where ${u_i} = {t_\nu }^{ - 1}\left( {F\left( {{x_i};\nu } \right)} \right)$,
where $t_\nu^{-1}$ is the quantile function of the student distribution with
shape parameter $\nu$.
$$
\tau \left( {{x_i},{x_j}} \right) = 4\int_0^1 {\int_0^1 {C\left( {{u_i},{u_j}} \right)dC\left( {{u_i},{u_j}} \right) - 1} }.
$$

$$
\varphi_R(u) = \prod\limits_{i = 1}^n {{\varphi_{\bar w{Z_i}}}} (u)
=  \exp{
   \left(
   \text{i}u\sum\limits_{j = 1}^d \bar\mu_j +
   \sum\limits_{j = 1}^d
   \left(
   \frac{\lambda_j}{2}
   \log{\left(\frac{\gamma}{\upsilon}\right)}
   +
   \log \left(
   \frac{K_{\lambda _j}(\bar\delta_j\sqrt{\upsilon})}{K_{\lambda_j}( \bar\delta_j \sqrt{\gamma})}
   \right)
   \right)
   \right)
   }
$$

where, $\gamma  = \bar \alpha _j^2 - \bar \beta _j^2$, $\upsilon  = \bar \alpha _j^2 - {({{\bar \beta }_j} + {\text{i}}u)^2}$,
and $(\bar \alpha_j, \bar \beta_j, \bar \delta_j, \bar \mu_j)$ are the scaled
versions of the parameters $(\alpha_{i}, \beta_{i}, \delta_{i}, \mu_{i})$
as shown in \eqref{eq:portfolio2}. The density may be accurately
approximated by FFT as follows,
$$
{f_R}(r) = \frac{1} {{2\pi }}\int_{ - \infty }^{ + \infty } {{e^{( - \text{i}u r)}}} \varphi_R
(u)\mathrm{d}u \approx \frac{1} {{2\pi }}\int_{ - s}^s {{e^{( - \text{i}u r)}}} \varphi_R
(u)\mathrm{d}u.
$$

$$
\begin{gathered}
  {M_{GH(\lambda ,\alpha ,\beta ,\delta ,\mu )}}(u) = {e^{\mu u}}{M_{GIG\left( {\lambda ,\delta \sqrt {{\alpha ^2} - {\beta ^2}} } \right)}}\left( {\frac{{{u^2}}}{2} + \beta u} \right), \hfill \\
   = {e^{\mu u}}{\left( {\frac{{{\alpha ^2} - {\beta ^2}}}{{{\alpha ^2} - {{(\beta  + u)}^2}}}} \right)^{\lambda /2}}\frac{{{K_\lambda }\left( {\delta \sqrt {{\alpha ^2} - {{(\beta  + u)}^2}} } \right)}}{{{K_\lambda }\left( {\delta \sqrt {{\alpha ^2} - {\beta ^2}} } \right)}} \hfill \\
\end{gathered}
$$

where $M_{GIG}$ represents the moment generating function of the Generalized Inverse
Gaussian which forms the mixing distribution in this variance-mean mixture subclass.
Powers of the MGF, $M_{GH}(u)^p$, only have the representation in \eqref{appendixI:eq:ghypmom}
for $p=1$, which means that GH distributions are not closed under convolution
with the exception of the NIG, and only in the case when the shape and skew parameters
are the same. The MGF of the NIG is,
$$
{M_{NIG(\alpha ,\beta ,\delta ,\mu )}}(u) = {e^{\mu u}}\frac{{{e^{\delta \sqrt {{\alpha ^2} - {\beta ^2}} }}}}{{{e^{\delta \sqrt {{\alpha ^2} - {{(\beta  + u)}^2}} }}}}.
$$
$$
NIG(\alpha ,\beta ,{\delta _1},{\mu _1}) \times...\times NIG(\alpha ,\beta ,{\delta _n},{\mu _n}) = NIG(\alpha ,\beta ,{\delta _1} + ... + {\delta _n},{\mu _1} + ... + {\mu _n}).
$$
$$
{\phi _{GH(\lambda ,\alpha ,\beta ,\delta ,\mu )}}(u) = {e^{\mu \text{i}u}}{\left( {\frac{{{\alpha ^2} - {\beta ^2}}}{{{\alpha ^2} - {{(\beta  + {\text{i}}u)}^2}}}} \right)^{\lambda /2}}\frac{{{K_\lambda }\left( {\delta \sqrt {{\alpha ^2} - {{(\beta  + {\text{i}}u)}^2}} } \right)}}{{{K_\lambda }\left( {\delta \sqrt {{\alpha ^2} - {\beta ^2}} } \right)}}.
$$

