STATIONARY PROCESSES THAT LOOK LIKE RANDOM WALKS - THE BOUNDED RANDOM WALK PROCESS IN DISCRETE AND CONTINUOUS TIME

Transcription

1 STATIONARY PROCESSES THAT LOOK LIKE RANDOM WALKS - THE BOUNDED RANDOM WALK PROCESS IN DISCRETE AND CONTINUOUS TIME João Nicolau Instituto Superior de Economia e Gestão Universidade Técnica de Lisboa O ce: Quelhas 406 Postal address: ISEG, Rua do Quelhas 6, Lisboa, Portugal nicolau@iseg.utl.pt Telephone: Fax: I would like to thank the referee, Carlos Braumann and Nuno Cassola for helpful comments. I am thankful to the Fundação para a Ciência e a Tecnologia, programme Praxis XXI, for partially sponsoring my research.

2 Abstract Several economic and nancial time series are bounded by an upper and lower nite limit (e.g., interest rates). It is not possible to say that these time series are random walks because random walks are limitless with probability one (as time goes to in nity). Yet, some of these time series behave just like random walks. In this article we propose a new approach that takes into account these ideas. We propose a discrete and a continuous time process (di usion process) that generate bounded random walks. These paths are almost indistinguishable from random walks, although they are stochastically bounded by an upper and lower nite limit. We derive for both cases the ergodic conditions and for the di usion process we present a closed expression for the stationary distribution. This approach suggests that many time series with random walk behaviour can in fact be stationary processes. 3

3 1 Introduction The study of stationary versus non-stationary time series has became a key issue in both time series and econometrics analysis. Their implications for economic theory are extremely important. Some time series seems to be non-stationary, such as industrial production, consumer prices, stock prices, among others [see Kwiatkowski et al.(1992)]. However there are others where there seems to be no consensus. For example, Perron (1989), could not reject the unit root hypothesis for the nominal interest rate. In the same direction, Chan et al. (1992) pointed out that the mean-reversion for the US interest rate is very weak which is a sign of unit root possibility. However, Dahlquist (1996) found some mean (linear) reversion e ects for interest rates in Denmark, Germany, Sweden and the UK. On the other hand, Aït-Sahalia (1996) concluded, that the mean linear reversion is not adequate for the 7-day Eurodollar deposit rate. He found that the drift (in nitesimal coe cient of the di usion process) of the spot rate process is essentially zero as long as the rate is between 4 and 17 percent, but pulls it strongly towards this middle region whenever it escapes. Thus, in the interval from 4 to 17 percent the process behaves like a random walk (RW) process (as the drift is zero) but is not a true RW as the process shows reversion effects whenever some high or low value is reached. Aït-Sahalia s interpretation seems to solve the puzzle: the interest rate behaves like a RW - so the usual test of stationarity is not able to reject the unit root - but the process is obviously bounded with reversion e ects at high and low levels, which eventually leads to stationarity and a mean reversion. Regarding the exchange rate, there is now a considerable amount of evidence 4

4 that real exchange rates do not have unit roots (see Rogo (1996) and Rose (1996) despite initial views (see Roll (1979) and Adler and Lehman (1983)). Therefore it is expected, in general, that real exchange rates are bounded in probability (i.e., does not diverge to 1) and they have long-run values to converge to. With regard to nominal exchange rates it can be argued from economic and statistical considerations that some nominal exchange rates should be bounded or be in a some kind of implicit target zone regime. Nicolau (1999) argues that the DEM/USD exchange rate is not a RW (at least in the last 15 years) despite the conclusions of the Dickey-Fuller test. That is, the DEM/USD behaves like a RW but cannot be a true RW as there is some evidence that this exchange rate is bounded 1 (we emphasize that the unit root process goes to +1 or 1 with probability one as time goes to +1 so a bounded process can not have a unit root). These ideas suggest that some economic and nancial time series can behave just like a RW (with some volatility pattern) but due to some economic reasons they are bounded processes (in probability, for instance) and even stationary processes. To build a model with such features it is necessary to allow RW behaviour during most of the time but force mean reversions whenever the processes try to escape from some interval. There is some evidence that the usual test of Dickey-Fuller under general speci cation of alternative hypothesis, has low power to detect stationary processes [Nicolau (1999) shows that power of the Dickey-Fuller test is extremely low when the alternative hypothesis is a stationary bounded random walk process (we present this model next), see also Kwiatkowski et al. (1992)]. In this article our aim is to present a new model in discrete and in continuous- 5

5 time that can generate paths with the following features: as long as the process is in the interval of moderate values, the process basically looks like a RW but there are reversion e ects towards the interval of moderate values whenever the process reaches some high or low values. As we will see, these processes can admit - relying on the parameters - stationary (or ergodic) distributions so we will come to the interesting conclusion: processes that are almost indistinguishable from the RW process can be, in e ect, stationary with ergodic distributions. The article consists of two main sections. In section 2 we introduce and discuss the main properties of the bounded random walk process in discrete-time. We do the same in section 3 for a continuous-time version of the bounded random walk. In the discrete-time version we admit a GARCH representation for the volatility, whereas in the continuous-time case we discuss an exponential form which could be appropriate for modelling a smile curve for volatility [Krugman and Miller (1992)]. In all cases we state the conditions under which the processes are ergodic 2. We present examples to make these models clear. 2 The Bounded Random Walk in Discrete Time The previous section shows that a bounded random walk model can be appropriate for modelling some economic and nancial time series. In this section we start to discuss some proprieties that a bounded random walk model should satisfy. 6

