A regression model that can accommodate non-normality (kurtosis and right or left skewness), and
autocorrelation and/or heteroskedasticity, is finally obtained by applying the transformation in equation (2)
to U*. When the error term vector U = (Y-Xβ), is autocorrelated, a first transformation is used to obtain a
non-autocorrelated error term U* = PU = (PY-PXβ) = (Y*-X*β). This is then transformed to a normal
error term vector V through equation (2). If U is only heteroskedastic, P is an n×n identity matrix. The
concentrated log-likelihood function to be maximized in order to estimate this model’s parameters is:
n n
(5) NNLL= -0.5×ln∣ψ∣ + Σ In(Gi) -Σ 0.5×⅛2 }; where:
i=1 i=1
Gi = G(Θ,μ)(1+Ri2)"1%; Hi = {sinh-1(Ri)∕Θ}-μ; Ri = {G(Θ,μ)ΘUi*∕σ}+F(Θ,μ);
and i=1,...,n refers to the observations, sinh-1(x) = ln{x+(1+x2)1/2} is the inverse hyperbolic sine function,
and F(Θ,μ) is as given in equation (2). The first and second terms in equation (5) are the natural logs of the
Jacobians of the first and second transformations, respectively. Hi is the inverse of the transformation to
normality in equation (2). If the error term is believed to be autocorrelated, P and ∣ψ∣ must be specified to
make equation (5) operational. Judge, et al. derives P and ∣ψ∣ for first- and higher-order autoregressive
processes. As before, in this autocorrelated specification, E[Y]= Xβ and Var[U]= σ2ψ. Thus, σ2 can still
be used to model systematic changes in the variance of U across the observations without affecting
skewness or kurtosis.
The multiple equation (SUR) equivalent of equation (5) is obtained by applying a set of normality
transformations [equation (2)] to a set of m “transformed” n×1 non-normal random errors, Uj* =
Yj*-Xj*βj (j=1,...,m), where Yj*=PjYj, Xj*=PjXj, and Yj and Xj are the original vectors and matrices of
dependent and independent variables, respectively. As in the single equation case, the transformed set of
random vectors Vj follow a multivariate normal distribution with means μj (j=1,...,m) and covariance
matrix ∑. The non-diagonal elements of ∑ (pjk) account for the correlation between the error terms of the m
equations. The concentrated log-likelihood function for this model is a straightforward multivariate
extension of equation (5):