Appendix 1
1.1 Stationarity Conditions
In this subsection we consider \(\Lambda\), defined in (5), a linear function.
Theorem 1
The marginal mean of \(Y_t\)in the log-symmetric-ARMAX(p, q) model is given by
$$\begin{aligned} \mathrm{E}[Y_t] = \Lambda ({\varvec{x}}_t^{\top }\varvec{\beta }), \end{aligned}$$
provided that \(\Phi (B):\mathbb {R}\rightarrow \mathbb {R}\)is an invertible operator (the autoregressive polynomial) defined by \(\Phi (B) = -\sum _{i=0}^{p}\kappa _i B^i\)with \(\kappa _0=-1\), and \(B^i\)is the lag operator, i.e., \(B^i y_t = y_{t-i}\).
Proof
Since \(\Lambda\) is linear, using (5) and (6) we have
$$\begin{aligned} Y_t&= \lambda _t+[Y_t-\lambda _t] \nonumber \\&= \Lambda ({\varvec{x}}_t^{\top }\varvec{\beta }) + \Lambda \left( \sum _{l=1}^p \kappa _l\, [Y_{t-l}- \Lambda ({\varvec{x}}_{t-l}^{\top }\varvec{\beta })] + \sum _{j=1}^q \zeta _j\, r_{t-j} + r_t \right) . \end{aligned}$$
(12)
Let \(\Theta (B) = \sum _{i=0}^{q}\xi _i B^i\) with \(\xi _0=1\), be the moving averages polynomial. Since \(\Theta (B)\Phi (B)^{-1}=\sum _{i=0}^{\infty }\psi _i B^i\) with \(\psi _0=1\), using (12), the log-symmetric-ARMAX(p, q) model can be rewritten as
$$\begin{aligned} w_t = \Lambda \left( \sum _{l=1}^{p}\kappa _l\, w_{t-l}+ \sum _{j=1}^{q}\zeta _j\, r_{t-j} + r_t \right) = \Lambda \big (\Theta (B)\Phi (B)^{-1} r_t\big ) = \Theta (B)\Phi (B)^{-1} \Lambda (r_t), \end{aligned}$$
(13)
where the error \(r_t=Y_t-\lambda _{t}\) is a MDS and \(w_t=Y_t-\Lambda ({\varvec{x}}_t^{\top }\varvec{\beta })\). Since \(\Lambda\) is linear and \(\mathrm{E}[r_t]=0\) for all t, we have \(\mathrm{E}[\Lambda (r_t)]=0\). Therefore, using (13), \(\mathrm{E}[w_t]=0\) for all t. Then
$$\begin{aligned} \mathrm{E}[Y_t] = \Lambda ({\varvec{x}}_t^{\top }\varvec{\beta })+\mathrm{E}[w_t] = \Lambda ({\varvec{x}}_t^{\top }\varvec{\beta }), \end{aligned}$$
whenever the series \(\Theta (B)\Phi (B)^{-1}r_t\) converges absolutely. \(\square\)
Theorem 2
Assuming that \(\Theta (B)\Phi (B)^{-1}=\sum _{i=0}^{\infty }\psi _i B^i\)and \(\Phi (B)\)is invertible, we have that the marginal variance of \(Y_t\)in the log-symmetric-ARMAX(p, q) model is given by
$$\begin{aligned} \mathrm {Var}[Y_t] = \sum _{i=0}^{\infty }\psi _i^2\, \mathrm{E}\big [\mathrm {Var}[\Lambda (Y_{t-i})|\mathcal {B}_{t-i-1}]\big ], \end{aligned}$$
where \({\mathcal {B}}_{t}=\sigma (\Lambda (Y_{t}),\Lambda (Y_{t-1}),\ldots ,)\)is the \(\sigma\)-field generated by the information up to time t.
