Processes with Volatility-Induced Stationarity. An Application for Interest Rates. João Nicolau

Transcription

1 Processes with Volatility-Induced Stationarity. An Application for Interest Rates João Nicolau Instituto Superior de Economia e Gestão/Univers. Técnica de Lisboa Postal address: ISEG, Rua do Quelhas 6, Lisboa, Portugal nicolau@iseg.utl.pt Abstract: In this paper we propose a re nement of the existing de nition of volatility-induced stationarity that allows us to distinguish between processes with drift and di usion induced stationarity and processes with pure volatility-induced stationarity. We also propose a classi cation of stationary processes with volatilityinduced stationarity according to the volatility that is needed to inject stationarity. Processes with volatility-induced stationarity are potentially applicable to interest rate time-series since, as has been acknowledged, mean-reversion e ects occur mainly in periods of high volatility. As such, we provide evidence that the logarithm of the Fed funds rate can be modelled as a local martingale with volatility-induced stationarity. Key Words and Phrases: stochastic di erential equations, di usion processes, parametric estimation, non-parametric estimation Running Head: Volatility-Induced Stationarity

2 Introduction Estimation of short-term interest rate processes has recently received much of attention. Several models have been proposed, ranging from discrete-time processes such as GARCH, stochastic volatility and regime-switching models, to di usion (with or without stochastic volatility) and jump-di usion processes (see Hong et al., 2004, for a recent overview). Most studies on short-term interest rate processes have shown at least two main facts. Firstly, the mean-reverting e ect is very weak (see, for example, Chan et al., 992 or Bandi, 2002). In fact, the stationarity of short-term interest rate processes is quite dubious. The usual unit root tests do not clearly either reject or accept the hypothesis of stationarity. Since interest rate processes are bounded by a lower (zero) and upper ( nite) value a pure unit root hypothesis seems impossible since a unit root process goes to or with probability one as time goes to. Some authors have addressed this question. The issue is how to reconcile an apparent absence of mean-reverting e ects with the fact that the interest rate is a bounded (and possibly stationary) process. Aït-Sahalia (996) using nonparametric methods suggests a non-linear drift coe cient. The drift is zero in some "central" region of the state space (so the process behaves like a martingale in this region) and mean-reverting at the edges of the range of the process, which is su cient to assure stationarity. Pritsker (998) and Chapman and Pearson (2000), however, have pointed out that the ndings of nonlinear drifts could be spurious due to the poor nite sample performance of the nonparametric estimators at the boundary region 2

3 of the interest rate data. Nevertheless, the martingale behaviour of the interest rate process seems clear over most of its range. While Aït-Sahalia (996) suggests that stationarity can be drift-induced, Conley et al. (997) (CHLS, henceforth) suggest that stationarity is primarily volatility induced. Their point is that what matters for reversion is a pull measure de ned as the ratio a (x) = (2b 2 (x)) ; where a and b are the drift and di usion in nitesimal coe cients, respectively. In fact, it has been observed that higher volatility periods are associated with mean reversion e ects. Thus, the CHLS hypothesis is that higher volatility injects stationarity in the data. The second (well known) fact is that the volatility of interest rates is mainly level dependent and highly persistent. The higher (lower) the interest rate is the higher (lower) the volatility. The volatility persistence can thus be partially attributed to the level persistence of the interest rate. The hypothesis of CHLS is interesting since volatility-induced stationarity can explain martingale behaviour (fact one), level volatility persistence (fact two), and mean-reversion. To illustrate these ideas and show how volatility can inject stationarity we present in gure a simulated path from the SDE dx t = + X 2 t dwt : () It is worth mentioning that the Euler scheme Y ti = Y ti + + Y 2 t i pti t i " ti ; " ti i.i.d.n (0; ) cannot be used since Y explodes as t i! (see Richter, 2002). For a method to 3

4 Figure : Simulated Path from the SDE dx t = ( + X 2 t ) dw t simulate X, see Nicolau (2005): Since the SDE () has zero drift, we could expect a random walk behaviour. Nevertheless, gure shows that the simulated trajectory of X exhibits reversion e ects towards zero, which is assured solely by the structure of the di usion coe cient. It is the volatility that induces stationarity. In the neighbourhood of zero the volatility is low so the process tends to spend more time in this interval. If there is a shock, the process moves away from zero and the volatility increases (since the di usion coe cient is + x 2 ) which, in turn, increases the probability that X crosses zero again. The process can reach extreme peaks in a very short time but quickly returns to the neighbourhood of zero. It can be proved, in fact, that X is a stationary process with stationary density p (x) = 2= ( + x 2 ) 2 and stationary moments E [X] = R xp (x) dx = 0 and V ar [X] = R x 2 p (x) dx = and that the conditional expectation decays exponentially fast to the stationary mean (see the example in appendix ). Thus, X is a stationary local martingale but not a martingale since 4

5 E [X t j X 0 ] converges to the stationary mean as t! and not to X 0 as would be required if X was a martingale (Bibby and Sørensen, 995, give another example of a stationary local martingale that is not a martingale). The main objectives of this paper are the following: to discuss volatility-induced stationarity (VIS), re ne the existing de nition of VIS (section 2) and illustrate that the logarithm of the Fed funds rate can be modelled as a local martingale process with VIS (section 3). 2 De nitions of Volatility-Induced Stationarity In this section we present a re nement of the existing VIS de nition. For simplicity, we always assume that the initial value of an ergodic process has a stationary density; thus, we do not distinguish between ergodic and stationary processes. To our knowledge, CHLS were the rst to discuss VIS ideas. Richter (2003) generalizes the de nition of CHLS. Basically, their de nition states that the stationary process X (solution of the stochastic di erential equation (SDE) dx t = a (X t ) dt + b (X t ) dw t ) has VIS at boundaries l = and r = if lim s X (x) < and lim s X (x) < (2) x!l x!r where s X is the scale density, s X (x) = exp Z x z 0 2a (u) b 2 (u) du (z 0 is an arbitrary value). 5

