The filter-based estimates for the unknown parameters in the HMRS-STAR model are derived using the Expectation Maximization, or EM, algorithm. Here we briefly present the idea of the EM algorithm. For details, one may refer to, for example, Elliott, Aggoun, and Moore (1995). Again the mathematical techniques used here follow from those in Elliott, Aggoun, and Moore (1995).

Let $\{{P}_{\theta}|\theta \in \mathrm{\Theta}\}$ be a family of probability measures on a measurable space $(\mathrm{\Omega},\mathcal{F})$ all absolutely continuous with respect to a fixed probability measure *P*_{0}, where *θ* is an unknown model parameter and Θ is the parameter space. Write *𝒴* for a sub-*σ*-field of *ℱ*. Then the likelihood function for computing an estimate of the unknown parameter *θ* given the available information in *𝒴* is:

$$L(\theta ):={{\textstyle \text{E}}}_{0}[\frac{d{P}_{\theta}}{d{P}_{0}}|\mathcal{Y}]\text{\hspace{1em}}.$$

Here E_{0} is the expectation under *P*_{0}. Then the maximum likelihood estimate (MLE) $\hat{\theta}$ of *θ* is given by:

$$\hat{\theta}:=\underset{\theta \in \mathrm{\Theta}}{\text{argmax}}L(\theta ).$$

It may be challenging to compute the MLE $\hat{\theta}$ directly. The EM algorithm provides an iterative approximation method to compute the MLE $\hat{\theta}$. It consists of the following four steps, [see, for example, Elliott, Aggoun, and Moore (1995)]:

**Step I:** Set the counter *p* = 0 and choose $\hat{\theta}$_{0}.

**Step II:** (E-step) Set ${\theta}^{\ast}={\hat{\theta}}_{p}$ and compute $Q(\cdot ,{\theta}^{\ast})$, where

$$Q(\theta ,{\theta}^{\ast}):={{\textstyle \text{E}}}_{{\theta}^{\ast}}[\mathrm{ln}(\frac{d{P}_{\theta}}{d{P}_{{\theta}^{\ast}}})|\mathcal{Y}]\text{\hspace{1em}},$$

where ${{\textstyle \text{E}}}_{{\theta}^{\ast}}$ is the expectation under ${P}_{{\theta}^{\ast}}$.

**Step III:** (M- step) Find

$${\theta}_{p+1}:=\underset{\theta \in \mathrm{\Theta}}{\text{argmax}}\text{\hspace{1em}}Q(\theta ,{\theta}^{\ast})\text{\hspace{1em}}.$$

**Step IV:** Replace *p* by *p* + 1 and repeat beginning with Step II until a certain stopping criterion is satisfied.

Note that the EM algorithm we applied here recursively converges only to a local maximum of log-likelihood. For a discussion of the EM algorithm and its convergent properties, one may refer to Baum et al. (1970), Dembo and Zeitouni (1986), and Elliott, Aggoun, and Moore (1995).

The set of parameters of interest can be described by the set *𝚯* defined as follows:

$$\mathbf{\Theta}:=\{({\pi}_{kl}{)}_{k,l=1,2,\dots ,N},(\mathit{\mu},\mathit{\theta},\mathit{\sigma}),r,\delta \}\text{\hspace{1em}},$$

where $\mathit{\mu}:=({\mu}_{1},{\mu}_{2},\dots ,{\mu}_{N}{)}^{\mathrm{\prime}}\in {\mathrm{\Re}}^{N}$; $\mathit{\theta}:=({\theta}_{1},{\theta}_{2},\dots ,{\theta}_{N}{)}^{\mathrm{\prime}}\in {\mathrm{\Re}}^{N}$; $\mathit{\sigma}:=({\sigma}_{1},{\sigma}_{2},\dots ,{\sigma}_{N}{)}^{\mathrm{\prime}}\in {\mathrm{\Re}}^{N}$.

Since *π*_{ji}’s are transition probabilities, we also require that

$$\sum _{j=1}^{N}{\pi}_{ji}=1\text{\hspace{1em}},\phantom{\rule{1em}{0ex}}i=1,2,\dots ,N\text{\hspace{1em}}.$$

Suppose now that the set of parameters 𝚯 is given and that the set of observed data described by the *σ*-field *𝒴*_{n} is known. We wish to determine a new set of parameters 𝚯(*n*) defined by:

$$\mathbf{\Theta}(n):=\{({\pi}_{kl}(n){)}_{k,l=1,2,\dots ,N},(\mathit{\mu}(n),\mathit{\theta}(n),\mathit{\sigma}(n)),r(n),\delta (n)\}\text{\hspace{1em}},$$

where

$$\begin{array}{rl}\mathit{\mu}(n)& :=({\mu}_{1}(n),{\mu}_{2}(n),\dots ,{\mu}_{N}(n){)}^{\mathrm{\prime}}\in {\mathrm{\Re}}^{N},\\ \mathit{\theta}(n)& :=({\theta}_{1}(n),{\theta}_{2}(n),\dots ,{\theta}_{N}(n){)}^{\mathrm{\prime}}\in {\mathrm{\Re}}^{N},\\ \mathit{\sigma}(n)& :=({\sigma}_{1}(n),{\sigma}_{2}(n),\dots ,{\sigma}_{N}(n){)}^{\mathrm{\prime}}\in {\mathrm{\Re}}^{N}.\end{array}$$

Such a new set of parameters is determined by maximizing the conditional log-likelihoods defined below. The basic idea is that we update one set of parameters at a time starting with the $[{\pi}_{kl}{]}_{k,l=1,2,\dots ,N}$. Using this method, it is known that a filter-based estimate ${\hat{\pi}}_{kl}(n)$ for *π*_{kl} given observed data described by the *σ*-field *𝒴*_{n} is given by:

$${\hat{\pi}}_{kl}(n)=\frac{{\textstyle \text{E}}[{J}_{n}^{kl}|{\mathcal{Y}}_{n}]}{{\textstyle \text{E}}[{O}_{n}^{l}|{\mathcal{Y}}_{n}]}=\frac{\sigma ({J}_{n}^{kl})}{\sigma ({O}_{n}^{l})}\text{\hspace{1em}}.$$

For derivation, interested readers may refer to Elliott, Aggoun, and Moore (1995), (see Chapter 2 therein).

