1. INTRODUCTION
The sample autocorrelogram functions (ACF) reported for macroeconomic series as purchasing power parity (PPP) deviations and for financial series concerning either financial returns, exchange rates, interest rates, or volatilities generally feature slow decay rate. This stylized fact gives rise to a large body of applied literature, where the dynamics of the process of interest is described by means of the so-called fractional or long-memory model. The standard form of this model introduced by Granger and Joyeux (1980) is
where L is the lag operator, d is the differencing order and the errors (εt) are independent and identically distributed (i.i.d.) with a nondegenerate distribution. This model provides a large range of (linear) serial dependence. For d ∈ (0, 0.5) the process (Yt) is both stationary and invertible but is nonstationary if d > 0.5. The unit root process is obtained for d = 1. When d is positive, the process exhibits long-memory features, because the spectral density at frequency 0 is infinite and the linear autocorrelations of (Yt) are not absolutely summable.
However, the assumption of i.i.d. error terms is restrictive and has important consequences on the properties of process (Yt). (i) For instance, the domain of the process is necessarily lower and upper unbounded. This latter property is not compatible with the interpretations or observed evolutions of a large number of macroeconomic or financial variables. Some variables such as risk-free interest rates or volatilities are nonnegative, the exchange rates between European countries entering in the Euro zone can be constrained to stay in a band, and the PPP deviations generally feature mean-reverting behavior. (ii) Moreover, the optimal prediction of Yt in model (1) involves a large (in fact infinite) number of lagged values of the process, whereas financial practitioners use (nonlinear) prediction formulas depending on a rather small number of lagged values.
Thus, there is a need for models that are compatible with the observed slow decay rate of linear ACF, boundary constraints, and prediction functions with a small number of lagged values. For this purpose, we can introduce nonlinear dynamics that allow for unit root in a given band and ensure nonexplosive behavior by introducing either a reflecting condition or a mean-reverting condition outside the band (see Brooks and Reveiz, 2002; Farnsworth and Bass, 2003). Reflecting Brownian motion is an example of a model including both a unit root characteristic and a boundary effect. However a model based on reflecting Brownian motion has the drawback of assuming a strict barrier. A more flexible specification is the BAND-TAR model introduced by Balke and Fomby (1997) and used by Bec, Ben Salem, and Carrasco (2004) for the analysis of PPP deviations. For an autoregressive dynamic model of order one, the recursive equation is
where the errors (εt) are i.i.d. such that
and the autoregressive coefficient ρ is smaller than 1 in absolute value. This model has a structural justification in a two-country one-sector model with transportation costs (see Sercu, Uppal, and Van Hulle, 1995): “The costs create a regime of no trade where the PPP relationship does not hold. Outside this area, international goods market arbitrage provides an error correction mechanism which brings national price levels back to equality, apart from a constant term reflecting the cost.” The regime dynamics is determined endogeneously according to the lagged value Yt−1.
The autoregressive model (2) with deterministic thresholds can be extended by considering a model with stochastic random walk regime intensity:
where πrw(Yt−1) gives the stochastic probability of being in the random walk regime.
(i) Model (2) corresponds to the choice of the indicator function πrw(y) = 1, if |y| < λ, πrw(y) = 0, otherwise.
(ii) Let us now assume for illustration that the regime switching is a consequence of intervention of central banks to change the autoregressive coefficient from 1 (explosive case) to ρ (stationary case). In model (2), deterministic interventions are implicitly assumed whenever |Yt−1| is larger than the given threshold λ. This model can be extended by considering that the central banks intervene when |Yt−1| is too large but not in a systematic way. Their behavior can be represented by a 0 − 1 variable It, where It = 1 if the intervention occurs and = 0 otherwise. A standard specification introduces a stochastic threshold (for a similar idea, see Enders and Granger, 1998) and assumes that It = 1 if |Yt−1 + ηt| > λ and = 0 otherwise, where (ηt) and (εt) are independent and (ηt) is a strong white noise with a symmetric continuous distribution. Then, we get model (3) with intensity πrw completely defined by the value of the threshold and by the distribution of ηt:
A special case has been studied recently by Rahbek and Shephard (2002) with μ1 = μ2 = 0,
and i.i.d. Gaussian errors.1
It can be checked that this model corresponds to a noise (ηt) with a twice differentiable c.d.f. G satisfying the differential equation
, for y > 0.
(iii) In fact, the stationarity of PPP deviation can be controlled even if the probability of intervention is not equal to 1 for large PPP deviation, that is, if intensity πrw is such that limy→∞ πrw(y) > 0. The existence of a small intervention probability for large PPP deviation does not necessarily imply the nonstationarity of the process. However, the distribution of the time spent in the unit root regime, the decay rate of the autocorrelation, and more generally the persistence properties depend on the pattern of intensity πrw and noise distribution F.
All the preceding models assume a symmetric probability of intervention (see Figures 1a corresponding to model (2), 1b corresponding to case (ii), and 1c corresponding to case (iii)). This assumption is approximately satisfied if there is a consensus between the two central banks to avoid PPP deviation, as was the case with European countries before the introduction of the Euro. However, an asymmetric probability of intervention can be more realistic if one of the countries is dominant. This is typically the case with the United States when the exchange rate of the Euro versus the dollar increases to facilitate U.S. exports. In this paper, we consider a model that mixes types (ii) and (iii) by considering a probability of intervention that is increasing and such that limy→∞ πrw(y) > 0 (see Figure 1d).
