GARCH models and all its derivatives have been widely used as shown in the literature review section. They consist of two key components i.e. the mean equation and the volatility equation ([1]) given by:

The mean equation

The variance equation

where: σ t 2 is the variance, ω is the constant, with the α i and β j coefficients of the ARCH(p) and GARCH(q) parameters on the model.

The GARCH model, in its natural state, offers a compact and interpretable state space representation, distinguishing it as a versatile basis for modeling time-varying volatility in financial time series. This representation facilitates the incorporation of traditional econometric methods with modern state space techniques, improving the general appreciation of the dynamics driving commodity volatility.

The GARCH model is intrinsically dynamic, capturing how current volatility is influenced by past squared returns and previous volatility estimates. This characteristic aligns impeccably with the state space framework, which is designed to model systems characterized by observable outputs (returns) and latent states (volatility). Consequently, the state space representation not only simplifies the modeling process but also allows for a more nuanced analysis of volatility in financial markets ([41]). The return equation is as in (1) while the variance equation is;

This representation allows for an efficient way to analyse the time varying nature of parameter and is predominantly useful in estimation and forecasting at the same time maintain the adaptability to various types time series data paving the way for KF recursions ([11]).

There are principally two approaches to tackling GARCH models within a state space framework. The first approach involves utilizing the nonlinear representation introduced by ([37]) and applying the Extended KF to estimate the model parameters. The second approach involves a linearization process through Taylor series expansion, which simplifies the estimation of the GARCH model by approximating the nonlinear dynamics ([21]).

Since the GARCH model is nonlinear as introduced by ([7]). ([37]) presented formulations for typical GARCH model with Gaussian perturbations for p, q ≥ 2; GARCH(p, 1), p ≥ 2 and GARCH(1, q), q ≥ 2. Earlier on in 2011, they derived a nonlinear state space representation of the GARCH(1, 1) model.

a) Basic Structure

where θ = ( ω , β 1 , α 1 ) are parameters that you estimate from historical data using the GARCH model. This part updates the state variables U 1 , U 1 , t and U 2 , U 2 , t Here, U 1 , t represents the conditional variance (volatility) at time t, and U 2 , U 2 , t is a normalized measure of the mean return.

b) Function of variables

is the function of the unobserved recursive state variable, U 1 , t = σ t 2 and

U 2 , t = μ t σ t ([37]). This function shows how the previous

volatility and returns influence current volatility. If past returns were high, it can suggest that future volatility might also be high.

To tackle the filtering problem involving nonlinear system dynamics (both state and observations), we consider the Extended Kalman Filter (EKF) as proposed by ([22]). For simplicity, we will assume that the system is free of external inputs. Given that the EKF provides approximations of the optimal parameter estimates, the nonlinearities in the system’s dynamics are effectively captured by a linearized representation of the nonlinear model, centered around the most recent state estimate ([51]). To achieve optimal results, the linearization process should closely approximate the nonlinear system within the bounds of the state estimates. The EKF algorithm refers to the works of citation. The prediction cycle and the filtering cycles are explained in detail. In that context, the EKF assesses the volatility of asset returns, considering volatility as a latent state variable shaped by past return data and fresh observations. The predict cycle focuses on forecasting future volatility based on established patterns, while the filtering cycle refines these predictions by incorporating actual return data, thereby enhancing accuracy. Both the EKF and the GARCH model address relationships that are not perfectly linear; volatility can exhibit unpredictable changes influenced by past returns and historical volatility. It is important to note that, if the assumptions underlying the model’s linearizations are inaccurate, the EKF may produce unreliable volatility estimates ([29]).

Linearization of GARCH type models refers to an approximation technique that simplifies the modeling and estimation process. To linearize GARCH models, the conditional variance equation is represented as a linear function of the model parameters and the lagged squared residuals, capturing the essential dynamics in a more tractable form. Linearization is generally accomplished by applying a first-order Taylor series expansion of the conditional variance equation around its unconditional mean ([43]). Consider a simple GARCH (1, 1) model whose conditional variance equation is given in (2), of course equating p = q = 1 . The resultant equation will have σ t 2 (volatility), ε t − 1 2 (sqaured innivations), ω (constant), β 1 and α 1 respective ARCH and GARCH parameters of the model.