$$
{\phi _{port}}(u) = \exp\Biggl\{\text{i}u \sum\limits_{j = 1}^d \biggl({\bar \mu }_j+ \frac{\lambda _j}{2}\log{\left( {\bar \alpha _j^2 - \bar \beta _j^2} \right)} - \frac{\lambda _j}{2} \log{\left( \bar \alpha _j^2 - ({\bar \beta }_j + {\text{i}}u)^2 \right)} +  \nonumber \\
\log{\left( K_{\lambda_j}\left( {\bar \delta }_j\sqrt{\bar \alpha _j^2 - ({\bar \beta }_j + \text{i}u)^2}\right) \right)} - \log{\left( K_{\lambda _j}\left({\bar \delta}_j \sqrt{\bar \alpha _j^2 - \bar \beta _j^2} \right) \right)} \biggr) \Biggr\}
$$

$$
\Delta X_{t}^{1} \\ \Delta X_{t}^{2}
$$

where $\Normal{b}{B}$ denotes the normal distribution with mean $b\in\mathbb{R}$ and variance $B\in\mathbb{R}^+$, and $\varepsilon_t$ and $\eta_t$ are independent.
The log-variance process $\hvec=(\hpar_1,\dots,\hpar_\leny)^\top$ is initialized by $\hpar_0\sim\Normal{\mupar}{\sigmapar^2/(1-\phipar^2)}$.
$\bm X=(\bm x_1^\top,\dots,\bm x_\leny^\top)^\top$ is an $\leny\times\nreg$ matrix containing in its $t$th row the vector of $\nreg$ regressors at time $t$.
The $\nreg$ regression coefficients are collected in $\bm\beta=(\beta_1,\dots,\beta_\nreg)^\top$.
We refer to $\svpars=(\mupar,\phipar,\sigmapar)$ as the SV parameters: $\mupar$ is the level, $\phipar$ is the persistence, and $\sigmapar$ (also called \emph{volvol}) is the standard deviation of the log-variance.

$$
y_i = x_i^\top \beta_i + u_i
\qquad (i = 1, \dots, n),
$$

Re-compiling KFAS
-  installing *source* package 'KFAS' ... (335ms)
   ** using staged installation
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   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  approx.f95 -o approx.o
   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  approxloop.f95 -o approxloop.o
   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  artransform.f95 -o artransform.o
   "C:/rtools40/mingw64/bin/"gcc  -I"D:/softApp/R/include" -DNDEBUG          -O2 -Wall  -std=gnu99 -mfpmath=sse -msse2 -mstackrealign -c cdistwrap.c -o cdistwrap.o
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   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  filter1stepnovar.f95 -o filter1stepnovar.o
   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  filtersimfast.f95 -o filtersimfast.o
   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  gloglik.f95 -o gloglik.o
   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  gsmoothall.f95 -o gsmoothall.o
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   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  isamplefilter.f95 -o isamplefilter.o
   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  kfilter.f95 -o kfilter.o
   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  kfilter2.f95 -o kfilter2.o
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   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  ldlssm.f95 -o ldlssm.o
   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  marginalxx.f95 -o marginalxx.o
   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  mvfilter.f95 -o mvfilter.o
   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  ngfilter.f95 -o ngfilter.o
   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  ngloglik.f95 -o ngloglik.o
   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  ngsmooth.f95 -o ngsmooth.o
   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  predict.f95 -o predict.o
   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  ptheta.f95 -o ptheta.o
   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  pytheta.f95 -o pytheta.o
   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  simfilter.f95 -o simfilter.o
   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  simgaussian.f95 -o simgaussian.o
   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  simgaussianuncond.f95 -o simgaussianuncond.o
   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  smoothonestep.f95 -o smoothonestep.o
   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  smoothsim.f95 -o smoothsim.o
   "C:/rtools40/mingw64/bin/"gfortran  -fno-optimize-sibling-calls    -O2  -mfpmath=sse -msse2 -mstackrealign -c  smoothsimfast.f95 -o smoothsimfast.o
   C:/rtools40/mingw64/bin/gcc -shared -s -static-libgcc -o KFAS.dll tmp.def approx.o approxloop.o artransform.o cdistwrap.o covmeanw.o filter1step.o filter1stepnovar.o filtersimfast.o gloglik.o gsmoothall.o init.o isample.o isamplefilter.o kfilter.o kfilter2.o kfstheta.o ldl.o ldlssm.o marginalxx.o mvfilter.o ngfilter.o ngloglik.o ngsmooth.o predict.o ptheta.o pytheta.o simfilter.o simgaussian.o simgaussianuncond.o smoothonestep.o smoothsim.o smoothsimfast.o -LD:/softApp/R/bin/x64 -lRlapack -LD:/softApp/R/bin/x64 -lRblas -lgfortran -lm -lquadmath -lgfortran -lm -lquadmath -LD:/softApp/R/bin/x64 -lR
   installing to C:/Users/ADMINI~1/AppData/Local/Temp/Rtmpo9CCsV/devtools_install_10a44fa971e2/00LOCK-KFAS/00new/KFAS/libs/x64
-  DONE (KFAS)
Writing NAMESPACE
Writing NAMESPACE
-- Building -------------------------------------------------------------------------------- KFAS --
Setting env vars:
* CFLAGS    : -Wall -pedantic -fdiagnostics-color=always
* CXXFLAGS  : -Wall -pedantic -fdiagnostics-color=always
* CXX11FLAGS: -Wall -pedantic -fdiagnostics-color=always
----------------------------------------------------------------------------------------------------
√  checking for file 'E:\devpackage\KFAS/DESCRIPTION' ...
-  preparing 'KFAS':
√  checking DESCRIPTION meta-information ...
-  cleaning src
-  installing the package to build vignettes (820ms)
   creating vignettes ...