Random walk regime intensities.
The model is described in Section 2, with special attention to constant and threshold intensity. In Section 3, we consider an exponential-type distribution for the noise. Indeed, in this framework it is possible to perform analytical computations and derive the stationary distribution of the process (Yt). Then, we discuss how the tails of this stationary distribution depend on intensity πrw. We provide in Section 4 ergodicity conditions for the process (Yt). We distinguish the case πrw = limy→∞ πrw(y) < 1, where the process is always stationary, and the case πrw = 1, in which joint constraints between πrw and the noise density f have to be introduced. The ergodicity conditions are especially important for statistical inference. They are also important in discussing the asymptotic behavior of the unit root tests or estimators of the fractional order d, usually built under specification (1), when this specification (1) is incorrect and the process satisfies a stochastic unit root model. Section 5 concludes. The proofs are gathered in Appendixes.
2. THE MODEL
2.1. Definition of the Process
The process (Yt) satisfies the stochastic unit root regime
where t ≥ 1, (εt) are i.i.d. zero mean random variables independent of Y0, and the random walk regime intensity πrw is a càdlàg increasing function with limy→−∞ πrw(y) = πrw and limy→∞ πrw(y) = πrw. We assume that εt has a density f strictly positive on the whole real line, that the variance of the noise distribution σ2 is finite, and that πrw > 0 and πrw < 1 to avoid the standard pure white noise or random walk cases. We denote by F the cumulative distribution function (c.d.f.) of εt. For expository purposes, we fix μ1 = μ2 = μ, ρ = 0, but the main results are valid for any μ1, μ2, and ρ, |ρ| < 1 (and will be systematically provided). This process will be called a dynamic switching (D.S.) Markov chain.
At any date t, there are two possible regimes corresponding to either a white noise, when Yt = μ + εt, or a random walk, when Yt = μ + Yt−1 + εt. This model can be considered as the reduced form of a dynamic model with a latent regime variable. The latent model is
where
and the variables (Zt) are i.i.d., independent of the noise (εt) with c.d.f. πrw. Thus, the latent variable Zt can take infinite values with a positive probability. In the example of PPP deviation, the indicator function Bt = 1 − It, where It denotes the indicator of intervention.
We get a linear autoregressive model with stochastic autoregressive coefficient Bt. The regime indicator is endogeneous, because its conditional distribution depends on the lagged value Yt−1. Therefore, the model differs from a model in which (Bt) is an underlying hidden Markov chain independent of the noise (εt) as in Hamilton (1989) or Granger and Swanson (1997).
Here (Yt) is a homogeneous Markov process, with a transition density corresponding to a mixture of distributions:
where fμ(y) = f (y − μ). The probability density function (p.d.f.) of the stationary distribution is denoted by l if it exists and is unique. The stationary density satisfies the integral equation
2.2. The Switching Regime Process
The process (Bt,t ≥ 1) provides the sequence of random walk regime indicators. The sequence of durations spent in the two admissible states is denoted by D10, D11, D20, D21,…; Dj0 is the duration of the jth spell in the white noise regime (called regime 0), whereas Dj1 is the duration of the jth spell in the random walk regime (called regime 1). This sequence may start by either a random walk or a white noise regime according to the initial value Y0 and the initial drawing B1 that follows conditional on Y0, a Bernoulli distribution with parameter πrw(Y0). As an illustration, Figure 2 provides a path of the D.S. Markov chain jointly with the path of the associated regime indicator process B. The selected stochastic intensity is πrw(y) = Φ(y), where Φ is the standard normal cumulative distribution function, μ = −0.2, whereas the noise is standard normal and the initial value is Y0 = 0. There is no excursion of process (Yt) in the set of large negative values. Indeed, when the process is negative, there is a high probability of staying in the white noise regime. When process (Yt) takes large positive values, the path is similar to the path of a random walk with a negative drift. In particular, this path does not come back to zero abruptly because of the high probability of staying in the random walk regime.
Paths of Y and B processes.
Figure 3 provides two paths of the process with different initial values Y0 = 12 (solid line) and Y0 = 7 (dotted line) and the same noise sequences (εt) and (Zt) (see eqn. (6)). Both paths are similar for t larger than the first time when the regime of the process with the largest value changes from the random walk regime to the white noise regime. In the example of Figure 3 where the process with the largest initial value has always been positive before changing of regime, any path with a smaller initial value joins this path at the same date.
Paths of Y for different initial values.
Let us now describe some distributional properties of the durations spent in the white noise and random walk regimes, respectively. Let us denote
, where Sk = ε1 + ··· + εk + kμ is the random walk process with drift. The probabilities pk, k ≥ 1, are strictly positive because πrw > 0 and f > 0. They define a monotonic nonincreasing sequence.
PROPOSITION 1.
(i) The durations spent in the random walk regime are a.s. finite if and only if limk→∞ pk = 0.
(ii) If limk→∞ pk = 0, the durations D10, D11, D20, D21,… are independent. Moreover, the durations spent in the white noise regime Dj0, j ≥ 2 (resp. Dj1, j ≥ 2) have identical distributions. The distribution is a geometric distribution for D20:
The distribution of D 2 1 is given by
Proof. See Appendix A.
