#### Abstract

Here, we present a method for a simple GARCH (1,1) model to fit higher order moments for different companies’ stock prices. When we assume a Gaussian conditional distribution, we fail to capture any empirical data when fitting the first three even moments of financial time series. We show instead that a mixture of normal distributions is needed to better capture the higher order moments of the data. To demonstrate this point, we construct regions (parameter diagrams), in the fourth- and sixth-order standardised moment space, where a GARCH (1,1) model can be used to fit moment values and compare them with the corresponding moments from empirical data for different sectors of the economy. We found that the ability of the GARCH model with a double normal conditional distribution to fit higher order moments is dictated by the time window our data spans. We can only fit data collected within specific time window lengths and only with certain parameters of the conditional double Gaussian distribution. In order to incorporate the nonstationarity of financial series, we assume that the parameters of the GARCH model can have time dependence. Furthermore, using the method developed here, we investigate the effect of the COVID-19 pandemic has upon stock’s stability and how this compares with the 2008 financial crash.

#### 1. Introduction

Modelling of financial time series is a very extensive area of research. One notable breakthrough in financial modelling is the discovery of the heteroskedasticity and conditional nature of volatility, manifesting itself in a slow stochastic process in the dynamics of price variance, in addition to the fast fluctuating price process itself. This motivates the development of the Autoregressive Conditional Heteroskedasticity models (ARCH) [1], which was later generalised (GARCH) [2]. The autoregressive processes allow a stochastic model to predict the price change probability density for a given time series. The level of return at a certain instance is described by a probability distribution (usually Gaussian) and the variance of the process. The variance of the process varies with time and is defined by both the variance and level of return at the previous time instance (*s*). However, GARCH is not limited to simply financial systems but to any system where this two scale stochastic processes is seen, for instance the study by Kumar et al. on atmospheric cycles in [3] or the study on pathogen growth by Ali in [4].

Extensive research has been undertaken to adapt the original Bollerslev GARCH model to fit empirical observations of time series [5–14]. Nevertheless, whilst these modifications of GARCH increase the accuracy of forecasting volatility, there is an increase in the complexity of the models and in the ambiguity of estimating model parameters. For example, fitting higher order statistical moments of financial series is an attractive approach for estimating model parameters [15, 16]. The original GARCH model allows us to obtain analytical relations between the statistical moments and the GARCH model parameters. In contrast, the modifications of the GARCH model lead to an increase in the complexity of the expressions for the higher order moments making the evaluation of the model parameters very challenging. Therefore, we wish to seek how effective the original GARCH model is at fitting higher order moments of empirical financial data series for different sectors of the economy. The focus of the study upon higher order moments is primarily due to the higher order moments’ ability to capture general aspects of a given distribution. It is well documented [17–20] that higher order moments are able to quantitatively represent the number of events that differ largely from the mean value of the process. In essence, they provide a different way to capture and effectively describe the statistical dynamics of the system.

There are several commonly used ways to estimate GARCH parameters [21, 22]. The most relevant and widely used is the Maximum Likelihood Estimation (MLE) [23, 24]. However, there are several pitfalls of such a task, the main one is the assumption of statistical properties within the empirical data. If this assumption is wrong, the estimated parameters are not reliable. Consequently, there has been significant work to adapt and implement the generalised method of moments (GMM) to the realm of financial studies [15, 16]. We can set a task to fit a certain set of statistical moments, for example , , and . As we have three parameters of the GARCH (1,1) model, we can solve this task using relations between the higher order moments and GARCH parameters derived in [2], and so we can undertake a GMM algorithm for a GARCH (1,1) model. We can ask if we can or cannot fit three empirically estimated moments of a chosen stock price series by three GARCH (1,1) parameters. The region where the parameters of the GARCH model can fit empirical moments shall be referred to throughout as the GARCH existence region or the “GARCHable” region. If we evaluate the time series and conclude that GARCH (1,1) parameters cannot fit empirical moments, then we can judge that the time series might no longer be purely stationary or a significant modification of the GARCH (1,1) model is needed to capture the dynamics.