$$
{{C}}_{t}+{{\left({\frac{P^E}{P}}\right)}}_{t}+\frac{1}{{{R^R}}_{t}}\, {\nu}\, {K}_{t}={{N}}_{t}\, {{\left({\frac{W}{P}}\right)}}_{t}+{{\left({\frac{P^E}{P}}\right)}}_{t}+{{\left({\frac{D^E}{P}}\right)}}_{t}+{\nu}\, {K}_{t-1}
$$

$$
y=\gamma \left( \delta x_1^{-\rho} + \left(1-\delta \right) x_2^{-\rho} \right)^{-\frac{\nu}{\rho}},
$$
where $y$ is the output quantity,
$x_1$ and $x_2$ are the input quantities,
and $\gamma$, $\delta$, $\rho$, and $\nu$ are parameters.
Parameter $\gamma \in [0,\infty)$ determines the productivity,
$\delta \in [ 0,1 ]$ determines the optimal distribution of the inputs,
$\rho \in [-1,0) \cup (0,\infty)$ determines the (constant) elasticity of substitution,
which is $\sigma = 1 \left/ \left( 1 + \rho \right) \right.$,
and $\nu \in [0,\infty)$ is equal to the elasticity of scale.

The CES function includes three special cases: for $\rho \rightarrow 0$,
$\sigma$ approaches $1$
and the CES turns to the Cobb-Douglas form;
for $\rho \rightarrow \infty$, $\sigma$ approaches $0$ and the CES turns
to the Leontief production function;  
and for $\rho \rightarrow -1$, $\sigma$ approaches infinity
and the CES turns to a linear function
if $\nu$ is equal to 1.
$$
\begin{align}
y = \; & \gamma \left( \sum_{i=1}^n \delta_i x_i^{-\rho} \right)^{-\frac{\nu}{\rho}}
\label{eq:n-ces-plain}\\
& \text{with } \sum_{i=1}^n \delta_i = 1,
\end{align}
$$

$$
y = \gamma \left[ \delta \left( \delta_1 x_1^{-\rho_1} + ( 1 - \delta_1 ) x_2^{-\rho_1} \right)^{\rho/\rho_1}
    +  ( 1 - \delta ) \left( \delta_2 x_3^{-\rho_2} + ( 1 - \delta_2 ) x_4^{-\rho_2} \right)^{\rho/\rho_2} \right]^{-\nu/\rho}.
$$
If $\rho_1 = \rho_2 = \rho$,
the four-input nested CES function defined in Equation~\ref{eq:4-ces-nested}
reduces to the plain four-input CES function defined in Equation

