Transmission of Macroeconomic Shocks to Risk Parameters:
Their uses in Stress Testing

To understand the transmission process of macroeconomic shocks to risk parameters, it is necessary to consider the characteristics and limitations of the agents that determine the dynamic behavior of the risk parameters. The series of risk parameters observed, derive from the relationship established between imperfect agents (in terms of rationality), which interact within an imperfect economic environment (in terms of information). For example, the agents that determine the dynamic behavior of the credit risk parameters are: the financial institution that grants the credits, the clients whom the credit is granted to and the financial regulator. All these agents have relevant limitations of rationality and information.

The imperfection of the economic environment conditions determines a slow processing of information by agents (behavioral or institutional inertia) and at the same time causes an unequal distribution of information among agents (asymmetry of information). The behavioral inertia and information asymmetry cause that macroeconomic shocks dynamically impact the risk parameters, that is, the impacts are propagated in time. The decay in the propagation of the impacts can be fast or slow; when the decay is slow we say that the propagation is of high persistence; and when the decay is fast, then the propagation is low. Thus, in inefficient environments the transmission process is characterized by persistence in the propagation. The nature and intensity of the persistence in the data of a specific portfolio need to be evaluated empirically and corroborated with some coherent economic criteria.

To empirically contrast the persistent nature of the impacts, we propose the use of two simple functions (Response and Diffusion):

The impact measurements presented in (1) and (3) describe the dynamic response of the risk parameter $Y$ to shocks in the macroeconomic variable $X$. In other words, they describe the way risk parameter absorbs the shocks, which is determined by the reaction of the agents to return to equilibrium. A slow decrease of the impact measures implies a high persistence at the absorption of shocks. Through these impact measures, two types of impact can be identified: Impacts with permanent effects and impacts with temporary effects. Shocks with permanent effects impact the stochastic trend of the risk parameter and their effects are measured by $\mathfrak{R}(j)$. Shocks with temporary effects impact the stationary fluctuations around the stochastic trend and their effects are measured by $\widehat{\mathfrak{D}}(j)$. For example, we can empirically identify these two types of impacts with the impact measurements referring to the credit risk data, see Figure 2.

The persistent nature of the shocks is observed in the decay rate of the impacts measured by $\widehat{\mathfrak{R}}(j)$. It is important to mention that due to the aggregate nature of the macroeconomic variables, the type of persistence needs to be contrasted with economic criteria that justify its nature. For example in the case of GDP and IDR, observing their corresponding Response Functions, see Figure 2, and considering that both macroeconomic variables are directly involved in the credit recovery policies and LGD calculation respectively, we can consider that the impact of the shocks of these two macroeconomic variables on the LGD of this portfolio are of low persistence. In the case of the macroeconomic variable Unemployment, its response function shows slow decay in the first lags, besides, it is a variable associated with labor laws and institutional policies that justify the high persistence of shocks in this macroeconomic variable. In this case, $\widehat{\mathfrak{D}}(j)$ not present a persistent pattern, that is, the shocks do not affect the variance of the risk parameter of persistence way, the only effect that the macroeconomic shocks do is to balance around their averages.

Let $X_{t}$ be the value of an macroeconomic, scalar variable $X$ and $Y_{t}$ the value of risk parameter at time $t$. A General Transfer Function Model (GTFM) for the dynamic effect of $X$ on the risk parameter $Y$, consistent with the empirical facts discussed in section (3), is defined as

	$\displaystyle Y_{t}$	$\displaystyle=\mathbf{F}_{t}^{\top}\mathbf{\theta}_{t}+v_{t}$		(7a)
	$\displaystyle\theta_{t}$	$\displaystyle=\mathbf{G}_{t}\theta_{t-1}+\psi_{t}X_{t}+\partial\theta_{t}$		(7b)
	$\displaystyle\psi_{t}$	$\displaystyle=\psi_{t-1}+\partial\psi_{t}$		(7c)

with terms described as follows: $\theta_{t}$ is an $\mathit{n}$-dimensional states vector (unobserved), $\mathbf{F}_{t}$ is a known $\mathit{n}$-vector (design vector), $\mathbf{G}_{t}$ a known transition matrix (evolution matrix) which determines the evolution of the states vector and, $v_{t}$ and $\partial\theta_{t}$ are observation and evolution noise terms. All these terms are precisely as in the standard Dynamic Linear Models (DLM), with the usual independence assumptions for the noise terms hold here. The term $\psi_{t}$ is an $\mathit{n}$-vector of parameters, evolving via addition of a noise term $\partial\psi_{t}$, assumed to be zero-mean normally distributed independently of $v_{t}$ (though not necessarily of $\partial\theta_{t}$).