Consider now another set of parameters (*𝝁*, *𝜽*). To change the set of parameters from (*𝝁*, *𝜽*) to ($\hat{\mathit{\mu}}$(*n*), $\hat{\mathit{\theta}}$(*n*)) while keeping other parameters (*𝝈*, *r*, *δ*) constant, we must consider the following factors, (*t* = 1, 2, … , *n*),

$$\begin{array}{rl}{\hat{\lambda}}_{t}^{1}& :=\mathrm{exp}\{\frac{1}{2{\u27e8\mathit{\sigma},{\mathbf{\text{X}}}_{t}\u27e9}^{2}}[{\u27e8\mathit{\mu}+\mathit{\theta}F(\frac{{Y}_{t-d}-r}{\delta}),{\mathbf{\text{X}}}_{t}\u27e9}^{2}-{\u27e8\hat{\mathit{\mu}}+\hat{\mathit{\theta}}F(\frac{{Y}_{t-d}-r}{\delta}),{\mathbf{\text{X}}}_{t}\u27e9}^{2}\\ & -2{Y}_{t}\u27e8\mathit{\mu}+\mathit{\theta}F(\frac{{Y}_{t-d}-r}{\delta}),{\mathbf{\text{X}}}_{t}\u27e9+2{Y}_{t}\u27e8\hat{\mathit{\mu}}+\hat{\mathit{\theta}}F(\frac{{Y}_{t-d}-r}{\delta}),{\mathbf{\text{X}}}_{t}\u27e9]\}\text{\hspace{1em}}.\end{array}$$

Note that, to simplify the notation, we write $\hat{\mathit{\mu}}$ and $\hat{\mathit{\theta}}$ for $\hat{\mathit{\mu}}$(*n*) and $\hat{\mathit{\theta}}$(*n*), respectively.

Write

$$\begin{array}{rl}{\hat{\mathrm{\Lambda}}}_{t}^{1}& :=\prod _{u=1}^{t}{\hat{\lambda}}_{u}^{1}\text{\hspace{1em}},\\ {\hat{\mathrm{\Lambda}}}_{0}^{1}& :=1\text{\hspace{1em}}.\end{array}$$

Then a new probability measure $\hat{P}$^{1} is defined so that the restriction of its Radon-Nikodym derivative $\frac{d{\hat{P}}^{1}}{dP}$ to *𝒴*_{n} is given by:

$$\frac{d{\hat{P}}^{1}}{dP}{|}_{{\mathcal{G}}_{n}}={\hat{\mathrm{\Lambda}}}_{n}^{1}\text{\hspace{1em}}.$$

It is then not difficult to see that under $\hat{P}$^{1}, the sequence defined by:

$$\frac{{Y}_{t}-\u27e8\hat{\mathit{\mu}}+\hat{\mathit{\theta}}F(\frac{{Y}_{t-d}-r}{\delta}),{\mathbf{\text{X}}}_{t}\u27e9}{\u27e8\mathit{\sigma},{\mathbf{\text{X}}}_{t}\u27e9}\text{\hspace{1em}},\phantom{\rule{1em}{0ex}}t=1,2,\dots ,n\text{\hspace{1em}},$$

is a sequence of *N*(0, 1) i.i.d. random variables.

Now

$$\begin{array}{rl}& \mathrm{ln}{\hat{\mathrm{\Lambda}}}_{n}^{1}\\ & =\sum _{t=1}^{n}\frac{1}{2{\u27e8\mathit{\sigma},{\mathbf{\text{X}}}_{t}\u27e9}^{2}}({\u27e8\mathit{\mu}+\mathit{\theta}F(\frac{{Y}_{t-d}-r}{\delta}),{\mathbf{\text{X}}}_{t}\u27e9}^{2}-{\u27e8\hat{\mathit{\mu}}+\hat{\mathit{\theta}}F(\frac{{Y}_{t-d}-r}{\delta}),{\mathbf{\text{X}}}_{t}\u27e9}^{2}\\ & -2{Y}_{t}\u27e8\mathit{\mu}+\mathit{\theta}F(\frac{{Y}_{t-d}-r}{\delta}),{\mathbf{\text{X}}}_{t}\u27e9+2{Y}_{t}\u27e8\hat{\mathit{\mu}}+\hat{\mathit{\theta}}F(\frac{{Y}_{t-d}-r}{\delta}),{\mathbf{\text{X}}}_{t}\u27e9)\\ & =\sum _{t=1}^{n}\sum _{i=1}^{N}\frac{1}{2{\sigma}_{i}^{2}}[({\mu}_{i}+{\theta}_{i}F(\frac{{Y}_{t-d}-r}{\delta}){)}^{2}-({\hat{\mu}}_{i}+{\hat{\theta}}_{i}F(\frac{{Y}_{t-d}-r}{\delta}){)}^{2}\\ & -2{Y}_{t}({\mu}_{i}+{\theta}_{i}F(\frac{{Y}_{t-d}-r}{\delta}))+2{Y}_{t}({\hat{\mu}}_{i}+{\hat{\theta}}_{i}F(\frac{{Y}_{t-d}-r}{\delta}))]\u27e8{\mathbf{\text{X}}}_{t},{\mathbf{\text{e}}}_{i}\u27e9\\ & =\sum _{t=1}^{n}\sum _{i=1}^{N}\frac{1}{2{\sigma}_{i}^{2}}[-({\hat{\mu}}_{i}+{\hat{\theta}}_{i}F(\frac{{Y}_{t-d}-r}{\delta}){)}^{2}+2{Y}_{t}({\hat{\mu}}_{i}+{\hat{\theta}}_{i}F(\frac{{Y}_{t-d}-r}{\delta}))]\u27e8{\mathbf{\text{X}}}_{t},{\mathbf{\text{e}}}_{i}\u27e9\\ & +R(\mathit{\mu},\mathit{\theta},\mathit{\sigma},r,\delta )\text{\hspace{1em}},\end{array}$$

