Following the EM algorithm-based estimation of the regime-switching model (2), we now proceed with the remaining set of parameters ${\widehat{\Theta}}_{2}$ by fitting a NIG distribution for each regime:

$X\mathrm{(}\text{Regime\hspace{0.17em}1}\mathrm{)}\sim {L}_{1}\mathrm{(}{\alpha}^{1}\mathrm{,}\text{\hspace{0.17em}}{\beta}^{1}\mathrm{,}\text{\hspace{0.17em}}{\delta}^{1}\mathrm{,}\text{\hspace{0.17em}}{\mu}^{1}\mathrm{)}$

and

$X\mathrm{(}\text{Regime\hspace{0.17em}2}\mathrm{)}\sim {L}_{2}\mathrm{(}{\alpha}^{2}\mathrm{,}\text{\hspace{0.17em}}{\beta}^{2}\mathrm{,}\text{\hspace{0.17em}}{\delta}^{2}\mathrm{,}\text{\hspace{0.17em}}{\mu}^{2}\mathrm{)}$

with *L*_{1} and *L*_{2} two separate NIG Lévy distributions.

The estimation of the distribution parameters is achieved by constrained maximum likelihood:

${\Phi}^{1}\mathrm{:}=\left\{{\alpha}^{1}\mathrm{,}\text{\hspace{0.17em}}{\beta}^{1}\mathrm{,}\text{\hspace{0.17em}}{\delta}^{1}\mathrm{,}\text{\hspace{0.17em}}{\mu}^{1}\right\}$

and

${\Phi}^{2}\mathrm{:}=\left\{{\alpha}^{2}\mathrm{,}\text{\hspace{0.17em}}{\beta}^{2}\mathrm{,}\text{\hspace{0.17em}}{\delta}^{2}\mathrm{,}\text{\hspace{0.17em}}{\mu}^{2}\right\}$

We denote by *r*_{t}:=log(*X*_{t})–log(*X*_{t–1}).

**Proposition 1** *The log-likelihood function of the sequence of log-returns with a NIG distribution* (*α*, *β*, *δ*, *μ*) *is given by*

$$L\mathrm{(}\alpha \mathrm{,}\text{\hspace{0.17em}}\beta \mathrm{,}\text{\hspace{0.17em}}\delta \mathrm{,}\text{\hspace{0.17em}}\mu \mathrm{)}=n\mathrm{log}\mathrm{(}\frac{\alpha}{\pi}\mathrm{)}+n\delta \gamma +{\displaystyle \sum _{t=0}^{n-1}}[\beta \delta {\tau}_{t}-\mathrm{log}{c}_{t}+\mathrm{log}{K}_{1}\mathrm{(}\alpha \delta {c}_{t}\mathrm{)}]\mathrm{,}$$(15)

*where* $\gamma =\sqrt{{\alpha}^{2}-{\beta}^{2}}.$ *For any* *t*=0, 1, …, *n*–1, ${\tau}_{t}=\frac{{r}_{t}-\mu}{\delta}$ *and* ${c}_{t}=\sqrt{1+{\tau}_{t}^{2}}.$

**Proof.** We propose to use the maximum likelihood method to estimate the set of parameters (*α*, *β*, *δ*, *μ*). Denote for all *t*=0, 1, … *n*–1, ${\tau}_{t}=\frac{{r}_{t}-\mu}{\delta}$ and ${c}_{t}=\sqrt{1+{\tau}_{t}^{2}}.$ Then, the log-likelihood function following (3) is given by

$\begin{array}{c}L\mathrm{(}\alpha \mathrm{,}\text{\hspace{0.17em}}\beta \mathrm{,}\text{\hspace{0.17em}}\delta \mathrm{,}\text{\hspace{0.17em}}\mu \mathrm{)}={\displaystyle \sum _{t=0}^{n-1}}\mathrm{log}{f}_{\text{NIG}}\mathrm{(}{r}_{t}\mathrm{;}\text{\hspace{0.17em}}\alpha \mathrm{,}\text{\hspace{0.17em}}\beta \mathrm{,}\text{\hspace{0.17em}}\delta \mathrm{,}\text{\hspace{0.17em}}\mu \mathrm{)}\\ ={\displaystyle \sum _{t=0}^{n-1}}\left[\mathrm{log}\mathrm{(}\frac{\alpha}{\pi}\mathrm{)}\alpha \gamma +\beta \delta {\tau}_{t}+\mathrm{log}\mathrm{(}\frac{{K}_{1}\mathrm{(}\alpha \delta {c}_{t}\mathrm{)}}{{c}_{t}}\mathrm{)}\right]\\ =n\mathrm{log}\mathrm{(}\frac{\alpha}{\pi}\mathrm{)}+n\delta \gamma +{\displaystyle \sum _{t=0}^{n-1}}[\beta \delta {\tau}_{t}-\mathrm{log}{c}_{t}+\mathrm{log}{K}_{1}\mathrm{(}\alpha \delta {c}_{t}\mathrm{)}]\mathrm{.}\end{array}$

