Solving the time fractional q-deformed tanh-Gordon equation: A theoretical analysis using controlled Picard's transform method

1.   Introduction

Asymmetric datasets modeling utilizing generalized continuous distributions is quite an interesting area of research in recent times, and quite a number of generalized continuous distributions have been proposed, established and utilized in different fields of study. Besides with the new progressions in probability distribution theory, more vigorous forms of continuous distributions have been proposed utilizing the exponentiated method, transformer-transformer method, beta-generated method and gamma-generated method, among others. Shockingly, these recent distributions have been valuable in resolving issues in different fields of study other than in GARCH-type volatility models. Along these lines, it is important to develop new conditional innovation distributions capable of handling both significant skewness and excess kurtosis present in financial asset returns. Jones (2001) proposed a simplified extension of the symmetric Student's-t distribution known as the skew-t distribution to handle real-life asymmetric datasets. Jones and Faddy (2003) described the distribution as a simplified skew-t distribution and provided some important structural properties. Some other forms of the skew-t distribution that have been developed can be found in the literature (Johnson et al., 1995; Sahu et al., 2003; Azzalini and Capitanio, 2003). More so, the statistical literature consists several extended forms of the skew-t distribution. For example, the generalized hyperbolic skew-t distribution (Aas and Haff, 2006), beta skew-t distribution (Shittu et al., 2014), Balakrishnan skew-t distribution (Shafiei and Doostparast, 2014), Kumaraswamy skew-t distribution (Khamis et al., 2017), exponentiated skew-t distribution (Dikko and Agboola, 2017), generalized alpha skew-t distribution (Altun et al., 2018), odd exponentiated skew-t distribution (Adubisi et al., 2021a), and type I half-logistic skew-t distribution (Adubisi et al., 2021b).

For any baseline distribution with parameter vector $\zeta$, Cordeiro et al. (2014) proposed the exponentiated half-logistic-G (EHL-G) family based on the T-X method (Alzaatreh et al., 2013). Given that $k\left(t\right)$ and $K\left(t\right)$ are the probability density function (pdf) and cumulative distribution function (cdf) of a random variable (r.v) $T\in \left[c, d\right]$ for $\left[c, d\right]\in \left(-\infty, +\infty \right)$, respectively. Let $R\left[H\left(y;\zeta \right)\right]$ be a function of the cdf of another r. v Y satisfying the following conditions: [i] $R\left[H\left(y;\zeta \right)\right]\in \left[c, d\right]$; [ii] $R\left[H\left(y;\zeta \right)\right]\to c$ as $y\to -\infty$ and $R\left[H\left(y;\zeta \right)\right]\to d$ as $y\to \infty$; and [iii] $R\left[H\left(y;\zeta \right)\right]$ is differentiable and monotonically non-decreasing. The cdf of the T-X method is given by

Setting $\begin{array}{cc}k\left(t\right) = \frac{2\alpha \varphi {e}^{-\alpha t}{\left[1-{e}^{-\alpha t}\right]}^{\varphi -1}}{{\left(1+{e}^{-\alpha t}\right)}^{\varphi +1}}, & t > 0, \end{array}$ where $\alpha, \varphi > 0$ are the shape parameters, and

$21R\left[H\left(y;\zeta \right)\right] = - \ \mathit{log} \ \left[1-H\left(y;\zeta \right)\right]$, the cdf of the EHL-G family is given by

and the corresponding pdf to Equation 2 is

where $\alpha, \varphi > 0$ are two additional shape parameters. $H\left(y;\zeta \right)$ and $h\left(y;\zeta \right)$ are the baseline cdf and pdf.

