**3.8 The exponential generalized autoregressive conditional heteroskedasticity (EGARCH) model**

The EGARCH model was proposed by Nelson [22] to overcome some weaknesses of the GARCH model in handling financial time series pointed out by [23], In particular, to allow for asymmetric effects between positive and negative asset returns, he considered the weighted innovation

$$\mathbf{g}(\varepsilon\_t) = \theta \varepsilon\_t + \mathbf{y}[|\varepsilon\_t| - E(|\varepsilon\_t|)],\tag{19}$$

By repeated substitutions, we have

*σ*2

*σ*2

discounting factor [14].

and *αi*, *γ<sup>i</sup>*

presentations:

**3.7 NGARCH(p, q) model**

Which can also be written as

**(EGARCH) model**

**106**

**3.6 TGARCH(p, q) model**

*<sup>t</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>β</sup>*<sup>1</sup> ð Þ *<sup>a</sup>*<sup>2</sup>

*Linked Open Data - Applications,Trends and Future Developments*

and a TGARCH(p, q) model is given by the following:

*<sup>t</sup>* <sup>¼</sup> *<sup>α</sup>*<sup>0</sup> <sup>þ</sup><sup>X</sup>

where *Nt*�*<sup>i</sup>* is an indicator for negative *at*�*<sup>i</sup>*, that is,

*σ*2

*ht* <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup><sup>X</sup>

*<sup>σ</sup><sup>t</sup>* <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup><sup>X</sup>

returns, he considered the weighted innovation

*q*

*αiε*<sup>2</sup>

*αiε*<sup>2</sup>

*i*¼1

*q*

*i*¼1

*p*

*i*¼1

*<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> *<sup>β</sup>*1*a*<sup>2</sup>

which is the well-known exponential smoothing formation with *β*<sup>1</sup> being the

The Threshold GARCH model is another model used to handle leverage effects,

*<sup>α</sup><sup>i</sup>* <sup>þ</sup> *<sup>γ</sup>* ð Þ *iNt*¼*<sup>i</sup> <sup>a</sup>*<sup>2</sup>

*Nt*�*<sup>i</sup>* <sup>¼</sup> 1 if *at*�*<sup>i</sup>* <sup>&</sup>lt; 0,

of GARCH models, [14]. When *p* ¼ 1, *q* ¼ 1, the TGARCH(1, 1) model becomes:

The Nonlinear Generalized Autoregressive Conditional Heteroskedasticity (NGARCH) Model has been presented variously in literature by the following scholars [19–21]. The following model can be shown to represent all the

> *<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup><sup>X</sup> *q*

> *<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup><sup>X</sup> *q*

**3.8 The exponential generalized autoregressive conditional heteroskedasticity**

The EGARCH model was proposed by Nelson [22] to overcome some weaknesses of the GARCH model in handling financial time series pointed out by [23], In particular, to allow for asymmetric effects between positive and negative asset

*i*¼1

Where *ht* is the conditional variance, and *ω*, *β* and *α* satisfy *ω* >0, *β* ≥ 0 and *α*≥ 0.

*i*¼1

*<sup>t</sup>* <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup> ð Þ *<sup>α</sup>* <sup>þ</sup> *<sup>γ</sup>Nt*�<sup>1</sup> *<sup>a</sup>*<sup>2</sup>

�

0 if *at*�*<sup>i</sup>* ≥0,

, and *β <sup>j</sup>* are nonnegative parameters satisfying conditions similar to those

*<sup>t</sup>*�<sup>2</sup> <sup>þ</sup> *<sup>β</sup>*<sup>2</sup> 1*a*3

> *<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup><sup>X</sup> *q*

*j*¼1 *β j σ*2 *t*�*j*

*:*

*<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> *βσ*<sup>2</sup>

*<sup>γ</sup>iε<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup><sup>X</sup>

*<sup>γ</sup>iε<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup><sup>X</sup>

*p*

*j*¼1

*p*

*j*¼1

*g*ð Þ¼ *ε<sup>t</sup> θε<sup>t</sup>* þ *γ ε<sup>t</sup>* j j � *E ε<sup>t</sup>* ½ � ð Þ j j , (19)

, (15)

*<sup>t</sup>*�<sup>1</sup> (16)

*β jht*�*<sup>j</sup>* (17)