6 2.1 Some Properties If a process is a random walk, the function E [X t j X t 1 = x] (where X t = X t X t 1 ) must be zero (for all x). On the other hand, if a process is bounded (in probability) and mean-reverting to (say), the function E [X t j X t 1 = x] must be positive if x is below and negative if x is above. Now consider a process that is bounded but behaves like a RW. What kind of function should E [X t j X t 1 = x] be? As the process behaves like a RW, (i) it must be zero in some interval and, since the process is bounded, (ii) it must be positive (negative) when x is low ( high ). Moreover we expect that: (iii) E [X t j X t 1 = x] is a monotonic function which, associated with (ii), means that the reversion e ect should be strong if x is far from the interval of reversion and should be weak in the opposite case; (iv) E [X t j X t 1 = x] is di erentiable (on the state space of X) in order to assure a smooth e ect of reversion. This kind of behaviour, for instance, with regard to interest and exchange rates, is implicit in Aït-Sahalia (1996), Stanton (1997) and Nicolau (1999) through the nonparametric estimation of E [X t j X t 1 = x]. A possible representation for E [X t j X t 1 = x] is drawn in gure 1 (we explain this gure in section 2.3). To satisfy (i)-(iv) we assume E [X t j X t 1 = x] = e k e 1(x ) e 2(x ) with 1 0, 2 0; k < 0. Let us x a (x) = e k e 1(x ) e 2(x ). As we will see, this function is su ciently general and exible in the sense that it can generate a vast range of bounded random walks in a stationary (or non-stationary) framework (since stationary processes are bounded in probability, it is quite natural to expect that bounded random walks are stationary). In addition, this function has good 7

7 properties, allowing the extension of our discrete-time model to a continuous-time framework. There may be other functions similar to a (x) that can generate bounded random walks, so the a (x) function is not unique. However, as far as we know, the only alternative models, in the literature, are the SETAR(3,1,1) [Self-Exciting Threshold Autoregressive, Tong (1990)] and the Regime-Switching model [Hamilton (1994)] [see however equation (11)]. In section 2.5, we show that our model has several important advantages over these two alternatives models. With our assumption about E [X t j X t 1 = x] we propose, therefore, the bounded random walk process (BRW) in discrete-time: X t = X t 1 + e k e 1(X t 1 ) e 2(X t 1 ) + t " t (1) ( 1 0, 2 0; k < 0) where f" t g is a sequence of i.i.d. random variables with E [" t ] = 0 and V ar [" t ] = 1; and t (volatility) belongs to the information set F t 1 = (X : t 1), for example 2 t can have a GARCH representation [Bollerslev (1986)]. 2.2 The Function a (x) We now analyse the function a (x) = e k e 1(x ) e 2(x ). It is evident that the case = 0 leads to the unit root (since it implies a (x) = 0; 8x). It is still obvious (in more general cases, for example, 1 > 0 and 2 > 0) that a () = 0, so X t must behave just like a random walk whenever X t 1 =. On the other hand, we can select the parameters k; 1 and 2 such that a (x) 0 whenever X t 1 is in neighbourhood of. The range of the interval where a (x) is approximately null depends on the parameters. Typically, the case 1 > 0, 2 > 0, 8

8 k < 0 and jkj is high with regard to 1 and 2 entails a (x) 0 over a large interval centred on. Suppose that X t 1 moves signi cantly away from, for example, X t 1 >. Then, a (x) turns out to be negative and the probability that X t decreases will be high. Thus, if X t 1 is high and far away from there will be reversion e ects that pull it towards lower values. However, near these reversion e ects will be almost null. Let us see, in more detail, the meaning of the parameters 1, 2 ; and k. The parameter k controls the range of the interval under which the process behaves like a random walk. When k < 0 and jkj is high (low) the range tends to be high (low). The parameter is a central measure of the process since, as we have seen, a () = 0. We should expected to be the mean of the process under the hypothesis 1 = 2 [we will turn again to this issue in section 3.3]. However, if a (x) is approximately zero over a large interval (centred in ) we should expect a reversion e ect towards a neighbourhood of (and not exactly to ). Finally, the 1 0 and 2 0 parameters measure the reversion e ect of the process whenever it escapes from the interval where the function a (x) is approximately zero. High values of these parameters imply a strong reversion e ect. It is easy to see that the 1 ( 2 ) parameter is linked to the reversion e ect when the process is low (high). We notice that the case 1 6= 2 leads to asymmetrical e ects. For instance, according to Gray (1996), among others, the interest rate process behaves typically like a random walk when the process is moderate or low, showing, however, a strong reversion e ect when it is very high. Therefore, we expect, 2 > 1 (what prevents the process from reaching the zero state is, actually, the very low volatility - see our discussion at the end of the section 9