In accordance with the empirical facts discussed in section (3), the GTFM defined in (7) establishes that $X$ impacts the stochastic trend of $Y$ (7b) and, at the same time, generates stationary fluctuations around that stochastic trend (7a). The parameters that determine the transfer can vary over time, allowing greater flexibility to the transmission process of shocks (7c). This model constitutes a small variation of the first model presented by Harrison and West (1999).

The states vector $\theta_{t}$ carries the effect of current and past values of the $X$ series to $Y_{t}$ in equation (7a); this is formed in (7b) as the sum of a linear function of past effects; $\theta_{t-1}$, and the current effect $\psi_{t}X_{t}$, plus a noise term. Notice that the parameterization (7) is more flexible than that of equation (5), allowing different stochastic interpretations for the effect of $X$ on $Y$.

The general model (7) can be rewritten in the standard DLM as follows. Define a 2$\mathit{n}$-dimensional state parameters vector $\tilde{\theta}_{t}$ by concatenating $\theta_{t}$ and $\psi_{t}$, giving $\tilde{\theta}_{t}=(\theta_{t}^{\top},\psi_{t}^{\top})$. Similarly, extend the $\mathbf{F}_{t}$ vector by concatenating an $\mathit{n}$-vector of zeros, giving a new $\tilde{\mathbf{F}}_{t}$ such that $\tilde{\mathbf{F}}_{t}=(\mathbf{F}_{t},0,\cdots,0)$. For the evolution matrix, define by

$\displaystyle\tilde{\mathbf{G}}_{t}=\begin{bmatrix}\mathbf{G}_{t}&X_{t}\mathbf{I}_{\mathit{n}}\\ \mathbf{0}&\mathbf{I}_{\mathit{n}}\end{bmatrix}$

where $\mathbf{I}_{\mathit{n}}$ is the $\mathit{n}$ x $\mathit{n}$ identity matrix. Finally, let $\mathbf{\omega}_{t}$ be the noise vector defined by $\omega_{t}^{\top}=(\partial\theta_{t}^{\top}+X_{t}\partial\psi_{t}^{\top},\partial\psi_{t}^{\top})$. Then the model (7) can be written as

$$\begin{split}Y_{t}&=\tilde{\mathbf{F}}^{\top}\tilde{\theta}_{t}+v_{t},\\ \tilde{\theta}_{t}&=\tilde{\mathbf{G}}\tilde{\theta}_{t-1}+\mathbf{\omega}_{t}.\end{split}$$

(8)

Thus the GTFM (7) is written in the standard DLM form (8) and the usual inferential analysis corresponding to State Space Models can be applied (see Durbin and Koopman (2012) and West and Harrison (2006)). Note that the generalization to include several macroeconomic variables in the model (7) is trivial.

In this section, we present some Transfer Functions underlying models that are specific cases of the GTFM.

Impact measures discussed in section 3.2 represent the empirical propagation of shocks. We will to use this qualitative information embedded in the impact measures to choose the Transfer Function. The functional form of the theoretical decay of the impacts (Dynamic Multiplier) must be consistent with the empirical propagation of the permanent impacts. Then, a criterion that we employed to choose the functional form of the Dynamic Multiplier of a macroeconomic variable, was that the functional form be consistent with the rapidity of decay of its Response Function $\widehat{\mathfrak{R}}(j)$. For example, if the Response Function $\widehat{\mathfrak{R}}(j)$ of a macroeconomic variable shows a high persistence or a behavior with wave decay, then the functional form proposed for its Dynamic Multiplier has to imitate that pattern, because it condenses the empirical shocks transmission.

It is important, as seen above, to point out that these measures are not used to specify the model with precision, as in the case of the autocorrelation function in the modeling of stationary series. Rather, due to the aggregate nature and the diffuse behavior (random walk) of the macroeconomic variables and the risk parameter, these measures are used to have a general approximation of the functional form of the impact decay that characterizes the transmission process.

Following the Bayesian paradigm, the specification of a model is complete after specifying the prior distribution of all parameters of interest. To complete the specification of the models described, we defined the prior distribution according to the declination pattern of the Response Function.