*β <sup>j</sup>σ<sup>t</sup>*�*<sup>j</sup>* (18)

*<sup>t</sup>*�<sup>3</sup> <sup>þ</sup> <sup>⋯</sup> � �, (14)

where *θ* and *γ* are real constants. Both *ε<sup>t</sup>* and *ε<sup>t</sup>* j j � *E ε<sup>t</sup>* ð Þ j j are zero-mean iid sequences with continuous distributions. Therefore, *E g*ð Þ *ε<sup>t</sup>* ½ �¼ 0. The asymmetry of *g*ð Þ *ε<sup>t</sup>* can easily be seen by rewriting it as

$$\mathbf{g}(\varepsilon\_{t}) = \begin{cases} (\theta + \chi)\varepsilon\_{t} - \chi E(|\varepsilon\_{t}|) & \text{if } \varepsilon\_{t} \ge \mathbf{0}, \\\\ (\theta - \chi)\varepsilon\_{t} - \chi E(|\varepsilon\_{t}|) & \text{if } \varepsilon\_{t} < \mathbf{0}. \end{cases} \tag{20}$$

An EGARCH(*m, s*) model, according to Dhamija and Bhalla [24] can be written as

$$\mathfrak{a}\_{t} = \sigma\_{t}\mathfrak{e}\_{t}, \ln\left(\sigma\_{t}^{2}\right) = \omega + \sum\_{i=1}^{s} a\_{i} \frac{|a\_{t-i}| + \theta\_{i}a\_{t-i}}{\sigma\_{t-i}} + \sum\_{j=1}^{m} \beta\_{j} \ln\left(\sigma\_{t-i}^{2}\right),\tag{21}$$

Which specifically results in EGARCH (1, 1) being written as

$$\mathfrak{a}\_{t} = \sigma\_{t}\mathfrak{e}\_{t}$$

$$\ln\left(\sigma\_{t}^{2}\right) = \omega + a\left(\left[|\mathfrak{a}\_{t-1}| - E(|\mathfrak{a}\_{t-1}|)\right]\right) + \theta\mathfrak{a}\_{t-1} + \beta\ln\left(\sigma\_{t-1}^{2}\right) \tag{22}$$

where j j *at*�<sup>1</sup> � *E a* ð Þ j j *<sup>t</sup>*�<sup>1</sup> are iid and have mean zero. When the EGARCH model has a Gaussian distribution of error term, then *<sup>E</sup> <sup>ε</sup><sup>t</sup>* ð Þ¼ j j ffiffiffiffiffiffiffi 2*=π* p , which gives:

$$\ln\left(\sigma\_t^2\right) = \alpha + a\left(\left[|a\_{t-1}| - \sqrt{2/\pi}\right]\right) + \theta a\_{t-1} + \beta \ln\left(\sigma\_{t-1}^2\right) \tag{23}$$

## **3.9 The absolute value GARCH (AVGARCH)**

An asymmetric GARCH (AGARCH), according to Ali [25] is simply

$$a\_t = \sigma\_t \varepsilon\_t; \sigma^2 = \alpha + \sum\_{i=1}^p a\_i |e\_{t-i} - b|^2 + \sum\_{j=1}^q \beta\_j \sigma\_{t-j}^2,\tag{24}$$

While the absolute value generalized autoregressive conditional heteroskedasticity (AVGARCH) model is specified as:

$$a\_t = \sigma\_t \mathbf{e}\_t; \sigma^2 = a + \sum\_{i=1}^p a\_i (|\mathbf{e}\_{t-i} + \mathbf{b}| - c \left(\mathbf{e}\_{t-i} + \mathbf{b}\right))^2 + \sum\_{j=1}^q \beta\_j \sigma\_{t-j}^2 \tag{25}$$

## **3.10 Nonlinear (asymmetric) GARCH, or N(a)GARCH or NAGARCH**

NAGARCH plays key role in option pricing with stochastic volatility because, as we shall see later on, NAGARCH allows you to derive closed-form expressions for European option prices in spite of the rich volatility dynamics. Because a NAGARCH may be written as

$$
\sigma\_{t+1}^2 = \alpha + a\sigma\_t^2 (\mathbf{z}\_t - \delta)^2 + \beta \sigma\_t^2 \tag{26}
$$