6 We note that a process with VIS is a stationary process (by de nition). In order to guarantee stationarity one has to assume that S X (l; x] = lim x!l R x x s X (u) du =, S X [x; r) = lim x2!r R x2 x s X (u) du = for x 2 (l; r) and R r l m X (x) dx < where m X (u) = (b 2 (u) s X (u)) (see appendix, assumptions A and A2). If these conditions hold with s X (x) nite then, according to previous de nition, X has VIS at the boundaries. The example we saw in the previous section, the solution of dx t = ( + X 2 t ) dw t ; has VIS at boundaries l = and r = since X is a stationary process and s X (x) = (in appendix we analyse this case). In contrast, some wellknown stationary di usions do not satisfy the VIS condition of CHLS and Richter. For example, the CIR process, introduced by Cox et al. (985), to model instantaneous spot rate, with in nitesimal coe cients, a (x) = ( x) and b (x) = p x ( > 0; > 0) does not verify the VIS condition since s X (x) = exp Z x z 0 2a (u) =b 2 (u) du = e2(x )=2 x 2=2 (z 0 = ) and s X (x)! as x! or x! 0 (in this case l = 0). Our point is that the VIS de nition of CHLS and Richter has the disadvantage that the source of stationarity is not clearly identi ed. To be more precise, consider the following example. According to the CHLS de nition, the solution of dx t = X t dt + p + X 4 t dw t has VIS at boundaries l = and r = ; since X is a stationary process (the A and A2 conditions in appendix are satis ed) and the limit of s X (x) = exp f=4 =4 arctan (=x 2 )g is nite when x! (lim x! s X (x) = exp (=4)): However, we cannot conclude that stationarity is only ensured by the 6

7 di usion coe cient, as the solution of dy t = Y t dt + dw t is stationary as well. Neither can we conclude that stationarity is ensured by the drift coe cient, as the solution of dy t = p + Y 4 t dw t is also stationary. In this case both in nitesimal coe cients independently assure stationarity. The example above shows that the problem with the CHLS de nition is that it does not exclude mean-reversion e ects and thus stationarity can also be driftinduced. Another way to con rm this idea is verifying that lim x! xa (x) < 0 (which implies mean-reversion e ects) is not ruled out by condition (2). To characterize this type of VIS, i.e. the case where both in nitesimal coe cients independently induce stationarity we state the following. Whenever a stationary process X satis es condition (2) and lim x! xa (x) < 0 we say that X has VIS of type one (VIS). The condition lim x! xa (x) < 0 implies mean-reversion e ects and this condition jointly with condition (2) implies that b 2 (x) dominates asymptotically a (x) in the sense that a (x) =b 2 (x)! 0 as jxj! : Otherwise, the integral R x z 0 2a (u) =b 2 (u) du is divergent. We now discuss other forms of VIS and propose a more intuitive version of VIS. Consider the following SDEs dx t = a (X t ) dt + b (X t ) dw t (3) dy t = a (Y t ) dt + dw t ; (4) where 0 < < " (" is a xed but controllable small value). We call Y the associated process of X: For example, if dx t = ( X t ) dt + p + X 2 t dw t ; the associated 7

8 process is the solution of dy t = ( Y t ) dt + dw t : Thus, both processes have the same drift. We say that a stationary process X has VIS of type two (VIS2) if the associated process Y does not possess a stationary distribution. In other words, X has VIS2 if X satis es the A and A2 conditions in appendix and Y does not satisfy at least one of these conditions. We now explain the intuition behind this de nition and why in the VIS2 de nition only the di usion coe cient induces stationarity. Although the process Y has the same drift as that of the process X, Y is nonstationary (by de nition) whereas X is stationary. That is, the substitution of by b (x) transforms a nonstationary process Y (equation (4)) into a stationary process (equation (3)). Thus, the stationarity of X can only be attributed to the role of the di usion coe cient (volatility) and in this case we have in fact a pure VIS process. The following is a simple criterion to identify VIS2, in the case l = and r =. We say that a stationary X process with boundaries l = and r = has VIS2 if lim xa (x) 0 or lim xa (x) 0: (5) x! x! It is easy to show that the solution of the associated process dy t = a (Y t ) dt + dw t ; under condition (5), does not possess a stationary distribution. Condition (5) implies an absence of mean-reversion of the process Y since when y is high ( low ) the drift a (y) is zero or positive (negative), and as a consequence the process moves further and further away from a central region. If the solution of dx t = a (X t ) dt + b (X t ) dw t 8

9 turns out to be stationary, where the drift satis es (5), this can only be due to the role of the instantaneous volatility b (X t ) : One example of VIS2 was presented in section (dx t = ( + Xt 2 ) dw t ). Another example is the following: dx t = ( + X t ) dt + + X 2 t dwt : The associated process Y; solution of the SDE dy t = ( + Y t ) dt + dw t ; is clearly nonstationary (the drift satis es condition (5)). Nevertheless, it can be proved (using either the conditions A and A2 in appendix or the proposition - (a), see below) that X is stationary. To appreciate the di erences between X and Y we present two simulated paths in gure 2. The initial condition is Y 0 = ; and in both cases the processes are driven by the same Wiener path W. The simulation was carried out following the ideas in Nicolau (2004). Figure 2 shows that the associated process is entirely dominated by the structure of the drift, which causes the process to reaches in nity as t! +: However, if the constant di usion coe cient is substituted by a function like b (x) = + x 2 ; the volatility prevents the process from moving away from its stationary mean. The mechanism to induce stationarity is very similar to that we described in section. We have so far stressed some di erences between the VIS de nition of CHLS and Richter and the proposed de nition VIS2. However, there are common features in these de nitions. In both cases, (a) X is stationary and (b) the condition lim x!l s X (x) < or lim x!r s X (x) < must be veri ed. Therefore, if a process 9