In this research work, a hybrid form of the skew-t distribution developed by Jones and Faddy (2003) using the exponentiated half logistic-G family of distributions is proposed. The new three-parameter model called the exponentiated half logistic skew t (EHL_ST) distribution appropriate for modeling right or left skewed and heavy-tailed datasets is developed. By inserting the skew-t cdf $H\left(y;\kappa \right) = \frac{1}{2}\left(1+\frac{y}{\sqrt{\kappa +{y}^{2}}}\right)$ into Equation 2 leads to

The corresponding pdf to Equation 4 is

where $\alpha > 0$ and $\varphi > 0$ are two shape parameters, and $\kappa$ denote the skew parameter. From now onward, $Y\sim EH{L}_{ST}\left(\alpha, \varphi, \kappa \right)$ denotes a random variable with pdf Equation 5. The hazard rate function (hrf) of Y is

Detailed properties of the EHL_ST model can be found in Adubisi et al. (2021c). Lately, there has been a key interest in the comparison of different estimation procedures for the parameter estimation of several distributions. For example, the extended exponential geometric (Louzada et al., 2016), Binomial exponential 2 (Bakouch et al., 2017), Poisson exponential (Rodrigues et al., 2018), type-I half-logistic Top-Leone (ZeinEldin et al., 2019), polynomial exponential (Chesneau et al., 2020), Fréchet (Ramos et al., 2020), odd exponential half-logistic exponential (Aldahlan and Afify, 2020), among others. The motivation in developing the new model is to create a more flexible heavy-tailed distribution with right-skewed, left-skewed, symmetric and unimodal features capable of handling the stylized features in financial datasets. The aim of this work is to estimate the parameters of the EHL_ST distribution using different frequentist estimation criterions such as the maximum likelihood, Anderson-Darling, maximum product of spacing, ordinary least squares, Cramer-von mises and weighted least squares. These estimation criterions using extensive Monte Carlo simulation are compared to discourse their performance. Also, Altun (2019) noted that the assumption of the conditional innovation density of the GARCH volatility model directly impacts on the accuracy of volatility predictions. Hence, the standardized form of the EHL_ST model is also derived to serve as an alternative conditional innovation density in modeling and forecasting the volatility of the inflation rates in sub-Sahara Africa, specifically Nigeria inflation rate using the GARCH-type volatility models. The rest of this research work is designed as follows. The quantile function of the EHL_ST is derived in Section 2. Six estimation criterions for estimating the parameters of the EHL_ST model are presented in Section 3. Simulation study to compare these criterions and real-life application are presented in Section 4. The application of the EHL_ST density in volatility modeling is illustrated via GARCH-type models in Section 5. The empirical results of the GARCH process are presented in Section 6. Conclusion of the work in Section 7.

Figures 1 and 2 provides plots of the pdf and hazard rate function of the EHL_ST distribution for selected values of the parameters. The plot in figure 2, shows that the hazard rate function can be J-shape, increasing and decreasing.

2.   Quantile function

The quantile function, $Q\left(u\right) = {F}^{-1}\left(u\right)$ of Y can be derived by inverting Equation 4. The quantile function of the EHL_ST distribution takes the form:

The quantile function in Equation 7 can define various quantile measures such as Bowley's skewness and Moor's kurtosis of the EHL_ST distribution. Figure 3 depicts the three-dimensional plots of these skewness and kurtosis measures. It is clear that the skewness was increasing in $\varphi$ and decreasing in $\alpha$ while the kurtosis is increasing in $\alpha$ and decreasing in $\varphi$, and were independent of $\kappa$.

3.   Methods of estimation

This section discusses six estimation criterions, these are the maximum likelihood (ML), maximum product of spacing (MPS), Anderson-Darling (ANDA), ordinary least squares (OLS), weighted least squares (WLS) and Cramer-von Mises (CVM) to estimate the new model parameters.

3.1.   Maximum likelihood

The ML estimation is considered the best for estimating distribution parameters given its excellent characteristics. Let ${y}_{1}, \dots, {y}_{T}$ be the observed values of sizes (T) from the EHL_ST distribution. The log-likelihood function derived from Equation 6 is given by