where *R*(*𝝁*, *𝜽*, *𝝈*, *r*, *δ*) does not involve $\hat{\mathit{\mu}}$ and $\hat{\mathit{\theta}}$ and it represents a quantity which may change from line to line.

Consequently,

$$\begin{array}{rl}& \mathrm{ln}{\hat{\mathrm{\Lambda}}}_{n}^{1}\\ & =\sum _{t=1}^{n}\sum _{i=1}^{N}\frac{1}{2{\sigma}_{i}^{2}}[-{\hat{\mu}}_{i}^{2}-{\hat{\theta}}_{i}^{2}{F}^{2}(\frac{{Y}_{t-d}-r}{\delta})-2{\hat{\mu}}_{i}{\hat{\theta}}_{i}F(\frac{{Y}_{t-d}-r}{\delta})\\ & +2{\hat{\mu}}_{i}{Y}_{t}+2{\hat{\theta}}_{i}{Y}_{t}F(\frac{{Y}_{t-d}-r}{\delta})]\u27e8{\mathbf{\text{X}}}_{t},{\mathbf{\text{e}}}_{i}\u27e9+R(\mathit{\mu},\mathit{\theta},\mathit{\sigma},r,\delta )\text{\hspace{1em}}.\end{array}$$

Define the following functions:

$$\begin{array}{rl}{f}_{d+1}^{1}({Y}_{t},{Y}_{t-1},\dots ,{Y}_{t-d})& :={Y}_{t}\text{\hspace{1em}},\\ {f}_{d+1}^{2}({Y}_{t},{Y}_{t-1},\dots ,{Y}_{t-d})& :=F(\frac{{Y}_{t-d}-r}{\delta})\text{\hspace{1em}},\\ {f}_{d+1}^{3}({Y}_{t},{Y}_{t-1},\dots ,{Y}_{t-d})& :={Y}_{t}F(\frac{{Y}_{t-d}-r}{\delta})\text{\hspace{1em}},\\ {f}_{d+1}^{4}({Y}_{t},{Y}_{t-1},\dots ,{Y}_{t-d})& :={F}^{2}(\frac{{Y}_{t-d}-r}{\delta})\text{\hspace{1em}},\end{array}$$

Consider the following functionals, for *i* = 1, 2, … , *N* and *j* = 1, 2, 3, 4:

$${L}_{d+1,n}^{i}({f}_{d+1}^{j}):=\sum _{t=1}^{n}{f}_{d+1}^{j}({Y}_{t},{Y}_{t-1},\dots ,{Y}_{t-d})\u27e8{\mathbf{\text{X}}}_{t},{\mathbf{\text{e}}}_{i}\u27e9\text{\hspace{1em}}.$$

Then

$$\begin{array}{rl}& \mathrm{ln}{\hat{\mathrm{\Lambda}}}_{n}^{1}\\ & =\sum _{i=1}^{N}\frac{1}{2{\sigma}_{i}^{2}}[-{\hat{\mu}}_{i}^{2}{O}_{n}^{i}-{\hat{\theta}}_{i}^{2}{L}_{d+1,n}^{i}({f}_{d+1}^{4})-2{\hat{\mu}}_{i}{\hat{\theta}}_{i}{L}_{d+1,n}^{i}({f}_{d+1}^{2})+2{\hat{\mu}}_{i}{L}_{d+1,n}^{i}({f}_{d+1}^{1})\\ & +2{\hat{\theta}}_{i}{L}_{d+1,n}^{i}({f}_{d+1}^{3})]+R(\mathit{\mu},\mathit{\theta},\mathit{\sigma},r,\delta )\text{\hspace{1em}}.\end{array}$$

Again *R*(*𝝁*, *𝜽*, *𝝈*, *r*, *δ*) is a quantity which does not depend on $\hat{\mu}$_{i} or $\hat{\theta}$_{i} and can change from line to line.