The path dependence of the random walk intensity does not imply the serial dependence of durations but only has an effect on their stationary distribution. For instance, if πrw and πrw* are such that πrw*(y) ≥ πrw(y), ∀y, then the distribution of duration D21 computed with πrw* stochastically dominates at order one the distribution of D21 computed with πrw, that is, P*[Dj1 ≥ d] ≥ P[Dj1 ≥ d], ∀d ≥ 1. Note also that the distribution of the duration spent in the white noise regime admits exponential tails, whereas the tail pattern for the duration spent in the random walk regime depends on intensity πrw.
The properties of durations Dj0 and Dj1 described in Proposition 1 can be used to simplify the analysis of process (Yt). More precisely, the discrete time process
is a renewal process, that is, it admits i.i.d. increments. In this case, the cycles {Yt+τ(k−1)}0≤t<τ(k)−τ(k−1), k ≥ 2, are also i.i.d. This means that (Yt) is a regenerative process, for which simplified properties have been given in the literature (Asmussen, 1987). Note also that the set
is a proper atom of the chain (Yt,Bt), that is,
for all
, and the times at which the chain enters the atom are just the regeneration times τ(k).
In the proposition that follows, we give sufficient conditions for a.s. finite durations spent in the random walk regime.
PROPOSITION 2. Sufficient conditions for a.s. finite durations spent in the random walk regime are
(i) πrw = limy→∞ πrw(y) < 1;
(ii) πrw = 1 and μ ≤ 0;
(iii) πrw = 1, μ > 0, and 1 − πrw(y) is not integrable at +∞.
Proof. Let us note that Ak = πrw(S1)πrw(S2) … πrw(Sk) is a nonincreasing sequence of nonnegative random variables. Thus, the a.s. convergence of Ak to zero implies the condition limk→∞ pk = 0, by Beppo–Levi's theorem (see Billingsley, 1995, p. 209). Let us now check for the convergence of Ak to zero.
(i) If πrw < 1, we get limk→∞ Ak ≤ limk→∞(πrw)k = 0.
(ii) Let us first assume that μ < 0. The random walk with drift Sn = ε1 + ··· + εn + nμ converges almost surely to −∞, when n increases and the result follows.
Let us now assume that μ = 0. Because τ−(1) = inf{n ≥ 1 : Sn ≤ 0} is a.s. finite (see Asmussen, 1987, Thm. 2.4, p. 169), the descending ladder epochs τ−(n + 1) = inf{k > τ−(n) : Sk ≤ Sτ−(n)} exist and (τ−(n + 1) − τ−(n)), n ≥ 1, are a.s. finite random variables. Then
because πrw(Sτ−(n)) ≤ πrw(0).
(iii) Finally, the last sufficient condition results from the law of iterated logarithm, which ensures that for any λ > 1 (see Billingsley, 1995, p. 155)
a finite number of times a.s. We deduce that a sufficient condition for the a.s. convergence of Ak to zero is
or equivalently
The result follows. █
In the proposition that follows, we exhibit lower and upper bounds for probabilities pn. These bounds can be used to transpose conditions written in terms of pn into conditions written in terms of πrw and f. When πrw > 0 and πrw < 1, we have (πrw)n ≤ pn ≤ (πrw)n and a geometric decay for pn. Some bounds of the probabilities pn can also be derived in the limiting case πrw = 1.
PROPOSITION 3. Let us assume πrw = 1 and the noise distribution with a Laplace transform existing in a neighborhood of 0.
(i) If πrw > 0, then, for any λ > 1, there exists a constant Cλ such that
(ii) If μ > 0, then, for any λ > 1, there exists a constant Dλ such that
(iii) If μ = 0, then, for any λ > 0, there exists a constant Dλ* such that
(iv) If μ < 0, there exist two scalars ϰ and θ < 1 such that
Proof. See Appendix B.
For example, let us assume that μ > 0 and
with ½ < τ ≤ 1, c > 0. The intensity is such that πrw = exp(−1) > 0 and πrw = 1. We can apply Proposition 3 (i) and (ii) with λ > 1.
1. If τ = 1, there exist two constants Cλ and Dλ such that
and (pn) has a hyperbolic rate of decay. Moreover, the mean of D21,
is finite if and only if c > μ.
2. If ½ < τ < 1, there exist two constants Cλ and Dλ such that
Because the survivor function of the durations spent in the random walk regime is proportional to probability pn (see Proposition 1), the bounds can be used to analyze the type of the tails of the durations. When πrw = 1 and μ > 0, they are Weibull or Pareto tails in the example (recall that we have thin tails if πrw < 1, or πrw = 1 and μ < 0).
2.3. Special Cases
(i) Constant intensity. When πrw(y) = πc is constant, we get exogenous regimes and a special case of hidden Markov process (see MacDonald and Zucchini, 1997). The regime indicators Bt are i.i.d., independent of the noise process. The durations Dj0, j ≥ 2, (resp. Dj1, j ≥ 2) have both geometric distributions P(D20 = n) = πc(1 − πc)n−1, (resp. P(D21 = n) = (1 − πc)πcn−1). Therefore, the assumption of exogenous regimes implies thin tails for both duration distributions (if 0 < πc < 1).
(ii) Threshold intensity. The process reduces to a threshold autoregressive process (see Tong, 1990) when πrw is the indicator function
. Indeed, we have
. This process is related to the Lindley process (Ut) defined by
and studied in Feller (1966).