9 3.3). Another example of the case 1 6= 2 is analysed in Nicolau (1999) in the study of the DEM/USD exchange rate. 2.3 An Example In gure 1 we draw a (x) = e k e 1(x ) e 2(x ) for the following values: k = 15, 1 = 2 = 3 and = 100. We see that in the interval I = (95; 105) the function a (x) is (approximately) zero, so in this interval X behaves like a random walk. Outside of I there are reversions towards I. *** Figure 1 *** In gure 2, we simulated two paths (t = 1; 2; :::; 1000): a BRW path from (1), using the values presented above and t = = :4 and a RW path from X t = X t 1 + :4" t : In both processes we used the same values of " t (we assumed that f" t g is a sequence of i.i.d. random variables with N (0; 1) distribution). We see that both trajectories are almost indistinguishable until t = 140 (approximately). Near the value 105 there are reversion e ects only in the BRW, that is a (x) < 0 (see gure 1) - which prevent the process from increasing beyond 105. After that the processes follow di erent trajectories. *** Figure 2 *** 2.4 Stationarity Firstly, consider an homogeneous stochastic discrete process X = fx t ; t = 0; 1; 2; :::g with initial value X 0 = x 0 (possibly random). Let us assume that X is a Markov 10

10 process governed by the function X t = f (X t 1 ; " t ) (2) where f (X t 1 ; " t ) = (X t 1 ) + (X t 1 ) " t (X t 1 ) = X t 1 + a (X t 1 ) and a : R! R is a nonlinear function of X t 1 and f" t ; t 1g is stochastic process. There are several approaches to check stationarity and ergodicity [see Borovkov (1998)]. For instance, consider the proposition below [adapted from Chang, Appendix 1 in Tong (1990), pp ]: Proposition 1 Assume that f is continuous everywhere and continuously di erentiable in a neighbourhood of the origin. Suppose that conditions A1-A7(see Appendix A) hold. Then X = f (X t 1 ; " t ) is geometrically ergodic. Proof. See Tong (1990), pp For nonlinear time series, it is very di cult to check the (very important) A2 condition (see Appendix A). This is also true in our model. Basically, A2 says (in state space R 1 ) that j t (x 0 )j Ke ct jx 0 j where K > 0; c > 0 and t (x 0 ) is the t-th iteration t (x 0 ) = ( (::: (x 0 ))) given the initial value x 0 : The problem is that it is generally impossible, for nonlinear time series, to get t (x 0 ) = ( (::: (x 0 ))) as a function of the initial value. However, in the case under analysis, with (x) = x + a (x) ; we consider the following conditions which are easy to check: H1 a (x) > 0 if x < 0 and a (x) < 0 if x > 0: 3 11

11 H2 a (x) < 2x if x < 0 and a (x) > 2x if x > 0: Proposition 2 The conditions H1 and H2 imply A2. Proof. See Appendix B. H1 says that when the process is, for example, above its equilibrium value = 0, that is, X t 1 > 0 there will be adjustments forcing X t to decrease, so we must observe E [X t j X t 1 ] = a (X t 1 ) < 0. The H2 condition settle that these adjustments must be moderate and progressive in order to preclude explosions. Consider now the BRW process (1). Without any loss of generality we x = 0 (this can be achieved by the transformation Y t = X t ): As we have seen, the BRW satis es the A1 condition. However, there is a potential problem: some adjustments can be explosive in the sense that every attempt to correct the path (when X t 1 is too low or too high, beyond some level) can be excessive and turn out to be explosive. The hypothesis H2, if satis ed, certainly avoids explosive behaviour. Unfortunately our function a (x) does not satisfy H2 for all x. In practice this is not a major problem - in practical applications we never expect that the case ja (x)j 2 jxj will happen. Nevertheless, we force H2 by coupling (1) with the additional regularity condition (in the case = 0): X t = X t 1 + t " t ; 0 < < 1 if ja (X t 1 )j 2 jx t 1 j (3) (we note that this condition is not necessary in the continuous-time case in order to analyse stationarity). We assume that the set fx : a (x) < 2x if x < 0 and a (x) > 2x if x > 0g has a positive Lebesgue measure. We state now: Proposition 3 Suppose that 1 > 0, 2 > 0, t > 0 (8t), " t satisfy A4 (in Appendix 12

12 A) and fu t = t " t ; t = 1; 2; :::g is covariance stationary. Then X (see equation (1)) with the condition (3) is geometrically ergodic. Proof. It is easy to see that the conditions H1-H2 as well A1-A7 (in Appendix A) are satis ed. What kind of stationary distributions can we expect from the BRW model? Since X behaves like a RW most of the time, say, in the interval I, we must expect a at distribution in the centre. On the other hand, outside of I there are strong reversions, so the tails of distribution must not be heavy. This is what we have observed in some nancial time series in levels. Notice that in the rst di erence sequence we observe a completely di erent pattern (heavy tails, peak distribution at the centre, high kurtosis). Therefore, under some conditions, including 1 > 0 and 2 > 0 the BRW process is geometrically ergodic, is bounded in probability and is not persistent in the sense that shocks do not have permanent e ects on the process. The implications for economic theory and policy actions are extremely important and are well described in the literature. For instance, policy actions are not so important in stationary models because shocks only have a transitory e ect. Nicolau (1999) found some evidence that the DEM/USD is a BRW (using daily observations from the last 14 years). Thus, there must be an implicit but e ective target zone regime that limits the size of exchange rate uctuations. This conclusion, applicable to the EURO/USD, is relevant for nancial markets and central banks. 13