Conditioning on *𝒴*_{n} under *P* gives:

$$\begin{array}{rl}& {\textstyle \text{E}}[\mathrm{ln}{\hat{\mathrm{\Lambda}}}_{n}^{1}|{\mathcal{Y}}_{n}]\\ & =\sum _{i=1}^{N}\frac{1}{2{\sigma}_{i}^{2}}[-{\hat{\mu}}_{i}^{2}{\hat{O}}_{n}^{i}-{\hat{\theta}}_{i}^{2}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{4})-2{\hat{\mu}}_{i}{\hat{\theta}}_{i}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{2})+2{\hat{\mu}}_{i}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{1})\\ & +2{\hat{\theta}}_{i}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{3})]+R(\mathit{\mu},\mathit{\theta},\mathit{\sigma},r,\delta )\text{\hspace{1em}}.\end{array}$$

Differentiating $\text{E}}[\mathrm{ln}{\hat{\mathrm{\Lambda}}}_{n}^{1}|{\mathcal{Y}}_{n}]$ with respect to $\hat{\mu}$_{i} and setting the derivative equal to zero gives:

$${\hat{\mu}}_{i}{\hat{O}}_{n}^{i}+{\hat{\theta}}_{i}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{2})={\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{1}).$$

Solving for $\hat{\mu}$_{i} then gives:

$${\hat{\mu}}_{i}=\frac{{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{1})-{\hat{\theta}}_{i}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{2})}{{\hat{O}}_{n}^{i}}=\frac{\sigma ({L}_{d+1,n}^{i}({f}_{d+1}^{1}))-{\hat{\theta}}_{i}\sigma ({L}_{d+1,n}^{i}({f}_{d+1}^{2}))}{\sigma ({O}_{n}^{i})},$$(6)

Note that the formula for $\hat{\mu}$_{i} depends on $\hat{\theta}$_{i}.

Differentiating $\text{E}}[\mathrm{ln}{\hat{\mathrm{\Lambda}}}_{n}^{1}|{\mathcal{Y}}_{n}]$ with respect to $\hat{\theta}$_{i} and setting the derivative equal to zero gives:

$${\hat{\theta}}_{i}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{4})+{\hat{\mu}}_{i}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{2})={L}_{d+1,n}^{i}({f}_{d+1}^{3}).$$

and this gives

$${\hat{\theta}}_{i}=\frac{{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{3})-{\hat{\mu}}_{i}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{2})}{{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{4})}=\frac{\sigma ({L}_{d+1,n}^{i}({f}_{d+1}^{3}))-{\hat{\mu}}_{i}\sigma ({L}_{d+1,n}^{i}({f}_{d+1}^{2}))}{\sigma ({L}_{d+1,n}^{i}({f}_{d+1}^{4}))}\text{\hspace{1em}},$$(7)

where the exact recursive formulae for evaluating $\sigma ({L}_{d+1,n}^{i}({f}_{d+1}^{j}))$ , *j* =1, 2, 3, 4, and $\sigma ({O}_{n}^{i})$ are given in Section 3, say Theorem 3.4 and Theorem 3.5. Note that the formula for $\hat{\theta}$_{i} depends on $\hat{\mu}$_{i}.

Consider now the set of parameters *σ*_{i}, *i* = 1, 2, … , *N*. To change the parameters from *σ*_{i} to $\hat{\sigma}$_{i}(*n*), *i* = 1, 2, … , *N*, while keeping (*𝝁*, *𝜽*, *r*, *δ*) fixed, we must consider factors, (*t* = 1, 2, ⋯ , *n*):

$${\hat{\lambda}}_{t}^{2}:=\frac{\u27e8\mathit{\sigma},{\mathbf{\text{X}}}_{t}\u27e9}{\u27e8\hat{\mathit{\sigma}},{\mathbf{\text{X}}}_{t}\u27e9}\frac{\mathrm{exp}[-\frac{1}{2{\u27e8\hat{\mathit{\sigma}},{\mathbf{X}}_{t}\u27e9}^{2}}({Y}_{t}-\u27e8\mathit{\mu}+\mathit{\theta}F(\frac{{Y}_{t-d}-r}{\delta}),{\mathbf{\text{X}}}_{t}\u27e9{)}^{2}]}{\mathrm{exp}[-\frac{1}{2{\u27e8\mathit{\sigma},{\mathbf{\text{X}}}_{t}\u27e9}^{2}}({Y}_{t}-\u27e8\mathit{\mu}+\mathit{\theta}F(\frac{{Y}_{t-d}-r}{\delta}),{\mathbf{\text{X}}}_{t}\u27e9){]}^{2}}\text{\hspace{1em}}.$$

Here we assume that $\hat{\sigma}$_{i}(*n*) > 0.

Again, we write

$$\begin{array}{rl}{\hat{\mathrm{\Lambda}}}_{t}^{2}& :=\prod _{u=1}^{t}{\hat{\lambda}}_{u}^{2}\text{\hspace{1em}},\\ {\hat{\mathrm{\Lambda}}}_{0}^{2}& :=1\text{\hspace{1em}}.\end{array}$$

A new probability measure $\hat{P}$^{2} can then be defined so that the restriction of its Radon-Nikodym derivative $\frac{d{\hat{P}}^{2}}{dP}$ to *𝒴*_{n} is given by:

$$\frac{d{\hat{P}}^{2}}{dP}{|}_{{\mathcal{G}}_{n}}={\hat{\mathrm{\Lambda}}}_{n}^{2}\text{\hspace{1em}}.$$