And if *zt* � *IIDN*ð Þ 0, 1 , *zt* is independent of *<sup>σ</sup>*<sup>2</sup> *<sup>t</sup>* as *σ*<sup>2</sup> *<sup>t</sup>* is only a function of an infinite number of past squared returns, it is possible to easily derive the long run, unconditional variance under NGARCH and the assumption of stationarity:

$$\begin{split} E\left[\sigma\_{t+1}^{2}\right] &= \overline{\sigma}^{2} = \alpha + aE\left[\sigma\_{t}^{2}\left(\mathbf{z}\_{t} - \boldsymbol{\delta}\right)^{2}\right] + \beta E\left[\sigma\_{t}^{2}\right] \\ &= \alpha + aE\left[\sigma\_{t}^{2}\right]E\left(\mathbf{z}\_{t}^{2} + \boldsymbol{\delta}^{2} - 2\delta \mathbf{z}\_{t}\right) + \beta E\left[\sigma\_{t}^{2}\right] \\ &= \alpha + a\overline{\sigma}^{2}\left(\mathbf{1} + \boldsymbol{\delta}^{2}\right) + \beta\overline{\sigma}^{2} \end{split} \tag{27}$$

Banking Supervision [29]. Backtesting is a statistical procedure where actual profits and losses are systematically compared to corresponding VaR estimates [30]. This book chapter adopted Backstesting techniques of [29]; The test was implemented in R using rugarch package and this test considered both the unconditional (Kupiec)

The unconditional (Kupiec) test also refer to as POF-test (Proportion of failure)

Here y is the number of exceptions and T is the number of observations and k is

<sup>½</sup><sup>1</sup> � *<sup>y</sup> p* � �*<sup>T</sup>*�*<sup>y</sup> <sup>y</sup> T* � �*<sup>y</sup>*

Under the null hypothesis that the model is correct and *LRPOF* is asymptotically chi-squared (*χ*2) distributed with degree of freedom as one (1). If the value of the *LRPOF* statistic is greater than the critical value (or p-value<0.01 for 1% level of significant or p-value<0.05 for 5% level of significant) the null hypothesis is

*T*

*T*�*y py* 1

CA*:*

*<sup>H</sup>*<sup>0</sup> : *<sup>p</sup>* <sup>¼</sup> *<sup>p</sup>*^ <sup>¼</sup> *<sup>y</sup>*

and conditional (Christoffersen) coverage tests for the correct number of

*LRPOF* ¼ �2 ln ð Þ <sup>1</sup> � *<sup>p</sup>*

0

B@

The Christoffersen's Interval Forecast Test combined the independence statistic with the Kupiec's POF test to obtained the joint test [30, 31]. This test examined the properties of a good VaR model, the correct failure rate and independence of exceptions, that is condition coverage (cc). the conditional

*LRcc* ¼ *LRPOF* þ *LRind*

1

3 7

<sup>5</sup> � 2 ln *<sup>p</sup>*ð Þ <sup>1</sup> � *<sup>p</sup> <sup>u</sup>*�<sup>1</sup> 1 *u* � � <sup>1</sup> � <sup>1</sup> *u* � �*<sup>u</sup>*�<sup>1</sup> !

CA

*ui* � �*ui*�<sup>1</sup>

Where *ui* is the time between exceptions I and i-1 while u is the sum of *ui*. If the value of the *LRcc* statistic is greater than the critical value (or p-value<0.01

for 1% level of significant or p-value<0.05 for 5% level of significant) the null

In this study we employed two innovations namely student t and skewed student t distributions they can account for excess kurtosis and non-normality in

� � <sup>1</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup>

*v* � ��ð Þ *<sup>v</sup>*þ<sup>1</sup>

2

; � ∞ < *y*< ∞

�2 ln *<sup>p</sup>*ð Þ <sup>1</sup> � *<sup>p</sup> ui*�<sup>1</sup> 1 *ui* � � <sup>1</sup> � <sup>1</sup>

hypothesis is rejected and that leads to the rejection of the model.

0

B@

exceedances (see details in [31, 32].

*Financial Time Series Analysis via Backtesting Approach*

*DOI: http://dx.doi.org/10.5772/intechopen.94112*

the confidence level. The test is given as

rejected and the model then is inaccurate.

coverage (cc) is given as

*LRind* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

*i*¼2

**3.14 Distributions of GARCH models**

The student t-distribution is given as

*f y*ð Þ¼ <sup>Γ</sup> *<sup>v</sup>*þ<sup>1</sup>

2 � � ffiffiffiffiffi *<sup>v</sup><sup>π</sup>* <sup>p</sup> <sup>Γ</sup> *<sup>v</sup>* 2

financial returns [28, 33].