Let ${\widehat{\alpha }}_{MLE}, {\widehat{\varphi }}_{MLE}$ and ${\widehat{\kappa }}_{MLE}$ be the ML estimates of $\alpha, \varphi$ and $\kappa$. These can be obtained numerically by maximizing $l\left(\alpha, \varphi, \kappa \right)$ or solving the following differential equations:

and

3.2.   Ordinary least squares

Let ${y}_{\left(1:T\right)}, {y}_{\left(2:T\right)}, \dots, {y}_{\left(T:T\right)}$ be the ordered sample of size (T) from the EHL_ST distribution with cdf in Equation 5. The OLS estimates ${\widehat{\alpha }}_{OLS}, {\widehat{\varphi }}_{OLS}$ and ${\widehat{\kappa }}_{OLS}$ of $\alpha, \varphi$ and $\kappa$ can be obtained numerically by minimizing

where $\wp (k, T) = \frac{\left(T+1-k\right)}{\left(T+1\right)}$, with respect to $\alpha, \varphi$ and $\kappa$.Also, the model estimates can be obtained by solving the following differential equations:

and

where ${z}_{(k:T)}$ is the order transformed value of ${z}_{k}$ given by Equation 9, and

and

3.3.   Weighted least squares

Applying the same symbolization as in the preceding subsection, the WLS estimates ${\widehat{\alpha }}_{WLS}, {\widehat{\varphi }}_{WLS}$ and ${\widehat{\kappa }}_{WLS}$ of $\alpha, \varphi$ and $\kappa$ can be obtained numerically by minimizing

where $\varPsi (k, T) = \frac{{\left(T+1\right)}^{2}\left(T+2\right)}{k\left(T-k+1\right)}$, relative to $\alpha, \varphi$ and $\kappa$. Also, the model estimates can be obtained by solving the following differential equations:

and

where ${z}_{(k:T)}$ and ${\varPhi }_{i}\left({y}_{(k:T)}\left|\alpha, \varphi, \kappa \right.\right),$ $i = \mathrm{1, 2}, 3$ are given by Eqs 9, 17, 18 and 19, respectively.

3.4.   Maximum product of spacing

Cheng and Amin(1979, 1983) proposed the MPS estimator for the estimation of unknown parameters with ordered sample ${y}_{\left(1:T\right)}, {y}_{\left(2:T\right)}, \dots, {y}_{\left(T:T\right)}$ from the EHL_ST distribution and uniform spacing for this random sample is given by:

where $F\left({y}_{\left(0:T\right)}\left|\alpha, \varphi, \kappa \right.\right) = 0$ and $F\left({y}_{\left(T+1:T\right)}\left|\alpha, \varphi, \kappa \right.\right) = 1$. The MPS estimates ${\widehat{\alpha }}_{MPS}$, ${\widehat{\varphi }}_{MPS}$ and ${\widehat{\kappa }}_{MPS}$ of $\alpha, \varphi$, and $\kappa$ can be obtained by maximizing

relative to $\alpha, \varphi$, and $\kappa$. Also, the estimates can be obtained by solving the following differential equations:

and

where ${\varPhi }_{i}(.\left|\alpha, \varphi, \kappa \right.),$ $i = \mathrm{1, 2}, 3$ are specified in Equation s 17, 18 and 19, respectively.

3.5.   Anderson-Darling

The ANDA estimates ${\widehat{\alpha }}_{ANDA}$, ${\widehat{\varphi }}_{ANDA}$ and ${\widehat{\kappa }}_{ANDA}$ of the EHL_ST parameters can be obtained by minimizing relative to $\alpha, \varphi$, and $\kappa$, the function

Also, estimates obtained by solving the following nonlinear equations:

and

where ${\varPhi }_{i}(.\left|\alpha, \varphi, \kappa \right.),$ $i = \mathrm{1, 2}, 3$ are specified in Eqs 17, 18 and 19, respectively.

3.6.   Cramer-von Mises

The CVM estimates ${\widehat{\alpha }}_{CVM}$, ${\widehat{\varphi }}_{CVM}$ and ${\widehat{\kappa }}_{CVM}$ of the EHLST parameters can be obtained by minimizing relative to $\alpha, \varphi$, and $\kappa$, the function

Solving the following nonlinear equations, the CVM estimates can also be obtained as follows:

and

where ${z}_{(k:T)}$ and ${\varPhi }_{i}\left({y}_{(k:T)}\left|\alpha, \varphi, \kappa \right.\right),$ $i = \mathrm{1, 2}, 3$ are given by Eqs 9, 17, 18 and 19, respectively.