Proof
Since \(\mathrm{E}[r_t|\mathcal {A}_{t-1}]=0\), a.s., for all t, and \(\mathrm {Cov}[r_s,r_t]=0\) for all \(t\ne s\), following the notation of Theorem 1, we have
$$\begin{aligned} \mathrm {Var}[Y_t]&= \mathrm {Var}[w_t] {\mathop {=}\limits ^{(13)}} \mathrm {Var}[\Theta (B)\,\Phi (B)^{-1} \Lambda (r_t)] = \mathrm {Var}\bigg [\sum _{i=0}^{\infty }\psi _i B^i \Lambda (r_t)\bigg ] \nonumber \\&= \sum _{i=0}^{\infty }\psi _i^2\, \mathrm{Var}[\Lambda (r_{t-i})]. \end{aligned}$$
(14)
On the other hand, the law of total variance states that
$$\begin{aligned} \mathrm {Var}[\Lambda (r_t)]&= \mathrm{E}\big [\mathrm {Var}[\Lambda (r_t)|\mathcal {B}_{t-1}]\big ] + \mathrm {Var}\big [\mathrm{E}[\Lambda (r_t)|\mathcal {B}_{t-1}]\big ] \nonumber \\&= \mathrm{E}\big [\mathrm {Var}[\Lambda (Y_t)|\mathcal {B}_{t-1}]\big ]. \end{aligned}$$
(15)
Combining (14) and (15), the proof follows. \(\square\)
Theorem 3
The covariance and correlation of \(Y_t\)and \(Y_{t-k}\)in the log-symmetric-ARMAX (p, q) model are given by
$$\begin{aligned} \mathrm {Cov}[Y_t,Y_{t-k}]&= \sum _{i=0}^{\infty } \psi _i \psi _{i-k}\, \mathrm{E}\big [\mathrm {Var}[\Lambda (Y_{t-i})|\mathcal {B}_{t-i-1}]\big ], \quad k>0, \\ \mathrm {Corr}[Y_t,Y_{t-k}]&= { \sum _{i=0}^{\infty } \psi _i \psi _{i-k}\, \mathrm{E}\big [\mathrm {Var}[\Lambda (Y_{t-i})|\mathcal {B}_{t-i-1}]\big ] \over \prod _{j\in \{0,k\}} \sqrt{ \sum _{i=0}^{\infty } \psi _i^2\, \mathrm{E}\big [\mathrm {Var}[\Lambda (Y_{t-j-i})|\mathcal {B}_{t-j-i-1}]\big ]} }, \end{aligned}$$
respectively, where \({\mathcal {B}}_{t}=\sigma (\Lambda (Y_{t}),\Lambda (Y_{t-1}),\ldots ,)\)is the \(\sigma\)-field generated by the information up to time t.
Proof
Since \(w_t=Y_t-\Lambda ({\varvec{x}}_t^{\top }\varvec{\beta })\) and \(\mathrm {Cov}[r_s,r_t]=0\) for all \(t\ne s\),
$$\begin{aligned} \mathrm {Cov}[Y_t,Y_{t-k}]&= \mathrm {Cov}[w_t,w_{t-j}] {\mathop {=}\limits ^{(13)}} \mathrm {Cov}\big [\Theta (B)\Phi (B)^{-1} \Lambda (r_t), \Theta (B)\Phi (B)^{-1} \Lambda (r_{t-k})\big ] \\&= \sum _{i=0}^{\infty } \psi _i \psi _{i-k}\, \mathrm {Var}[\Lambda (r_{t-i})]. \end{aligned}$$
Using (15) the expression on the right side is equal to \(\sum _{i=0}^{\infty } \psi _i \psi _{i-k}\, \mathrm{E}\big [\mathrm {Var}[\Lambda (Y_{t-i})|\mathcal {B}_{t-i-1}]\big ],\) and the proof follows. \(\square\)
Appendix 2
The Hessian matrix can be determined by the following matrix
$$\begin{aligned} \ddot{\varvec{\ell }}(\varvec{\theta }^*) = \begin{bmatrix} \displaystyle {\partial ^2 \ell _{0,1}\over \partial \beta _{r}^2}(\varvec{\theta }^*) &{}\displaystyle {\partial ^2 \ell _{0,1}\over \partial \beta _{r} \partial \tau _s}(\varvec{\theta }^*) &{}\displaystyle {\partial ^2 \ell _{0,1}\over \partial \beta _{r} \partial \kappa _l}(\varvec{\theta }^*) &{}\displaystyle {\partial ^2 \ell _{0,1}\over \partial \beta _{r} \partial \zeta _j}(\varvec{\theta }^*) \\ \displaystyle {\partial ^2 \ell _{0,1}\over \partial \tau _s \partial \beta _{r}}(\varvec{\theta }^*) &{}\displaystyle {\partial ^2 \ell _{0,1}\over \partial \tau _s^2}(\varvec{\theta }^*) &{}\displaystyle {\partial ^2 \ell _{0,1}\over \partial \tau _s \partial \kappa _l}(\varvec{\theta }^*) &{}\displaystyle {\partial ^2 \ell _{0,1}\over \partial \tau _s \partial \zeta _j}(\varvec{\theta }^*) \\ \displaystyle {\partial ^2 \ell _{0,1}\over \partial \kappa _l \partial \beta _{r}}(\varvec{\theta }^*) &{}\displaystyle {\partial ^2 \ell _{0,1}\over \partial \kappa _l\partial \tau _s}(\varvec{\theta }^*) &{}\displaystyle {\partial ^2 \ell _{0,1}\over \partial \kappa _l^2}(\varvec{\theta }^*) &{}\displaystyle {\partial ^2 \ell _{0,1}\over \partial \kappa _l \partial \zeta _j}(\varvec{\theta }^*) \\ \displaystyle {\partial ^2 \ell _{0,1}\over \partial \zeta _j \partial \beta _{r}}(\varvec{\theta }^*) &{}\displaystyle {\partial ^2 \ell _{0,1}\over \partial \zeta _j\partial \tau _s}(\varvec{\theta }^*) &{}\displaystyle {\partial ^2 \ell _{0,1}\over \partial \zeta _j\partial \kappa _l}(\varvec{\theta }^*) &{}\displaystyle {\partial ^2 \ell _{0,1}\over \partial \zeta _j^2}(\varvec{\theta }^*) \end{bmatrix}, \end{aligned}$$
where \(\varvec{\theta }^*=(\beta_r,\tau_s,\kappa_l,\zeta_j); \ r=0,\ldots ,k\); \(s=0,\ldots , l\); \(l=1,\ldots ,p\) and \(j=1,\ldots ,q\). Since the function \(\ell _{0,1}(\varvec{\theta }^*)\) has continuous second partial derivatives at a given point \({\varvec{\theta }^*}\) in \(\mathbb {R}^{4}\), by Schwarz’s Theorem follows that the partial differentiations of this function are commutative at that point, that is,
$$\begin{aligned} {\partial ^2 \ell _{0,1}\over \partial a \partial b}(\varvec{\theta }^*) = {\partial ^2 \ell _{0,1}\over \partial b \partial a}(\varvec{\theta }^*), \quad \text {for} \ a\ne b \ \text {in} \ \{\beta _r,\tau _s,\kappa _l,\zeta _j\}. \end{aligned}$$
It can easily be seen that the first order partial derivatives of \(\ell _{0,1}(\varvec{\theta }^*)\) are
$$\begin{aligned} \begin{array}{llllll} \displaystyle {\partial \ell _{0,1}\over \partial a} (\varvec{\theta }^*) = {1\over g(z^2_t)}\, {\partial g(z^2_t)\over \partial a}, \ a\in \{\beta _r,\kappa _l,\zeta _j\}, \quad&\displaystyle {\partial \ell _{0,1}\over \partial \tau _{s}} (\varvec{\theta }^*) = -{1\over 2\phi _t}\, {\partial \phi _t\over \partial \tau _s}\, + {1\over g(z^2_t)}\, {\partial g(z^2_t)\over \partial \tau _s}, \end{array} \end{aligned}$$
(16)
the second order partial derivatives are
$$\begin{aligned} {\partial ^2 \ell _{0,1}\over \partial a^2} (\varvec{\theta }^*)&= - {1\over [g(z^2_t)]^2}\, {\partial g(z^2_t)\over \partial a} + {1\over g(z^2_t)}\, {\partial ^2 g(z^2_t)\over \partial a^2}, \quad a\in \{\beta _r,\kappa _l,\zeta _j\}, \nonumber \\ {\partial ^2 \ell _{0,1}\over \partial \tau _{s}^2} (\varvec{\theta }^*)&= {1\over 2\phi _t^2}\, \left( {\partial \phi _t\over \partial \tau _s}\right) ^2\, - {1\over 2\phi _t}\, {\partial ^2 \phi _t\over \partial \tau _s^2}\, - {1\over [g(z^2_t)]^2}\, \Big [{\partial g(z^2_t)\over \partial \tau _s}\Big ]^2 + {1\over g(z^2_t)}\, {\partial ^2 g(z^2_t)\over \partial \tau ^2_s}, \end{aligned}$$