For the Resilience Parameter $\phi\in(0,1)$, that corresponds to Response functions with monotone decay pattern, we specified a $\mathcal{B}eta(a,b)$ distribution; and for $\phi\in(-1,0)$, that corresponds to Response functions with wave decay pattern, it is specified by setting $\phi=-\phi^{*}$ where $\phi^{*}\sim\mathcal{B}eta(c,d)$. We chose to work with the normal distribution for the regressors parameters, however, we decided to restrict the support according to the Response Function observed at the data, that is, considering the dynamic relationship between the risk parameter and the macroeconomic series.

• •

  If the relation present by the Response Function of the $j$-th macroeconomic variable is positive,

  $\displaystyle\beta_{j}\sim\textrm{Half-Normal}(\mu_{0},\sigma_{0}^{2})\hskip 28.45274pt\textrm{with}\hskip 28.45274pt\beta_{j}\in[0,\infty).$

• •

  If the relation of the Response Function of the $j$-th macroeconomic variable is negative,

  $\displaystyle\beta_{j}\sim\textrm{Half-Normal}(\mu_{0},\sigma_{0}^{2})\hskip 28.45274pt\textrm{with}\hskip 28.45274pt\beta_{j}\in(-\infty,0].$

Finally, the prior distribution for $\sigma_{E}$, $\eta$ and $\sigma_{\phi}$ are specified as,

	$\displaystyle\sigma_{E}$	$\displaystyle\sim\textrm{Inv-Gamma}(p_{1},q_{1})$
	$\displaystyle\sigma_{\phi}$	$\displaystyle\sim\textrm{Inv-Gamma}(p_{2},q_{2})$
	$\displaystyle\eta$	$\displaystyle\sim\textrm{Gamma}(\tau,\varpi).$

There are several approximation methods based on simulations to obtain samples of the posterior distribution. The most widely used is the Markov Chain Monte Carlo Methods (MCMC), described in Robert et al. (2010). Within the MCMC family stand out the algorithms: Metropolis (Metropolis et al., 1953), Metropolis-Hastings (Hastings, 1970), Gibbs Sampling (Gelfand and Smith, 1990) and Hamiltonian Monte Carlo (Neal et al., 2011). Great advances have been made recently with Hamiltonian Monte Carlo algorithms (HMC, see for example Girolami and Calderhead (2011)) for the estimation Bayesian models. The HMC has an advantage over the other algorithms mentioned, since it avoids the random walk behavior, presents smaller correlations structure and converges with less simulations. In this paper we used the HMC to obtain samples of the posterior distribution.

One of the challenges in the modeling process is choosing the model that best represents the data structure and that is parsimonious. A widely used metric, in the case of Bayesian approach, is the Deviance Information Criterion (DIC) proposed by Spiegelhalter et al. (2002), which is defined as,

$\displaystyle DIC=-2\log p\big{(}\mathbf{Y}|\hat{\theta}_{Bayes}\big{)}+2\textrm{p}_{DIC},$

where $\mathbf{Y}=(Y_{1},\dots,Y_{T})$ are the data, $\theta$ is the vector of parameters of the model, $\hat{\theta}_{Bayes}=E[\theta|\mathbf{Y}]$ and $\textrm{p}_{DIC}$ is a penalizing term given by,

$\displaystyle\textrm{p}_{DIC}=2\Big{(}\log p(\mathbf{Y}|\hat{\theta}_{Bayes})-E_{\theta|\mathbf{Y}}\big{[}\log p(\mathbf{Y}|\theta)\big{]}\Big{)}.$

Then, given M simulations from the posterior distribution of $\theta$, $p_{DIC}$ can be approximated as,

$\displaystyle\hat{\textrm{p}}_{DIC}=2\Bigg{(}\log p\big{(}\mathbf{Y}|\hat{\theta}_{Bayes}\big{)}-\frac{1}{M}\sum_{m=1}^{M}\log p\big{(}\mathbf{Y}|\theta^{m}\big{)}\Bigg{)}$

Additionally, in this paper, we used the Watanabe Information Criterion (WAIC) described by Watanabe (2010). The WAIC does not depend on Fisher’s asymptotic theory; therefore, it is not responsible for later displacement to a single point, as well as it is an alternative to more advanced models (see Watanabe (2013), Vehtari et al. (2015) and Vehtari et al. (2017)). It is defined as,

$$\textrm{elpd}_{WAIC}=\textrm{lpd}-\textrm{p}_{WAIC}$$

(18)

where $\textrm{p}_{WAIC}=\sum_{i=1}^{T}V_{\theta|\mathbf{Y}}\big{(}\log p(Y_{i}|\theta)\big{)}$ penalizes for the effective number of parameters and $\textrm{lpd}=\sum_{i=1}^{T}\log p(Y_{i}|\mathbf{Y})$ is the log pointwise predictive density, as defined in Watanabe (2010), where,