**109**

2 6 4

Where

with its null hypothesis given as

Where *<sup>σ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>E</sup> <sup>σ</sup>*<sup>2</sup> *t* � � and *E σ*<sup>2</sup> *t* � � <sup>¼</sup> *<sup>E</sup> <sup>σ</sup>*<sup>2</sup> *t*þ1 � � because of stationary. Therefore

$$
\sigma^2 \left[ \mathbf{1} - a \left( \mathbf{1} + \delta^2 \right) + \beta \right] = \alpha \Rightarrow \overline{\sigma}^2 = \frac{a}{\mathbf{1} - a \left( \mathbf{1} + \delta^2 \right) + \beta} \tag{28}
$$

Which exists and positive if and only if *<sup>α</sup>* <sup>1</sup> <sup>þ</sup> *<sup>δ</sup>*<sup>2</sup> � � <sup>þ</sup> *<sup>β</sup>* <sup>&</sup>lt;1. This has two implications:

i. The persistence index of a NAGARCH(1,1) is *<sup>α</sup>* <sup>1</sup> <sup>þ</sup> *<sup>δ</sup>*<sup>2</sup> � � <sup>þ</sup> *<sup>β</sup>* and not simply *α* þ *β*;

ii. a NAGARCH(1,1) model is stationary if and only if *<sup>α</sup>* <sup>1</sup> <sup>þ</sup> *<sup>δ</sup>*<sup>2</sup> � � <sup>þ</sup> *<sup>β</sup>* <sup>&</sup>lt;1.

See details in [22].

## **3.11 Persistence**

The low or high persistency in volatility exhibited by financial time series can be determined by the GARCH coefficients of a stationary GARCH model. The persistence of a GARCH model can be calculated as the sum of GARCH (*β*1) and ARCH (*α*1) coefficients that is *α* þ *β*1. In most financial time series, it is very close to one (1) [26, 27]. Persistence could take the following conditions:

If *α* þ *β*<sup>1</sup> <1: The model ensures positive conditional variance as well as stationary.

If *α* þ *β*<sup>1</sup> ¼ 1: we have an exponential decay model, then the half-life becomes infinite. Meaning the model is strictly stationary.

If *α* þ *β*<sup>1</sup> >1: The GARCH model is said to be non-stationary, meaning that the volatility ultimately detonates toward the infinitude [27]. In addition, the model shows that the conditional variance is unstable, unpredicted and the process is nonstationary [28].

## **3.12 Half-life volatility**

Half-life volatility measures the mean reverting speed (average time) of a stock price or returns. The mathematical expression of half-life volatility is given as

$$Half - Lif\hat{e} = \frac{\ln\left(0.5\right)}{\ln\left(a\_1 + \beta\_2\right)}$$

It can be noted that the value of *α* þ *β*<sup>1</sup> influences the mean reverting speed [27], which means that if the value of *α* þ *β*<sup>1</sup> is closer to one (1), then the volatility shocks of the half-life will be longer.

#### **3.13 Backtesting**

Financial risk model evaluation or backtesting is an important part of the internal model's approach to market risk management as put out by Basle Committee on *Financial Time Series Analysis via Backtesting Approach DOI: http://dx.doi.org/10.5772/intechopen.94112*

Banking Supervision [29]. Backtesting is a statistical procedure where actual profits and losses are systematically compared to corresponding VaR estimates [30]. This book chapter adopted Backstesting techniques of [29]; The test was implemented in R using rugarch package and this test considered both the unconditional (Kupiec) and conditional (Christoffersen) coverage tests for the correct number of exceedances (see details in [31, 32].