(17)
and the mixed partial derivatives are given by
$$\begin{aligned} {\partial ^2 \ell _{0,1}\over \partial \beta _r\partial a} (\varvec{\theta }^*)&= -{1\over [g(z^2_t)]^2}\, {\partial g(z^2_t) \over \partial \beta _r}\, {\partial g(z^2_t)\over \partial a} + {1\over g(z^2_t)}\, {\partial ^2 g(z^2_t)\over \partial \beta _r \partial a}, \quad a\in \{\tau _s,\kappa _l,\zeta _j\}, \\ {\partial ^2 \ell _{0,1}\over \partial \tau _s\partial b} (\varvec{\theta }^*)&= -{1\over [g(z^2_t)]^2}\, {\partial g(z^2_t) \over \partial \tau _s}\, {\partial g(z^2_t)\over \partial b} + {1\over g(z^2_t)}\, {\partial ^2 g(z^2_t)\over \partial \tau _s \partial b}, \quad b\in \{\kappa _l,\zeta _j\}, \\ {\partial ^2 \ell _{0,1}\over \partial \kappa _l\partial \zeta _{j}} (\varvec{\theta }^*)&= -{1\over [g(z^2_t)]^2}\, {\partial g(z^2_t) \over \partial \kappa _l}\, {\partial g(z^2_t)\over \partial \zeta _j} + {1\over g(z^2_t)}\, {\partial ^2 g(z^2_t)\over \partial \kappa _l \partial \zeta _j}. \end{aligned}$$
Let
$$\begin{aligned} \eta _t:={z_t\over \sqrt{\phi _t}} = {[\log (y_t)-\log (\lambda _t)]\over \phi _t}, \quad t=m+1,\ldots ,n. \end{aligned}$$
The first order partial derivatives of g are
$$\begin{aligned} \begin{array}{llllll} \displaystyle {\partial g(z^2_t)\over \partial a} = - {2}\, {\eta _t\over \lambda _t} {\partial \lambda _t\over \partial a}\, g'(z^2_t), \quad a\in \{\beta _r,\kappa _l,\zeta _j\},&\qquad \displaystyle {\partial g(z^2_t)\over \partial \tau _s} = -\eta _t^2\, {\partial \phi _t\over \partial \tau _s}\, g'(z^2_t), \end{array} \end{aligned}$$
(18)
the second order partial derivatives are, for each \(a\in \{\beta _r,\kappa _l,\zeta _j\}\),
$$\begin{aligned} {\partial ^2 g(z^2_t)\over \partial a^2}&= 2 \Big [ {1\over \lambda _t^2} \left( {1\over \phi _t}+\eta _t\right) \left( {\partial \lambda _t \over \partial a} \right) ^2 - {\eta _t\over \lambda _t} {\partial ^2 \lambda _t \over \partial a^2} \Big ] g'(z_t^2) + 4\left( {\eta _t\over \lambda _t}\right) ^2 \left( {\partial \lambda _t \over \partial a}\right) ^2 g''(z_t^2), \nonumber \\ {\partial ^2 g(z^2_t)\over \partial \tau ^2_s}&= \eta _t^2 \Big [ {2\over \phi _t} \left( {\partial \phi _t\over \partial \tau _s }\right) ^2 - {\partial ^2\phi _t\over \partial \tau _s^2 } \Big ] g'(z^2_t) + \eta _t^4 \left( {\partial \phi _t\over \partial \tau _s}\right) ^2 g''(z^2_t), \end{aligned}$$
(19)
and the mixed partial derivatives are given by