PROPOSITION 4. When Y0 = U0 = 0, the process
is such that Ut = Yt+ is a Lindley process.
Proof. The proof is by recursion. The result is obvious for t = 0. If it is valid for t − 1, we get
We deduce Yt+ = (Ut−1 + μ + εt)+ = Ut. █
The durations Dj0, j ≥ 2, have identical distributions: P(D20 = n) = (1 − p)pn−1, where p = P(S1 ≤ 0). The durations Dj1, j ≥ 2, are such that P(D21 = n) = P(τ = n + 1|τ ≥ 2), where
Here τ is the time to the first visit of a random walk with drift in the set of negative values. When μ < 0, the first moment of the duration D21 can be derived by applying random walk theory (see, e.g., Asmussen, 1987, p. 176). We have
Because P(τ = 1) = p, we deduce
3. EXPONENTIAL-TYPE DISTRIBUTION
Analytical computations can be performed when the noise admits an exponential-type distribution. This is the aim of this section. Let us assume that
where α, β are positive. The mean of this distribution is μ = (α−1 − β−1)/2.
PROPOSITION 5. If the intensity πrw is differentiable on
, if the stationary distribution of the D.S. Markov chain exists, and if its density l is twice differentiable on
, then it satisfies the second-order linear differential equation
Proof. See Appendix C.
Closed form stationary densities can be derived in special cases.
(i) Constant intensity. When πrw(y) = πc is constant, we have
We get
where
The terms γ and δ are the solutions of the equation Ψε(u) = (πc)−1, where
is the Laplace transform of the noise distribution.
Thus, for constant intensity πrw(y) = πc and exponential-type noise distribution, the stationary distribution of the D.S. Markov chain also has exponential tails. However the left and right tail parameters are related in a complicated way to the tail parameters of the noise α, β and to the random walk regime intensity πc. For symmetric tails of the noise distribution α = β, we get symmetric tails for the stationary distribution with
. The stochastic random walk regime implies a tail increase when we pass from the noise distribution to the stationary distribution. This result is compatible with the tail increase already observed for conditionally heteroskedastic models (see Engle, 1982) and time deformed processes (see Mandelbrot and Taylor, 1967; Clark, 1973).
(ii) Threshold intensity. When
, we also get differential equations with constant coefficients on the positive (resp. negative) real line:
This differential system admits a density function as solution if and only if μ < 0, which is the stationarity condition of the Lindley process. The solution is given by
with
The tails of the stationary distribution are still of exponential type.
(iii) Model without drift. If α = β = 1, the drift is zero, and the differential equation becomes
The right tail of the stationary distribution is necessarily such that l′′/l is decreasing, with value in [0,1]. As an illustration, we consider subsequently various admissible patterns of the stationary distribution (restricted to the positive real line).
[bull ] Exponential tail. If l′′(y) = γ2A exp(−γy)[1 + o(1)], 0 < γ < 1, for large y, we get l(y) = A exp(−γy)[1 + o(1)] by using l'Hospital's rule. Then, l′′(y)/l(y) = γ2[1 + o(1)] and the intensity πrw tends to 1 − γ2 = πrw < 1 at infinity. For instance, an admissible pair is
where
.
[bull ] Exponential tail with hyperbolic effect. But we can have l′′(y)/l(y) = γ2[1 + o(1)] without strict exponential tails. For instance, an admissible pair is
where
.
[bull ] Weibull tail. If l′′(y) = Ac2τ2y2(τ−1) exp(−cyτ)[1 + o(1)], 0 < τ < 1, and c > 0, for large y, we have l(y) = A exp(−cyτ)[1 + o(1)] and 1 − πrw(y) = c2τ2y2(τ−1)[1 + o(1)]. For instance, an admissible pair is
with 0 < τ < 1, 0 < c < 1, and b > 0.
[bull ] Pareto tail. If l′′(y) = Aa(a + 1)y−(a+2)[1 + o(1)], a > 1, for large y, then l(y) = Ay−a[1 + o(1)] and 1 − πrw(y) = a(a + 1)y−2[1 + o(1)]. For instance, an admissible pair is
where a > 1, d2 > α(α + 1), and b > 0.
[bull ] Gompertz tail. If l(y) = A exp(−exp y) y > 0, we get l′′(y)/l(y) = exp[y(exp y − 1)][1 + o(1)], which is not between 0 and 1. Thus, the stationary distribution cannot admit Gompertz tail.
The preceding examples show that
- stationary distributions with exponential tails cannot be obtained if πrw = 1 and μ = 0 but there is a large variety of admissible tails;
- stationary distributions with very thin tails such as Gompertz tails can never be obtained.
4. ERGODICITY AND PERSISTENCE OF THE PROCESS
In Section 2, we mentioned that the D.S. Markov chain is regenerative, that is, it can be split up into independent cycles, if the time spent in the random walk regime is a.s. finite. Roughly speaking, the process does not explode, and a stationary distribution may exist. In this section, we use Markov chain theory to formalize this discussion, and we study the stochastic properties of the process.