4.   Simulation study

The performance of the six estimators defined in Section 5 are compared using Monte Carlo simulations. The parameter combinations (Comb), i.e., Comb 1 $\left(\alpha = 1.0, \varphi = 1.3, \kappa = 0.5\right),$ Comb 2 $\left(\alpha = 1.3, \varphi = 1.7, \kappa = 1.0\right),$ Comb 3 $\left(\alpha = 2.0, \varphi = 2.5, \kappa = 1.5\right),$ and Comb 4 $\left(\alpha = 2.5, \varphi = 3.0, \kappa = 2.0\right)$. The datasets are produced from the EHL_ST distribution under these combinations by selecting n = 10, 50,100 and 200. For each process, 1000 random samples from the EHL_ST distribution are produced using the quantile function in Eq 6. The parameter estimates performance are evaluated through the average values (AVEs), average absolute biases (AVABs), mean square errors (MSEs) and root mean square errors (RMSEs) for the different sample sizes (n). For the estimates, the AVAB, MSE and RMSE are computed using

where, $\widehat{\vartheta } = \left(\widehat{\alpha }, \widehat{\varphi }, \widehat{\kappa }\right)$ and $\vartheta = (\alpha, \varphi, \kappa)$. The partial and overall ranks of the six estimators for the combinations to determine the best method for estimating the EHL_ST parameters are provided in Table 1. Based on the results, it is realistic to use the ML and MPS methods for estimating the unknown parameters of the EHL_ST distribution.

Tables 2–5 provides the AVEs, AVABs, MSEs and RMSEs of the six estimators. More so, these tables show the rank of each estimator among all the estimators in each row, the superscripts are the rank indicators and $\sum ranks$ is the partial sum of the ranks for each column and sample size (n).

The results in Tables 2–5 show that the mean square errors (MSEs) decrease for all parameter combinations as the sample size (n) increases, that is, all the estimators are quite consistent since their RMSE tend to zero for large sample size (n). Hence, the estimates of the EHL_ST parameters provide credible MSEs and low AVABs for all the parameter combinations using the six estimation methods for large sample size (n). Additionally, the average absolute biases (AVABs) approach zero as the sample size (n) increase, proving that all the estimators are quite asymptotically unbiased.

The model evaluation in relation to some existing distributions from the same family of distributions using the inflation rates (INF) monthly dataset is performed. The few existing distributions from the same family are the exponentiated half logistic Lomax (EHLL), exponentiated half logistic Fréchet (EHLF) and exponentiated half logistic log-logistic (EHLLL). The model parameters are estimated using the maximum likelihood and maximum product of spacing estimation procedures. Using the R-software package "AdequacyModel", the following performance measures: negative log-likelihood, Akaike Information Criterion (AIC), Kolmogorov-Smirnov (K-S) statistic and its p-value including the ML estimates (MLEs) and standard errors (SEs) of the parameters are provided in Table 6 while the MPS estimates (MPSEs), log-likelihood and other measures are provided in Table 7. The results in Tables 6 and 7 show that the new model is quite competitive with the existing four-parameter distributions in fitting the inflation rates dataset.

5.   Standardized exponentiated half logistic skew-t distribution

The standardized form of the EHL_ST density function is derived using the transformed random variable $z = \frac{⟨y-\mu ⟩}{\sqrt{{h}^{2}}}$, where ${E}\left(z\right) = 0$ and $\mathit{var}\left(z\right) = 1$. Thus, the standardized EHL_ST density function with $z = \frac{\varepsilon }{\sqrt{{h}^{2}}}$ and $\frac{\partial z}{\partial \varepsilon } = \frac{1}{\sqrt{{h}^{2}}}$ is given by

The log-likelihood function (LL) is given by

where $\xi = \left(\alpha, \varphi, \kappa, {h}_{t}^{2}\right)$ and ${h}_{t}^{2}$ denote the GARCH-type volatility models with unknown vector parameters.

