) when and the model is .