$\displaystyle\log p(Y_{i}|\mathbf{Y})=\log\int_{\Theta}p(Y_{i}|\theta)p(\theta|\mathbf{Y})d\theta.$

Similar to DIC, given M simulated from posterior distribution, the two terms in (18) are estimated as,

lpd

$\displaystyle=\sum_{i=1}^{T}\log\Bigg{(}\frac{1}{M}\sum_{m=1}^{M}p(Y_{i}|\theta^{m})\Bigg{)}$

and

$\displaystyle\textrm{p}_{WAIC}$

$\displaystyle=\sum_{i=1}^{T}V_{m=1}^{M}\Big{(}\log p(Y_{i}|\theta)\Big{)}.$

where $V_{m=1}^{M}$ denotes the sample variances of log $p(Y_{i}|\theta^{(1)}),\ldots,p(Y_{i}|\theta^{(M)})$. Thus, WAIC is defined as,

$\displaystyle WAIC=-2\textrm{elpd}_{WAIC}.$

The log pointwise predictive density can also be estimated using approximate leave-one-out cross-validation (LOOIC) as,

$\displaystyle\textrm{lpd}_{LOOIC}$

$\displaystyle=\sum_{i=1}^{T}\log p(Y_{i}|Y_{-i})=\sum_{i=1}^{T}\log\int_{\Theta}p(Y_{i}|\theta)p(\theta|Y_{-i})d\theta,$

where $Y_{-i}$ denotes the data vector with the $i$th observation deleted. Vehtari et al. (2015) introduced an efficient approach to compute LOOIC using Pareto-Smoothed Importance Sampling (PSIS). The lower the value of the selection criteria, the better the model is considered. It is worth mentioning that the estimates for WAIC and LOOIC are obtained as the sum of independent components, so it is possible to calculate the approximate standard errors for the estimated predictive errors.

Without loss of generality, we will assume a univariate case (only a macroeconomic variable). We want to apply our model analysis to a new set of data, where we observed the new covariate $X_{T+1}$ corresponding to one of the stress scenarios, and we wish to predict the corresponding outcome $Y_{T+1}$. From a Bayesian perspective, this process can be performed considering the posterior predictive distribution.

$$p(Y_{T+1}|D_{T})=\int p(Y_{T+1}|\gamma,D_{T})p(\gamma|Y_{T},D_{T})d\gamma,$$

(19)

where $D_{T}$ is all the information until the instant $T$ and $\gamma=(E_{T+1},\sigma_{E})^{\top}$. However, since there is no analytical determination of this distribution, we adopted the following procedure.

	$\displaystyle\phi_{T+1}^{(m)}$	$\displaystyle\sim\mathcal{N}\Big{(}\phi_{T}^{(m)},\sigma_{\phi}^{2(m)}\Big{)}$
	$\displaystyle E_{T+1}^{(m)}$	$\displaystyle\sim\mathcal{N}\Big{(}\phi_{T}^{(m)}E_{T}^{(m)}+X_{T+1}^{\top}\beta^{(m)},\sigma_{E}^{2(m)}\Big{)}$
	$\displaystyle Y_{T+1}^{(m)}$	$\displaystyle\sim\mathcal{N}\Big{(}E_{T+1}^{(m)},\sigma_{v}^{2(m)}\Big{)},$

where $m=1,\ldots,M$ are MCMC samples of the posterior distribution. Then, $\Big{(}Y_{t+1}^{(1)},\ldots,Y_{t+1}^{(M)}\Big{)}$ are an i.i.d. sample from $p(Y_{T+1}|D_{T})$. The process can be repeated until $h$-steps-ahead, considering $X_{T+2},\ldots,X_{T+h}$, to obtain the forecast to $Y_{T+2},\ldots,Y_{T+h}$.

In this section we present the results of the two steps of the simulation study.

According the Table 1, the hit rate of the response function is satisfactory. With this we can be use the response function it to evaluate the decay of the shocks of the macroeconomic variables. Note that when the value of one of the betas is double the other, its decay hit rate is higher than the which one with of the lowest weight, but yet is satisfactory to evaluate the decay.

To the step two we evaluate the estimations considering two configurations and compare the Response Function Mean of the estimates with the Expected decay from of each variables.