13 2.5 Alternative Models We now address the issue of alternative models. As far as we know, only the SE- TAR(3,1,1) [Tong (1990)] and regime-switching model (with two regimes) [Hamilton (1994)] can generate similar (but not identical) behaviour to the BRW. In the SETAR model the E [X t j X t 1 = x] function is not di erentiable at the threshold parameters and implies sudden transition of regimes as soon as some threshold parameter is crossed by the process. In the regime-switching model the Markov chain must depend on the past information of X (in the sense that, if the process is in the random walk regime and if it crosses some high or low value, the probability of entering in the stationary regime must be high). We think that our model has some advantages over the two above mentioned alternative models in modeling bounded processes that behave like a RW. It is a simpler model based on a single regime (without suddenly switching regimes) and is the easier to estimate, for instance through Pseudo (or Quasi) Maximum Likelihood. Furthermore, under some weak conditions, it converges weakly to a continuous time process (more precisely to a di usion process) as the interval of time between observations goes to zero. We notice that the Pseudo Maximum Likelihood (or even the Maximum Likelihood if we want to assume a distribution for the innovations) is immediately applicable to the BRW (with white noise or GARCH innovations) in a stationary framework. However, we should point out that the model is not identi able in the case = 0 (it can be shown that the Fisher information matrix is singular). Actually, the case = 0 leads to the RW model. Therefore, in the estimation procedure it is suitable to restrict the parameters 1 and 2 to the stationarity region, i.e., imposing 14

14 1 > 0 and 2 > 0. 3 The Bounded Random Walk Di usion Model 3.1 A Convergence Result In the previous section we assumed that the interval between observations was xed and equal to one (say = 1). Now we consider the case where X is a continuoustime process. There are some advantages by assuming this hypothesis. For instance, continuous-time processes are frequently preferred in nance theory. Moreover, in a continuous-time framework it is, in general, easier to nd limit properties such as stationary moments and distributions and in general laws of probability governing the process than in the discrete-time version. On the other hand, continuous-time processes are more di cult to estimate when the observations are discrete, but even in this situation (and this is always the case) it can be argued that continuoustime formulation is closer to the way that the data are actually generated: [:::] the economy does not move in regular discrete jumps corresponding to the observations - it is adjusting in between observations and it can change at any point of time [:::], Bergstrom (1993). How can we de ne a bounded RW in continuous-time? One way consists of analysing the limit process of the stochastic di erence equation (1) as the length of the discrete-time intervals between observations goes to zero. So, let us consider (1) in a more convenient notation, assuming, for simplicity that t is constant: X ti = X ti 1 + e k e 1(X ti 1 ) e 2(X ti 1 ) + " ti ; X 0 = c: (4) We have: t i are the instances at which the process is observed, (0 t 0 t 1 ::: T ); 15

15 is the interval between observations, = t i t i 1; k and are parameters depending on (if 1, we consider k = k and = ) and f" ti ; i = 1; 2; :::g is a sequence of i.i.d. random variables with E [" ti ] = 0 and V ar [" ti ] = 1. We notice that when is changing some parameters in (4) must change accordingly. We are concerned with the following problem: which process must (4) converge to when # 0? We are, actually, concerned with the convergence of the sequence X t formed as a step function from X ti, that is X t = X ti if t i t < t i+1. It can be proved, under the conditions k = k + log and = p that the sequence X t converges weakly (i.e., in distribution) as # 0 to the X t process de ned by the stochastic integral equation X t = c + Z t 0 e k e 1(X s ) e 2(X s ) ds + Z t 0 dw s (5) where W is a standard Brownian motion, independent of c Two Di usion Models Firstly, we suppose that X is governed by the following stochastic di erential equations (SDE) dx t = e k e 1(X t ) e 2(X t ) dt + dw t ; X t0 = c (6) where c is a constant and W is a standard Wiener process (t t 0 ) 5. As in the discrete version, it is evident that the case a (x) = 0 (for all x) leads to the Wiener process (which can be understood as the random walk process in continuous-time, but having some special features - see Arnold (1974), chapter 3). It is still obvious that a () = 0, so X t must behave just like a Wiener process when 16