10 Figure 2: A Path of Y and X where Y: dy t = ( + Y t ) dt+dw t ; X: dx t = ( + X t ) dt+ ( + X 2 ) dw t (the initial condition is Y 0 = and both the processes are driven by the same Wiener path W ) X Y time satis es the VIS2 condition it also satis es the VIS condition of CHLS and Richter. The behaviour of the Y process corresponding to a VIS2 process can be quite distinct. For example, consider the stochastic di erential equations in table, below. All processes satisfy the VIS2 condition. They are stationary and satisfy (5). Nevertheless, the associated processes (all nonstationary) exhibit a quite distinct behaviour. The process dy t = Y 2 t dt + dw t is explosive (reaches in nity in nite time with probability one) and non-recurrent; the third one (dy t = ( + Y t ) dt + dw t ) is still non-recurrent but the expected time to reach is in nity; the fth (dy t = dw t ) is the Wiener process (which is recurrent). In gure 3 we show the typical behaviour of these three processes: an explosive process (path A), a process that drifts away from a central region eventually reaching in nity, but without blowing out in nite time (path B) and a Wiener process (path C). 0

11 Figure 3: Three Paths from the following processes: A- dy t = Y 2 t dt + dw t ; B- dy t = ( + Y t ) dt + dw t ; C- Y t = W t (the initial condition is Y 0 = 0 and all the processes are driven by the same Wiener path W ) A B C 5 0 time 5 It seems obvious that the volatility that is needed to inject stationarity varies according to the nature of the associated process. Typically, it has to be higher when the associated process is explosive (path A in gure 3) than when the associated process is only null-recurrent (path C in gure 3). For example, consider the SDE dx t = ( 0 + X t + 2 Xt 2 ) dt + ( + Xt 2 ) dw t ; > 0; > 0: Following the conditions for stationarity presented in appendix we have the following. If < 0 and 2 = 0 the drift induces mean-reversion e ects and it is not necessary to impose a restriction on in order that X is stationary. If 0 = = 2 = 0 the associated process is null-recurrent and one has to impose > =4: If 0 > 0; > 0 and 2 = 0 we have to impose > =2 or = =2 and > 2: Finally, if 0 > 0; > 0 and 2 > 0 the associated process is explosive and one has to impose > 3=4 or = 3=4 and > 2 p 2: Thus, roughly speaking, the lower is the mean-reversion e ect the

12 greater the volatility needed to inject stationarity. To distinguish the volatility that is needed to inject stationarity we consider three sub-types of VIS2 according to the behaviour of the associated Y process: from explosive processes ( rst case) to null-recurrent processes (third case). Consider again SDEs (3) and (4) and the following functionals (see Karlin and Taylor, 98) related to the associated Y process: n where s Y (x) = exp 2a(u) z 0 2 S Y [x; x 0 ] = Z x x 0 s Y (u) du Z x0 Z Y (l) = lim S Y [; u] m Y (u) du; Y (r) = lim S Y [u; ] m Y (u) du #l "r x 0 R o x du (z 0 is an arbitrary value) and m Y (u) = : 2 s Y (u) We say that the stationary X process has volatility-induced stationarity of type 2a, 2b and 2c (VIS2-a, VIS2-b and VIS2-c) if it has, respectively: VIS2-a Y (l) < or Y (r) < : VIS2-b S Y (l; x] < or S Y [x; r) < and Y (l) = Y (r) = : VIS2-c S Y (l; x] = S Y [x; r) = ; R r l m Y (u) du = : Figure 3 shows possible behaviours of the associated processes under conditions VIS2-a, VIS2-b and VIS2-c, respectively paths A, B and C. In appendix 2 we discuss these conditions in detail and we provide some more examples. Also, see table. We notice that if l = then S Y (l; x] should be interpreted as R x l s Y (u) du = lim k!l R x k s Y (u) du (likewise for S Y [x; r) if r = ). 2

13 The following proposition establishes su cient conditions for the three sub-types of VIS2 in the case where the state space is R: Proposition Let R be the state space of X. Suppose that functions a and b have continuous derivatives and that b (x) > 0 for all x 2 ( ; ) : If (a) lim x! x a(x) < b 2 (x) and lim 2 x! x a(x) b 0 (x) < then X is ergodic and the stationary density is b 2 (x) b(x) 2 p (x) = m X (x) = R m X (x) dx where m X (x) = = (s X (x) b 2 (x)) is the speed density. Furthermore, (b) if a (x) = O (jxj ) ; > and lim x! a (x) = ; lim x! a (x) = then X has VIS of type 2a (VIS2-a); (c) if a (x) = O (jxj ) ; and lim x! xa (x) > 0 then X has VIS of type 2b (VIS2-b); (d) if lim x! xa (x) = 0 then X has VIS of type 2c (VIS2-c); Proof: See appendix 3. The application of proposition is straightforward, as table shows. 3

14 Table : Examples of application of proposition Stoch. Di. Equations () (2) (3) (4) (5) (6) (7) VIS2- dx t = X 2 t dt + ( + X 4 t ) dw t a dx t = Xt dt + p + X 2 +Xt 2 t dw t b dx t = ( + X t ) dt + ( + X 2 t ) dw t b dx t = X t dt + p + X 2 t dw t () dx t = ( + X 2 t ) dw t c dx t = dt + ( + X 2 +Xt 2 t ) =3 dw t /3-2/3 c () : a (x) = O (jxj ) ; (2) lim x! xa (x) ; (3) lim x! xa (x) ; (4) lim x! x a(x) b 2 (x) (5) lim x! x a(x) b 2 (x) ; (6) lim x! x a(x) b 2 (x) b 0 (x) ; (7) lim b(x) x! x a(x) b 2 (x) b 0 (x) b(x) () b if 0 < < 2 3 Modelling the Fed funds Rate with VIS Processes with VIS are potentially applicable to interest rate time-series since, as has been acknowledged, reversion e ects (towards a central measure of the distribution) occur mainly in periods of high volatility. CHLS were the rst (to the best of my knowledge) to propose a VIS speci cation for interest rates. To model the Fed funds rate CHLS consider the following general speci cation for the in nitesimal coe cients: LX a (x) = i x i ; b 2 (x) = x ; ; > 0 (6) i= k With this speci cation it is obvious that the drift cannot be zero (i.e., i = 0, i = k; :::; L); otherwise the left boundary (zero) would be attracting (i.e. S X (0; x] < 4