In this case,
the parameters of
the four-input nested CES function defined in Equation~\ref{eq:4-ces-nested}
(indicated by the superscript $n$)
and the parameters of
the plain four-input CES function defined in Equation~\ref{eq:n-ces-plain}
(indicated by the superscript $p$)
correspond in the following way:
where $\rho^p = \rho_1^n = \rho_2^n = \rho^n$,
$\delta_1^p = \delta_1^n \; \delta^n$,
$\delta_2^p = ( 1 - \delta_1^n ) \; \delta^n$,
$\delta_3^p = \delta_2^n \; ( 1 - \delta^n )$,
$\delta_4^p = ( 1 - \delta_2^n ) \; ( 1 - \delta^n )$,
$\gamma^p = \gamma^n$,
$\delta_1^n = \delta_1^p / ( \delta_1^p + \delta_2^p )$,
$\delta_2^n = \delta_3^p / ( \delta_3^p + \delta_4^p )$, and
$\delta^n = \delta_1^p + \delta_2^p$.

$$
y = \gamma \left[ \delta \gamma_1^{-\rho} \left( \delta_1 x_1^{-\rho_1} + (1-\delta_1) x_2^{-\rho_1}\right)^{\rho/\rho_1}
+ ( 1 - \delta ) x_3^{-\rho} \right]^{-\nu/\rho}.
$$
However, adding the term $\gamma_1^{-\rho}$ does not increase the flexibility
of this function
as $\gamma_1$ can be arbitrarily normalised to one;
normalising $\gamma_1$ to one
changes $\gamma$ to
$\gamma \left( \delta \gamma_1^{-\rho} + ( 1 - \delta ) \right)^{-( \nu / \rho )}$
and changes $\delta$ to
$\left. \left( \delta \gamma_1^{-\rho} \right) \right/
\left( \delta \gamma_1^{-\rho} + ( 1 - \delta ) \right)$,
but has no effect on the functional form.
Hence, the parameters $\gamma$, $\gamma_1$, and $\delta$
cannot be (jointly) identified in econometric estimations
(see also explanation for the four-input nested CES function
above Equation

approximation of the classical two-input CES production function
that could be estimated by ordinary least-squares techniques.
$$
\begin{align}
\ln y = & \ln \gamma + \nu \; \delta \ln x_1 + \nu
    \left( 1 - \delta \right) \ln x_2
   \label{eq:kmenta}\\
  &- \frac{\rho \, \nu}{2} \;
      \delta \left( 1 - \delta \right) \left( \ln x_1 - \ln x_2 \right)^2
   \nonumber
\end{align}
$$
While \citet{kmenta67} obtained this formula by logarithmising the CES function
and applying a second-order Taylor series expansion to
$\ln \left( \delta x_1^{-\rho} + ( 1 - \delta ) x_2^{-\rho} \right)$
at the point $\rho = 0$,
the same formula can be obtained
by applying a first-order Taylor series expansion
to the entire logarithmised CES function

$$
\begin{align}
\ln y =& \alpha_0 + \alpha_1 \ln x_1 + \alpha_2 \ln x_2
   \label{eq:kmentaTranslog}\\
   & + \frac{1}{2} \; \beta_{11} \left( \ln x_1 \right)^2
   + \frac{1}{2} \; \beta_{22} \left( \ln x_2 \right)^2
   + \beta_{12} \, \ln x_1 \ln x_2,
   \nonumber
\end{align}
$$
where the two restrictions are
$$
\beta_{12} = - \beta_{11} = - \beta_{22}.
$$
If this is the case,
a simple $t$-test for the coefficient $\beta_{12} = - \beta_{11} = - \beta_{22}$
can be used to check if the Cobb-Douglas functional form is an acceptable
simplification of the Kmenta approximation of the CES function.
$$
\begin{align}
\gamma &= \exp( \alpha_0 )
   \label{eq:kmentaTranslogGamma}\\
\nu &= \alpha_1 + \alpha_2
   \label{eq:kmentaTranslogNu}\\
\delta &= \frac{ \alpha_1 }{ \alpha_1 + \alpha_2 }
   \label{eq:kmentaTranslogDelta}\\
\rho &= \frac{ \beta_{12} \left( \alpha_1 + \alpha_2 \right) }{
   \alpha_1 \cdot \alpha_2 }
   \label{eq:kmentaTranslogRho}
\end{align}
$$