16 X t crosses. However, it is possible, by selecting adequate values for k; 1 and 2 to have a Wiener process behaviour over a large interval centred on (that is, such that a (x) 0 over a large interval centered on ). Nevertheless, whenever X t escapes from some levels there will always be reversion e ects towards the : A possible drawback of model (6) is that the di usion coe cient is constant. In the exchange rate framework and under a target zone regime, we should observe a volatility of shape \ with respect to x (maximum volatility at the central rate) [see Krugman and Millca (1992)]. On the other hand, under a free oating regime, it is common to observe a smile volatility [see Krugman and Millca (1992)]. For both possibilities, we allow the volatility to be of shape \ or [ by assuming a n speci cation like exp + (x ) 2o : Depending on the we will have volatility of \ or [ form. Naturally, = 0 leads to constant volatility. This speci cation, with > 0, can also be appropriate for interest rates [see Nicolau (1998), Gray (1996)]. We propose, therefore, dx t = e k e 1(X t ) e 2(X t ) dt + e =2+=2(Xt )2 dw t ; X t0 = c (7) 3.3 Properties of the Models Let us rst consider the constant volatility model (6). The rst question to ask is whether this process has exactly one continuous global solution over the entire interval [t 0 ; 1). According to Arnold (1974) (theorem 6.37, p. 114), if the in nitesimal coe cients a (drift) and b (di usion) are continuously di erentiable then X has a unique local solution that is de ned up to a random explosion time in the interval t 0 < 1. Therefore, the main question is to know if P [ = 1] = 1. We now state 17

17 the following: Proposition 4 The solution of the SDE (6) with 1 = 0; 2 > 0 or 1 > 0; 2 = 0 or 1 > 0, 2 > 0 verify P [ = 1] = 1. Proof. See Appendix B. So, (6) has exactly one continuous global solution over the entire interval [t 0 ; 1). Proposition 5 The solution of the SDE (6) with 1 = 0; 2 > 0 or 1 > 0; 2 = 0 or 1 > 0; 2 > 0 is ergodic and has a stationary density of the form p (x) _ 2 exp Proof. See Appendix B. (!) 2e k e 1(x ) 2 + e2(x ) : (8) 1 2 The boundaries l = 1 and r = 1 are natural so they are not attracting and cannot be attained [see Karlin and Taylor (1981)] starting the process at X 0 = x where l < x < r. This is an important di erence from the discrete version of model (1). In e ect, whereas the discrete-time version recursion of (1) cannot start from an arbitrary value in R; the continuous-time version does not have such a restriction (we only avoid that the initial points be l = 1 or r = 1). Intuitively, if X is continuous, an oscillation explosive behaviour in the bounded RW is precluded. For example, if the initial value X 0 = x is far from the equilibrium point, there will be strong reversion e ects towards. As soon as X t starts to approximate the reversion e ect will decrease and eventually will stop when is reached. In discrete-time, if X 0 = x is far from the equilibrium and if the condition (3) is not used, any tendency to return to equilibrium is made by explosive oscillations that are further and further from. 18

18 To exemplify this model we consider the case k = 2, 1 = 2 = 2, = 100 and = 4. In the neighbourdhood of = 100 the function a (x) is (approximately) zero, so X behaves as a Wiener process (or a random walk in continuous-time). In e ect, if a (x) = 0, we have dx t = dw t (or X t = X 0 + W t ). In gure 3 we simulate two trajectories in the period t 2 [0; 20], with X 0 = 100. We compare the Wiener process ( unbounded random walk ) with the bounded random walk, solution of (6), which was simulated by the Platen and Wagner discretization method [or scheme of order see Kloeden and Platen (1992)]. We draw two arbitrary lines to show that the bounded random walk almost never crosses these lines. On the other hand, within the bands the BRW behaves like a pure RW. *** Figure 3 *** Density can be very at near as gure 4 shows. As we have already pointed out, it is necessary to distinguish the distribution in levels from the distribution in the rst di erences sequence. This latter is usually leptokurtic. Financial time in levels are usually integrated or near integrated, therefore, over a large interval of the state space, we do not expected to nd values more likely than others, so, actually, the distribution in levels (if existing 6 ) must be at at the centre. *** Figure 4 *** In the above example, we have dp () =dx = 0 so is a central measure. In the case 1 = 2 it seems clear that must be the stationary mean (notice that the 19

19 stationary density is symmetrical around and the tails of distributions fall abruptly on the x axis which must assure the existence of several moments - actually, this can be proved). Let us now consider the exponential random walk model. We state now about (7): Proposition 6 If > 0 then P [ = 1] = 1: If < 0 and > 0 then P [ = 1] = 1: Proof. See Appendix B. So, under the conditions of the previous proposition, (7) has exactly one continuous global solution on the entire [t 0 ; 1) : Proposition 7 If > 0 the process (7) is ergodic and has a stationary density of the form p (x) _ exp (x ) 2 + A 1 [A 2 f 1 (x) A 3 f 2 (x)] p (9) where, A 1 = p e k 1 2 ; A 2 = e ( 1+ 2 ) ; A 3 = e ( 1+ 2 ) ; 2 (x ) + 1 f 1 (x) = erf 2 p 2 (x ) 2 f 2 (x) = erf 2 p erf (x) = p 2 Z x e u2 du: 0 ; ; 20