6.   Volatility applicatıon

The applicability of the standardized EHL_ST density function as the conditional innovation density in GARCH-type volatility models are demonstrated using the monthly all-items Nigeria inflation rates (year on change) dataset from January 2003 to August 2021. The dataset can be sourced from the Central Bank of Nigeria website at www.cbn.gov.ng.

6.1.   GARCH-type models

In financial time series, volatility modeling is a dynamic direction for both advisors and financial expert. Engle (1982) and Bollerslev (1986) introduced the autoregressive conditional heteroscedasticity (ARCH) and the generalized ARCH (GARCH) models for volatility modeling. The log-return is denoted as ${r}_{t}$, and the GARCH (1, 1) model is expressed as follows:

where $\omega > 0$, ${\eta }_{1} > 0$, ${\eta }_{2} > 0$, ${z}_{t}$ is the conditional innovation distribution with ${E}\left({z}_{t}\right) = 0$ and $\mathit{var}\left({z}_{t}\right) = 1$. The conditional variance and mean are denoted as ${h}_{t}^{2}$ and $\mu$, respectively. In this research work, the GARCH model (Bollerslev, 1986) and GJRGARCH model (Glosten et al., 1993) are considered. The GARCH model is used as the benchmark in this research. The GJRGARCH (1, 1) model conditional variance functional form is expressed as:

where ${\eta }_{3}$ is the leverage effect, and ${I}_{t-1}$ is an indicator function expressed as:

The GJRGARCH model checks for leverage effects and when the parameter ${\eta }_{3}$ is positive it implies that negative shocks have a larger effect on volatility than positive shocks. The common conditional innovations are provided in Appendix A.

6.2.   Evaluation of volatility predictions

6.2.1.   Model selection criteria

The accuracy of both models is appraised using the modified information criteria of Brooks and Burke (2003). The modified Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), criteria are given by

where $T$ is the number of observations, $c$ is number of estimated parameters, and $LL$ denotes the log-likelihood value. The model with the least AIC and BIC values is regarded as the finest model in terms of the specified conditional innovation density.

6.2.2.   Prediction evaluation

The predictive ability of the GARCH-type models is evaluated using the mean square error (MSE), root mean square root (RMSE), and mean absolute error (MAE). The MSE, RMSE, and MAE for the volatility forecasts are given by

where ${h}_{t}$ and ${\widehat{h}}_{t}$ represent the realized volatility and predicted volatility, and $T$ is the sample size. The model with the least MSE, RMSE, and MAE values is regarded as the most suitable for predicting the volatility of the financial dataset.

7.   Empirical finding

7.1.   Data report

To assess the ability of the GARCH-type volatility models using the EHLST density in relation to the commonly existing densities in modeling and predicting volatility, the all-items Nigeria inflation rates is utilized. The utilized data consist of 224 inflation rates from January 2003 to August 2021. Table 8 presents the summary statistics of 223 monthly Inflation rate (INF) log-returns. Negative skewness and excess kurtosis are quite evident, leading to large Jarque-Bera (JB) test statistic value (p < 0.001) indicating that the monthly log-returns are non-normally distributed. More so, the Lagrange-multiplier (LM) test specifies the prevalence of ARCH effects in the conditional variance. The graphical plots of the log-returns, squared log-returns and absolute log-returns including their autocorrelation functions are depicted in Figure 4.

The plots in Figure 4_{(b, c)} show evidence of volatility clustering; that is low volatility values followed by low volatility values and high volatility values followed by high volatility values. More so, the monthly log-returns show no clear evidence of serial correlation, but the squared and absolute log-returns are positively autocorrelated as depicted in Figures 4_{(e, f).}

6.2.   Estimation

The GARCH-type volatility models are estimated under the normal (NORM), Student-t (STD), generalized error (GED) and their skewed versions, generalized hyperbolic (GHYP), Johnson (SU) reparametrized (JSU), generalized hyperbolic skew-Student (GHST), Normal inverse Gaussian (NIG) and EHL_ST innovation densities. To acquire the parameter estimates of both models under NORM, STD, GED, SNORM, SSTD, SGED, GHYP, JSU, GHST and NIG densities, the rugarch package in R-software is used. The Optim function in R-software is utilized for the estimation of the model's parameter with the EHL_ST innovation density. Table 9 shows the GARCH (1, 1) and GJRGARCH (1, 1) models parameter estimates under eleven innovation densities.