15 ) and the model would be incorrect. To impose stationarity via volatility, CHLS consider a number of restrictions on i and : Some of these restrictions are explicitly de ned to avoid the process reaching the left boundary with positive probability. The model of CHLS follows a VIS2-a speci cation. In this section we discuss an alternative model to the CHLS for interest rates. The main objective is to illustrate VIS ideas and provide further evidence to support VIS. 3. Data and Identi cation We consider monthly sampling of the Fed funds rate between January 962 and December 2002 (see gure 4). The source for the data is the H-5 Federal Reserve Statistical Release. At high frequencies (for example, daily frequency) the Fed funds rate series exhibits strong microstructure e ects re ecting the institutional features of the operational framework, including the beginning and the end of the reserve maintenance period and the open market operations allotment and settlement days. These e ects almost disappear at monthly frequency. We now present evidence that supports the speci cation dx t = exp =2 + =2 (X t ) 2 dw t (7) where X t = log r t and r represents the Fed funds rate (original data was divided by 00) (see gure 4). We denote fr ti g i and fx ti g i = flog r ti g i the data (observed at equidistant instants t ; t 2 ; :::; t n ): 5

16 Figure 4: Fed Funds and log Fed Funds in the Period Jan 62 - Dez 02 Monthly Fed Funds (Jan 62 Dez 02) Monthly log Fed Funds (Jan 62 Dez 02) Jan 62 Jan 66 Jan 70 Jan 74 Jan 78 Jan 82 Jan 86 Jan 90 Jan 94 Jan 98 Jan 02 Jan 62 Jan 66 Jan 70 Jan 74 Jan 78 Jan 82 Jan 86 Jan 90 Jan 94 Jan 98 Jan 02 We observe that the state space of r is (0; ) and X is ( ; ) : That is, X can assume any value in R: This transformation preserves the state space of r, since r t = exp (X t ) > 0: By Itô s formula, equation (7) implies a VIS2-a speci cation for interest rates 2 rt ) dr t = r t e+(log dt + r t e =2+=2(log rt )2 dw t : (8) 2 A direct application of proposition -(a) to equation (7) allows us to conclude that X has non-attracting boundaries and the speed density function m X is integrable in ( ; ) (in fact, lim x! xa (x) =b 2 (x) = 0 and lim x! x (a (x) =b 2 (x) b 0 (x) =b (x)) = ): Thus X is an ergodic process with stationary density p (x) = m p X (x) R mx (x) dx = p e (x )2 (9) i.e. X = log r N (; = (2)). By the continuous mapping theorem, r = exp (X) 6

17 is an ergodic process. Furthermore, it has a log-normal stationary density. There is some empirical evidence that supports model (7). It is based on three facts. () The empirical marginal distribution of X t = log r t matches the (marginal) distribution that is implicit in model (7). If models (7) and (8) are correct then the empirical marginal (or unconditional) distribution of X t = log r t should be Gaussian (or approximately Gaussian). Table 2 and gure 5 show that the hypothesis that the Fed funds rate logarithm is normally distributed cannot be rejected by the data in the period considered. Thus, the empirical marginal distribution of X t = log r t matches the (marginal) distribution that is implicit in model (7). It is worth mentioning that the rst di erence sequence of r and log r (respectively, rti r and ti log r i ti log r ti ) presents a completely di erent picture: fat i tails, strong mean-reversion e ects, etc. Table 2: Summary Statistics for log interest rates flog r ti g i Mean Variance Kurtosis Skewness Bera-Jarque (pvalue) (2) The results of Dickey-Fuller tests are compatible with a zero drift function for X; as speci ed in model (7). The logarithm of the Fed funds rate displays high persistence and the Augmented Dickey-Fuller tests do not reject an integrated process at all conventional levels (see table 3). This pattern of persistence is similar to the Fed funds rate. As we pointed out in section, a pure unit root process is impossible so the Dickey-Fuller test results can be misleading. We conjecture that 7

18 Figure 5: Kernel Estimates of the log interest rates vs. Normal Density normal 0.9 kernel these tests have very low power against certain forms of stationary local martingales. Although the Dickey-Fuller results are unreliable in deciding whether X is stationary or not, we consider that they give strong evidence of a zero drift di usion. In e ect, the null hypothesis of the Dickey-Fuller test is compatible with the integral equation X ti = X ti + R t i t i b (X u ) dw u (zero drift). Table 3: Augmented Dickey-Fuller for log Fed funds flog r ti g i Autoregr.param. ADF t-statistic 0% CV random walk without drift random walk with drift random walk with drift and trend (3) Nonparametric estimates of a (x) and b 2 (x) do not reject speci cation (7). The conditional in nitesimal dynamics X = log (r) can be analyzed through a nonparametric approach. This approach helps to identify the in nitesimal coe cients. 8

19 Figure 6: Nonparametric Estimates of Drift and Di usion Coe cients based on log (r) data (with 95% Con dence Band) Nonparametric Drift Estimates Nonparametric Diffusion Estimates We use the following nonparametric estimators P n i=0 K x Xti (Xti+ X ti ) h ^a (x) = P n i=0 K x Xti ; ^b2 (x) = h P n i=0 K x Xti (Xti+ X ti ) 2 h P n i=0 K x Xti h for a (x) and b 2 (x) respectively, where = (t i t i ) = =2 is the discretization step, K is the Gaussian kernel and h = (4=3) 0:2 n =5 ( is the standard deviation of the X). The properties of these estimators are discussed in Bandi and Phillips (2003) and Nicolau (2003). In gure 6 we present nonparametric estimates of the in nitesimal coe cients using log (r) data. It seems clear that a zero drift exits for the logarithm of interest rates. This has already been suggested by Dickey-Fuller tests. The presence of an exponential 9