20 Let + = jj. If < 0 and > 0 the process (7) is ergodic and has a stationary density of the form ( p (x) _ exp + (x ) 2 A 4 [ ) A 5 f 3 (x) + A 6 f 4 (x)] p + (10) where, A 4 = p e k ; A 5 = e 1(4 + ( +)+ 1) 4 + ; Proof. See Appendix B. A 6 = e 2(4 + ( )+ 2) 4 + ; f 3 (x) = er 2+ (x ) 1 2 p + f 4 (x) = er 2+ (x ) p + er (x) = p 2 Z x e u2 du: 0!! ; ; From the previous proposition 7 we see that the solution of the particular case dx t = e =2+=2(Xt )2 dw t, > 0 (11) turns out to be stationary, despite the drift s nullity. At rst sight, as the drift is null, the process should not have any attraction towards a stable point. Finally, we point out a new form of random walk behaviour in a stationary framework. However, the process drifts to as t! 1 (it can be proved that is the stationary mean). Moreover, E [X t j X s ] = X s (t s) but the process is not integrated in the usual econometric sense because integrated processes diverge (almost surely). Intuitively we can explain it as follows: when the process is near the instantaneous 21

21 n volatility, b (x) = exp =2 + =2 (x ) 2o, is low and the process tends to remain near. If X drifts away from, volatility increases. Now X is much more irregular so there is a positive probability that the process crosses again. It is the volatility that pushes the process towards a steady point 7. On the other hand, in this model, large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes. In e ect, when X is near small changes tend to be followed by small changes as volatility is low; when X is far from large changes tend to be followed by large changes as volatility is high. Therefore, roughly speaking, it is also possible to have random walk behaviour (with a null drift) in a stationary framework as long as the volatility pushes the process towards a steady point as described in the last paragraph. This, ultimately, makes the process bounded [see an application for interest rates in Nicolau (1998); also, in this paper, we address the estimation issue in a continuous time model]. 22

22 Footnotes 1 There are two major reasons why DEM/USD exchange rate should not reach any arbitrary large value. First, the in ation rate and the GNP growth of the two economies have not shown strong and persistent di erences for, at least, the last 15 years. Second, the G-7 council of economic ministers agreed on a set of policies with regard to exchange rates. In e ect, by the end of 1986, the G-7 council considered that the USD had depreciated too much. The Plaza and Louvre Accord (1985 and 1987) agreed to stabilize exchange rates. This meant limiting the size of exchange rate uctuations with the use of coordinated central bank intervention. So in practice it is possible that the DEM/USD has been in an implicit target zones regime. 2 If X is geometrically ergodic then X is asymptotically stationary exponentially fast (despite the initial value). Furthermore, if X 0 (initial value) has stationary distribution ; X is strictly stationary and covariance stationary if R x 2 (dx) < 1. 3 It would be possible to replace this condition by: there exist a K > 0 and a M > 0 such that, a (x) > K if x < M and a (x) < K if x > M: 4 We apply the theorem 2.2 in Nelson (1990). 5 Obviously, (6) is equivalent to (5). 6 We are talking about stationary distributions so these distributions only make sense when the process is stationary. 7 Technically, when the process is in its natural scale (see the following notation in the Appendix B), i.e., s (x) = 1, the quantity m (x) " 2 is of the order of the expected time the process spends in the interval (x "; x + ") given X 0 = x before departure thereof [see Karlin e Taylor (1981), pp ; in e ect E [T x ";x+" j X 0 = x] = 23

23 m (x) " 2 where T a;b = min ft a ; T b g and T a is hitting time of a, so T a;b is the rst time n the process reaches either a or b]. It can be proved that m (x) = exp (x ) 2o, so m (x) is maximum when x =. That is, the process spends more time in the interval ( "; + ") than in any other interval (with xed "). 24

24 References Adler, M. & Lehman, B. (1983) Deviations from Purchasing Power Parity in the Long Run. Journal of Finance 38, Aït-Sahalia, Y. (1996) Testing Continuous-Time Models of the Spot Interest Rate. The Review of Financial Studies 9(2), Arnold, L. (1974) Stochastic Di erential Equations: Theory And Application. John Wiley & Sons. Bergstrom, A. R. (1993) The ET Interview: A.R. Bergstrom In P.B. Phillips (ed.), Models, Methods and Applications of Econometrics, B. Blackwell, Cambridge, Mass. Bollerslev, T. (1986) Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics 31, Borovkov, A. (1998) Ergodicity and Stability of Stochastic Processes. Wiley Series in Probability and Statistics, John Wiley and Sons. Chan, K. & Karolyi, G. & Longsta, F. & Sanders, A. (1992) An Empirical Comparison of Alternative Models of the Short-Term Interest Rate. The Journal of Finance XLVII (3), Dahlquist, M. (1996) On Alternative Interest Rate Processes. Journal of Banking and Finance 20, Gray, S. (1996) Modeling the Conditional Distribution of Interest Rates as a Regime- Switching Process. Journal of Financial Economics 42,