Now,

$$\begin{array}{rl}\mathrm{ln}{\hat{\mathrm{\Lambda}}}_{n}^{2}& =\sum _{t=1}^{n}[-\mathrm{ln}\u27e8\hat{\mathit{\sigma}},{\mathbf{\text{X}}}_{t}\u27e9-\frac{1}{2{\u27e8\hat{\mathit{\sigma}},{\mathbf{\text{X}}}_{t}\u27e9}^{2}}({Y}_{t}-\u27e8\mathit{\mu}+\mathit{\theta}F(\frac{{Y}_{t-d}-r}{\delta}),{\mathbf{X}}_{t}\u27e9{)}^{2}]\\ & +R(\mathit{\mu},\mathit{\theta},\mathit{\sigma},r,\delta )\\ & =\sum _{t=1}^{n}\sum _{i=1}^{N}[-\mathrm{ln}{\hat{\sigma}}_{i}-\frac{1}{2{\hat{\sigma}}_{i}^{2}}({Y}_{t}-{\mu}_{i}-{\theta}_{i}F(\frac{{Y}_{t-d}-r}{\delta}){)}^{2}]\u27e8{\mathbf{\text{X}}}_{t},{\mathbf{\text{e}}}_{i}\u27e9\\ & +R(\mathit{\mu},\mathit{\theta},\mathit{\sigma},r,\delta )\\ & =\sum _{t=1}^{n}\sum _{i=1}^{N}[-\mathrm{ln}{\hat{\sigma}}_{i}-\frac{1}{2{\hat{\sigma}}_{i}^{2}}({Y}_{t}^{2}-2{Y}_{t}{\mu}_{i}-2{\theta}_{i}{Y}_{t}F(\frac{{Y}_{t-d}-r}{\delta})+{\mu}_{i}^{2}\\ & +{\theta}_{i}^{2}{F}^{2}(\frac{{Y}_{t-d}-r}{\delta})+2{\mu}_{i}{\theta}_{i}F(\frac{{Y}_{t-d}-r}{\delta}){)}^{2}]\u27e8{\mathbf{\text{X}}}_{t},{\mathbf{\text{e}}}_{i}\u27e9\\ & +R(\mathit{\mu},\mathit{\theta},\mathit{\sigma},r,\delta )\text{\hspace{1em}}.\end{array}$$

Define

$${f}_{d+1}^{5}({Y}_{t},{Y}_{t-1},\cdots ,{Y}_{t-d}):={Y}_{t}^{2}\text{\hspace{1em}},$$

and

$${L}_{d+1,n}^{i}({f}_{d+1}^{5})=\sum _{t=1}^{n}{f}_{d+1}^{5}({Y}_{t},{Y}_{t-1},\cdots ,{Y}_{t-d})\u27e8{\mathbf{\text{X}}}_{t},{\mathbf{\text{e}}}_{i}\u27e9\text{\hspace{1em}}.$$

Again conditioning on *𝒴*_{n} under *P* gives:

$$\begin{array}{rl}& {\textstyle \text{E}}[\mathrm{ln}{\hat{\mathrm{\Lambda}}}_{n}^{2}|{\mathcal{Y}}_{n}]\\ & =\sum _{i=1}^{N}[-\mathrm{ln}{\hat{\sigma}}_{i}{\hat{O}}_{n}^{i}-\frac{1}{2{\hat{\sigma}}_{i}^{2}}({\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{5})-2{\mu}_{i}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{1})-2{\theta}_{i}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{3})+{\mu}_{i}^{2}{\hat{O}}_{n}^{i}\\ & +{\theta}_{i}^{2}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{4})+2{\mu}_{i}{\theta}_{i}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{2}))]+R(\mathit{\mu},\mathit{\theta},\mathit{\sigma},r,\delta )\text{\hspace{1em}}.\end{array}$$

Differentiating $\text{E}}[\mathrm{ln}{\hat{\mathrm{\Lambda}}}_{n}^{2}|{\mathcal{Y}}_{n}]$ with respect to $\hat{\sigma}$_{i} and setting the derivative equal to zero gives:

$$\begin{array}{rl}{\hat{\sigma}}_{i}^{2}& =\frac{1}{{\hat{O}}_{n}^{i}}({\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{5})-2{\mu}_{i}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{1})-2{\theta}_{i}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{3})+{\mu}_{i}^{2}{\hat{O}}_{n}^{i}+{\theta}_{i}^{2}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{4})\\ & +2{\mu}_{i}{\theta}_{i}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{2}))\\ & =\frac{1}{\sigma ({O}_{n}^{i})}(\sigma ({L}_{d+1,n}^{i}({f}_{d+1}^{5}))-2{\mu}_{i}\sigma ({L}_{d+1,n}^{i}({f}_{d+1}^{1}))-2{\theta}_{i}\sigma ({L}_{d+1,n}^{i}({f}_{d+1}^{3}))+{\mu}_{i}^{2}\sigma ({O}_{n}^{i})\\ & +{\theta}_{i}^{2}\sigma ({L}_{d+1,n}^{i}({f}_{d+1}^{4}))+2{\mu}_{i}{\theta}_{i}\sigma ({L}_{d+1,n}^{i}({f}_{d+1}^{2}))).\end{array}$$(8)

By construction, these estimates are greater than zero. Note that the formula for ${\hat{\sigma}}_{i}^{2}$ depends on *μ*_{i} and *θ*_{i}.

Finally, we consider the set of parameters (*r*, *δ*). In this case, we consider a specific form of the smooth transition function *F*. For illustration, we consider the “normal” smooth transition function given in Example 1. That is,

$$F(x)=\mathrm{\Phi}(x)={\int}_{-\mathrm{\infty}}^{x}\frac{1}{\sqrt{2\pi}}{e}^{-\frac{1}{2}{y}^{2}}dy\text{\hspace{1em}}.$$

In this case, we consider a Laplace series expansion for Φ(*x*) as follows:

$$\begin{array}{rl}\mathrm{\Phi}(x)& =\frac{1}{2}+\frac{1}{\sqrt{2\pi}}\sum _{n=0}^{\mathrm{\infty}}(-1{)}^{n}\frac{{x}^{2n+1}}{n!{2}^{n}(2n+1)}\\ & =\frac{1}{2}+\frac{1}{\sqrt{2\pi}}(x-\frac{{x}^{3}}{6}+\frac{{x}^{5}}{40}-\cdots )\\ & =\frac{1}{2}+\frac{1}{\sqrt{2\pi}}x+R({x}^{3})\text{\hspace{1em}}.\end{array}$$

