Part A. Let

. Then

We start with

and find an asymptotic representation of the components

that appear in

. Using Euler summation (which justifies (1), which follows) and partial integration (which justifies (2)) we obtain an asymptotic series representation as follows:

To show that (3) is a valid asymptotic series, we can ignore the O(1) term, and by a further application of partial integration to the second term we see that the remainder is proportional to

Let L = n^α for some α ∈ (0,1) and write

But

It follows from (4)–(6) that

so that

and thus

showing that (3) is a valid asymptotic series.

In a similar way we may establish that

is a valid asymptotic series. Combining (8) and (10), we get

observing that we have to go to the third-order terms in expansion (12) to avoid degeneracy. We can obtain higher order asymptotic expansions by taking the preceding process to further terms, leading to the following explicit expression to order n/ln¹⁰ n:

It is immediately apparent from the size of the coefficients in (14) that very large values of n are required before these approximations can be expected to work well. Of course, such approximations are hardly necessary because

is amenable to direct calculation or to direct approximation using the logarithmic integrals given in (1) and (9). The latter can be evaluated from the following well known series representation of the logarithmic integral (e.g., Gradshteyn and Ryzhik, 1967, 8.213.2, p. 926):

where γ is Euler's constant. In the present case we have

Observe that

is the same as the error in the ordinary least squares (OLS) estimator of β in the regression

where {u_t} is a martingale difference sequence with respect to the natural filtration

. The persistent excitation condition (13) holds in this regression, and so

is strongly consistent.

Now observe that

and so

.

Next, turn to the asymptotic distribution of

. We have

Let y_nt = z_tu_t /s_n; then y_n2, y_n3,…, y_nn are independent random variables with zero means and variances that sum to σ². To apply the Liapounoff central limit theorem for

, we need to show that

But

and so

Hence

Note that

Thus, the term

dominates the asymptotics of

, and we deduce that

In view of (16) and (17), we have the following joint asymptotics:

Remarks

1. The limit distribution (18) is singular, and the components

are perfectly negatively correlated as n → ∞. Observe that the rate of convergence of

exceeds that of

, by virtue of the fact that the signal from the intercept is stronger than the signal from the evaporating logarithmic regressor ln⁻¹ t.

2. Result (18) gives first-order asymptotics. As is apparent from (14), asymptotic series expansions in the present model involve factors of 1/ln n in contrast to the more usual

, and, correspondingly, they deliver only very slow improvements in the first-order asymptotics. The asymptotic variance (σ² ln⁴ n)/n and higher order approximations based on (14) are therefore poor approximations to the variance of

even for quite large n. To illustrate, for values of the sample size n ∈ [10²,10⁴] and for σ² = 1, Figure 1 provides graphs of the exact variance of

, the asymptotic variance, the two-term and three-term series approximations to the variance based on the first few terms of (14), and direct calculation of the logarithmic integral representations of the variance obtained from (11) and (15). Apparently, only the latter are adequate for sample sizes n in this range.

and asymptotic and integral approximations.

Part B. Using the Phillips–Solo (1992) device, we have

where

. Then, we can write

It follows that

Note that the first term satisfies the limit theory given in the earlier part. If we prove that the last three terms converge strongly to zero, then we are done with the strong consistency of

. To obtain the asymptotic distribution, we need to consider

. We prove that the last three terms, so normalized, are o_p(1), and then the asymptotic distribution is determined by the first term. The following three results are given first.

Proof. These are immediate because

. █

Proof. Note that z_n is bounded and for any δ > 0

so

and the first result follows. The second follows because

. █

Proof. Note that

has finite fourth moment because

Here we have employed the fact that

, which follows from the assumption that

.

Next,

Therefore, for any δ > 0,

Because

we have

and so

. Similarly,

The last equality follows as

. Hence

. █

Combining results (a)–(c), we have

and then

With (19), (20), and

(e.g., Phillips and Solo, 1992), the previous arguments can now be repeated to obtain the strong consistency and the asymptotic distribution of

. Specifically