25 Hamilton, J. (1994) Time Series Analysis. Princeton University Press. Ikeda, N. & Watanabe, S. (1981) Stochastic Di erential Equations and Di usion Processes. North-Holland Mathematical Library. Karlin, S. & Taylor, H. (1981) A Second Course in Stochastic Processes. Academic Press. Kloeden, P. & Platen, E. (1992) Numerical Solution of Stochastic Di erential Equations. Springer-Verlag. Krugman, P. & Miller, M. (1992) Exchange Rate Targets and Currency Bands. Centre for Economic Policy Research, Cambridge University Press. Kwiatkowski, D. & Phillips, P. & Schmidt, P. & Shin Y. (1992) Testing the Null Hypothesis of Stationarity Against the Alternative of a Unit Root - How Sure are We That Economic Time Series Have a Unit Root? Journal of Econometrics, 54, Nelson, D. (1990) ARCH Models as Di usion Approximations. Journal of Econometrics 45, Nicolau, J. (1998) Simulated Likelihood Estimation of Non-Linear Di usion Processes through Non-Parametric Procedure with an Application to the Portuguese Interest Rate. WP 4-99, Bank of Portugal. Nicolau, J. (1999) Modelling the DEM/USD Exchange Rate as an Ergodic Stationarity Continuous Time Process. Working Paper n o 2-99, CEMAPRE - Instituto Superior de Economia e Gestão. 26

26 Perron, P. (1989) The Great Crash, the Oil Price Shock, and the Unit Root Hypothesis. Econometrica 57, Rogo, K. (1996) The Purchasing Power Parity Puzzle. Journal of Economic Literature 34, Roll, R. (1979) Violations of Purchasing Power Parity and their implication for e cient international commodity markets In M. Sarnat & G. Szego (eds.), International Finance and Trade, Ballinger, Cambridge. Rose, A. (1996) International Finance and Macroeconomics, NBER Reporter, Winter, 1-6. Skorokhod, A. (1989) Asymptotic Methods in the Theory of Stochastic Di erential Equation, Translation of Mathematical Monographs 78, American Mathematical Society. Stanton, R. (1989) A Nonparametric Model of Term Structure Dynamics and the Market Price of Interest Rate Risk. Journal of Finance 52, Tong, H. (1990) Non-Linear Time Series - A Dynamical System Approach, Oxford Statistical Science Series 6. Appendix A The A1-A7 Conditions A1 0 2 R is an equilibrium state in the sense that (0) = 0. 27

27 A2 0 2 R is exponentially asymptotic stable, that is, 9K,c > 0 such that 8t 0, and x 0 2 R k t (x 0 )k Ke ct kx 0 k, where kxk is the Euclidean norm of x and t (x 0 ) is the t-th iteration t (x 0 ) = ( (::: (x 0 ))) given the initial value x 0 (observe that t (x 0 ) = k ( s (x t s k )), t > k > s). A3 8x 2 R and for all neighbourdhoods V of 0 2 R there is a non-null conditionally probability of (X t 1 ) " t being in V given X t 1 = x. A4 The distribution of " t has an absolutely continuous component (with respect to the Legesgue measure) with positive probability density function over some open interval ( ; ). (0; 0) =@x 6= 0. A6 is Lipschtiz continuous over R, that is, 9M > 0 such that 8x; y 2 R k (x) (y)k M kx yk : A7 8x 2 R, 0 < E [k (X t 1 ) " t kj X t 1 = x] < 1. B Proofs Proof of Proposition 2. The joint condition of H1 and H2 is 0 < a (x) < 2x if x < 0 and 2x < a (x) < 0 if x > 0. Therefore, we have, in both cases, 2x < a (x) < 0, jx + a (x)j < jxj 0 < a (x) < 2x, jx + a (x)j < jxj so, we can write, j (x)j = jx + a (x)j < jxj or j (x)j jxj where 0 < < 1: Now 28

28 j (x 0 )j jx 0 j j 2 (x 0 )j = j ( (x 0 ))j j (x 0 )j 2 jx 0 j ::: j t (x 0 )j t jx 0 j so A2 holds with c = log > 0:. The remaining proofs are based on the following. Let dx t = a (X t ) dt + b (X t ) dw t, t > t 0 ; be a di usion process, I = (l; r) the n R o z state of space of X process, s (z) = exp z 0 2a (u) =b 2 (u) du the scale density function, where z 0 is an arbitrary point inside I; and m (u) = b 2 (u) s (u) 1 the speed density [see Karlin and Taylor (1981)]. Let S (l; x] = lim x1!l R x x 1 s (u) du and S [x; r) = lim x2!r R x2 x s (u) du where, l < x 1 < x < x 2 < r: According to Arnold (1974), p. 114, if the in nitesimal coe cient a and b have continuous derivatives with respect to x, then there exists a unique continuous process de ned until the random moment expulsion in the interval t 0 < 1: Ikeda and Watanable (1981), pp have proved that if S (l; x] = S [x; r) = 1 then P [ = 1j X 0 = x] = 1: Now, it is known that if S (l; x] = S [x; r) = 1 and R r l m (x) dx < 1 then X is ergodic and the invariant distribution P 0 has density p (x) = m (x) = R r l m (u) du with respect to the Lebesgue measure [see Skorokhod (1989), theorem 16]. Proof of Proposition 4. We must prove S (l; x] = S [x; r) = 1 where l = 1 and r = 1 and a (x) = 29