For the traditional two-input CES function, these partial derivatives are:$$\begin{align}
\frac{\partial y}{\partial \gamma } =\;&
   e^{\lambda \, t} \, \left( \delta x_1^{-\rho} + ( 1 - \delta ) x_2^{-\rho} \right)^{-\frac{ \nu }{\rho}}
   \label{eq:derivYGamma}\\
\frac{\partial y}{\partial \lambda } =\;&
    \gamma \; t \; \frac{\partial y}{\partial \gamma }
   \label{eq:derivYLambda}\\
\frac{\partial y}{\partial \delta } =\;&
   -  \gamma \, e^{\lambda \, t} \, \frac{\nu }{ \rho } \left( x_1^{-\rho} - x_2^{-\rho} \right)
   \left( \delta x_1^{-\rho} + ( 1 - \delta ) x_2^{-\rho} \right)^{-\frac{\nu}{\rho} - 1}
   \label{eq:derivYDelta}\\
\frac{\partial y}{\partial \rho } =\;&
   \gamma \, e^{\lambda \, t} \, \frac{ \nu }{ \rho^2 } \;
   \ln \left( \delta x_1^{-\rho} + ( 1 - \delta ) x_2^{-\rho} \right)
   \left( \delta x_1^{-\rho} + ( 1 - \delta ) x_2^{-\rho} \right)^{-\frac{ \nu }{\rho}}
   \label{eq:derivYRho}\\
   & + \gamma \, e^{\lambda \, t} \, \frac{ \nu }{ \rho }
   \left( \delta \ln( x_1 ) x_1^{-\rho} + ( 1 - \delta ) \ln( x_2 ) x_2^{-\rho} \right)
   \left( \delta x_1^{-\rho} + ( 1 - \delta ) x_2^{-\rho} \right)^{-\frac{\nu}{\rho} -1}
   \nonumber\\
\frac{\partial y}{\partial \nu } =\;&
   - \gamma \, e^{\lambda \, t} \, \frac{ 1 }{ \rho } \;
   \ln \left( \delta x_1^{-\rho} + ( 1 - \delta ) x_2^{-\rho} \right)
   \left( \delta x_1^{-\rho} + ( 1 - \delta ) x_2^{-\rho} \right)^{-\frac{ \nu }{\rho }}
   \label{eq:derivYNu}
\end{align}
$$

$$
\begin{align}
\frac{\partial y}{\partial \gamma } =\;&
   e^{\lambda \, t} \, x_1^{\nu \, \delta} \; x_2^{\nu \, ( 1 - \delta )}
   \exp \left( - \frac{\rho}{2} \, \nu \, \delta \, ( 1 - \delta )
      \left( \ln x_1 - \ln x_2 \right)^2
   \right)
   \label{eq:derivYGammaApprox}\\
\frac{\partial y}{\partial \delta } =\;&
   \gamma \, e^{\lambda \, t} \, \nu \, \left( \ln x_1 - \ln x_2 \right)
   x_1^{ \nu \, \delta } \; x_2^{ \nu ( 1 - \delta ) }
   \label{eq:derivYDeltaApprox}\\
   & \left( 1 - \frac{ \rho }{ 2 }
      \big[ 1 - 2 \, \delta + \nu \, \delta ( 1 - \delta )
         \left( \ln x_1 - \ln x_2 \right) \big]
      \left( \ln x_1 - \ln x_2 \right)
   \right)
   \nonumber\\
\frac{\partial y}{\partial \rho } =\;&
   \gamma \, e^{\lambda \, t} \, \nu \, \delta \, ( 1 - \delta ) \,
   x_1^{ \nu \, \delta } \; x_2^{ \nu ( 1 - \delta ) }
   \bigg( - \frac{1}{2} \left( \ln x_1 - \ln x_2 \right)^2
   \label{eq:derivYRhoApprox}\\
      & + \frac{\rho}{3} ( 1 - 2 \, \delta ) \left( \ln x_1 - \ln x_2 \right)^3
      + \frac{\rho}{4} \, \nu \, \delta ( 1 - \delta )
         \left( \ln x_1 - \ln x_2 \right)^4
   \bigg)
   \nonumber\\
\frac{\partial y}{\partial \nu } =\;&
   \gamma \, e^{\lambda \, t} \, x_1^{ \nu \, \delta } \; x_2^{ \nu ( 1 - \delta ) }
   \bigg( \delta \ln x_1 + ( 1 - \delta ) \ln x_2
      \label{eq:derivYNuApprox}\\
      & - \frac{\rho}{2} \, \delta ( 1 - \delta )
      \left( \ln x_1 - \ln x_2 \right)^2
      \left[ 1 + \nu
         \left( \delta \ln x_1 + ( 1 - \delta ) \ln x_2 \right)
      \right]
   \bigg)
   \nonumber
\end{align}
$$
If $\rho$ is zero or close to zero,
the partial derivatives with respect to $\lambda$ are calculated
also with Equation~\ref{eq:derivYLambda},
but now $\partial y / \partial \gamma$ is calculated
with Equation

