5.a Additional assumptions
Our additional assumptions with respect to the errors and disturbances are the
non-autocorrelation assumptions:
Assumption (B1):
E(vi0tviθ)=0KK,t6= θ,
Assumption (C1): E(uituiθ)=0, t6= θ.
Sometimes, the following weaker assumptions, allowing for some autocorrelation,
will be sufficient:
|
Assumption (B2): |
E(vi0tviθ)=0KK, |
∖t — θ∖ > τ. |
|
Assumption (B3): |
E(vi0tviθ) is invariant to t, θ, |
t 6= θ, |
|
Assumption (C2): |
E(uituiθ) = 0, |
∖t — θ∖ >τ |
|
Assumption (C3): |
E(uituiθ) is invariant to t, θ, |
t 6= θ, |
of which (B2) and (C2) allow for a (vector) moving average (MA) structure up to
order τ (≥ 1), and (B3) and (C3) allow for time invariance of the autocorrelation.
The latter will, for example, be satisfied if the measurement errors and the distur-
bances have individual components, say vit = v1i + v2it , uit = u1i + u2it , where v 1i ,
v2it, u1i, and u2it are independent IID processes.
Our additional assumptions with respect to the distribution of the latent regres-
sor vector ξit are:
Assumption (D1):
Assumption (D2):
Assumption (E):
E(ξit)
E(αiξit)
rank(E[ξi0p(∆ξitθ)]) = K
is invariant to t,
is invariant to t,
for some p, t, θ different
Assumptions (D1) and (D2) hold when ξit is mean stationary for all i. Assump-
tion (E) imposes non-IID and some form of autocorrelation or (covariance) non-
stationarity on ξit . It excludes, for example, the case where ξit has an individual
component, so that ξit = ξ1i + ξ2it, where ξ1i and ξ2it are independent (vector) IID
processes.
Assumptions (A) - (E) do not go very far in structuring the distributions of
the variables of the model. This has both its pros and cons. It may be possible
13
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