To change the parameters (*r*, *δ*) to ($\hat{r}$(*n*), $\hat{\delta}$(*n*)) while keeping other parameters (*𝝁*, *𝜽*, *𝝈*) fixed, we consider the following factors, (*t* = 1, 2, ⋯ , *n*):

$$\begin{array}{rl}{\hat{\lambda}}_{t}^{3}& :=\mathrm{exp}\{\frac{1}{2{\u27e8\mathit{\sigma},{\mathbf{\text{X}}}_{t}\u27e9}^{2}}[{\u27e8\mathit{\mu}+\mathit{\theta}\mathrm{\Phi}(\frac{{Y}_{t-d}-r}{\delta}),{\mathbf{\text{X}}}_{t}\u27e9}^{2}-{\u27e8\mathit{\mu}+\mathit{\theta}\mathrm{\Phi}(\frac{{Y}_{t-d}-\hat{r}}{\hat{\delta}}),{\mathbf{\text{X}}}_{t}\u27e9}^{2}\\ & -2{Y}_{t}\u27e8\mathit{\mu}+\mathit{\theta}\mathrm{\Phi}(\frac{{Y}_{t-d}-r}{\delta}),{\mathbf{\text{X}}}_{t}\u27e9+2{Y}_{t}\u27e8\mathit{\mu}+\mathit{\theta}\mathrm{\Phi}(\frac{{Y}_{t-d}-\hat{r}}{\hat{\delta}}),{\mathbf{\text{X}}}_{t}\u27e9]\}\text{\hspace{1em}}.\end{array}$$

Write

$$\begin{array}{rl}{\hat{\mathrm{\Lambda}}}_{t}^{3}& :=\prod _{u=1}^{t}{\hat{\lambda}}_{u}^{3}\text{\hspace{1em}},\\ {\hat{\mathrm{\Lambda}}}_{0}^{3}& :=1\text{\hspace{1em}}.\end{array}$$

Again a new probability measure $\hat{P}$^{3} is defined so that the restriction of its Radon-Nikodym derivative $\frac{d{\hat{P}}^{3}}{dP}$ to *𝒴*_{n} is given by:

$$\frac{d{\hat{P}}^{3}}{dP}{|}_{{\mathcal{G}}_{n}}={\hat{\mathrm{\Lambda}}}_{n}^{3}\text{\hspace{1em}}.$$

Consequently,

$$\begin{array}{rl}& \mathrm{ln}{\hat{\mathrm{\Lambda}}}_{n}^{3}\\ & =\sum _{i=1}^{N}\sum _{i=1}^{N}\frac{1}{2{\sigma}_{i}^{2}}[({\mu}_{i}+{\theta}_{i}\mathrm{\Phi}(\frac{{Y}_{t-d}-r}{\delta}){)}^{2}-({\mu}_{i}+{\theta}_{i}\mathrm{\Phi}(\frac{{Y}_{t-d}-\hat{r}}{\hat{\delta}}){)}^{2}\\ & -2{Y}_{t}({\mu}_{i}+{\theta}_{i}\mathrm{\Phi}(\frac{{Y}_{t-d}-r}{\delta}))+2{Y}_{t}({\mu}_{i}+{\theta}_{i}\mathrm{\Phi}(\frac{{Y}_{t-d}-\hat{r}}{\hat{\delta}}))]\u27e8{\mathbf{\text{X}}}_{t},{\mathbf{\text{e}}}_{i}\u27e9\\ & =\sum _{t=1}^{n}\sum _{i=1}^{N}\frac{1}{2{\sigma}_{i}^{2}}[-({\mu}_{i}+{\theta}_{i}\mathrm{\Phi}(\frac{{Y}_{t-d}-\hat{r}}{\hat{\delta}}){)}^{2}+2{Y}_{t}{\theta}_{i}\mathrm{\Phi}(\frac{{Y}_{t-d}-\hat{r}}{\hat{\delta}})]\u27e8{\mathbf{\text{X}}}_{t},{\mathbf{\text{e}}}_{i}\u27e9\\ & +R(\mathit{\mu},\mathit{\theta},\mathit{\sigma},r,\delta )\\ & =\sum _{t=1}^{n}\sum _{i=1}^{N}\frac{1}{2{\sigma}_{i}^{2}}[-{\theta}_{i}^{2}{\mathrm{\Phi}}^{2}(\frac{{Y}_{t-d}-\hat{r}}{\hat{\delta}})+2({Y}_{t}-{\mu}_{i}){\theta}_{i}\mathrm{\Phi}(\frac{{Y}_{t-d}-\hat{r}}{\hat{\delta}})]\u27e8{\mathbf{\text{X}}}_{t},{\mathbf{\text{e}}}_{i}\u27e9\\ & +R(\mathit{\mu},\mathit{\theta},\mathit{\sigma},r,\delta )\\ & \approx \sum _{t=1}^{n}\sum _{i=1}^{N}\frac{1}{2{\sigma}_{i}^{2}}\{-{\theta}_{i}^{2}[\frac{1}{4}+\frac{1}{\sqrt{2\pi}}(\frac{{Y}_{t-d}-\hat{r}}{\hat{\delta}})+\frac{1}{2\pi}(\frac{{Y}_{t-d}-\hat{r}}{\hat{\delta}}{)}^{2}]\\ & +2({Y}_{t}-{\mu}_{i}){\theta}_{i}[\frac{1}{2}+\frac{1}{\sqrt{2\pi}}(\frac{{Y}_{t-d}-\hat{r}}{\hat{\delta}})]\}\u27e8{\mathbf{\text{X}}}_{t},{\mathbf{\text{e}}}_{i}\u27e9\\ & +R(\mathit{\mu},\mathit{\theta},\mathit{\sigma},r,\delta )\\ & =\sum _{t=1}^{n}\sum _{i=1}^{N}\frac{1}{2{\sigma}_{i}^{2}}[(-\frac{1}{4}{\theta}_{i}^{2}+\frac{{\theta}_{i}^{2}\hat{r}}{\sqrt{2\pi}\hat{\delta}}-\frac{{\theta}_{i}^{2}{\hat{r}}^{2}}{2\pi {\hat{\delta}}^{2}}-{\mu}_{i}{\theta}_{i}+\frac{2{\mu}_{i}{\theta}_{i}\hat{r}}{\sqrt{2\pi}\hat{\delta}})\\ & +(\frac{{\theta}_{i}^{2}\hat{r}}{\pi {\hat{\delta}}^{2}}-\frac{{\theta}_{i}^{2}}{\sqrt{2\pi}\hat{\delta}}-\frac{2{\mu}_{i}{\theta}_{i}}{\sqrt{2\pi}\hat{\delta}}){Y}_{t-d}+({\theta}_{i}-\frac{2\hat{r}{\theta}_{i}}{\sqrt{2\pi}\hat{\delta}}){Y}_{t}+\frac{2{Y}_{t}{Y}_{t-d}{\theta}_{i}}{\sqrt{2\pi}\hat{\delta}}\\ & -\frac{{\theta}_{i}^{2}}{2\pi {\hat{\delta}}^{2}}{Y}_{t-d}^{2}]\u27e8{\mathbf{\text{X}}}_{t},{\mathbf{\text{e}}}_{i}\u27e9+R(\mathit{\mu},\mathit{\theta},\mathit{\sigma},r,\delta )\text{\hspace{1em}},\end{array}$$