20 quadratic speci cation for the di usion is, however, not so clear. But we cannot reject it either (in the sense that we can select ; and such that the speci cation b 2 (x) = exp + (x ) 2 matches approximately the nonparametric estimates and is inside the con dence band). Nonparametric estimates for the di usion also suggest a quadratic type speci cation for log r. There is perhaps a more convincing argument to support the exponential quadratic speci cation. A zero drift coe cient of a stationary di usion process governed by a SDE dx t = b (X t ) dw t has a stationary density proportional to b 2 (x) (this is due to the fact that a (x) = 0 ) s X (x) = ) m X (x) = b 2 (x) ; and thus if X is a stationary process then p (x) is proportional to m X (x) = b 2 (x) - see appendix ). For example, if we choose a quadratic speci cation for b 2 (x) (i.e. b 2 (x) = + (x ) 2 ; ; > 0), the solution of the SDE dx t = b (X t ) dw t will have a Cauchy stationary density. If we choose an exponential quadratic speci cation for b 2 (x) (i.e. b 2 (x) = exp + (x ) 2 ; > 0), the solution of the SDE dx t = b (X t ) dw t will have a Gaussian stationary density (see equation (9)). Since, as we pointed out, the data give evidence of a Gaussian (marginal) distribution for X = log r; the most appropriate choice for b 2 (x) is the exponential quadratic speci cation. We observe that the logarithmic transformation induces the volatility to rise when the interest rate is low and to decrease when the interest rate is high. This explains why non-parametric estimates of the logarithm of Fed funds volatility has a "U" form (smiling volatility). 20

21 3.2 Parametric Estimation The transition (or conditional) densities of X required to construct the exact likelihood function are unknown. Several estimation approaches have been proposed under these circumstances (see Nicolau, 2002, for a brief survey). To estimate the parameters of equation (7) we considered the simulated maximum likelihood estimator suggested in Nicolau (2002) (with N = 20 and S = 20). The method proposed by Aït-Sahalia (2002) with J = (Aït-Sahalia s notation for the order of expansion of the density approximation) gives similar results (to apply Aït-Sahalia s method we use a program developed by Aït-Sahalia, which is available on his homepage). The approximation of the density based on J = 2 is too complicated to implement (it involves dozens of intricate expressions that are di cult to evaluate). The quasi-mle based on the Euler transition density approximation also gives similar results. As we have pointed out, although X is a stationary process, X cannot be simulated by standard discretized methods (e.g. Euler, Milstein or Platen) since these schemes are explosive. Thus, simulation-based methods using discretizated solutions such as Pedersen s method cannot, in principle, be used. In table 4 we report the estimation results. Table 4: Estimates of (7) (Simulated MLE) Estimate 3:49 :59 2:92 St.Error

22 We observe that the parametric (and non-parametric) estimates of a (r) = r r )2 e+(log 2 are approximately zero when r 2 (0:02; 0:2) : Thus, in this interval, a quasi-martingale behaviour for r is allowed by the model. The minimum point of a (r) is exp (2 ) : 2 An estimate of this quantity (based on parameter estimates) is 0.04 (4%) and the minimum value is a (0:04) = 0:0007. In the neighbourhood of r = 4% the drift is approximately zero and only when log r t moves signi cantly away from ; that is, when the quantity (log r ) 2 is large, does the drift increase. The drift increases especially sharply when r approaches zero since in this case the quantity (log r ) 2 becomes arbitrarily large. This behaviour prevents r from reaching the zero boundary. Our model (7) compares extremely favourably with other proposed one-factor continuous-time models. In table 5 we compare the proposed model with other relevant models for interest rates. Only the proposed method was estimated by us. Remaining information was obtained from table VI of Aït-Sahalia (999). For comparison purposes the proposed model was estimated using the same method applied to the other models (we considered the density approximation proposed by Aït-Sahalia, 2002, with J = ; in the period January-63 to December-98). Table 5 indicates that the proposed model outperforms the others in terms of accuracy and parsimony. 22

23 Table 5: Log-Likelihood of some Parametric Models for the Monthly Federal Funds Data, Models ln L no. of parameters dr t = ( r t ) dt + dw t 569:9 3 dr t = ( r t ) dt + p r t dw t 692:6 3 dr t = r t ( ( 2 ) r t ) dt + r 3=2 t dw t 80:9 3 dr t = ( r t ) dt + r t dw t 802:3 4 dr t = r t r t + 4 r 2 t dt + r 3=2 t dw t 802:7 5 dr t = r t 2 e+(log rt )2 dt + r t e =2+=2(log rt )2 dw t 805: 3 Source ( rst ve models): table VI of Aït-Sahalia (999). In Aït-Sahalia s table VI, ln L means ln L=n: 4 Conclusion In this paper we stress the role of volatility in ensuring stationarity. We suggest a re nement of the existing de nition of volatility-induced stationarity that allows us to distinguish between processes with drift and di usion induced stationarity and processes with pure volatility-induced stationarity. We also emphasize that the volatility needed to inject stationarity varies according to the nature of the associated process. Typically, it has to be higher when the associated process is explosive than when the associated process is only null-recurrent. This remark leads to a further classi cation of volatility-induced stationarity. As an application we consider the Fed funds rate. We provide evidence that the 23

24 logarithm of the Fed funds rate can be modelled as a local martingale with volatilityinduced stationary. We believe that there are other economic applications where volatility can inject stationarity, either as in VIS where both in nitesimal coe cients induce stationarity, or as in VIS2 where only the di usion coe cient induces stationarity. 24

25 References Aït-Sahalia, Y. (996), Testing Continuous-Time Models of the Spot Interest Rate, The Review of Financial Studies 9, Aït-Sahalia, Y. (999), Transition Densities for Interest Rate and Other Nonlinear Di usions, The Journal of Finance LIV, Aït-Sahalia, Y. (2002), Maximum Likelihood Estimation of Discretely Sampled Di usions: a Closed-Form Approximation Approach, Econometrica 70, Arnold, L. (974), Stochastic Di erential Equations: Theory And Application, John Wiley & Sons, New York. Bandi, F. (2002), Short-Term Interest Rate Dynamics: A Spatial Approach, Journal of Financial Economics 65, Bandi, F. and P. Phillips (2003), Fully Nonparametric Estimation of Scalar Diffusion Models, Econometrica 7, Bibby, B. and M. Sørensen (995), Martingale Estimation Function for Discretely Observed Di usion Process, Bernoulli, Chan, K., G. Karolyi, F. Longstaff and A. Sanders (992), An Empirical Comparison of Alternative Models of the Short-Term Interest Rate, The Journal of Finance XLVII,