Allen-Uzawa elasticity of substitution
$$
\sigma_{i,j} = \begin{cases}
        (1 - \rho )^{-1}
        & \text{for } i = 1,2; \; j=3 \\
        & \\
        \dfrac{(1-\rho_1)^{-1} - (1-\rho)^{-1}}
        {\delta \left( \dfrac{y}{
        B_1^{-\frac{1}{\rho_1}}}\right)^{1+\rho}}+(1-\rho)^{-1}
        & \text{for } i=1; \; j=2
   \end{cases}
$$

Hicks-McFadden elasticity of substitution
$$
\sigma_{i,j} = \begin{cases}
        \dfrac{\dfrac{1}{\theta_i} + \dfrac{1}{\theta_j}}
        {(1-\rho_1) \left( \dfrac{1}{\theta_i} - \dfrac{1}{\theta^*} \right)
        + (1-\rho_2) \left( \dfrac{1}{\theta_j} - \dfrac{1}{\theta} \right)
        + (1-\rho) \left( \dfrac{1}{\theta^*} - \dfrac{1}{\theta} \right) }
        & \text{for } i=1,2; \; j=3 \\
        & \\
        (1-\rho_1)^{-1}
        & \mbox{text } i=1; \; j=2
   \end{cases}
$$
with
$$
\begin{align}
& \theta^* = \delta
            B_1^{\frac{\rho}{\rho_1}} \cdot y^\rho \\
& \theta = (1-\delta) x_3^{-\rho} \cdot y^{\rho} \\
& \theta_1 = \delta \delta_1 x_1^{-\rho_1}
            B_1^{-\frac{\rho_1 - \rho}{\rho_1}}
            \cdot y^{\rho} \\
& \theta_2 = \delta (1 - \delta_1) x_2^{-\rho_1}
            B_1^{-\frac{\rho_1 - \rho}{\rho_1}}
            \cdot y^{\rho} \\
& \theta_3 = \theta
\end{align}
$$

$$f_{\delta}\left(\rho\right)=h_{\delta}\left(\rho\right)\,\exp\left(-g_{\delta}\left(\rho\right)\right)
$$

$$
\frac{\partial y}{\partial\nu}=-\gamma \, e^{\lambda \, t} \, \frac{1}{\rho}\,\ln\left(\delta\, x_{1}^{-\rho}+\left(1-\delta\right)\, x_{2}^{-\rho}\right)\,\left(\delta\, x_{1}^{-\rho}+\left(1-\delta\right)\, x_{2}^{-\rho}\right)^{-\frac{\nu}{\rho}}$$
Now we define the function $f_{\nu}\left(\rho\right)$

$$
\begin{eqnarray}
f_{\nu}\left(\rho\right) & = & \frac{1}{\rho}\,\ln\left(\delta\, x_{1}^{-\rho}+\left(1-\delta\right)\, x_{2}^{-\rho}\right)\,\left(\delta\, x_{1}^{-\rho}+\left(1-\delta\right)\, x_{2}^{-\rho}\right)^{-\frac{\nu}{\rho}}\\
& = & \frac{1}{\rho}\,\ln\left(\delta\, x_{1}^{-\rho}+\left(1-\delta\right)\, x_{2}^{-\rho}\right)\,\exp\left(-\frac{\nu}{\rho}\,\ln\left(\delta\, x_{1}^{-\rho}+\left(1-\delta\right)\, x_{2}^{-\rho}\right)\right)\end{eqnarray}
$$

so that we can approximate $\partial y/\partial\nu$ by using the
first-order Taylor series approximation of $f_{\nu}\left(\rho\right)$:
$$
\begin{eqnarray}
\frac{\partial y}{\partial\nu} & = & -\gamma \, e^{\lambda \, t} \, f_{\nu}\left(\rho\right)\\
& \approx & -\gamma \, e^{\lambda \, t} \, \left(f_{\nu}\left(0\right)+\rho f_{\nu}'\left(0\right)\right)\end{eqnarray}$$