From this state space representation, we can extract and estimate the GARCH model parameters using statistical techniques, such as MLE-parameters that maximize the likelihood of the observed data given the model or the KF-algorithm iteratively updates the estimates of the state variables (volatility and returns) and can be utilized to derive estimates of the parameters.

Another method would be to use the first-order Taylor series ([42]) of the conditional variance equation around its unconditional mean:

By applying the first-order Taylor series expansion, we create a linear approximation of the conditional variance (volatility) equation. This means we can break down our complex volatility model into something more manageable, which allows us to use standard statistical methods.

Yielding a linearized equation:

Once we have this simpler linear form, we can use the KF to estimate model parameters. In the context of commodity returns, it helps us continuously update our estimates of volatility as new price information comes in ([42]).

The Kalman filter can be used to estimate unobserved states (like volatility) over time. The outputs from the Kalman filter (e.g., estimated volatility) can be used to compute the likelihood function. This approach leverages the dynamic nature of the Kalman filter, allowing for state estimation that evolves with the data. The methodology provides a recursive way to update estimates as new data arrives, which can better capture changes in volatility over time. Computing programs such as R and Python have excellent functionality to handle these algorithms.

The tractable MLE for SS_GARCH models is supported through theoretical derivations that leverage properties of the KF and the structure of SSMs. SS_GARCH models are capable of incorporating non-Gaussian error distributions (e.g., STD, SSTD), which is crucial for modeling the fat tails and skewness observed in commodity returns ([2]). This model can be expressed in a state space framework, which allows us to describe the dynamics of both the observed variable (returns) and the unobservable state variable (volatility) in a convenient form given by Equations (1) and (3), where ϵ t | F t − 1 ~ N ( 0 , 1 ) with ϵ t an independent standard normal.

Proof

To derive the MLE in a tractable form, we first notice that the joint distribution of observations can be expressed recursively. Below, we express the log-likelihood function;

Substituting the conditional distribution of the observed returns (Gaussian) ([12]), we get:

This formulation shows that the likelihood can be computed if we can derive σ t 2 through its state space representation. By utilizing the KF’s updating mechanism, we can iteratively compute σ t 2 efficiently, facilitating the efficient maximization of the log-likelihood function to derive the necessary model parameters for the GARCH framework as required.

Given the structure of a GARCH model, we deal with latent volatility states that are not directly observable. In an SS_GARCH model, the estimation process is meticulously structured into three integral steps. Initially, the process begins with initialization, where initial estimates for both the state and variances are established to provide a foundation for subsequent calculations. Following this, the prediction phase employs the state equation to forecast the next state, particularly focusing on the anticipated volatility of the financial series.

Finally, the model enters the update step, where the most recent observation is integrated to refine the state estimates and update the likelihood, thus enhancing the precision of the model’s forecasts. This methodical approach enhances the modeling of time-varying volatility, enabling a thorough representation of the dynamic traits intrinsic to financial returns.

In this section, we demonstrate that SS_GARCH models can effectively accommodate nonstationary processes. This is achieved by defining the state transition matrix to encapsulate the dynamics of GARCH processes. In the scenario of a nonstationary GARCH process, the model integrates parameters that vary over time or utilizes a time index within the state equations to accurately capture the evolving nature of the underlying volatility, thereby enhancing the model’s adaptability to fluctuations in the data.

The independence of ϵ t enables inference about the entire sequence of σ t and thus about the likelihood of the observed data to be performed despite potential nonstationarity, within the foundational data-generating mechanism. This is mathematically substantiated by employing the innovation process:

where r ^ t | t − 1 represents the forecasted value derived from previous observations. and the Kalman gain adjusts the impact of ϵ t . The tractable representation through the KF allows for efficient optimization of the likelihood function, ensuring that both stationary and nonstationary GARCH models can be estimated accurately.