The configuration 1 we considered is: $\alpha=3$, $\phi=0.4$, $\beta_{1}=-0.4$, $\beta_{2}=0.4$ and $\sigma_{v}=0.1$.

As present in the Table 2, the estimates obtained are close to the true values of the parameters. Also we observated that the standard deviation of $\alpha$ and the difference in scale of the $\sigma$ parameter is slightly higher than the others presented.

In the Figure 6 we presented the Reponse Function Mean of the $M=500$ simulations and present the Expected decay of the variables used to construct the data series. As shown the average empirical functional form of decay is in accordance with the expected form of theoretical decay of the variables.

The configuration 1 we considered is: $\alpha=3$, $\phi=0.7$, $\beta_{1}=-0.8$, $\beta_{2}=0.4$ and $\sigma_{v}=0.01$.

Like in configuration 1, the the estimates obtained in configuration 2, showing in the Table 3, are close to the true values of the parameters. We also note that the standard deviation of all estimates has decreased considerably. We believe it is due to the resilience parameter.

As observed in both configuration the functional form of average decay that is the qualitative information that will be used to build the priori, present in Figures 6 and 7 , are in agreement with the form of theoretical decay already defined by construction of the data series. It is worth mentioning that, since Bayesian inference was used, the estimates presented are based on the mean and standard deviation of a posteriori means.

The Model I can be represented as,

	$\displaystyle Y_{t}$	$\displaystyle=E_{t}+v_{t},\hskip 119.50148ptv_{t}\sim\mathcal{N}(0,\sigma_{Y}^{2})$
	$\displaystyle E_{t}$	$\displaystyle=\phi E_{t-1}+X_{t}^{\top}\beta+\partial E_{t},\hskip 56.9055pt\partial E_{t}\sim\mathcal{N}(0,\sigma_{E}^{2}),$		(20)
	$\displaystyle\textrm{Initial State:}\hskip 56.9055ptE_{1}$	$\displaystyle\sim\mathcal{N}(Y_{1},\sigma_{E}^{2}),$

where, $\eta=0$, $X_{t}^{\top}=(1,GDP_{t},IDR_{t},Unemp_{t},Unemp_{t-1},Unemp_{t-2},Unemp_{t-3})$ and $\beta=(\alpha,\beta_{1},\ldots,\beta_{6})^{\top}$. We defined the following prior distributions,

	$\displaystyle\alpha$	$\displaystyle\sim\mathcal{N}(1.5,0.5),\hskip 14.22636pt\textrm{with}\hskip 14.22636pt\alpha\in\mathbb{R}$
	$\displaystyle\beta_{j}$	$\displaystyle\sim\textrm{Half-Normal}(0,1),\hskip 14.22636pt\textrm{with}\hskip 14.22636pt\beta_{j}\in(-\infty,0]\,\,\textrm{and}\,\,j=1$
	$\displaystyle\beta_{j}$	$\displaystyle\sim\textrm{Half-Normal}(0,1),\hskip 14.22636pt\textrm{with}\hskip 14.22636pt\beta_{j}\in[0,\infty)\,\,\textrm{and}\,\,j\in\{2,3,4,5,6\}$
	$\displaystyle\phi$	$\displaystyle\sim\mathcal{B}eta(2,2),\hskip 14.22636pt\textrm{with}\hskip 14.22636pt\phi\in(0,1)$
	$\displaystyle\sigma_{E}$	$\displaystyle\sim\textrm{Inv-Gamma}(2,0.1),\hskip 14.22636pt\textrm{with}\hskip 14.22636pt\sigma_{E}\in[0,\infty).$

The results of this model are presented below,

The Model II can be represented as,

	$\displaystyle Y_{t}$	$\displaystyle=E_{t}+v_{t},\hskip 119.50148ptv_{t}\sim\mathcal{N}(0,\sigma_{Y}^{2})$
	$\displaystyle E_{t}$	$\displaystyle=\phi E_{t-1}+X_{t}^{\top}\beta+\partial E_{t},\hskip 56.9055pt\partial E_{t}\sim\mathcal{N}(0,\sigma_{E}^{2}),$		(21)
	$\displaystyle\textrm{Initial State:}\hskip 56.9055ptE_{1}$	$\displaystyle\sim\mathcal{N}(Y_{1},\sigma_{E}^{2}),$

where $X_{t}^{\top}=(1,GDP_{t},IDR_{t},Unemp_{t},Unemp_{t-1},Unemp_{t-2},Unemp_{t-3})$ and $\beta=(\alpha,\beta_{1},\ldots,\beta_{6})^{\top}$. We defined the following prior distributions,

