2. MAIN RESULTS

For the nonnormal error term ε_t in (1), we assume that it has the following moments m_j = E(ε_t^j):

where γ₁ and γ₂ are the Pearson's measures of skewness and kurtosis of the distribution and these and γ₃,…, γ₆ can be regarded as measures for deviation from normality. For a normal distribution, the parameters γ₁,…, γ₆ are all zero. Note that these γ's can also be expressed as cumulants of ε_t; e.g., γ₁ and γ₂ represent the third and fourth cumulants. We assume that the start-up observation y₀ of the AR(1) process has been observed and it can be either fixed or stochastic (and independent of ε_t, t > 0). We restrict that for stochastic y₀, it has bounded moments up to the sixth order, i.e., E(y₀ⁱ) = O(1), i ≤ 6, and for the case of nonstochastic start-up value, y₀ = c = O(1).

Let

denote the LS estimator of ρ based on a sample of T observations: y₁,…, y_T; then

where y₋₁ = (y₀,y₁,…, y_T−1)′, y = (y₁,y₁,…, y_T)′, ε = (ε₁,…, ε_T)′, F is a T × 1 vector with the tth element being ρ^t−1, and C is a strictly lower triangular T × T matrix with the tt′th lower off-diagonal element being ρ^t−t′−1. Note that y₋₁ = y₀ F + Cε.

The LS-based s-period-ahead forecast is given by (2), and therefore the fore-cast error is

From (5), the forecast bias is

and the MSFE is

The preceding exact expectation in (6) was evaluated by Malinvaud (1970), Fuller and Hasza (1980), and Dufour (1984, 1985), and the exact expectations in (7) were evaluated by Hoque et al. (1988). Although the exact forecast bias was shown to be zero for symmetrically distributed error terms, the exact MSFE was more complicated, and the result was only available under normally distributed error terms. See Ullah (2004, Ch. 6) for a comprehensive discussion on this and the approximate techniques under normality. Now we consider the approximate forecast bias and MSFE with the nonnormal error terms characterized by (3).

Because E(D) = y₀²F′F + σ² tr(C′C) = O(T), V [D − E(D)] = V(D) = E(D²) − [E(D)]² = O(T), E(N) = 0, and V(N) = O(T), we have N = O_P(T^1/2), D − E(D) = O_p(T^1/2), [D − E(D)][E(D)]⁻¹ = O_p(T^−1/2), and we can implement a Nagar (1959) type expansion on the ratio N/D, which is N/D = N [E(D)]⁻¹{1 + [D − E(D)][E(D)]⁻¹}⁻¹ = N [E(D)]⁻¹{1 − [D − E(D)][E(D)]⁻¹ + [D − E(D)]²[E(D)]⁻²} + o_P(T^−3/2), so that we can write

as follows:

where a_−i/2 = O_P(T^−i/2), i = 1,2,3, and they are

Similarly, we can write any positive power of

as the sum of a deterministic part and terms of decreasing order of the sample size. More specifically, for ρ ≠ 0 and q > 0,

where a_−i/2^(q) = O_P(T^−i/2), i = 1,2,3, and they are

Note that the expansion (10) includes (8) as a special case by setting q = 1. For the case of ρ = 0,

which is actually the same as (10) if we define 0⁰ = 1. So in what follows, we use the expansion (10). Substituting (10) into (6) and (7), we have

If we define a T × 1 vector e = (0,…,0,1)′, then y_T = e′y = y₀ ρe′F + ρe′Cε + e′ε. Given y_T rewritten as such, (4), and (9), the bias and MSFE results in (13) essentially involve expectations of quadratic forms in the nonnormal vector ε.³

In particular, the bias expression involves expectations of quadratic forms up to order 3, and the MSFE involves expectations of quadratic forms up to order 4. Numerically, we can always calculate the expectations of all the quadratic forms using Bao and Ullah (2006), substitute them back into (13), and evaluate the approximate forecast bias and MSFE. So in principle, the s-period-ahead forecast bias, up to O(T⁻¹), should depend on the nonnormal parameters, up to the seventh cumulant, and the MSFE, up to O(T^−3/2), should depend on the nonnormal parameters, up to the eighth cumulant, of the error distribution. On the other hand, as it turns out, many terms in the expansions for a_−1/2 y_T, a₋₁ y_T, and a_−1/2²y_T in the bias expression are of order lower than O_P(T⁻¹), and many terms in the expansions for a_−1/2²y_T², a_−1/2 a₋₁ y_T², and a_−1/2³y_T² in the MSFE expression are of order smaller than O_P(T^−3/2). For example, E(a_−1/2 y_T) = [E(D)]⁻¹E [(y₀ F′ε + ε′Cε)(y₀ ρe′F + ρe′Cε + e′ε)] = [E(D)]⁻¹E [(y₀ ρε′Fe′Cε + y₀ε′Fe′ε)] noting that E(ε′Cε) = 0 and E(ε′Cεε) = 0. But E(ε′Fe′Cε) = σ²e′CF = o(1), E(ε′Fe′ε) = σ²e′F = o(1), and [E(D)]⁻¹ = O(T⁻¹), so E(a_−1/2 y_T) = o(T⁻¹). Similarly, we can verify E(a₋₁ y_T) = o(T⁻¹) and E(a_−1/2²y_T) = o(T⁻¹). Therefore, up to order O(T⁻¹), the s-period-ahead forecast is unbiased, regardless of the error distribution:

As for the MSFE, straightforward albeit tedious algebra leads to E(a_−1/2³y_T²) = o(T^−3/2) and the following expressions:

where

Noting that tr(C′C) = T(1 − ρ²)⁻¹ − (1 − ρ²)⁻² + o(T⁻¹), e′CC′e = (1 − ρ²)⁻¹ + o(T⁻¹), upon substitution, we have

To summarize, for the pure AR(1) model, the LS-based forecast is unbiased, up to order O(T⁻¹), and it is robust against the error distribution, as indicated in (14). The MSFE is also robust against the error distribution, up to order O(T^−3/2), as indicated in (15). Interestingly, there is in fact no O(T^−3/2) term appearing in (15), which makes it also the asymptotic MSFE under normality given by Box and Jenkins (1970) and Yamamoto (1976).

Note that if we go beyond the first two moments, the robustness results may no longer hold. For example, for the third moment

, it is obvious that the term

involves the skewness coefficient γ₁.