It is convenient to express the Dickey–Fuller (DF) statistic as

(see Phillips, 1987). To prove that the asymptotic distribution of Z_T under the nonnegativity constraint is standard, we need to prove that

and that

.

The weak convergence result can be obtained by showing that the constrained random walk {X_t} can be expressed as a functional of the simple random walk

. This key result is given in the following lemma, which is adapted from Proposition 4 in Cavaliere (2002).

LEMMA 1. For each t ≥ 0, X_t = S_t + L_t, where

.

Proof. First, note that for each t the process can be written as X_t = X_t−1 + ε_t + l_t, where

Hence

, where

. We need to show that L_t = −min_i≤t{S_i}, which can be proved by induction. For t = 0, S₀ = 0, L₀ = 0 and the relation is therefore satisfied. Assume that the relation is satisfied at time t, i.e., L_t = −min_i≤t{S_i}. At time t + 1, if X_t + ε_t+1 ≥ 0, then l_t+1 = 0, L_t+1 = L_t = −min_i≤t{S_i} = −min_i≤t+1{S_i}, and the relation is satisfied. Conversely, if X_t + ε_t+1 < 0, then l_t+1 = −(X_t + ε_t+1) = −(S_t + L_t + ε_t+1), which implies −L_t+1 = S_t+1. But because l_t+1 > 0, −L_t+1 = S_t+1 < −L_t = min_i≤t{S_i}, which gives −min_i≤t+1{S_i} = S_t+1 = −L_t+1, and the relation is proved. █

The mapping from {S_t} to {X_t} is explained graphically in Figure 1, where {X_t} is plotted along with {S_t} and {−L_t}. It is interesting to notice that the cumulated effect of the nonnegativity constraint, i.e., the differential between the unconstrained and the constrained random walks, S_t − X_t, is given by the running minimum min_i≤t{S_i} =: − L_t.

Construction of the constrained random walk: Xt (solid line), St (short dashed line), and −Lt (dashed line), T = 50, σ = 1.

From Lemma 1, the invariance principle, and the continuous mapping theorem, as T ↑ ∞

uniformly for all s ∈ [0,1] . The probability result given by the last distributional equality in (4) is well known in the probability literature (see, e.g., Harrison, 1985). Note that it also implies

and hence

The proof is completed by showing that the nonnegativity constraint does not affect the estimation of the variance σ². Because

({ε_t} is Gaussian and independently and identically distributed [i.i.d.]), we only need to prove that

. Let l_t, 0 ≤ l_t ≤ |ε_t|, be defined as in (3). As (ΔX_t)² = (ε_t + l_t)² = ε_t² + l_t(l_t + 2ε_t) ≤ ε_t² + 3l_t|ε_t|,

But

({ε_t} is i.i.d. and Gaussian); hence, the random variable on the right-hand side of (5) is o_p(1), and the desired result follows.

Final Remark. The asymptotics obtained show that the nonnegativity constraint does not affect the asymptotic distribution of the Dickey–Fuller unit root test. However, the same property does not automatically apply to other unit root tests (e.g., tests that employ deterministic corrections). See Cavaliere (2001) for further insights.