where *R*(*𝝁*, *𝜽*, *𝝈*, *r*, *δ*) does not involve $\hat{r}$ and $\hat{\delta}$.

Write

$$\begin{array}{rl}{f}_{d+1}^{6}({Y}_{t},{Y}_{t-1},\cdots ,{Y}_{t-d})& ={Y}_{t-d}\text{\hspace{1em}},\\ {f}_{d+1}^{7}({Y}_{t},{Y}_{t-1},\cdots ,{Y}_{t-d})& ={Y}_{t-d}^{2}\text{\hspace{1em}},\\ {f}_{d+1}^{8}({Y}_{t},{Y}_{t-1},\cdots ,{Y}_{t-d})& ={Y}_{t}{Y}_{t-d}\text{\hspace{1em}},\end{array}$$

and for *j* = 6, 7, 8,

$${L}_{d+1,n}^{i}({f}_{d+1}^{j})=\sum _{t=1}^{n}{f}_{d+1}^{j}({Y}_{t},{Y}_{t-1},\cdots ,{Y}_{t-d})\u27e8{\mathbf{\text{X}}}_{t},{\mathbf{\text{e}}}_{i}\u27e9\text{\hspace{1em}}.$$

Consequently,

$$\begin{array}{rl}& \mathrm{ln}{\hat{\mathrm{\Lambda}}}_{n}^{3}\\ & =\sum _{i=1}^{N}\frac{1}{2{\sigma}_{i}^{2}}[(-\frac{1}{4}{\theta}_{i}^{2}+\frac{{\theta}_{i}^{2}\hat{r}}{\sqrt{2\pi}\hat{\delta}}-\frac{{\theta}_{i}^{2}{\hat{r}}^{2}}{2\pi {\hat{\delta}}^{2}}-{\mu}_{i}{\theta}_{i}+\frac{2{\mu}_{i}{\theta}_{i}\hat{r}}{\sqrt{2\pi}\hat{\delta}}){O}_{n}^{i}\\ & +(\frac{{\theta}_{i}^{2}\hat{r}}{\pi {\hat{\delta}}^{2}}-\frac{{\theta}_{i}^{2}}{\sqrt{2\pi}\hat{\delta}}-\frac{2{\mu}_{i}{\theta}_{i}}{\sqrt{2\pi}\hat{\delta}}){L}_{d+1,n}^{i}({f}_{d+1}^{6})+({\theta}_{i}-\frac{2\hat{r}{\theta}_{i}}{\sqrt{2\pi}\hat{\delta}}){L}_{d+1,n}^{i}({f}_{d+1}^{1})\\ & +\frac{2{\theta}_{i}}{\sqrt{2\pi}\hat{\delta}}{L}_{d+1,n}^{i}({f}_{d+1}^{8})-\frac{{\theta}_{i}^{2}}{2\pi {\hat{\delta}}^{2}}{L}_{d+1,n}({f}_{d+1}^{7})]+R(\mathit{\mu},\mathit{\theta},\mathit{\sigma},r,\delta )\text{\hspace{1em}}.\end{array}$$

