2. THE ASYMPTOTIC BEHAVIOR OF [sum ]₀^T exp(σZ_t)

We prove the following theorem.

THEOREM 1. Suppose the increments (ε_i)₁^∞ are i.i.d. and satisfy

Then for all γ > 0

Proof. Clearly

so it is sufficient to prove

which is equivalent to

which in turn follows from

for some constant A. Consider the term

where F_t,T denotes the distribution function of M_T − Z_t. In the following discussion we make repeated use of the reversal procedure described on page 394 of Feller (1971). We have

where

. The proof of Theorem 1 relies entirely on the estimate

where now and in the discussion that follows a and b denote generic constants whose values may change at each appearance. As

converges weakly to the maximum of a Brownian motion on [0,1], (12) would follow immediately if a Berry–Esséen type result were available for the distribution of

. We have not found such a result in the literature so we now give a proof that relies on properties of the random walk in

as given in Chapter XII of Feller (1971). For the sake of simplicity we assume that the random variables Z_t have a continuous distribution function, in which case the distinction between weak and strict variables is unnecessary (see Feller, 1971, pp. 392–393). We consider the case of strict variables only; the arguments for weak variables are to be found in Chapter 12 of Feller (1971). We have

where

To estimate q_j in (15) we note that the assumptions of the theorem imply

for some finite constant c (see Feller, 1971, p. 612). From this we deduce

(see Feller, 1971, pp. 418–419).

We turn to estimates for H_j(x) of (14). We note first that

As Z_n − Z_n/2 is independent of Z_n/2 the Berry–Esséen theorem for

gives

and on substituting this into (18) we obtain

where we have used the argument leading to (17). The second estimate for H_n(x) that we require is

where ψ(x) is the renewal function of the first ladder height (see the duality lemma in Feller, 1971, p. 395). The inequality in (20) follows from the renewal theorem (Feller, 1971, pp. 358–366).

Applying the inequalities (17), (19), and (20) in (13) we obtain

where the first estimate follows from (17) and (20) and the second from (17) and (19). This finally proves (12), and on using this in (11) we obtain

From (21) we conclude

This proves (9) with A = a and with it the theorem. █

3. THE ASYMPTOTIC BEHAVIOR OF THE DICKEY–FULLER STATISTIC

We turn to the asymptotic behavior of the Dickey–Fuller statistic when applied to the Y_t.

THEOREM 2. Suppose the increments (ε_i)₁^∞ are i.i.d. and satisfy

Then

Proof. The lower bound of −T follows directly from (2) on noting that

is always positive. To obtain the upper bound we require an upper bound for the sum S₁(T). This requires an analysis of the path (t,Z_t)₀^T with Z₀ = 0. We denote the path by B(T) and write S₁(B(T)) = S₁(T). If part of the path is constant, that is, Z_t = Z_t+1 = ··· = Z_t+s for some s ≥ 1, then the points (t + 1,Z_t+1),…,(t + s,Z_t+s) can be removed without changing the value of S₁. Without loss of generality we can therefore suppose that the path has no intervals where it is constant. We consider the effect of removing the local maxima from the path B(T). As the path contains no intervals of constancy there is a local maximum at the point t if

We suppose that Z_t−1 ≤ Z_t+1 < Z_t. A similar argument works if Z_t+1 ≤ Z_t−1 < Z_t. A new path of length T − 1 is constructed by setting

. We denote this new path by

and set Δ = Z_t − Z_t+1 and γ + Δ = Z_t − Z_t−1 with γ and Δ both nonnegative. We have

We note that the value of the final point of the path is not altered, i.e.,

. Under the conditions of the theorem

for all a > 0 so that the global maximum is an interior local maximum with probability tending to one as T tends to infinity. We reserve the Δ in (23) for the point corresponding to the global maximum. We now proceed to remove all local maxima and finally obtain a path

of length T − k with no local maxima and that satisfies

As

has no local maxima it follows that the global maximum of this path is located either at 0 or T − k. As the removal of local maxima does not alter the value of the last point we have

, and hence

As

for any sequence a_T with

it follows that

, which yields

This combined with the result of Theorem 1 implies

To complete the proof of the theorem it suffices to show that

where Δ* = min{ε_t, − ε_t+1} on M_T = Z_t. Conditioning on M_T = Z_t we have for ε_t

and

As the ε_j are assumed to be independently distributed the inequalities (27) have no effect on the conditional distribution of ε_t. The inequalities (26) may be written in the form

On conditioning on ε₁,…,ε_t−1 and using the fact that

for any random variable X and for any x and a ≥ 0 we deduce

Similarly we have

These latter inequalities imply

from which we may deduce (25). This completes the proof of the theorem. █

4. THE SIMPLE RANDOM WALK

In the case of the simple random walk a more precise result is possible. In fact we state the theorem somewhat more generally as we need only the properties (28) and (29), which follows and they are satisfied for the simple random walk. In particular we do not require that the ε_i are independently or identically distributed.