The parameter estimates of the conditional variance for the GARCH-EHL_ST are statistically significant at various standard levels including the leverage effect parameter ${\eta }_{3}$. Hence, the impact of the shocks is asymmetric in nature which implies that the impact of positive shocks is higher than negative shocks of the same magnitude on volatility. Tables 10 provides vital statistics for model selection, which shows that the GARCH-EHL_ST has the highest log-likelihood, and least AIC and BIC values relative to other models. Hence, the GARCH-EHL_ST model is selected as the best model for modeling the volatility of the financial data series.

Table 11 provides the diagnostic analysis tests for the GARCH-type volatility models which indicates that the GARCH-EHL_ST model did well in describing the conditional variance dynamics as well as other models. Basically, the Ljung-Box statistic (p-value) indicates no sign of serial-correlation in the squared standardized residuals, and the ARCH-LM statistic indicates no extra ARCH processes in the standardized residuals which implies that the conditional variance equations are well-defined.

6.3.   Forecasts

The in-sample volatility predictive measures for the models are provided in Table 12, and Table 13 provides the partial and total ranking of the innovation densities based on the three predictive performance measures. The results show that the GARCH (1, 1) and GJRGARCH (1, 1) models has the least MSE, RMSE and MAE values under the EHL_ST innovation density. Overall, the results indicate that the GARCH-EHL_ST model is statistically effective in forecasting the volatility of the inflation log-returns relative to other models based on the MSE and RMSE results while the GJRGARCH-EHL_ST model is statistically effective based on the MAE result. Furthermore, the EHL_ST innovation density has the least total rank value based on the performance measures among the eleven innovation densities considered in this study.

Overall, the comparison between the volatility models in this study basically supports the use of the symmetric GARCH-EHL_ST model for estimating and forecasting the volatility of the inflation rates. More so, the proposed EHL_ST density seems generally the best innovation density for both the GARCH (1, 1) and GJRGARCH (1, 1) models.

7.   Conclusions

In this research, the estimation of the exponentiated half logistic skew-t (EHL_ST) distribution parameters using six classical estimation procedures, namely the maximum likelihood, maximum product of spacing, least squares, Anderson-Darling, weighted least squares and Cramer-von mises is considered. Given that the theoretical comparison between these classical estimation procedures is not quite practicable, an extensive Monte Carlo simulation study is performed to compare the estimators in terms of average absolute biases, mean square error of each estimate and average parameter estimates. The results show that the criterions performance ordering from finest to poorest, using the overall ranks is MLE, MPS, WLS, OLS, ANDA, and CVM for all the parameter combinations. The MLE outperform all the other criterions with an overall rank of 32. Consequently, the performance of the EHL_ST model in fitting the inflation rates, showed that the EHL_ST model is quite competitive against the existing four-parameter models from the same family of distributions. Furthermore, volatility modeling of the inflation log-returns using the GARCH (1, 1) and GJRGARCH (1, 1) models with EHL_ST innovation density relative to the normal, student-t, generalized error and their skewed versions, generalized hyperbolic, Johnson (SU) reparametrized, generalized hyperbolic skew-Student, Normal inverse Gaussian innovation densities in terms of predictive performance using three forecast performance measures was carried-out. The research findings confirm that GARCH (1, 1) and GJRGARCH (1, 1) models with EHL_ST innovation density is optimally the best model based on model selection criteria and forecasts performance measures among other models. Overall, the results validate the superiority of the GARCH (1, 1) and GJRGARCH (1, 1) models with EHL_ST innovation density in both in-and-out samples performance over other models for inflation rates volatility modeling.

Acknowledgements

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sector.

Conflict of interest

All authors declare no conflicts of interest in this paper.