4.1. Ergodicity
General results on Markov processes can be applied to the D.S. Markov chain. Let us denote by ν the Lebesgue measure and by
the Borel σ-algebra on
. For any measurable set A, we define the time to entry in A:
and the number of visits to A:
(i) The first property that can be expected from a Markov chain is that any subset of the space can be reached, for any starting value. A Markov chain is ν-irreducible if any set of positive Lebesgue measure can be reached in finite time starting from any initial value. This condition can be written as
The D.S. Markov chain is ν-irreducible because the density of ε1 is strictly positive and
(ii) A Markov chain is unstable when it returns only a finite number of times to a given set of “reasonable size.” Two definitions of stable chains are commonly used:
(a) the Markov chain (Yt) is ν-recurrent if it is ν-irreducible and for any set of positive Lebesgue measure the expected number of returns to this set is infinite starting from any initial value. This condition can be written as
(b) The Markov chain (Yt) is Harris-recurrent if it is ν-irreducible and for any set of positive Lebesgue measure the chain returns to the set infinitely often starting from any element in the set. This condition can be written as
A Harris-recurrent Markov chain is ν-recurrent, and, if the chain is ν-recurrent, then it admits a unique invariant measure. Let us now consider the D.S. Markov chain.
LEMMA 1. If the duration spent in the random walk regime is a.s. finite, the D.S. Markov chain is Harris-recurrent.
Proof. Indeed, under this condition, the durations spent in the two regimes are a.s. finite, and there is an infinite number of cycles {Yt+τ(k−1)}0≤t<τ(k)−τ(k−1). Then, we have
where the first inequality uses the independence between cycles corresponding to k ≥ 2 and the starting value. Because ετ(k−1)+1, k ≥ 2, are i.i.d. and
we conclude by the Borel–Cantelli lemma that
Sufficient conditions for Harris-recurrence of the D.S. Markov chain are directly deduced from Proposition 2.
(iii) The Markov chain (Yt) is positive Harris-recurrent if it is Harris-recurrent and the invariant measure is finite.
Positive Harris-recurrence can be analyzed for the D.S. Markov chain. If the expected duration spent in the random walk regime is finite, the regenerative process (Yt) admits a limiting probability distribution (see Asmussen, 1987, Thm. 1.2, p. 126) and the Markov process is positive Harris-recurrent. Therefore, sufficient conditions for the positive Harris-recurrence of the D.S. Markov chain are directly deduced from Proposition 3.
For instance, we get immediately the corollary that follows.
COROLLARY 1. Let us assume that πrw = 1.
(i) The D.S. Markov chain is Harris-recurrent if one of the following conditions holds:
(a) μ ≤ 0;
(b) μ > 0 and 1 − πrw(y) is not integrable at +∞.
(ii) The D.S. Markov chain is positive Harris-recurrent if one of the following conditions holds:
(a) μ < 0;
(b) μ ≥ 0 and limy→∞[1 − πrw(y)]y = c ∈ (0;∞] and c > μ.
Proof.
(i) It is deduced from Proposition 2.
(ii) If μ < 0, from Proposition 3(iv) we deduce that
If μ > 0 and limy→∞[1 − πrw(y)]y = c is finite, then, for large n, pn < Dλ n−c/λμ (see Proposition 3(ii)). Let us choose 1 < λ < cμ−1. Then, we get
. A similar discussion can be done if c is not finite or if μ = 0. █
In the following proposition, we give additional sufficient conditions for positive Harris-recurrence when πrw = 1 and μ = 0 and for null-recurrence2
A Markov chain is null-recurrent if it is Harris-recurrent but not positive Harris-recurrent.
PROPOSITION 6.
(i) Let us assume that πrw = 1 and μ = 0. The D.S. Markov chain is positive Harris-recurrent if limy→∞[1 − πrw(y)]y2 = b ∈ (0;∞] and b > σ2.
(ii) Let us assume that πrw = 1, μ > 0, and E|εt|η < ∞ for some η > 1. The D.S. Markov chain is null-recurrent if limy→∞[1 − πrw(y)]y = c > 0 and μ > c.
Proof. See Appendix D.
Condition (i) of Proposition 6 holds for the exponential-type noise distribution without drift (α = β = 1) discussed in Section 3(iii). It is clearly satisfied if the solution has exponential or Weibull tail. Let us now consider Pareto tail: l(y) = Ay−a[1 + o(1)], where a > 1; the intensity πrw satisfies
This example shows that the sufficient condition of Proposition 6(i) is rather accurate.
The preceding results can be extended to a situation of stationary autoregressive regime with 0 < ρ < 1 instead of the white noise regime. In this case, the sufficient conditions for the D.S. Markov chain to be positive Harris-recurrent become3
The proofs are based on Lyapunov functions and are available on http://www.crest.fr/pageperso/gouriero/gouriero.htm.
- μ < 0;
- μ > 0, limy→∞[1 − πrw(y)]y = c ∈ (0;∞], and c(1 − ρ) > μ;
- μ = 0, limy→∞[1 − πrw(y)]y2 = b ∈ (0;∞], and b(1 − ρ2) > σ2.
(iv) The Markov chain (Yt) is geometric ergodic if there exists a scalar r > 1 such that
where Y is a random variable with the stationary distribution.
PROPOSITION 7. If πrw < 1, the D.S. Markov chain is geometrically ergodic. Moreover, if the distribution of Y0 is the stationary distribution, the process is β-mixing4
Let us consider a strictly stationary process (Yt,t ≥ 0). Let Bi,j = σ{Yi,…,Yj} be the σ-algebra generated by Yi ,…, Yj. The β-mixing coefficients are defined by
A process is β-mixing if limk→∞ β(k) = 0 (see Doukhan, 1994).