To generalize the SS_GARCH model for non-Gaussian errors, we replace the Gaussian assumption with a more flexible distribution that captures skewness and kurtosis (Bianchi et al. 2011). Common candidates for non-Gaussian distributions include the STD, SSTD, GED and PIVD among many others. The extension involves allowing ϵ t ~ S k e w − t ( γ , ε , ϑ ) , γ is the degrees of freedom, ε represents the skewness and ϑ is the scale parameter.

Proposition1: Given a non-Gaussian distribution for ϵ t , the likelihood function needs to be articulated in relation to the probability density function (PDF) of the selected distribution. For instance, using the SSTD, the density function can be defined as:

Proof.

Since L, the Likelihood function given above, is;

This equation represents the log-likelihood function, where f ( r t | σ t ) is the probability density function, the pertubations of a chosen distribution ([12]).

Substituting the skew-t density function gives:

The KF needs to be adapted to handle the non-Gaussian innovations. Instead of using the Gaussian innovation, we need to re-evaluate the Kalman gain and the state update equations to account for the PDF of the selected non-Gaussian distribution ([44]). Firstly, the innovation process is modified as in ([44]):

Parameters are estimated using the modified log-likelihood function resulting from the chosen non-Gaussian distribution.

To address nonstationarity, the state transition equations can include time-varying parameters or local trends:

Here, φ 0 ( t ) can be represented as a function of time or other variables, enabling the model to adjust to changes in the underlying data distribution over time.

The theoretical groundwork provided by the two propositions justifies the application of state space estimation techniques for GARCH models. The ability to leverage KF algorithms facilitates the efficient handling of latent states and parameter estimation processes, demonstrating the attractiveness of this approach in both stationary and nonstationary contexts ([46]).

By modifying the observation equation to capture non-Gaussian behavior and incorporating suitable probability distributions, we can develop a SS_GARCH model capable of accommodating skewness and heavier tails. More recent theory around these distributions was presented by ([32]; [35]) as in proposition 2 below.

Proposition 2: Given a non-Gaussian distribution for the error term ϵ t the likelihood function L ( θ ) is defined as the product of the probability density functions (PDFs) of the chosen distribution evaluated at each observation, conditional on the past information ([9]):

where, θ is the vector of parameters, the state space GARCH model to be estimated, F t − 1 : θ is the ([18]) conditional PDF of ϵ t given the information set F t − 1 and parameters θ and F t − 1 represents the information set available up to time t − 1 , typically including past, ([47]). Recent studies have used the skewed t-distribution to capture the skewness and heavy tails often observed in financial time series.

The dataset consists of 5722 daily closing prices spanning from January 2, 2001, to October 26, 2023. This data was partitioned into an in-sample portion of 80%, while the remaining 20% was designated as out-of-sample data for model calibration. Additionally, commodity price data from 2001 to 2023 was similarly divided in a 4:1 ratio between in-sample and out-of-sample segments.

Actual return is a return which has occurred, calculated based on the historical data ([5]). To calculate the actual return of the stock from each sample by using daily commodity price;

where R i t is the i t h commodity return on day t with price P i , t , see ([5]).

The KFAS package in R is a powerful tool for estimating state space models, including GARCH models, which are particularly useful for modeling volatility in financial time series, such as commodity returns.

The KFAS package in R provides a basis for estimating state space representations of GARCH models, mostly useful for modeling commodity returns as a combination of underlying volatility processes and observation equations. Central to KFAS is the KF algorithm. It iteratively estimates latent state variables from observable returns and updates its estimates as new data becomes available. The estimation process begins with initializing the model using initial guesses for the state and variances, followed by making predictions based on state space equations. As new commodity return data is replicated, the model refines its parameter estimates through optimization procedures ([25]).

The state space formulation of a GARCH model was presented in (1) and (2) in Section 1. Below, we present Kalman filter predictions and update formulations.