As in the case of stocks and shares, the global economic climate and external factors are major stressors when determining the price of a given stock. Such a complex dependence of factors leads to a very fluid economic environment. Higher order moment analysis can determine the behaviour of the time series in response to this economic environment. However, economic factors affect the individual time series on different time scales. As such, the time window we analyse for the financial series will show different signatures for these different time scales as well as economic cycles and general tendencies. Therefore, it is plausible that truncating the time series into different time windows, we will gain different sets of model parameters for each time interval. The information about the changing behaviour of time series can manifest itself in a variation of the GARCH parameters and can identify changing economic factors and trends, including crisis periods [25]. The idea of truncating empirical data to relatively short time windows can be challenged by the necessity to have long data sampling to accurately estimate higher order statistical moments, especially, if the moment values significantly exceed the corresponding Gaussian values. This motivates us to develop a practical procedure by combining empirical studies of higher order moments within medium to long time windows with reasonable accumulated statistics and evaluating GARCH parameters from asymptotic analytical expressions of higher order moments derived in [2]. In this study, we analysed data for the period of 6 October 2000 to 6 October 2018, in most cases we use a subset of this data set. For example, this can be divided into a pre-crisis, post-crisis, and crisis period. This division is extremely valuable in deducing the statistical features that are inherent to an economic crisis. This will be reflected in the results we gain from evaluating certain statistical moments in the years from 2000 to 2018.

The study is organised as follows: in Section 2, we initially analyse the sixth-order moment for several companies and discuss the economic environments. In Appendix E, we present the findings for quarterly truncated time windows. In Section 3, we discuss the methods we will be using and how we have created the parameter diagrams in higher order moment space where GARCH can describe empirical data. Section 4 presents our findings for a GARCH model with a Gaussian conditional probability distribution (we will proceed calling these GARCH-normal models) for empirical time-series fitting, whilst also showing the failure of the GARCH-normal models to describe higher order moments of financial time series. In Section 5, we discuss a GARCH model with a double Gaussian conditional probability distribution (GARCH-double-normal models) to account for this shortfall. We also show how with the assumption of time-dependent parameters, the data we analyse can be described by nonstationary GARCH-double-normal models. In Section 6, we discuss problems faced when using the likelihood method for an empirical data set, with given fourth- and sixth-order standardised moments. In Section 7, we perform analysis upon data series from the COVID-19 pandemic crisis period. Finally, Section 8 concludes our study.

#### 2. Raw Data Analysis

In order to determine the behaviour of the moments of financial time series, we first highlight the time dependence of the sixth-order moment for several companies and a government bond (gilt) through the financial crisis of 2008. To do this, we use the daily closing price of each trading day over 6 month periods for 8 years, 2002–2010. We then use the following equation to calculate the nth-order moment:where we take to be 6 months, so the period we average over is 126 days (due to trading exclusion dates) and each time step a trading day. Here, we define as the logarithm of price change: is the closing price at day . In this study, we also consider the average over a longer time history, for example when , in essence, we consider an 18-year time horizon:where corresponds to 6th October 2000. We also evaluate empirical standardised moments for higher order moments:

Figure 1 shows the behaviour of the sixth-order statistical moment in response to a shifting time window. To create the plot, we take an 18-year time series, 2000–2018, and move a six-month long window from the start to finish, with nonoverlapping segments. We then calculate the maximum and minimum values of our error intervals, as described in Appendix A. For the time series between 2000 and 2007, we see a flat response of the higher order moment with respect to time. However, when we move the window over the 2008 financial crash, we see a large increase in the value of the sixth-order moment.

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We undertake this analysis for four banking securities and one commodity. For the banking stocks, we can see the value of the sixth-order moment increases within the region of the financial crisis and then once the crisis is over, we see the value of the moment return to this pre-crisis level. The behaviour is not seen in the commodity security, Gold ETFs. What we instead see, is a small increase throughout the financial crisis period, a small deviation from its level before. However, we do see a slight change in the behaviour of the moment of the security before the financial crash, perhaps a pre-cursor to the turmoil to come. It can thus be seen from this simple analysis, that the banking securities have a very distinct behaviour.