The unconditional (Kupiec) test also refer to as POF-test (Proportion of failure) with its null hypothesis given as

$$H\_0: p = \hat{p} = \frac{\mathcal{Y}}{T}$$

Here y is the number of exceptions and T is the number of observations and k is the confidence level. The test is given as

$$LR\_{POF} = -2\ln\left(\frac{(1-p)^{T-\mathcal{Y}}p^{\mathcal{Y}}}{[1-\left(\frac{\mathcal{Y}}{p}\right)^{T-\mathcal{Y}}\left(\frac{\mathcal{Y}}{T}\right)^{\mathcal{Y}}}\right).$$

Under the null hypothesis that the model is correct and *LRPOF* is asymptotically chi-squared (*χ*2) distributed with degree of freedom as one (1). If the value of the *LRPOF* statistic is greater than the critical value (or p-value<0.01 for 1% level of significant or p-value<0.05 for 5% level of significant) the null hypothesis is rejected and the model then is inaccurate.

The Christoffersen's Interval Forecast Test combined the independence statistic with the Kupiec's POF test to obtained the joint test [30, 31]. This test examined the properties of a good VaR model, the correct failure rate and independence of exceptions, that is condition coverage (cc). the conditional coverage (cc) is given as

$$LR\_{\infty} = LR\_{POF} + LR\_{ind}$$

Where

*E σ*<sup>2</sup> *t*þ1

*t* � � and *E σ*<sup>2</sup>

Where *<sup>σ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>E</sup> <sup>σ</sup>*<sup>2</sup>

*α* þ *β*;

See details in [22].

**3.11 Persistence**

stationary [28].

**3.12 Half-life volatility**

of the half-life will be longer.

**3.13 Backtesting**

**108**

implications:

� � <sup>¼</sup> *<sup>σ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup> *<sup>α</sup><sup>E</sup> <sup>σ</sup>*<sup>2</sup>

*Linked Open Data - Applications,Trends and Future Developments*

<sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup> *<sup>α</sup><sup>E</sup> <sup>σ</sup>*<sup>2</sup>

*t* � � <sup>¼</sup> *<sup>E</sup> <sup>σ</sup>*<sup>2</sup>

(1) [26, 27]. Persistence could take the following conditions:

infinite. Meaning the model is strictly stationary.

*t* � �*E z*<sup>2</sup>

<sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup> *ασ*<sup>2</sup> <sup>1</sup> <sup>þ</sup> *<sup>δ</sup>*<sup>2</sup> � � <sup>þ</sup> *βσ*<sup>2</sup>

*<sup>σ</sup>*<sup>2</sup> <sup>1</sup> � *<sup>α</sup>* <sup>1</sup> <sup>þ</sup> *<sup>δ</sup>*<sup>2</sup> � � <sup>þ</sup> *<sup>β</sup>* � � <sup>¼</sup> *<sup>ω</sup>* ) *<sup>σ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>ω</sup>*

Which exists and positive if and only if *<sup>α</sup>* <sup>1</sup> <sup>þ</sup> *<sup>δ</sup>*<sup>2</sup> � � <sup>þ</sup> *<sup>β</sup>* <sup>&</sup>lt;1. This has two

i. The persistence index of a NAGARCH(1,1) is *<sup>α</sup>* <sup>1</sup> <sup>þ</sup> *<sup>δ</sup>*<sup>2</sup> � � <sup>þ</sup> *<sup>β</sup>* and not simply

ii. a NAGARCH(1,1) model is stationary if and only if *<sup>α</sup>* <sup>1</sup> <sup>þ</sup> *<sup>δ</sup>*<sup>2</sup> � � <sup>þ</sup> *<sup>β</sup>* <sup>&</sup>lt;1.

The low or high persistency in volatility exhibited by financial time series can be determined by the GARCH coefficients of a stationary GARCH model. The persistence of a GARCH model can be calculated as the sum of GARCH (*β*1) and ARCH (*α*1) coefficients that is *α* þ *β*1. In most financial time series, it is very close to one

If *α* þ *β*<sup>1</sup> <1: The model ensures positive conditional variance as well as stationary. If *α* þ *β*<sup>1</sup> ¼ 1: we have an exponential decay model, then the half-life becomes

If *α* þ *β*<sup>1</sup> >1: The GARCH model is said to be non-stationary, meaning that the volatility ultimately detonates toward the infinitude [27]. In addition, the model shows that the conditional variance is unstable, unpredicted and the process is non-

Half-life volatility measures the mean reverting speed (average time) of a stock

It can be noted that the value of *α* þ *β*<sup>1</sup> influences the mean reverting speed [27], which means that if the value of *α* þ *β*<sup>1</sup> is closer to one (1), then the volatility shocks