26 Chen, X. Hansen and M. Carrasco (998), Nonlinearity and Temporal Dependence. Unpublished. Chapman, D. and N. Pearson (2000), Is the Short Rate Drift Actually Nonlinear? Journal of Finance 55, Conley, T., L. Hansen, E. Luttmer and J. Scheinkman (997), Short-term interest rates as subordinated di usions, The Review of Financial Studies 0, Cox, J., J. Ingersoll and S. Ross (985), A theory of the Term Structure of Interest Rates, Econometrica 53, Hansen, L. and J. Scheinkman (995), Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes, Econometrica 63, Hong, Y., L. Haitao and F. Zhao (2004), Out-of-sample performance of spot interest rate models, Journal Business and Economic Statistics 22, Ikeda, N. and S. Watanabe (98), Stochastic Di erential Equations and Di usion Processes, North-Holland Mathematical Library, Amsterdam. Karlin, S. and H. Taylor (98), A Second Course in Stochastic Processes, Academic Press, New York. 26

27 Nicolau, J. (2002), A New Technique for Simulating the Likelihood of Stochastic Di erential Equations, The Econometrics Journal 5, Nicolau, J. (2003), Bias Reduction in Nonparametric Di usion Coe cient Estimation, Econometric Theory 9, Nicolau, J. (2005), A Method for Simulating Non-Linear Stochastic Di erential Equations in R ; Journal of Statistical Computation and Simulation, Forthcoming. Pritsker, M. (998), Nonparametric Density Estimation and Tests of Continuous Time Interest Rate Models, Review of Financial Studies,, Richter, M. (2002), A Study of Stochastic Di erential Equations with Volatility Induced Stationarity. Unpublished. Skorokhod, A. (989), Asymptotic Methods in the Theory of Stochastic Di erential Equation, Translation of Mathematical Monographs 78, American Mathematical Society, Providence, Rhode Island. 27

28 Appendix - Stationary and Ergodic Conditions for Univariate Di usions Let X = fx t ; t 0g be a di usion process, with state space I = (l; r), governed by the stochastic di erential equation dx t = a (X t ) dt + b (X t ) dw t, X 0 = x where fw t ; t 0g is a (standard) Wiener process, a and b are the in nitesimal coe - cients and x is either a constant value or a random value F 0 -mensurable independent of W t. We assume that a and b have continuous derivatives. n R o z Let s X (z) = exp z 0 2a (u) =b 2 (u) du be the scale density function (z 0 is an arbitrary point inside I) and m X (u) = (b 2 (u) s X (u)) the speed density function. Let S X (l; x] = lim x!l R x x s X (u) du and S X [x; r) = lim x2!r R x2 x s X (u) du where, l < x < x < x 2 < r. To characterize the X process we present four assumptions. The rst one is: A S X (l; x] = S X [x; r) = for x 2 I. According to Arnold (974, page 4), if the in nitesimal coe cient a and b have continuous derivatives with respect to x, then there exists a unique continuous process that is de ned up to a random explosion time in the interval t 0 <. The A condition assures that P [ = j X 0 = x] = (Ikeda and Watanabe, 98, pp ). Furthermore, the boundaries l and r are neither attracting, nor attainable (see Karlin and Taylor, 98, chapter 5) and the process is recurrent, i.e. 28

29 P [T y < j X 0 = x] = for every x; y 2 I where T y = inf ft 0; X t = yg (Ikeda and Watanabe, 98, Theorem 3., Chapter VI). Roughly speaking, the boundaries l and r are never attained although every nite point can be reached with probability one in nite time. Global Lipschitz and growth conditions, which fail to be satis ed for many interesting models in economics in nance (Aït-Sahalia, 996), are not needed in the presence of the previous assumptions. The A condition is not very strong: for example, the standard Brownian motion satis es the A condition (Karlin and Taylor, 98, page 228). Actually, every process with zero drift a (x) = 0 (and b (x) > 0) satis es the A condition. A2 R r l m X (x) dx <. A and A2 conditions assure that X is ergodic and the invariant distribution P 0 has density p (x) = m X (x) = R r l m X (u) du with respect to the Lebesgue measure. They are also necessary conditions (Skorokhod, 989, theorem 6). The expression p (x) is usually denoted as stationary density. A3 X 0 = x has distribution P 0. Assumption A3, together with A-A2 implies that X is stationary (Arnold, 974). Assumption A3 can, in some cases, be replaced by: X 0 is a random variable with mean and variance 2 such that R xp (x) dx = < and R (x ) 2 p (x) dx = 2 <. In this case, X is a covariance-stationary process. Even if the A3 condition does not hold, it is known that when t is su ciently large, the distribution of the ergodic process X t is well approximated by the distribution with density p (x). 29

30 A4 lim x!r sup a(x) b(x) b 0 (x) < 0, lim 2 x!l sup a(x) b(x) b 0 (x) > 0: 2 These conditions are discussed in Chen et al. (998) and are similar to ones proposed by Hansen and Scheinkman (995). Under the A4 assumption the process is - mixing (see Chen et al., 998). Technically, for a Markov process, the notion of - mixing requires the conditional expectations operator for any interval of time to be a strong contraction for all functions with zero mean and nite 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 R f (x) p (x) dx = 0 and R f 2 (x) p (x) dx < (see Hansen and Scheinkman, 995, proposition 8). We notice that even if the drift is zero or converges to zero, the A4 assumption can hold, provided that volatility grows at least linearly. As an example, consider the SDE discussed in section : dx t = ( + X 2 ) dw t. Thus, a (x) = 0 and b (x) = + x 2 and the state space of X is (l; r) = ( ; ) : We have, Z r l s X (z) = exp S X (l; x] = lim x! Z z 0 Z x 2a (u) =b 2 (u) du x s X (u) du = lim x! m X (x) = b 2 (x) s X (x) = ( + x 2 ) 2 ; m X (x) dx = Z ( + x 2 ) 2 dx = 2 < : = ; Z x x du = ; Z x2 S X [x; r) = lim du = x 2! x Therefore, the stationary density is p (x) = m X (x) R r m l X (x) dx = (+x 2 ) 2 = ( + x 2 ) 2