Proof. See Appendix D.
4.2. Stationary Distribution
When it exists, the survivor function of the stationary distribution of the D.S. Markov chain can be written as an infinite weighted sum of the survivor functions of Sn, n varying.
PROPOSITION 8. If the D.S. Markov chain admits a stationary distribution, we have
where
, and
, where
denotes the expectation of πrw(Y) with respect to the stationary distribution. Moreover, if the density of the stationary distribution exists, we have
5The nth convolution p.d.f. is defined recursively by
where the weights qn(y) satisfy the recursion
Proof. See Appendix E.
The tail behavior of the stationary distribution of the D.S. Markov chain depends not only on the tail behavior of the noise distribution but also on the stochastic intensity through these weights.
For constant regime intensity πrw(y) = πc, the weights are constant rn(y) = qn(y) = (1 − πc)πcn−1, and the stationary distribution of the D.S. Markov chain is a compound distribution:
The following proposition gives a sufficient condition for the existence of moments of the stationary distribution.
PROPOSITION 9. Assume that μ > 0, limy→∞[1 − πrw(y)]y = c > μ and that there exists α > 2 such that
. Let k ≤ α and k < c/μ. Then
.
Proof. See Appendix E.
The existence of the moments of the stationary distribution depends on the existence of moments of the noise distribution (by mean of α) and of moments of the duration spent in the random walk regime. Indeed, if μ > 0 and limy→∞[1 − πrw(y)]y = c, the distribution of the duration spent in the random walk regime D1 has a Pareto tail with index c/μ (see Proposition 3(iii) and the example following this proposition). In particular, the stationary distribution does not necessarily admit moment at any order even if the noise does.
When the white noise regime is replaced by a stationary autoregressive regime with μ1 = μ2 = μ and 0 < ρ < 1, the conditions of Proposition 9 have to be replaced by the conditions k ≤ α and c(1 − ρk) > kμ. Therefore, the stationary distribution has fatter tails when ρ increases.
4.3. Persistence
By construction, the joint process (Yt,Bt) is a Markov process, whose history can be summarized by Yt−1 only. In this section, we consider the persistence properties of the quantitative and qualitative components (Yt) and (Bt), respectively. Let us first note that (Yt) is a Markov process with respect to its own history, whereas the qualitative process alone is not Markovian. Because process (Bt) is a renewal process, its persistence properties are related to the tail behavior of the distribution of the duration spent in the random walk regime. Typically, if this distribution has Pareto tail, that is, if (pn) features a hyperbolic rate of decay, the autocorrelogramm of (Bt) will also feature asymptotically a hyperbolic rate of decay, which can create persistence effect (see, e.g., Heath, Resnick, and Samorodnitsky, 1998; Granger and Terasvirta, 1999; Gouriéroux and Jasiak, 2001).
The persistence properties of (Yt) are related to the persistence properties of (Bt). They can be studied by comparing the conditional distribution of Yt given Y0 to the marginal one. We first provide the expression of the conditional distribution.
PROPOSITION 10. The conditional distribution of Yt given Y0 = y0 is given by
Proof. See Appendix E.
When πrw < 1, we have
and the conditional distribution of Yt given Y0 converges to the stationary distribution at a geometric rate. A stronger degree of persistence may be achieved when πrw = 1. As an illustration, Figure 4 displays the paths of both processes when
, whereas the noise distribution is N(0,1) and the initial value is Y0 = 0.
Paths of processes (Yt) and (Bt).
The marginal and cross-autocorrelograms of both processes are provided in Figure 5, where a hyperbolic rate of decay is observed.6
These theoretical autocorrelograms have been derived by simulations over a large number of replications (10,000).
Marginal and cross-autocorrelograms.
Histogram of the duration variable.
The degree of nonlinear persistence of a stationary process may be measured by the way the autocorrelation coefficients of nonlinear functions of the process
approach zero. The proposition that follows provides an asymptotic equivalent of the autocovariance coefficients of (πrw(Yt)) when the noise distribution admits a Laplace transform in a neighborhood of 0 and limy→∞[1 − πrw(y)]y = c > μ > 0.
PROPOSITION 11. Let us assume that πrw > 0, the noise distribution admits a Laplace transform in a neighborhood of 0, and limy→∞[1 − πrw(y)]y = c is finite such that c > μ > 0. If the distribution of Y0 is the stationary distribution, then
Proof. See Appendix F.
From Proposition 3, we deduce that we have found a nonlinear function of the D.S. Markov chain with hyperbolic rate of decay of the autocorrelogram (for another example, see Davydov, 1969). Because (pn /p1) is the survivor function of D21, the asymptotic equivalent is proportional to
, which is a standard measure of the tail magnitude
7is a standard measure of heaviness of the tail of D21 (see Embrechts, Klüppelberg, and Mikosch, 1997). For instance it corresponds to the tailVaR or expected shortfall proposed to analyze a risk.
The autocorrelation coefficients (12) may not be precise enough to define appropriately the degree of nonlinear persistence of a stationary process. It appears to be better measured by the ρ-mixing coefficients:
where Bi,j = σ{Yi,…,Yj} is the σ-algebra generated by Yi,…,Yj. The corollary that follows shows that the D.S. Markov chain may have a very high degree of nonlinear dependence.