I)PredictionStep

Equation (17) predicts the conditional variance based on past observations. This equation is essential in predicting the state.

Predicttheobservation:

After presicting the state, we predict the observation as given in Equation (18).

II)UpdateStep

Equation (19) calculates the innovation or prediction error, which is crucial for updating the state estimates.

Equation (20) is used to update the estimation of the state variance:

where K is the Kalman gain, which can be calculated based on the current estimate of the state and the error as in section 3.2.2.

While the KF provides a mechanism for updating the state estimates, parameters ω , α 1 and β 1 are then estimated using optimization techniques that minimize the likelihood function derived from the recursive updates and observations.

Table1 shows descriptive statistics, split into three categories as follows: the measures of central tendency, skewness and kurtosis, independents, normality, and stationary tests. A general increasing trend in commodity prices is evidenced by all positive return means. Skewed distributions and volatility clustering are again evident on all commodity return indices. Heavy tailed distributions are recommended.

Table 1. Summary statistics for commodity returns.

The ADF test and JB test for normality give significant p-values for all five commodities return series. This implies that the normality assumption is not met at all levels of significance. The Ljung-Box test, at Q(5), Q(10) and the squared values indicate that autocorrelation is insignificant for returns but highly significant in squared returns indicative of persistence in the volatility process of commodity returns. Heavy tailed asymmetric GARCH-type models are recommended under these circumstances.

The augmented Dickey-Fuller (ADF) and Phillips and Perron (PP) tests rejected the unit root hypothesis while the Kwiatkowski failed to reject the unit root hypothesis of stationarity. Hence commodity price returns are largely stationary in mean.

Figures 1-4 show plots of the time series, return series, box plots, Q-Q plots and ACF of return of square returns for the five commodities. The returns of all the indices exhibit volatility clustering. In the same vein, all the ACFs and PACFs of the returns series the absence of long memory. All the QQ plots show that the tails of all the commodity indices’ returns are heavier than the tails of normal distribution. They indicate the presence of heavy-tailed distributions and asymmetric dispersion of all the returns.

Figure 1. Composite analysis of commodity prices and returns: Left_top: Daily Price; Right_top: Gold_Returns; Left_bottom: Returns_QQ; Right-bottom: Returns_ACF for Gold.

Figure 2. Composite analysis of commodity prices and returns: Left_top: Daily Price; Right_top: Crude_Returns; Left_bottom: Returns_QQ; Right_bottom: Returns_ACF for crude oil.

Figure 3. Composite analysis of commodity prices and returns: Left_top: Cotton_Price; Right_top: Cotton_Returns; Left_bottom: Returns_QQ; Rihgt_bottom: Returns_ACF for Cotton.

Figure 4. Composite analysis of commodity prices and returns: Top-left: Daily Price; Top-right: daily Returns; Bottom-left: Returns QQ; Bottom-right: Returns Box plot for Lithium.

Based on the Lo Modified R/S test, there is no evidence of long memory among all the for commodity returns. We failed to reject the null hypothesis of no persistence (Table2).

Table2. Lo Modified R/S test.

Note: t-value ranges are: 90%: [0.861, 1.747], 95%: [0.809, 1.862] and 99%: [0.721, 2.098].

In financial modeling, Monte Carlo simulation is a powerful tool for evaluating complex statistical methods ([13]). Simulation is useful as it mimic reality and allows different analytical methods to be tested and different scenarios to be tried before using empirical data. By creating artificial datasets that mimic actual financial data, we can meticulously evaluate the efficacy and plausibility of our models.

In this section, we simulate a return series using specified simulation parameters. This return series is then used to estimate the GARCH parameters using both maximum likelihood estimation (MLE) and state space techniques. We will compare the methods to determine which approach most correctly reproduces the theoretical parameters used to simulate the series.

Table3. Model performance with simulated GARCH (1, 1) data under normality assumptions (Sample size n = 1000).

Table4. Model performance with simulated GARCH (1, 1) data under STD assumptions (Sample size n = 1000).