31 and the stationary (or unconditional) moments are Z Z E [X] = xp (x) dx = 0; V ar [X] = x 2 p (x) dx = : R R In this example the A4 assumption still holds. Proposition (a) establishes su cient conditions (easy to verify) for A and A2. We can use proposition (a) in the previous example. Since lim x x! lim x a (x) x! b 2 (x) b 0 (x) b (x) a (x) b 2 (x) = 0 < 2 = lim x x! 2x = 2 < + x 2 2 proposition (a) guarantees that X has a stationary distribution. 3

32 Appendix 2 - VIS: discussion and examples Consider SDEs (3) and (4) and the following functionals (see Karlin and Taylor, 98) of the associated Y process: S Y [x; x 0 ] = Z x x 0 s Y (u) du Z x0 Z Y (l) = lim S X [; u] m Y (u) ; Y (r) = lim S Y [u; ] m Y (u) #l "r x 0 n where s Y (x) = exp R x 2a(u) z 0 2 o du and m Y (u) = 2 s Y : Let l < < < r; (u) T = inf ft 0 : Y t = g and T ^ T = min ft ; T g : We say that the stationary process X has volatility-induced stationary of type 2a, 2b and 2c (VIS2-a, VIS2-b and VIS2-c) if it has, respectively: VIS2-a Y (l) <, lim! E [T ^ T j X 0 = x] < or Y (r) <, lim! E [T ^ T j X 0 = x] < VIS2-b S Y (l; x] <, lim! P x [T T j X 0 = x] > 0 or S Y [x; r) <, lim! P x [T T j X 0 = x] > 0 and Y (l) = Y (r) = VIS2-c S Y (l; x] = S Y [x; r) = ; R r l m Y (u) du = We brie y analyze these conditions and relate them to VIS. In VSI2-a at least one of the boundaries is attainable, i.e. the expected time for one of the boundaries to be reached is nite (notice that Y (l) < implies S Y (l; x] < ). When l = and r = ; this case characterizes explosive processes (the expected time to reach l = or r = is nite). Furthermore, the process 32

33 is not recurrent. As an example, consider dy t = Y 3 t dt + dw t : It can be proved that Y (l) < and Y (r) < : Any trajectories of this process quickly go to in nity in a nite expected time. Further, the conditions Y (l) < and Y (r) < imply P [ < j Y 0 = y] = where is the random explosion time of Y de ned in the interval t 0 < (Ikeda and Watanabe, 98, theorem 3.2). That is, the process blows up in nite time with probability one. When one or both boundaries are nite one has absorbing phenomenon (re ecting or sticky barrier is possible if further assumptions regarding the behaviour at the boundaries is speci ed; in any case, this would imply a discontinuity of the trajectory). The solution of dx t = X 3 t dt + ( + X 4 t ) dw t is an example of VIS of type 2a, since it can be proved that S X ( ; x] = S X [x; ) = ; R m X (u) du < : Another example of VIS2-a is dx t = ( + X t + X 2 t ) dt + X 4 t dw t ; with state space (0; ) : This case is analyzed in CHLS. To explain the VIS2-b case, suppose S Y [x; r) < and r = : Then the boundary r = can be reached prior to reaching an arbitrary state with positive probability from any interior starting point x < ; although not in a nite expected time, as by hypothesis, Y (r) = : This condition can characterize processes that drift away from a central region in the state space, eventually reaching the boundaries but without blowing out in nite time with probability one. Consider for example the SDE dy t = Yt +Yt 2 dt + dw t ; Y 0 = 0 with state space ( ; ) : It can be proved that S Y (l; x] < ; S Y [x; r) < and Y (l) = Y (r) = : Any realization of Y tends to move further and further away from Y 0 = 0 (since a (x) > 0 if x > 0 and a (x) < 0 33

34 if x < 0). These trajectories are accompanied by a reduction of the drift magnitude. Despite a (x)! 0 as jxj! ; we have jy t j! as t! with positive probability, although the random explosion time occurs in in nite time with probability one. The expected time to reach r = is in nite, i.e. Y (r) =. As in the previous case, these processes are not recurrent. This second case can be thought of as a moderate version of case one: although non-recurrent, the expected time to reach or is in nite. The solution of dx t = can be shown S X (l; x] = ; S X [x; r) = and R r l Xt dt + p + X 2 +Xt 2 t dw t is an example of VIS2-b since it m X (u) du < : Another example of VIS2-b is the solution of dx t = ( + X t ) dt + ( + X 2 t ) dw t : In fact, while X is an ergodic process, the solution of dy t = ( + Y t ) dt + dw t has boundaries that are attracting, although the expected time to reach them is in nite. In the third case (VIS2-c) Y is a null-recurrent process. If > x, the recurrence (every nite point can be reached with probability one in nite time) follows from P [T < j X 0 = x] P [T < T l j X 0 = x] = (we notice that S Y (l; x] = is equivalent to P [T < T l j X 0 = x] = ): By similar arguments, if < x, then S Y [x; r) = implies P [T < j X 0 = x] = :The boundaries cannot be reached from the interior of the state space (observe that S Y [x; r) = ) Y (r) = ). The case R r l m Y (u) du = can occur either with M Y (l; x] = or M Y [x; r) = : The quantity M Y (l; x] can be interpreted as a measure of the speed of the process near l (see Karlin and Taylor, 98, page 23). The higher this quantity is, the stickier the boundary. Generally speaking, the condition R r l 34 m Y (u) du = can be thought

35 of as an absence of a general drifting towards a central set for large excursions of the stochastic paths. The solution of dx t = p + Xt 2 dw t is an example of VIS2-c, since the X solution admits an ergodic distribution and the associated Y process (the Brownian motion) is a null-recurrent process and nonstationary. 35

