A Derivations of Results in Section 3

In the A-GARCH model under Laplace errors, eq. (7) is deduced from

Equation (9) is obtained by noting that

so that using eq. (7) leads to

Also, since

then we have

and hence

which leads to the recursive formula (10). Besides, eq. (12) is by induction with

In the NA-GARCH model under Laplace errors, eq. (14) is due to

while eq. (15) is a result of

and eq. (16) is from

In the GJR-GARCH model under Laplace errors, eq. (20) results from

and eq. (21) is implied by

Similarly, eq. (22) is a result of

and thus

In the E-GARCH model under Laplace errors, eq. (26) is deduced by

so that we require 0≤β<1 for the limit factor to converge, and due to

we have that for 0≤k<1, |ψ|<2−α is needed for the MGF of g(ϵt) to converge. Proof of the convergence of the infinite product is given in the next appendix. Likewise, eq. (28) is derived as

where |ψ|<2/2−α is required to ensure that the MGF converges for 0≤k<2. Furthermore, for l≥1, since

we have the expression (29). The forecast eq. (33) is merely calculated from

B Proof of Convergence of Infinite Product in eq. (26)

Although Nelson (1991) verified that convergent MGF of errors automatically leads to convergent unconditional variance in the E-GARCH model, we expressly give a proof in the case of Laplace errors. We attempt to prove that 0≤β<1 with |ψ|<2−α is also sufficient for the infinite product in eq. (26) to converge. To do this, note that a sufficient and necessary condition for an infinite product ∏i=1∞ai with positive terms to converge to a nonzero finite number is that the infinite series ∑i=1∞lnai converges to a finite number.

Further, notice that ∑i=1∞|lnai|<∞ guarantees ∑i=1∞lnai<∞. Also, it is necessary that limi→∞ai=1 for both ∑i=1∞|lnai| and ∑i=1∞|ai−1| to converge. According to the comparison test of positive infinite series, if this condition is satisfied,

and then ∑i=1∞|ai−1| converges if and only if ∑i=1∞|lnai| converges.

In Appendix A we have shown that, with k=βi−1, for i=1,2,... and 0≤β<1, the MGF of g(ϵt) in the E-GARCH model under Laplace errors converges to

provided that |ψ|<2−α. Hence, in the convergence problem of ∏i=1∞mg(βi−1), it suffices to show that ∑i=1∞|mg(βi−1)−1|<∞.

Clearly, for a new sequence (βi−1),

which says that the series ∑i=1∞|mg(βi−1)−1| has the same convergence rate as ∑i=1∞βi−1. With 0≤β<1, ∑i=1∞βi−1=1/(1−β)<∞ leads to ∑i=1∞|mg(βi−1)−1|<∞ as well and therefore ∏i=1∞mg(βi−1)<∞ as desired.

C Selected Moment Formulas under Normal or Student Errors

Recall that the error distribution does not affect the unconditional variance in each model but changes the kurtosis. The following expressions for kurtosis have been used in our analysis of constraints as well as empirical modeling for comparison. They directly follow from generalizing E[ϵt4]=κϵ in Section 3. Recall that normal errors have κϵ=3 while Student-5 errors and Student-7 errors have κϵ=9 and κϵ=5, respectively. Similar versions can also be found in Tsay (2005) .

Assume that p has the same value as that in the corresponding Laplace model. In the A-GARCH model, if p2+(κϵ−1)α2<1, then

in the NA-GARCH model, if p2+α2((κϵ−1)+(κϵ−2)θ2)<1, then

in the GJR-GARCH model, if p2+(κϵ−1)α(α+ϕ)+(2κϵ−1)/4ϕ2<1, then

For the E-GARCH, it is useful to study the MGF of g(ϵt) in the presence of normal or Student errors. Using the fact that the standard normal density function is n(x)=1/2πe−x2/2 with x∈R and that E[|ϵt|]=2/π, the MGF with rate k≥0 is hence calculated as

where N(⋅) is the the standard normal cumulative distribution function. Indeed, under normal errors, mg(k)<∞ as long as k<∞. On the other hand, when errors have a Student distribution, they have indeterminate MGF (see, e.g., Walck (2007)). As such, the MGF of g(ϵt) cannot exist unless |ψ|<−α, in which case the exponential is nonincreasing in ϵt>0 for any given t. This constraint is unrealistically inappropriate by requiring that the returns impact level α must be negative. As illustrated in Section 5, such a constraint is hardly satisfied in practice^[14].