29 e k e 1(x ) e 2(x ), b (x) =. First, consider the case 1 > 0, 2 > 0. We have (!) 2e k e 1(x ) s (x) = exp 2 + e2(x ) : 1 Now, if 1 > 0, 2 > 0 then s (x)! 1 as x! 1 or x! 1: Then, S (l; x] = S [x; r) = 1. In the case 1 > 0; 2 = 0 (the analysis of 1 = 0, 2 > 0 is similar) 2 we have s (x) = exp ( 2e k 2 e 1(x ) 1 + x!) : and so s (x)! 1 as x! 1 or x! 1: Then, S (l; x] = S [x; r) = 1. Proof of Proposition 5. We must prove R r l m (x) dx < 1. First, consider the case 1 > 0, 2 > 0. We have m (x) = 2 exp (!) 2e k e 1(x ) 2 + e2(x ) : 1 2 It is easy to conclude that there are some positive constants k 1 ; k 2 such that m (x) f (x) = exp k 1 k 2 x 2, 8x 2 I: Now R R f (x) < 1 ) R R m (x) dx < 1. In the case 1 > 0; 2 = 0 (the analysis of 1 = 0, 2 > 0 is similar) we have m (x) = 2 exp (!) 2e k e 1(x ) x : As Z R m (x) dx = Z 0 1 Z 0 1 m (x) dx + Z 1 0 m (x) dx exp k 1 k 2 x 2 dx + Z 1 0 exp f k 3 xg dx < 1 then R r l m (x) dx < 1 (for k 1 ; k 2 ; k 3 > 0). 30

30 Proof of Proposition 6. We must prove S (l; x] = S [x; r) = 1 where l = 1 and r = 1 and a (x) = e k e 1(x ) e 2(x ), b (x) = e =2+=2(x )2. Consider the case > 0. We have, s (x) = exp A1 [A 2 f 1 (x) + A 3 f 2 (x)] p where, A 1 = p e k 1 2 ; A 2 = e ( 1+ 2 ) ; A 3 = e ( 1+ 2 ) ; 2 (x ) + 1 f 1 (x) = erf 2 p 2 (x ) 2 f 2 (x) = erf 2 p erf (x) = p 2 Z x e u2 du: It is not possible to determine S (x) = R x s (u) du, but it can be proved that S (x) is not integrable. The function s (x) is of type 0 ; ; exp fc 1 erf (c 2 + c 3 x) + c 4 erf (c 5 + c 6 x)g where c i are parameters. As 1 erf (x) 1 it follows that s (x) does not tend to zero as jxj! 1. Therefore, S (l; x] = S [x; r) = 1. Now consider the case < 0 and let + = jj. We have s (x) = exp ( A 4 [ ) A 5 f 3 (x) + A 6 f 4 (x)] p + where, A 4 = p e k ; 31

31 A 5 = e 1(4 + ( +)+ 1) 4 + ; The function s (x) is of type exp ( A 6 = e 2(4 + ( )+ 2) 4 + ; f 3 (x) = er 2+ (x ) 1 2 p + f 4 (x) = er 2+ (x ) p + c 1 er c 2 x + c 3 1 p + er (x) = p 2 Z x e u2 du:! 0!! + c 4 er c 2 x + c p + ; ;!) where c 1 ; c 2 ; c 4 > 0 and c 3 2 R. As er (x) is not bounded, this case needs to be analysed carefully. Let us study the function (x) = c 1 er (x + k 1 ) + c 4 er (x + k 2 ) c 1 ; c 2 > 0. The derivative of is 0 (x) = 2 p c 1 e (x+k 1) 2 + c 4 e (x+k 2) 2 : If k 2 > k 1 then 8 >< > 0 x > x 0 0 (x) = >: < 0 x < x 0 for some x 0, and, under these conditions, (x) increases as jxj increases (to any arbitrary large value). Without any loss of generality, let us take s (x) = e (x) where k 2 = 2 and k 1 = 1. Therefore the condition, > 0 ensures that s (x)! 1 and so S (l; x] = S [x; r) = 1: Proof of Proposition 7. 32

32 We must prove R r l m (x) dx < 1. First, consider the case > 0. We have m (x) _ exp (x ) 2 + A 1 [A 2 f 1 (x) A 3 f 2 (x)] p where A 1 ; A 2 ; A 3 ; f 1 and f 2 are those given in proof of proposition (6). The function m (x) is of type exp c 1 + c 2 x c 3 x 2 + f (x) where c 2 ; c 3 > 0 and f (x) satis ed jf (x)j < L < 1 (notice that f (x) depends on constants and on the erf function). Therefore it is possible to nd k 1 ; k 2 ; k 3 > 0 such that m (x) exp k 1 + k 2 x k 3 x 2 so R r l m (x) dx 1. In the case < 0 and using the ideas of proof (6) it can be shown that log m (x) tends quickly to 1 as jxj! 1. So m (x) is integrable. 33

33 Figure 1: Curve a (x) x 120 Figure 2: Bounded Random Walk vs. Random Walk 115 R a n d o m W a l k B R W ( D i s c r e t e T i m e ) t 34

34 Figure 3: Bounded Randow Walk vs. Wiener Process W i e n e r B R W ( C o n t i n u o u s T i m e ) t 0,07 Figure 4: Stationary Density 0,06 0,05 0,04 0,03 0,02 0, x 35