THEOREM 3. Suppose that the process Z_t satisfies

and

for all a > 0. Then

where σ is as in (1).

Proof. Without loss of generality we may set Z₀ = 0, and to ease the notation we consider only the case σ = 1. Given a level j we consider those times t, if any, for which Z_t = j and Z_t+1 = j + 1. These are the set of up-crossings from j to j + 1. Similarly we define the set of down-crossings from j + 1 to j as those points t, if any, for which Z_t = j + 1 and Z_t+1 = j. Consider first a path with Z_T ≥ 1. Then for all j ≥ Z_T and for all j ≤ −1 the number of up-crossings from j to j + 1 is equal to the number of down-crossings from j + 1 to j. For 0 ≤ j ≤ Z_T − 1 the number of up-crossings from j to j + 1 exceeds the number of down-crossings from j + 1 to j by 1. Similarly if Z_T ≤ 0 then for j ≥ 0 and j ≤ Z_T − 1 the number of up-crossings from j to j + 1 is equal to the number of down-crossings from j + 1 to j. For Z_T ≤ j ≤ −1 the number of down-crossings from j + 1 to j exceeds the number of up-crossings from j to j + 1 by 1. For each level j let N(j,T) denote the number of up-crossings from j to j + 1 that can be paired with a down-crossing from j + 1 to j. Further let E(j,T) denote the excess of up-crossings from j to j + 1 over the down-crossings from j + 1 to j. The preceding argument shows that E(j,T) is either −1, 0, or 1.

The contribution of an up-crossing from j to j + 1 to the sum S₁(T) of (3) is

Similarly the contribution to S₁(T) of a down-crossing from j + 1 to j is

We may therefore write

Similarly for the sum S₀(T) we have

Suppose now that Z_T > 0. Then the argument given previously shows that E(j,T) = 0 for j ≥ Z_T and consequently

Similarly if Z_T ≤ 0 then

On combining these last two inequalities we obtain

Consider a random walk satisfying

It follows from the arguments given previously that N(j,T) ≥ 1 for j = M_T − 1. This implies

by (33) and consequently

From (29) it follows that there exists a sequence (a_T)₁^∞ satisfying lim_T→∞ a_T = ∞ such that

Using this in (34) gives

from which we obtain with (31)

and with (32)

On using these asymptotic equivalences in (3) we have

which proves the theorem. █

5. SIMULATIONS

Figure 1 shows the results of a simulation of a simple random walk with the ε_i i.i.d. and taking the values 1 and −1 with probability 0.5, σ = 0.01, and T = 100,000. Part (a) shows the path behavior of the Z_t-process, (b) shows the corresponding process Y_t, and (c) shows the path behavior of the Dickey–Fuller statistic. The final value of the statistic is −5.163337e − 05, which compares well with the theoretical slope given by

(a) The zt process for the simple random walk with σ = 0.01 and T = 100,000. (b) The corresponding yt process. (c) The path behavior of the Dickey–Fuller statistic.

Figure 2 shows the corresponding results when the ε_i are i.i.d. standard Gaussian random variables. The variability of the individual paths is very high. To overcome this 500 paths of length 20,000 were simulated with σ = 0.01 and the ε_i being standard Gaussian white noise. The mean value of the Dickey–Fuller statistic was calculated at each of the time points 1,000, (1,000), 20,000. The results are shown in Figure 3, where a linear trend is apparent. This is consistent with the main theorem. It makes plausible the conjecture that the probability limit plim_T→∞ DF(T)/T exists and is negative and finite also in the Gaussian case

(a) The zt process for standard Gaussian increments with σ = 0.01 and T = 100,000. (b) The corresponding yt process. (c) The path behavior of the Dickey–Fuller statistic.

The value of the Dickey–Fuller statistic averaged over 500 simulated paths.

with d > 0 and finite. We offer no proof of this. There well may be other distributions for which (36) holds true, and in this case one can expect the constant d to be dependent on the underlying distribution of the ε_i.