Table5. Model performance with simulated GARCH (1, 1) data under SSTD assumptions (Sample size n = 1000).

Table6. Model performance with simulated GARCH (1, 1) data under PIVD assumptions (Sample size n = 1000).

The simulation results presented in Tables3-6 highlight the parameter estimates obtained from both the MLE GARCH (1, 1) and the State Space GARCH (1, 1) methods, alongside the theoretical parameters used for simulation with different innovations assumptions among them, the STD, the SSTD and the PIVD.

Estimates for the parameters ω , α 1 , and β 1 from both methods are compared to the theoretical values used in the simulation. For instance, the estimated ω from MLE is significantly higher than the theoretical value (0.1000), indicating a potential overestimation in the model’s volatility baseline. In contrast, the SSG method produces an estimate of 0.1441, which is also considerably lower and can be rounded to the theoretical value. The p-values for both methods are uniformly low (<0.0001), indicating that the estimated parameters are statistically significant. This suggests that both models can reliably capture the underlying volatility structure. The z-values further reinforce this significance, with both methods showing strong statistical evidence for their parameter estimates.

The findings indicate that the State Space method outperforms Maximum Likelihood Estimation (MLE) for parameter estimation in GARCH (1, 1) models at different error distributions. Notably, heavy-tailed distributions, such as the STD, SSTD and PIVD, demonstrated superior performance in terms of the AIC (−2.7582, −2.7926 and −2.6834, respectively) and the accuracy of parameter estimates relative to the theoretical values used during data simulation for the SSG. These results underscore the State Space approach as a more robust choice for time series analysis in financial contexts, where precise volatility estimation is essential, especially in the presence of heavy-tailed data characteristics. Overall, the simulation results validate the efficacy of using both MLE and state space methodologies for estimating GARCH model parameters.

After analyzing the simulated data, focus is now on empirical results. This section deals with validation procedures applied to real-life commodity data to assess the practicality and robustness of the coupled skewed GARCH and state space GARCH models across various error distributions. By examining actual market data, we aim to gain insights into how effectively our models capture the complexities of financial returns, including volatility clustering and asymmetry, thus enhancing our understanding of their performance in real-world scenarios.

Table7. Parameter estimates for the GARCH model, Crude oil returns.

Table8. Parameter estimates for the GARCH model, Gold returns.

From Table7, on Crude oil, the SSTD_SS_GARCH (1, 1) model exhibits all parameter estimates as statistically significant and achieves the lowest Akaike Information Criterion (AIC) value, indicating its superior performance compared to the other models.

Table9. Parameter estimates for the GARCH model, Lithium returns.

Table 10. Parameter estimates for the GARCH model, Cotton returns.

Table8 exhibits parameter estimates for Gold. The STD_SS_GARCH (1, 1) model demonstrated notable effectiveness for Gold, outperforming all other models assessed. Its superior performance is evidenced by its statistical significance across all parameter estimates and its favorable performance metrics, which collectively highlight its reliability and robustness compared to the alternatives.

Table9 shows parameter estimates for Lithium. The STD_MLE_GARCH (1, 1) model exhibited strong performance, surpassing the other models evaluated. Its effectiveness is reflected in the statistical significance of its parameter estimates and favorable performance metrics, which collectively underscore its reliability and robustness.

As depicted in Table10, the PIVD_SS_GARCH (1, 1) model outperformed all other models for Cotton returns. Its superior performance is highlighted by the statistical significance of its parameter estimates and advantageous performance metrics ([19]).