COROLLARY 2. Let us assume that πrw > 0, the noise distribution admits a Laplace transform in a neighborhood of 0, and limy→∞[1 − πrw(y)]y = c is finite such that c > μ > 0. Then ρt = 1, for all t ≥ 1.
Proof. The ρ-mixing coefficients of a Markov chain decay to zero at an exponential rate or are identically one (see Bradley, 2005, Thm. 3.3(1)). We have ρt ≥ ct ≥ 0, and, by Proposition 11, the sequence (ct)t≥0 has an hyperbolic rate of decay. Therefore, ρt = 1 for all t ≥ 0. █
5. CONCLUDING REMARKS
Slow decay rates of the linear autocorrelogram are often observed for macroeconomic and financial series. This stylized fact is generally taken into account by considering fractional processes with strong white noise. However, this type of model is not compatible with the domain restrictions on the variables and contradicts the prediction practices. The aim of our paper was to study carefully a class of nonlinear short-memory models able to reconcile stylized facts and practices. The stochastic unit root model is a dynamic model with both a random walk and a stationary regime. Its persistence and fat tail properties depend on the pattern of the random walk regime intensity as a function of the past or equivalently of the tail of the distribution of the duration spent in the random walk regime. The stochastic unit root model is flexible, because it is possible to increase the autoregressive orders in both regimes and to allow unconstrained either the noise distribution or the random walk regime intensity. Moreover, the example of PPP deviation shows that the stochastic unit root model can admit structural interpretations, which is not the case with the standard fractional process. Finally, this model will be very useful for discussing the asymptotic behavior of the unit root tests or estimators of the fractional order d, usually built under specification (1), when this specification (1) is incorrect and the process satisfies a stochastic unit root model (see Shimotsu, 2005).
APPENDIX A: Distribution of the Switching Regime Process (Proof of Proposition 1)
Let us first consider an initial condition Y0, B1 = 0 (say), which ensures that the first spell corresponds to the white noise regime. The distribution of the 2j first durations (conditional on Y0, B1 = 0) is
Conditional on the noise (εt), the regime indicators are independent with different Bernoulli distributions. We deduce that
Therefore, we get
A similar computation can be performed conditional on Y0, B1 = 1.
The multiplicative decomposition of the joint duration distribution provides the independence of the durations and their stationary distributions. Finally, the probability that the duration spent in the random walk regime is finite is given by
This probability is equal to one if and only if limk→∞ pk = 0, which is condition (i). █
APPENDIX B: Bounds for the Probabilities to Stay in the Random Walk Regime (Proof of Proposition 3)
Lower Bound.
Let us assume that πrw > 0. The law of iterated logarithm ensures that for any λ > 1
a finite number of times a.s. Let A be the random set of indices for which the previous inequalities are jointly satisfied and a be the cardinal number of A. Because intensity πrw is a nondecreasing function, we get
where
is strictly positive because a is finite a.s.
Upper Bound.
Let us recall some lemmas.
LEMMA A.1 (Feller, 1966, p. 398). Let X1,X2,… be i.i.d. random variables and Σn = X1 + ··· + Xn. Let Hn = {i : i ≤ n,Σi ≥ 0}, Cn = card(Hn) be the number of nonnegative terms among Σ1,…, Σn. Then
where ak = P(Σ1 > 0,…, Σk > 0), bk = P[Max(Σ1,…, Σk) ≤ 0].
Because Cn increases with n, Cn converges a.s. to a limit C, possibly with infinite value. The term C counts the number of nonnegative terms among Σ1,Σ2…. We have
where X is independent of X1,X2,… and has the same distribution as Xn. The random variable Max(0,Σ1,…, Σn−k−1) has the same distribution as Un−k−1, where (Un) is the Lindley process defined by Un = (Un−1 + Xn)+ (cf. Asmussen, 1987, Prop. 1.1, p. 181). If
, Max(0,Σ1,…, Σn−k−1) weakly converges to a nondegenerate positive distribution as n tends to infinity. Because P(Xn < 0) > 0 and limn→∞ P[Max(0,Σ1,…, Σn−k−1) = 0] > 0, we deduce that limn→∞ bn−k = b > 0, and the limit C is such that
LEMMA A.2 (Chernoff). (See, e.g., Billingsley, 1995, p. 151). Let X1,X2,… be i.i.d. random variables satisfying
. Let Σn = X1 + ··· + Xn. Then
where ρ = inft ΨX(t) and ΨX(t) is the Laplace transform of X1.
The preceding lemmas are used to derive the upper bound for pn according to the sign of μ.
(i) Let us first consider the case μ > 0 and denote Xi = εi − (λ − 1)μ, where λ > 1. Note that
. We have
Then, we get the upper bound:
It remains to prove that
is finite. For this purpose, let us first consider the Laplace transform of C. From Lemma A.1, we have
From Lemma A.2, we get
where 1 > ρλ′ > ρλ = inft[Ψε(t)e−t(λ−1)μ]. We deduce that
is finite if t < −log ρλ′.
Then, we have
for any integer N0. The bound is finite if N0 is sufficiently large, for instance, if N0 = inf{i : −log πrw(iλμ) < −log ρλ′}, which is finite because πrw = 1.
(ii) In the case μ = 0, we select Xi = εi − λ, where λ > 0, and conclude as in (i).