36 Appendix 3 - Proof of Proposition (a) If the in nitesimal coe cients a and b have continuous derivatives with respect to x, then there is a unique continuous process that is de ned up to a random explosion time in the interval t 0 < (Arnold, 974, page 4). This random explosion time is in nite with probability one if S X (l; x 0 ] = S X [x 0 ; r) = for x 0 2 R (Ikeda and Watanabe, 98, pp ) where S X [c; d] = S X (d) S X (c) and S X (x) = Z x x 0 s X (u) du: Firstly, we show that lim x! x a(x) b 2 (x) < 2 implies S X (l; x] = S X [x; r) = : Consider Z Z S X [x 0 ; r) = lim s X (u) du = lim exp! x! 0 x 0 Z u z 0 2 a (s) b 2 (s) ds du If there exists some x > x 0 such that exp Z x z 0 2 a (s) b 2 (s) ds > x Z x, 2 a (s) ds > log x, z 0 b 2 (s) Z x z 0 2 a (s) ds < log x b 2 (s) (z 0 is an arbitrary value) for all x > x > 0 then S X [x 0 ; r) = (observe that lim! R x 0 x dx = ): Equivalently, if lim x! R x z 0 2 a(s) b 2 (s) ds log x < then s X is not integrable. Using L Hôpital rule we have R x z lim 0 2 a(s) ds 2 a(x) b 2 (s) b <, lim 2 (x) x! log x x! x <, lim x! a (x) b 2 (x) x < 2 : Thus, if a(x) b 2 (x) x < 2 as x! ; then S X [z 0 ; r) = : Using similar arguments (with a change of variable y = x) we conclude that if a(x) b 2 (x) x < 2 as x! then S X (l; z 0 ] = : 36

37 We now show that lim x! x a(x) b 2 (x) b 0 (x) b(x) < 2 implies M X [x 0 ; r) < and M X (l; x 0 ] <, i.e. R m X (x) dx < where m X (x) = s X (x)b 2 (x) : Consider n Z Z R o u exp z 0 2 a(s) ds b M X [x 0 ; r) = lim du = lim 2 (s) du! x 0 s X (u) b 2 (u)! x 0 b 2 (u) If there exists some x > x 0 and > such that n R o x exp z 0 2 a(s) ds b 2 (s) < Z x b 2 (x) x, 2 a (s) z 0 b 2 (s) ds log b2 (x) < log x R x z, 0 2 a(s) ds log b 2 (s) b2 (x) < log x for all x > x > 0 then M X [x 0 ; r) < : Equivalently, if lim x! R x z 0 2 a(s) b 2 (s) ds log x log b2 (x) < then M X [x 0 ; r) < : Using L Hôpital rule we have lim x! Thus lim x! a(x) b 2 (x) R x z 0 2 a(s) b 2 (s) ds log x log b2 (x) <, lim 2 a(x) b 2 (x) x! a (x), lim x! b 2 (x) x b 0 (x) b (x) 2 b0 (x) b(x) < x < 2 b 0 (x) x < is a su cient condition for M b(x) 2 X [x 0 ; r) < : Using similar arguments, one can conclude that a (x) lim x! b 2 (x) b 0 (x) x < b (x) 2 ) M X (l; x 0 ] < : Note that for any constants x 0 and x we have R m X (x) dx = M X (l; x 0 ]+M X [x 0 ; x ]+ M X [x 0 ; ) : Since a and b are continuous functions and b (x) > 0 we always have M X [x 0 ; x ] < : Therefore, M X [x 0 ; r) < and M X (l; x 0 ] < imply R m X (x) dx < 37

38 : Now, it is known that if S X (l; x] = S X [x; r) = and R m X (x) dx < then X is ergodic and the invariant distribution P 0 has density p (x) = m X (x) = R r l with respect to the Lebesgue measure (see Skorokhod, 989, theorem 6). m X (u) du (b) It is su cient to show that Y () = Y ( ) < with respect to the process dy t = X t dt + dw t (without loss of generality we x = ). We have where f (x) = 2 Z Z Y () = lim S Y [u; ] m Y (u) = lim f (u) du! x! 0 x e 2x+ + + (gamma ; h gamma ; 2x and gamma (a; x) = R x ta e t : Simple but cumbersome calculations show that if then f (x) > =x which implies that Y () = and that if > there i ) exists a > such that f (x) < =x which implies that Y () < : Similar arguments can be used to conclude that implies that Y ( ) = and > implies that Y () < : (c) Following the arguments in (b) we conclude that a (x) = O (jxj ) ; implies that Y () = Y ( ) < : It remains to be shown that S Y (l; x 0 ] < and S Y [x 0 ; r) < : Consider Z Z S Y [x 0 ; r) = lim s Y (u) du = lim exp! x! 0 x 0 2 If there exists some x > x 0 and > such that exp 2 Z x z 0 2a (s) ds < x, 2 Z x z 0 Z u z 0 2a (s) ds du 2a (s) ds > log x 38

39 for all x > x then s Y is integrable on (x 0 ; r), i.e. S Y [x 0 ; r) < : Equivalently, if lim x! R x 2 z 0 2a (s) ds log x > then s Y is integrable on (x 0 ; r). Using L Hôpital rule we have lim x! R x 2 z 0 2a (s) ds log x >, lim x! 2a (x) 2 x, lim x! xa (x) > 2 2 : > Since is a small xed controlled parameter, it follows that lim x! xa (x) > 0 implies S Y [x 0 ; r) < : Similar argument, allow us to conclude that lim x! xa (x) > 0 implies S Y (l; x 0 ] < : (d) We sketch the proof since it is similar to the above ones. We have 2 lim xa (x) < x! 2 ) S Y [z 0 ; r) = and S Y (l; z 0 ] = lim a (x) x > 2 x! 2 ) M Y [x; r) = and M Y (l; x 0 ] = : Thus lim x! xa (x) = 0 is su cient to assure the results stated. 39