Estimating GARCH (1, 1) models for commodity returns through both state space and maximum likelihood estimation (MLE) techniques offer significant insights into the dynamics of these markets. MLE relies on several critical assumptions concerning the disturbances distribution and the model’s functional structure. If these assumptions do not hold, the resulting estimates can be biased or inconsistent. Additionally, MLE is sensitive to the choice of initial parameter values; poorly chosen initial values may cause the optimization algorithm to converge to local minima, leading to suboptimal estimates ([3]). The results obtained for the four commodity returns analyzed in this study acme the significance of understanding the characteristics of the data underlying data when selecting an estimation method. The adaptability and robustness of the state space method make it particularly suitable for modeling commodities with long memory properties ([26]). In contrast, MLE may be more effective in simpler, stationary contexts. As commodity markets continue to evolve, ongoing research into advanced modeling techniques that can accommodate various characteristics will be essential for accurate risk assessment, estimation, and forecasting.

To address the limitations of state space GARCH models, it’s essential to consider several key factors. These models often exhibit computational complexity, requiring sophisticated estimation techniques that can lead to longer processing times, especially with large datasets ([7]). They are also sensitive to initial conditions, where inadequate starting values may cause convergence issues and affect reliability. Additionally, the risk of model specification errors can result in biased estimates, highlighting the need for thorough diagnostics ([16]). While state space models provide flexibility, their interpretability may be hindered by complexity, limiting practical application ([7]). Finally, they generally require larger datasets for reliable estimation, which can compromise performance in data-limited scenarios. Addressing these considerations offers a more comprehensive understanding of state space GARCH models and their trade-offs.

4.7.1. Evaluation of Estimates

In Table11, the best-performing model’s performance and reliability were assessed using out of sample data to ensure that they accurately capture the commodity volatility patterns and can generalize well to new data. The comparable estimates across these models indicate that they did not suffer from overfitting, a common issue where a model captures noise instead of underlying patterns, leading to poor generalization to new data ([15]; [14]).

Table 11. Model validation using out of sample data.

These models maintained consistent estimates in out-of-sample applications, demonstrating an effective balance between complexity and generalization. Models that are overly complex may excel in training but underperform on new data, characterized by high variance and low bias ([34]). In contrast, overly simplistic models risk underfitting by failing to capture essential trends.

The findings suggest that these models are reliable for practical applications, showcasing robustness against overfitting while effectively generalizing to new datasets. This aligns with best practices in model validation and mitigates the risk of overfitting ([50]).

4.7.2. Forecasts Evaluation

The forecasting evaluation metrics presented in Table12 compare the performance of the State Space GARCH (SS_GARCH) and Maximum Likelihood Estimation GARCH (MLE_GARCH) models across different commodities: Crude oil, Gold, Lithium, and Cotton.

The SS_GARCH model consistently shows lower MSE values for most commodities compared to the MLE_GARCH model, particularly for Crude oil (0.0100 vs. 0.0102) and Lithium (0.0015 vs. 0.0012). Lower MSE indicates that the SS_GARCH model generally provides more accurate forecasts, which is significant for risk management and investment decisions ([39]). The same pattern was observed in MAE, particularly in Cotton, the SS_GARCH MAE (0.0108) is slightly lower than that of MLE_GARCH (0.0115). Regarding the DM Test, the p-values below 0.05 (e.g., for Crude oil, Gold and Lithium) suggest that the SS_GARCH model significantly outperforms the MLE_GARCH model in forecast accuracy for those commodities, reinforcing the validity of the SS_GARCH approach. The R² values indicate the proportion of variance explained by the model. While both models show similar R² values, the SS_GARCH model’s slightly lower values may reflect its more dynamic nature in capturing time-varying volatility ([14]). The other metrics in Table12 tell the same story.

Table 12. Forecasting evaluation metrics of the SS_GARCH and MLE_GARCH (1, 1) models.

4.7.3. Model Selection

In Table13, the SSTD_SS_GARCH (1, 1) model is particularly effective for crude oil and gold, as evidenced by its performance across the metrics. The PIVD_SS_GARCH (1, 1) model performs well for cotton. Although the STD_MLE_GARCH (1, 1) model is well-suited for lithium, all models exhibit different levels of effectiveness in capturing the intricate volatility patterns of commodity returns.

4.7.4. Sensitivity Analysis

Below we present sensitivity analysis of the best fitted model.

Table13. Top performing models out of sample test.