(iii) In the case μ < 0, let us select Xi = εi + κμ, where 0 < κ < 1. Note that
. Then
For any i ≥ 1 and t > 0, πrw [i(1 − κ)μ] ≤ πrw(t), because πrw is nondecreasing. Therefore:
As in part (i),
is finite if −log πrw(t) < ρκ = inft[Ψε(t)etκμ]. Let N1 = inf{j ≥ 1 : −log πrw(j) < ρκ}. We get
The result follows with
. █
APPENDIX C: Stationary Distribution for Exponential Noise (Proof of Proposition 5)
The stationary density l satisfies the integral equation
Let us denote h(y) = exp(αy)l(y). For positive y, the equation becomes
By differentiating both sides of the equation, we get
We deduce
which is equivalent to
A similar approach can be applied to nonpositive values with h(y) = exp(−βy)l(y). █
APPENDIX D: Ergodicity Property
Proof of Proposition 6. From Theorem 11.0.1 of Meyn and Tweedie (1993), the Markov chain is positive Harris-recurrent if there exist a petite set C, a constant b < ∞, and a Lyapunov function V ≥ 0, finite at some
, such that
Recall that a set C is petite if the chain satisfies
for any y ∈ C, any
, where
is a nondegenerate measure on
. Because intensity πrw is nondecreasing, for any positive R there exists a constant η(R) such that, for any y ≤ R and any Borel set A,
This means that every set of the form (−∞; R] is petite.
Let us consider the function V(y) = ay2. We have
If b is finite, let us consider δ > 0 such that b − σ2 − δ > 0 and define a−1 = b − σ2 − δ. If b is not finite, let us choose a = 1. There exists R > 0 such that for y > R, ΔV(y) ≤ −1. Then we have
where C = {y,y ≤ R} and R = inf{z : ∀y ≥ z, ΔV(y) ≤ −1}.
Null-recurrence.
From Theorem 11.5.1 of Meyn and Tweedie (1993), the Markov chain is null-recurrent if there exist a petite set C, a Lyapunov function V ≥ 0 such that
Let us consider the function
then
Because
we have
Because
for some η > 1, the last terms of the decomposition are asymptotically negligible:
Thus we can focus on the other terms to find the petite set C = (−∞; R]. If limy→∞[1 − πrw(y)]y = c > 0 and μ > c, then limy→∞ ΔV(y) = μ − c > 0. There exists R > 0 such that for y > R, ΔV(y) ≥ 0.
Let us now focus on the second condition for a null-recurrent chain. We have
and
. The result follows. █
Proof of Proposition 7. It is easily seen that the Markov process is aperiodic by using Proposition A1.2 of Tong (1990). Let us now prove that the Markov process is geometrically ergodic. According to Theorem 15.0.1 of Meyn and Tweedie (1993), we have to find a petite set C, scalars b < ∞, β > 0, and a Lyapunov function V ≥ 1 such that
Let us consider the function
we get
and then for 0 < θ < 1:
where
is the set defined by
The set C is not empty, because
, and it is easily seen that it is petite. Therefore, there exist scalars c < ∞ and r > 1 such that
In particular, because the Lyapunov function V is finite, the condition for geometric ergodicity is satisfied. Moreover, condition (D.2) implies
Finally, let us now consider the β-mixing condition. It is well known that a stationary, geometrically ergodic Markov chain is β-mixing with mixing coefficients decaying at a geometric rate (see Doukhan, 1994). █
APPENDIX E: Marginal and Conditional Distributions
Stationary Distribution (Proof of Proposition 8). Let us assume that the distribution of Y0 is the stationary distribution. Then, for any t > 0:
By the dominated convergence theorem, we have
The stationary density can also be decomposed as
The weights qn can be computed by recursion:
which finishes the proof. █
Existence of Moments (Proof of Proposition 9). Note that the D.S. Markov chain is ν-irreducible and aperiodic. Let g ≥ 1 be a function on
. From Theorem 14.0.1 of Meyn and Tweedie (1993),
is finite if there exist a petite set C, a constant b < ∞, and a nonnegative function V satisfying V(y0) < ∞ for some
, and
Let us consider the power function V(y) = |y|k, 2 < k ≤ α. Then, we have for y > 0
Let c > 1. By derivating [k] times the function
and by using fy(j)(0) = 0, j ≤ [k], it is easily seen that for any x ≥ −y
We deduce that
where
, and
Because ε1 admits moments of order j ≤ α with α > 2, then
. Now let us remark that
by the dominated convergence theorem. Then for large y, there exists a constant d > 0 such that
Let us recall that limy→∞(1 − πrw(y))y = c. Then the previous inequality can be written for large y:
with c − kμ > η > 0. We conclude by the same arguments as in the proof of Proposition 6(i).
Conditional Distribution (Proof of Proposition 10). The expression of the conditional distribution is deduced from the decomposition that follows:
APPENDIX F: Proof of Proposition 11
Let p0 = 1. From Proposition 10, we have
Then, we get
or equivalently
Let us introduce the positive increasing function
From Proposition 8, we have
and then
. Moreover, by using the same arguments as for Proposition 3(i) and (ii), we can show that, for any
,
Therefore, gt can be considered a cumulative weighting function, which weights +∞ for large t. We deduce
Therefore
The sequence
is the convolution of the sequences (pt)t≥0 and (ct)t≥0. From Proposition 3,
have a hyperbolic rate of decay and
. Consequently, the asymptotic behavior of the convolution is given by the asymptotic behavior of the sequence with the heavier “tail.” We deduce that
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