Financial risk model evaluation or backtesting is an important part of the internal model's approach to market risk management as put out by Basle Committee on

ln *α*<sup>1</sup> þ *β*<sup>2</sup> ð Þ

price or returns. The mathematical expression of half-life volatility is given as

*Half* � *Life* <sup>¼</sup> ln 0ð Þ *:*<sup>5</sup>

*t*þ1

*<sup>t</sup>*ð Þ *zt* � *<sup>δ</sup>* <sup>2</sup> h i

*<sup>t</sup>* <sup>þ</sup> *<sup>δ</sup>*<sup>2</sup> � <sup>2</sup>*δzt* � � <sup>þ</sup> *<sup>β</sup><sup>E</sup> <sup>σ</sup>*<sup>2</sup>

<sup>þ</sup> *<sup>β</sup><sup>E</sup> <sup>σ</sup>*<sup>2</sup> *t* � �

� � because of stationary. Therefore

*t* � �

<sup>1</sup> � *<sup>α</sup>* <sup>1</sup> <sup>þ</sup> *<sup>δ</sup>*<sup>2</sup> � � <sup>þ</sup> *<sup>β</sup>* (28)

(27)

$$LR\_{ind} = \sum\_{i=2}^{n} \left[ -2\ln\left(\frac{p(1-p)^{u\_i-1}}{\left(\frac{1}{u\_i}\right)\left(1-\frac{1}{u\_i}\right)^{u\_i-1}}\right) \right] - 2\ln\left(\frac{p(1-p)^{u-1}}{\left(\frac{1}{u}\right)\left(1-\frac{1}{u}\right)^{u-1}}\right)$$

Where *ui* is the time between exceptions I and i-1 while u is the sum of *ui*.

If the value of the *LRcc* statistic is greater than the critical value (or p-value<0.01 for 1% level of significant or p-value<0.05 for 5% level of significant) the null hypothesis is rejected and that leads to the rejection of the model.

#### **3.14 Distributions of GARCH models**

In this study we employed two innovations namely student t and skewed student t distributions they can account for excess kurtosis and non-normality in financial returns [28, 33].

The student t-distribution is given as

$$f(\boldsymbol{y}) = \frac{\Gamma\left(\frac{\boldsymbol{v}+1}{2}\right)}{\sqrt{\boldsymbol{v}\pi}\Gamma\left(\frac{\boldsymbol{v}}{2}\right)} \left(\mathbb{1} + \frac{\boldsymbol{y}^2}{\boldsymbol{v}}\right)^{-\frac{(\boldsymbol{v}+1)}{2}}; -\infty < \boldsymbol{y} < \infty$$

The Skewed student t-distribution is given as

$$f(\boldsymbol{y};\boldsymbol{\mu},\boldsymbol{\sigma},\boldsymbol{\nu},\boldsymbol{\lambda}) = \begin{cases} bc \left( \mathbf{1} + \frac{1}{\boldsymbol{\nu} - 2} \left( \frac{b \left( \frac{\boldsymbol{\gamma} - \boldsymbol{\mu}}{\boldsymbol{\sigma}} \right) + d}{1 - \boldsymbol{\lambda}} \right)^2 \right)^{-\frac{\boldsymbol{\nu} + 1}{2}}, \text{if } \boldsymbol{\gamma} < -\frac{d}{b} \\\\ bc \left( \mathbf{1} + \frac{1}{\boldsymbol{\nu} - 2} \left( \frac{b \left( \frac{\boldsymbol{\gamma} - \boldsymbol{\mu}}{\boldsymbol{\sigma}} \right) + d}{1 + \boldsymbol{\lambda}} \right)^2 \right)^{-\frac{\boldsymbol{\nu} + 1}{2}}, \text{if } \boldsymbol{\gamma} \ge -\frac{d}{b} \end{cases}$$

Where *v* is the shape parameter with 2 <*v*< ∞ and *λ* is the skewness parameter with �1<*λ*<1. The constants a, b and c are given as

$$a = 4\lambda c \left(\frac{v-2}{v-1}\right); b = 1 + \mathfrak{z}(\lambda)^2 - a^2; c = \frac{\Gamma\left(\frac{v+1}{2}\right)}{\sqrt{\pi(v-2)\Gamma\left(\frac{v}{2}\right)}}$$

Where *μ* and *σ* are the mean and the standard deviation of the skewed student t distribution respectively.