	$\displaystyle\alpha$	$\displaystyle\sim\mathcal{N}(1.5,0.5),\hskip 14.22636pt\textrm{with}\hskip 14.22636pt\alpha\in\mathbb{R}$
	$\displaystyle\beta_{j}$	$\displaystyle\sim\textrm{Half-Normal}(0,1),\hskip 14.22636pt\textrm{with}\hskip 14.22636pt\beta_{j}\in(-\infty,0]\,\,\textrm{and}\,\,j=1$
	$\displaystyle\beta_{j}$	$\displaystyle\sim\textrm{Half-Normal}(0,1),\hskip 14.22636pt\textrm{with}\hskip 14.22636pt\beta_{j}\in[0,\infty)\,\,\textrm{and}\,\,j\in\{2,3,4,5,6\}$
	$\displaystyle\phi$	$\displaystyle\sim\mathcal{B}eta(2,2),\hskip 14.22636pt\textrm{with}\hskip 14.22636pt\phi\in(0,1)$
	$\displaystyle\sigma_{E}$	$\displaystyle\sim\textrm{Inv-Gamma}(2,0.1),\hskip 14.22636pt\textrm{with}\hskip 14.22636pt\sigma_{E}\in[0,\infty)$
	$\displaystyle\eta$	$\displaystyle\sim\textrm{Gamma}(10,1),\hskip 14.22636pt\textrm{with}\hskip 14.22636pt\eta\in[0,\infty).$

The results of this model are presented below,

The Model III can be represented as,

	$\displaystyle Y_{t}$	$\displaystyle=E_{t}+v_{t},\hskip 123.76965ptv_{t}\sim\mathcal{N}(0,\sigma_{Y}^{2})$
	$\displaystyle E_{t}$	$\displaystyle=\phi_{t}E_{t-1}+X_{t}^{\top}\beta+\partial E_{t},\hskip 56.9055pt\partial E_{t}\sim\mathcal{N}(0,\sigma_{E}^{2})$
	$\displaystyle\phi_{t}$	$\displaystyle=\phi_{t-1}+\partial\phi_{t},\hskip 101.00737pt\partial\phi_{t}\sim\mathcal{N}(0,\sigma_{\phi}^{2}),$		(22)
	$\displaystyle\textrm{Initial State:}\hskip 56.9055ptE_{1}$	$\displaystyle\sim\mathcal{N}(Y_{1},\sigma_{E}^{2}),$
	$\displaystyle\phi_{0}$	$\displaystyle\sim\mathcal{B}eta(2,2),$

where, $\eta=0$, $X_{t}^{\top}=(1,GDP_{t},IDR_{t},Unemp_{t},Unemp_{t-1},Unemp_{t-2},Unemp_{t-3})$ and $\beta=(\alpha,\beta_{1},\ldots,\beta_{6})^{\top}$. Note that the resilience parameter varies over time according to random walk process; generating stochastic transmissions. We defined the following prior distributions,

	$\displaystyle\alpha$	$\displaystyle\sim\mathcal{N}(1.5,0.5),\hskip 14.22636pt\textrm{with}\hskip 14.22636pt\alpha\in\mathbb{R}$
	$\displaystyle\beta_{j}$	$\displaystyle\sim\textrm{Half-Normal}(0,1),\hskip 14.22636pt\textrm{with}\hskip 14.22636pt\beta_{j}\in(-\infty,0]\,\,\textrm{and}\,\,j=1$
	$\displaystyle\beta_{j}$	$\displaystyle\sim\textrm{Half-Normal}(0,1),\hskip 14.22636pt\textrm{with}\hskip 14.22636pt\beta_{j}\in[0,\infty)\,\,\textrm{and}\,\,j\in\{2,3,4,5,6\}$
	$\displaystyle\sigma_{\phi}$	$\displaystyle\sim\textrm{Inv-Gamma}(2,0.1),\hskip 14.22636pt\textrm{with}\hskip 14.22636pt\sigma_{\phi}\in[0,\infty)$
	$\displaystyle\sigma_{E}$	$\displaystyle\sim\textrm{Inv-Gamma}(2,0.1),\hskip 14.22636pt\textrm{with}\hskip 14.22636pt\sigma_{E}\in[0,\infty).$

The results of this model are presented below,