$$\begin{aligned} {\partial ^2 g(z^2_t)\over \partial \beta _r \partial \tau _s}&= {2\eta _t\over \lambda _t} \Big [ {1\over \phi _t} g'(z_t^2) + \eta _t^2 g''(z_t^2) \Big ] {\partial \lambda _t\over \partial \beta _r} {\partial \phi _t\over \partial \tau _s}, \\ {\partial ^2 g(z^2_t)\over \partial \beta _r \partial a}&= {2\over \lambda _t} \Big \{ \Big [ {1\over \lambda _t} \left( {1\over \phi _t}+\eta _t\right) {\partial \lambda _t\over \partial \beta _r } {\partial \lambda _t\over \partial a } - \eta _t {\partial ^2 \lambda _t\over \partial \beta _r\partial a} \Big ] g'(z_t^2) + 2{\eta _t^2\over \lambda _t} {\partial \lambda _t\over \partial \beta _r} {\partial \lambda _t\over \partial a} g''(z_t^2) \Big \}, \quad a\in \{\kappa _l,\zeta _j\}, \\ {\partial ^2 g(z^2_t)\over \partial \tau _s \partial b}&= {2\eta _t\over \lambda _t} \Big [ {1\over \phi _t} g'(z_t^2) + \eta _t^2 g''(z_t^2) \Big ] {\partial \phi _t\over \partial \tau _s} {\partial \lambda _t\over \partial b}, \quad b\in \{\kappa _l,\zeta _j\}, \\ {\partial ^2 g(z^2_t)\over \partial \kappa _l \partial \zeta _j}&= {2\over \lambda _t} \Big \{ \Big [ {1\over \lambda _t} \left( {1\over \phi _t}+\eta _t\right) {\partial \lambda _t\over \partial \kappa _l } {\partial \lambda _t\over \partial \zeta _j } - \eta _t {\partial ^2 \lambda _t\over \partial \kappa _l\partial \zeta _j} \Big ] g'(z_t^2) + 2{\eta _t^2\over \lambda _t} {\partial \lambda _t\over \partial \kappa _l} {\partial \lambda _t\over \partial \zeta _j} g''(z_t^2) \Big \}, \end{aligned}$$
with
$$\begin{aligned} \begin{array}{llll} \displaystyle {\partial \phi _t\over \partial \tau _s} = w_{ts} \Lambda '(\varvec{w}^{\top }_{t}\varvec{\tau }),&\qquad \displaystyle {\partial ^2 \phi _t\over \partial \tau _s^2} = w^2_{ts} \Lambda ''(\varvec{w}^{\top }_{t}\varvec{\tau }). \end{array} \end{aligned}$$
By (8), the first order partial derivatives of \(\lambda _t\) are
$$\begin{aligned} {\partial \lambda _t\over \partial \beta _r}&= \left( x_{tr} - \sum \limits _{l=1}^p \kappa _l\, x_{(t-l)r} - \sum \limits _{j=1}^q\zeta _j\, {\partial \lambda _{t-j} \over \partial \beta _r} \right) \Lambda '({\varvec{x}}_t^{\top }\varvec{\beta } + \varrho _{t}), \end{aligned}$$
(20)
$$\begin{aligned} {\partial \lambda _t\over \partial \kappa _l}&= \left( y_{t-l} - \sum _{i=0}^{k} \beta _i\,x_{(t-l)i} - \sum \limits _{j=1}^q\zeta _j\, {\partial \lambda _{t-j}\over \partial \kappa _l} \right) \Lambda '({\varvec{x}}_t^{\top }\varvec{\beta } + \varrho _{t}), \end{aligned}$$
(21)
$$\begin{aligned} {\partial \lambda _t\over \partial \zeta _j}&= \left( r_{t-j}-\lambda _{t-j} - \sum \limits _{\tilde{j}=1}^q\zeta _{\tilde{j}}\, {\partial \lambda _{t-\tilde{j}}\over \partial \zeta _j} \right) \Lambda '({\varvec{x}}_t^{\top }\varvec{\beta } + \varrho _{t}), \end{aligned}$$
(22)
the second order partial derivatives are given by
$$\begin{aligned}&{\partial ^2 \lambda _t\over \partial \beta _r^2} = - \sum \limits _{j=1}^q\zeta _j\, {\partial ^2 \lambda _{t-j} \over \partial \beta _r^2} \, \Lambda '({\varvec{x}}_t^{\top }\varvec{\beta } + \varrho _{t}) + \left( x_{tr} - \sum \limits _{l=1}^p \kappa _l\, x_{(t-l)r} - \sum \limits _{j=1}^q\zeta _j\, {\partial \lambda _{t-j} \over \partial \beta _r} \right) ^2 \Lambda ''({\varvec{x}}_t^{\top }\varvec{\beta } + \varrho _{t}) , \\&{\partial ^2 \lambda _t\over \partial \kappa _l^2} = -\sum \limits _{j=1}^q\zeta _j\, {\partial ^2 \lambda _{t-j}\over \partial \kappa _l^2} \Lambda '({\varvec{x}}_t^{\top }\varvec{\beta }+\rho _t) + \left( y_{t-l} - \sum _{i=0}^{k} \beta _i\,x_{(t-l)i} - \sum \limits _{j=1}^q\zeta _j\, {\partial \lambda _{t-j}\over \partial \kappa _l} \right) ^2 \Lambda ''({\varvec{x}}_t^{\top }\varvec{\beta } + \varrho _{t}) , \\&{\partial ^2 \lambda _t\over \partial \zeta _j^2} = -\left( 2{\partial \lambda _{t-j}\over \partial \zeta _j} + \sum \limits _{\tilde{j}=1}^q \zeta _{\tilde{j}}\, {\partial ^2 \lambda _{t-\tilde{j}}\over \partial \zeta _j^2} \right) \Lambda '({\varvec{x}}_t^{\top }\varvec{\beta }+\rho _t) + \left( r_{t-j}-\lambda _{t-j} - \sum \limits _{\tilde{j}=1}^q\zeta _{\tilde{j}}\, {\partial \lambda _{t-\tilde{j}}\over \partial \zeta _j} \right) ^2 \Lambda ''({\varvec{x}}_t^{\top }\varvec{\beta } + \varrho _{t}) , \end{aligned}$$
with mixed partial derivatives
$$\begin{aligned} {\partial ^2 \lambda _t\over \partial \beta _r \partial \kappa _l}&= - \left( x_{(t-l)r} + \sum \limits _{j=1}^q\zeta _j\, {\partial ^2 \lambda _{t-j}\over \partial \beta _r\partial \kappa _l} \right) \Lambda '({\varvec{x}}_t^{\top }\varvec{\beta }+\rho _t) \\&\qquad +\left( x_{tr} - \sum \limits _{l=1}^p \kappa _l\, x_{(t-l)r} - \sum \limits _{j=1}^q\zeta _j\, {\partial \lambda _{t-j} \over \partial \beta _r} \right) \left( y_{t-l} - \sum _{i=0}^{k} \beta _i\,x_{(t-l)i} - \sum \limits _{j=1}^q\zeta _j\, {\partial \lambda _{t-j}\over \partial \kappa _l} \right) \Lambda ''({\varvec{x}}_t^{\top }\varvec{\beta } + \varrho _{t}), \\ {\partial ^2 \lambda _t\over \partial \beta _r \partial \zeta _j}&= - \left( 2{\partial \lambda _{t-j}\over \partial \beta _r} + \sum \limits _{\tilde{j}=1}^q\zeta _{\tilde{j}}\, {\partial ^2 \lambda _{t-\tilde{j}}\over \partial \beta _r\partial \zeta _j} \right) \Lambda '({\varvec{x}}_t^{\top }\varvec{\beta } + \varrho _{t}) \\&\qquad + \left( r_{t-j}-\lambda _{t-j} - \sum \limits _{\tilde{j}=1}^q\zeta _{\tilde{j}}\, {\partial \lambda _{t-\tilde{j}}\over \partial \zeta _j} \right) \left( x_{tr} - \sum \limits _{l=1}^p \kappa _l\, x_{(t-l)r} - \sum \limits _{j=1}^q\zeta _j\, {\partial \lambda _{t-j} \over \partial \beta _r} \right) \Lambda ''({\varvec{x}}_t^{\top }\varvec{\beta } + \varrho _{t}), \\ {\partial ^2 \lambda _t\over \partial \kappa _l \partial \zeta _j}&= - \left( 2{\partial \lambda _{t-j}\over \partial \kappa _l} + \sum \limits _{\tilde{j}=1}^q\zeta _{\tilde{j}}\, {\partial ^2 \lambda _{t-\tilde{j}}\over \partial \kappa _l\partial \zeta _j} \right) \Lambda '({\varvec{x}}_t^{\top }\varvec{\beta } + \varrho _{t}) \\&\qquad + \left( r_{t-j}-\lambda _{t-j} - \sum \limits _{\tilde{j}=1}^q\zeta _{\tilde{j}}\, {\partial \lambda _{t-\tilde{j}}\over \partial \zeta _j} \right) \left( y_{t-l} - \sum _{i=0}^{k} \beta _i\,x_{(t-l)i} - \sum \limits _{j=1}^q\zeta _j\, {\partial \lambda _{t-j}\over \partial \kappa _l} \right) \Lambda ''({\varvec{x}}_t^{\top }\varvec{\beta } + \varrho _{t}). \end{aligned}$$