Now we define the helper function $g_{\nu}\left(\rho\right)$
$$\begin{eqnarray}
g_{\nu}\left(\rho\right) & = & \frac{1}{\rho}\,\ln\left(\delta\, x_{1}^{-\rho}+\left(1-\delta\right)\, x_{2}^{-\rho}\right)\\
& = & \frac{1}{\rho}\,\ln\left(g\left(\rho\right)\right)\end{eqnarray}$$

with first and second derivative$$\begin{eqnarray}
g_{\nu}'\left(\rho\right) & = & \frac{\rho\frac{g'\left(\rho\right)}{g\left(\rho\right)}-\ln\left(g\left(\rho\right)\right)}{\rho^{2}}\\
& = & \frac{1}{\rho}\,\frac{g'\left(\rho\right)}{g\left(\rho\right)}-\frac{\ln\left(g\left(\rho\right)\right)}{\rho^{2}}\\
g_{\nu}''\left(\rho\right) & = & -\frac{1}{\rho^{2}}\,\frac{g'\left(\rho\right)}{g\left(\rho\right)}+\frac{1}{\rho}\,\frac{g''\left(\rho\right)}{g\left(\rho\right)}-\frac{1}{\rho}\,\frac{\left(g'\left(\rho\right)\right)^{2}}{\left(g\left(\rho\right)\right)^{2}}+2\frac{\ln\left(g\left(\rho\right)\right)}{\rho^{3}}-\frac{1}{\rho^{2}}\frac{g'\left(\rho\right)}{g\left(\rho\right)}\\
& = & \frac{-2\rho\,\frac{g'\left(\rho\right)}{g\left(\rho\right)}+\rho^{2}\,\frac{g''\left(\rho\right)}{g\left(\rho\right)}-\rho^{2}\,\frac{\left(g'\left(\rho\right)\right)^{2}}{\left(g\left(\rho\right)\right)^{2}}+2\ln\left(g\left(\rho\right)\right)}{\rho^{3}}\end{eqnarray}$$

and use the function $f\left(\rho\right)$ defined above so that$$
f_{\nu}\left(\rho\right)=g_{\nu}\left(\rho\right)\,\exp\left(f\left(\rho\right)\right)$$
and$$\begin{eqnarray}
f_{\nu}'\left(\rho\right) & = & g_{\nu}'\left(\rho\right)\,\exp\left(f\left(\rho\right)\right)+g_{\nu}\left(\rho\right)\,\exp\left(f\left(\rho\right)\right)\, f'\left(\rho\right)\end{eqnarray}$$

Now we can calculate the limits of $g_{\nu}\left(\rho\right)$, $g_{\nu}'\left(\rho\right)$,
and $g_{\nu}''\left(\rho\right)$ for $\rho\rightarrow0$ by
$$\begin{eqnarray}
g_{\nu}\left(0\right) & = & \lim_{\rho\rightarrow0}g_{\nu}\left(\rho\right)\\
& = & \lim_{\rho\rightarrow0}\frac{\ln\left(g\left(\rho\right)\right)}{\rho}\\
& = & \lim_{\rho\rightarrow0}\frac{\frac{g'\left(\rho\right)}{g\left(\rho\right)}}{1}\\
& = & -\delta\,\ln x_{1}-\left(1-\delta\right)\,\ln x_{2}\end{eqnarray}$$