#### 3. Stochastic Model

In this section, we focus upon a GARCH-normal (1,1) model. We can see from Bollerslev’s work [2] that for such a model is a random variable with zero mean and possesses the conditional variance, . We define . Here, is a random process with zero mean and variance equal to one. Depending on the system, we wish to model the variable can be described by different probability distributions, see for example [8, 26–29]. Following [17], we first assume the conditional probability to be Gaussian, we will refer to such models as the GARCH-normal model. The GARCH (1,1) processes are defined via the relation:where, , , and are the parameters, whilst refers to the previous value of the conditional variance and the previous value of the innovation . If we know the exact probability density of a process, we could write the definition of moments by . However, we do not know the analytical expression for the probability distribution of the GARCH process. To resolve this problem, Bollerslev [2] proposed the recurrence relations for moments of the GARCH-normal (1,1) model:where

Therefore, we can derive equations for the unconditional variance, the fourth-order and sixth-order standardised moments [17]:

The relation (6) and (7) define all moments if we fix the three GARCH parameters, , , and . For a moment to exist, we should have . Solving , we obtain the functions . In doing so, we can create Figure 2 [2]. In this figure, we see the different curves of = where takes the value: 2, 4, 6, 8, 10, and 12. When the corresponding moments and standardised moments have finite values, whilst for , these moments diverge. Since the particular lines , below called divergence lines, separate the region of parameters where the -th moment exists and where it does not, we can interpret this as a parameter diagram in model parameter space, [2]. For the second-, fourth-, and sixth-order divergence lines, we are able to gain analytical expressions for the divergence lines, see Appendix B. In Figure 2, we present a filled area that shows the region of existence of the sixth-order moment. The red circle in this figure represents a set of parameter values that allow for the existence of the second, fourth, and sixth moments but not the eighth or higher, while for the black square in Figure 2, only the second- and fourth-order moments are finite.

Whilst the present problem of finding the GARCH moments knowing the three model parameters is straightforward, the inverse problem to estimate the three GARCH parameters, if three moments are known, is much more complicated and reduces to a set of transcendental equations which are hard to solve.

#### 4. GARCH-Normal Models

##### 4.1. Company Trajectories

Here, we will consider the situation of when we need to fit only the second- and fourth-order moments, or equivalently, fitting the unconditional variance and fourth-order standardised moment . Since the GARCH-normal (1,1) model has three parameters, we can conclude that we can express two GARCH parameters, for instance, and , as a function of the third parameter . To do so, we use equations (8) and (9) to fit the empirical values of variance and the fourth-order standardised moment, , for a certain company, such that and . In doing so, we derive:

It is clear from these equations that for any value of and , we can find a family of one-parametric GARCH models, corresponding to different values of . So, we obtain the parametric curves; in (, ) space. Such curves represent the “company trajectories” with already fixed (empirical) variance and empirical fourth-order standardised moments .

In Figure 3, we see an extension of Figure 2 for a banking stock, a commodity, a pharmaceutical, and a mining company, respectively. The dotted lines represent the parameters of the GARCH-normal model for the given company’s trajectory. They allow us to see the “stability” of the time series, in essence, which statistical moments can exist for the GARCH description of the empirical data of a certain company. It is evident, for the longest time period (18 years) that apart from the gold ETFs (Exchange Traded Funds), trajectories of all other companies lie above the divergence line of the sixth-order moment. This implies that the empirical values of the second- and fourth-order empirical statistical moments do not allow for any higher order moments to be fitted via a GARCH-normal model.

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If we decrease the time window of data collection, for example a year, 6 October 2000 to 6 October 2001, or even six months, 6 October 2000 to 6 April 2001, then we can see the migration of the company’s trajectory to deeper inside the stability region in the (, ) plane, where higher moments are finite (see Figures 3(a)–3(d)). We have also examined the time windows of nine months, fifteen months, and three years. In these figures (Figures 3(a)–3(d)), it is clear that the Rio Tinto 6-month time series allows the largest number of higher order moments to exist for its description within the corresponding GARCH-normal (1,1) model. In general, the shorter a time series we take, the more moments exist for a GARCH-normal (1,1) model.

As we traverse a company’s trajectories in (, ) space, we can work out the value of the sixth-order standardised moment generated from the GARCH-normal (1,1) model for these specific and values. In Table 1, we see the minimum and maximum of which can be achieved. We can see that does not vary significantly along the company’s trajectory, resulting in a problem to fit diverse values of the empirical sixth-order standardised moments.