Conditioning on *𝒴*_{n} under *P* then gives:

$$\begin{array}{rl}& {\textstyle \text{E}}[\mathrm{ln}{\hat{\mathrm{\Lambda}}}_{n}^{3}|{\mathcal{Y}}_{n}]\\ & =\sum _{i=1}^{N}\frac{1}{2{\sigma}_{i}^{2}}[(-\frac{1}{4}{\theta}_{i}^{2}+\frac{{\theta}_{i}^{2}\hat{r}}{\sqrt{2\pi}\hat{\delta}}-\frac{{\theta}_{i}^{2}{\hat{r}}^{2}}{2\pi {\hat{\delta}}^{2}}-{\mu}_{i}{\theta}_{i}+\frac{2{\mu}_{i}{\theta}_{i}\hat{r}}{\sqrt{2\pi}\hat{\delta}}){\hat{O}}_{n}^{i}\\ & +(\frac{{\theta}_{i}^{2}\hat{r}}{\pi {\hat{\delta}}^{2}}-\frac{{\theta}_{i}^{2}}{\sqrt{2\pi}\hat{\delta}}-\frac{2{\mu}_{i}{\theta}_{i}}{\sqrt{2\pi}\hat{\delta}}){\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{6})+({\theta}_{i}-\frac{2\hat{r}{\theta}_{i}}{\sqrt{2\pi}\hat{\delta}}){\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{1})\\ & +\frac{2{\theta}_{i}}{\sqrt{2\pi}\hat{\delta}}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{8})-\frac{{\theta}_{i}^{2}}{2\pi {\hat{\delta}}^{2}}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{7})]+R(\mathit{\mu},\mathit{\theta},\mathit{\sigma},r,\delta )\text{\hspace{1em}}.\end{array}$$

Differentiating with respect to $\hat{r}$ and setting the derivative equal to zero gives:

$$\begin{array}{rl}\hat{r}& =\frac{\sum _{i=1}^{N}\frac{1}{2{\sigma}_{i}^{2}}[[\frac{{\theta}_{i}^{2}}{\sqrt{2\pi}\hat{\delta}}+\frac{2{\mu}_{i}{\theta}_{i}}{\sqrt{2\pi}\hat{\delta}}]{\hat{O}}_{n}^{i}+\frac{{\theta}_{i}^{2}}{\pi {\hat{\delta}}^{2}}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{6})-\frac{2{\theta}_{i}}{\sqrt{2\pi}\hat{\delta}}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{1})]}{\sum _{i=1}^{N}\frac{1}{2{\sigma}_{i}^{2}}\frac{{\theta}_{i}^{2}}{\pi {\hat{\delta}}^{2}}{\hat{O}}_{n}^{i}}\\ & =\frac{\sum _{i=1}^{N}\frac{1}{2{\sigma}_{i}^{2}}[[\frac{{\theta}_{i}^{2}}{\sqrt{2\pi}\hat{\delta}}+\frac{2{\mu}_{i}{\theta}_{i}}{\sqrt{2\pi}\hat{\delta}}]\sigma ({O}_{n}^{i})+\frac{{\theta}_{i}^{2}}{\pi {\hat{\delta}}^{2}}\sigma ({L}_{d+1,n}^{i}({f}_{d+1}^{6}))-\frac{2{\theta}_{i}}{\sqrt{2\pi}\hat{\delta}}\sigma ({L}_{d+1,n}^{i}({f}_{d+1}^{1}))]}{\sum _{i=1}^{N}\frac{1}{2{\sigma}_{i}^{2}}\frac{{\theta}_{i}^{2}}{\pi {\hat{\delta}}^{2}}\sigma ({O}_{n}^{i})}.\end{array}$$(9)

Note that the formula for $\hat{r}$ depends on $\hat{\delta}$, *θ*_{i}, *μ*_{i} and *σ*_{i}.

Differentiating with respect to $\hat{\delta}$ and setting the derivative equal to zero gives:

$$\begin{array}{rl}\hat{\delta}& =\frac{\sum _{i=1}^{N}\frac{{\theta}_{i}^{2}}{2\pi {\sigma}_{i}^{2}}[-{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{7})+2{\hat{r}\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{6})-{\hat{r}}^{2}{\hat{O}}_{n}^{i}]}{\sum _{i=1}^{N}\frac{1}{2{\sigma}_{i}^{2}}[{a}_{i}{\hat{O}}_{n}^{i}+{b}_{i}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{6})+{c}_{i}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{1})+{d}_{i}{\hat{L}}_{d+1,n}^{i}({f}_{d+1}^{8})]}\\ & =\frac{\sum _{i=1}^{N}\frac{{\theta}_{i}^{2}}{2\pi {\sigma}_{i}^{2}}[-\sigma ({L}_{d+1,n}^{i}({f}_{d+1}^{7}))+2\hat{r}\sigma ({L}_{d+1,n}^{i}({f}_{d+1}^{6}))-{\hat{r}}^{2}\sigma ({O}_{n}^{i})]}{\sum _{i=1}^{N}\frac{1}{2{\sigma}_{i}^{2}}[{a}_{i}\sigma ({O}_{n}^{i})+{b}_{i}\sigma ({L}_{d+1,n}^{i}({f}_{d+1}^{6}))+{c}_{i}\sigma ({L}_{d+1,n}^{i}({f}_{d+1}^{1}))+{d}_{i}\sigma ({L}_{d+1,n}^{i}({f}_{d+1}^{8}))]},\end{array}$$(10)

where

$$\begin{array}{rl}{a}_{i}& :=-\frac{{\theta}_{i}^{2}\hat{r}}{\sqrt{2\pi}}-\frac{2{\mu}_{i}{\theta}_{i}\hat{r}}{\sqrt{2\pi}}\text{\hspace{1em}},\\ {b}_{i}& :=\frac{{\theta}_{i}^{2}}{\sqrt{2\pi}}+\frac{2{\mu}_{i}{\theta}_{i}}{\sqrt{2\pi}}\text{\hspace{1em}},\\ {c}_{i}& :=\frac{2\hat{r}{\theta}_{i}}{\sqrt{2\pi}}\text{\hspace{1em}},\\ {d}_{i}& :=-\frac{2{\theta}_{i}}{\sqrt{2\pi}}\text{\hspace{1em}}.\end{array}$$

Note that these coefficients depend on $\hat{r}$, *μ*_{i}, *θ*_{i} and *σ*_{i}.

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