■

To compute the standard errors of our estimators, we give the first partial derivatives of each parameter:

**Proposition 2** *We have that*

$\begin{array}{l}\frac{\partial L}{\partial \gamma}=\frac{2n\gamma}{{\alpha}^{2}}+n\delta -{\displaystyle \sum _{t=0}^{n-1}}\mathrm{(}\frac{\gamma \delta}{\alpha}{c}_{t}{R}_{1}\mathrm{(}\alpha \delta {c}_{t}\mathrm{)}\mathrm{)}\\ \frac{\partial L}{\partial \delta}=n\gamma +\frac{n}{\delta}-{\displaystyle \sum _{t=0}^{n-1}}\mathrm{(}\frac{\alpha}{{c}_{t}}{R}_{1}\mathrm{(}\alpha \delta {c}_{t}\mathrm{)}\mathrm{)}\\ \frac{\partial L}{\partial \beta}=2n\frac{\beta}{{\alpha}^{2}}+{\displaystyle \sum _{t=0}^{n-1}}\mathrm{(}\delta {\tau}_{t}-\frac{\beta \delta}{\alpha}{c}_{t}{R}_{1}\mathrm{(}\alpha \delta {c}_{t}\mathrm{)}\mathrm{)}\\ \frac{\partial L}{\partial \mu}=-n\beta +{\displaystyle \sum _{t=0}^{n-1}}\mathrm{(}\alpha \frac{{\tau}_{t}}{{c}_{t}}{R}_{1}\mathrm{(}\alpha \delta {c}_{t}\mathrm{)}\mathrm{)}\end{array}$

*where for all* *x*>0,

${R}_{1}\mathrm{(}x\mathrm{)}=\frac{{K}_{2}\mathrm{(}x\mathrm{)}}{{K}_{1}\mathrm{(}x\mathrm{)}}$

To compute the standard errors of the maximum likelihood estimators, we have to evaluate the Fisher matrix 𝒥 which is given by:

$\mathcal{J}=\mathbb{E}\left[\frac{\partial L\mathrm{(}X\mathrm{;}\text{\hspace{0.17em}}\theta \mathrm{)}}{\partial \theta}\frac{\partial L\mathrm{(}X\mathrm{;}\text{\hspace{0.17em}}\theta \mathrm{)}}{\partial {\theta}^{\prime}}\right]$

where *θ* and *θ*′ are both parameters of the NIG distribution [i.e. (*α*, *β*, *δ*, *μ*)]. We have the following equality:

$\mathbb{E}\left[\frac{\partial L\mathrm{(}X\mathrm{;}\text{\hspace{0.17em}}\theta \mathrm{)}}{\partial \theta}\frac{\partial L\mathrm{(}X\mathrm{;}\text{\hspace{0.17em}}\theta \mathrm{)}}{\partial {\theta}^{\prime}}\right]=-\mathbb{E}\left[\frac{\partial {L}^{2}\mathrm{(}X\mathrm{;}\text{\hspace{0.17em}}\theta \mathrm{)}}{\partial \theta \partial {\theta}^{\prime}}\right]$

Thus, to estimate the Fisher matrix with regard to the time series, we can evaluate the empirical average of the Hessian matrix:

$\widehat{\mathcal{J}}=\frac{1}{n}{\displaystyle \sum _{t=0}^{n-1}}\left[\frac{\partial L\mathrm{(}{r}_{i}\mathrm{;}\text{\hspace{0.17em}}\widehat{\theta}\mathrm{)}}{\partial \theta}\frac{\partial L\mathrm{(}{r}_{i}\mathrm{;}\text{\hspace{0.17em}}\widehat{\theta}\mathrm{)}}{\partial {\theta}^{\prime}}\right]$

Finally, the diagonal terms of the inverse matrix of $\widehat{\mathcal{J}}$ give the estimations of the variance of our maximum likelihood estimators.

Moreover, to take into account the constraints on the NIG parameters (i.e. the positivity of *γ* and *δ*), the optimization is performed according to the following variables (*γ*′, *δ*′, *β*, *μ*) with ${\gamma}^{\prime}=\mathrm{log}\mathrm{(}\sqrt{{\alpha}^{2}-{\beta}^{2}}\mathrm{)}$ and *δ*′=log(*δ*).

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