$$\begin{eqnarray}
g_{\nu}'\left(0\right) & = & \lim_{\rho\rightarrow0}g_{\nu}'\left(\rho\right)\\
& = & \lim_{\rho\rightarrow0}\left(\frac{1}{\rho}\,\frac{g'\left(\rho\right)}{g\left(\rho\right)}-\frac{\ln\left(g\left(\rho\right)\right)}{\rho^{2}}\right)\\
& = & \lim_{\rho\rightarrow0}\frac{\rho\,\frac{g'\left(\rho\right)}{g\left(\rho\right)}-\ln\left(g\left(\rho\right)\right)}{\rho^{2}}\\
& = & \lim_{\rho\rightarrow0}\frac{\frac{g'\left(\rho\right)}{g\left(\rho\right)}+\rho\,\frac{g''\left(\rho\right)g\left(\rho\right)-\left(g'\left(\rho\right)\right)^{2}}{\left(g\left(\rho\right)\right)^{2}}-\frac{g'\left(\rho\right)}{g\left(\rho\right)}}{2\rho}\\
& = & \lim_{\rho\rightarrow0}\frac{g''\left(\rho\right)g\left(\rho\right)-\left(g'\left(\rho\right)\right)^{2}}{2\left(g\left(\rho\right)\right)^{2}}\\
& = & \frac{g''\left(0\right)g\left(0\right)-\left(g'\left(0\right)\right)^{2}}{2\left(g\left(0\right)\right)^{2}}\\
& = & \frac{1}{2}\left(\delta\left(\ln x_{1}\right)^{2}+\left(1-\delta\right)\left(\ln x_{2}\right)^{2}-\left(-\delta\,\ln x_{1}-\left(1-\delta\right)\,\ln x_{2}\right)^{2}\right)\\
& = & \frac{1}{2}\biggl(\delta\left(\ln x_{1}\right)^{2}+\left(1-\delta\right)\left(\ln x_{2}\right)^{2}-\delta^{2}\left(\ln x_{1}\right)^{2}\\
&  & -2\delta\left(1-\delta\right)\,\ln x_{1}\ln x_{2}-\left(1-\delta\right)^{2}\left(\ln x_{2}\right)^{2}\biggr)\nonumber \\
& = & \frac{1}{2}\biggl(\left(\delta-\delta^{2}\right)\left(\ln x_{1}\right)^{2}+\left(\left(1-\delta\right)-\left(1-\delta\right)^{2}\right)\left(\ln x_{2}\right)^{2}\\
&  & -2\delta\left(1-\delta\right)\,\ln x_{1}\ln x_{2}\biggr)\nonumber \\
& = & \frac{1}{2}\biggl(\delta\left(1-\delta\right)\left(\ln x_{1}\right)^{2}+\left(1-\delta\right)\left(1-\left(1-\delta\right)\right)\left(\ln x_{2}\right)^{2}\\
&  & -2\delta\left(1-\delta\right)\,\ln x_{1}\ln x_{2}\biggr)\nonumber \\
& = & \frac{\delta\left(1-\delta\right)}{2}\left(\left(\ln x_{1}\right)^{2}-2\,\ln x_{1}\ln x_{2}+\left(\ln x_{2}\right)^{2}\right)\\
& = & \frac{\delta\left(1-\delta\right)}{2}\left(\ln x_{1}-\ln x_{2}\right)^{2}\end{eqnarray}
$$

so that we can calculate the limit of $f_{\nu}\left(\rho\right)$and
$f_{\nu}'\left(\rho\right)$ for $\rho\rightarrow0$ by

$$
\begin{eqnarray}
f_{\nu}\left(0\right) & = & \lim_{\rho\rightarrow0}f_{\nu}\left(\rho\right)\\
& = & \lim_{\rho\rightarrow0}\left(g_{\nu}\left(\rho\right)\,\exp\left(f\left(\rho\right)\right)\right)\\
& = & \lim_{\rho\rightarrow0}g_{\nu}\left(\rho\right)\,\lim_{\rho\rightarrow0}\exp\left(f\left(\rho\right)\right)\\
& = & \lim_{\rho\rightarrow0}g_{\nu}\left(\rho\right)\,\exp\left(\lim_{\rho\rightarrow0}f\left(\rho\right)\right)\\
& = & g_{\nu}\left(0\right)\,\exp\left(f\left(0\right)\right)\\
& = & \left(-\delta\,\ln x_{1}-\left(1-\delta\right)\,\ln x_{2}\right)\exp\left(\nu\left(\delta\,\ln x_{1}+\left(1-\delta\right)\,\ln x_{2}\right)\right)\\
& = & -\left(\delta\,\ln x_{1}+\left(1-\delta\right)\,\ln x_{2}\right)x_{1}^{\nu\delta}x_{2}^{\nu\left(1-\delta\right)}\end{eqnarray}
$$