##### 4.2. Methods of Parameter Fitting

If we want to fit the second, fourth, and sixth moments, the values of the parameters must be below the divergence curve: , which does not cover all parameter space for the existence of the fourth and second order moments. This can result in some values of the fourth and second moments, or fourth-order standardised moment and the second-order moment being unreachable for GARCH modelling, see Appendix C.

Let us consider the algorithms we can use to fit empirical values of and which can be reformulated in terms of the variance as well as the fourth- and sixth-order standardised moments and , respectively. In the first approach, we present and as a function of , that is, and , from equations (11) and (12), then numerically solve the equation:

To find the value of , this method is inspired by the trajectory analysis we use in the previous section. We search for by traversing the trajectory and trying to fit the empirical sixth-order standardised moment. However, if is lower than the minimum or larger than the maximum of possible stated in Table 1, this equation cannot be solved, indicating that the GARCH-normal model with such a value of the empirical sixth-order standardised moment does not exist.

In the second approach to fit empirical values of , and , we first fit the empirical fourth- and sixth-order standardised moments using the fact that and do not depend on , see equations (9) and (10). Therefore, we can reduce the problem to two equations:

Allowing us to evaluate values of and reserve to the fitting of variance: . The set of equations (14) can be further reduced to one equation by eliminating using the first equation of the set namely

And substitute it to the second equation of (14). This enables us to write the one-variable equation:

Note, we similarly can exclude , resulting in equations for .

##### 4.3. Phase Diagram

Equations (14) can only be solved for some region, in standardised moment space, (, ), which is the region between the black lines in Figure 4(a). This is the region of phase space where the respective values of the fourth- and sixth-order standardised moments can be fitted by a GARCH-normal model. For example, the first point is inside the “GARCHable” region. However, the second point is outside of the “GARCHable” region, highlighting that these moment values cannot be fitted by a GARCH-normal model. Therefore, no solution is possible to equations (14).

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To evaluate the appropriateness of a GARCH-normal (1,1) model for the fitting of higher order moments in stock market data, we shall be investigating time series for companies of different sectors of the economy by estimating their empirical values of the fourth- and sixth-order standardised moments and comparing with the GARCH-normal (1,1) parameter region in (, ) space, Figure 4(a). To see the effect of the length of the time window on the ability of the GARCH-normal (1,1) model to fit empirical moments, we study data in different economic periods. We start by taking a time window of of the overall time series and increment in percents up to its full length (see Figure 5), an example of this can be seen for the several stocks in Figure 4(a), [30], ignoring time windows shorter than 30 days. We then overlap these data points on top of the “GARCHable” region detailed above. It can be seen from Figure 4(b), the distribution of empirical data points highlights a distinct corridor where the data sits. We generate much more information by this method than working out the errors using a standard error procedure. In fact, we see distinct areas in the parameter space where more empirical data points reside than others. We further show by this method there are only very specific regions where the empirical data lies, and it does not span all of the parameter space, seen by the grid of points in Figure 4(b).

We do not see the empirical data inside of the GARCH-normal (1,1) phase region for the time period analysed. Therefore, we can say that a GARCH-normal (1,1) model is unable to simultaneously fit three even higher order moments of the empirical time series we have studied.

#### 5. GARCH-Double-Normal Models

Since we cannot fit the fourth- and sixth-order standardised moments with the GARCH-normal (1,1) model, we consider a GARCH model with the more flexible double Gaussian conditional distribution. Such GARCH-double-normal models have been extensively researched [31–33], where the authors use them for volatility and exchange rate modelling. The conditional double Gaussian distribution can be written as

In addition to an obvious normalisation condition:we also have constraints on the second moment:

Due to the requirement that the conditional distribution for a GARCH process should have variance equal to one. We can introduce two more convenient parameters (the 4th and 6th moments of the conditional distribution) which fully define the distribution in (17):

The parameterisation (20) and (21) of the double Gaussian distribution allows us to generalise Bollerslev’s equation (9). The second-order moment is not affected and is still determined by equation (8), while the fourth- and sixth-order standardised moments for GARCH with double-Gaussian distribution can be written as

Using the methods described prior and based on the existence of solutions of the set of equations (14) with the corresponding standardised moments, defined by equations (22) and (23), we create a family of phase diagrams parameterised by and . To understand which empirical values are achievable using a GARCH-double-normal model, we need to understand restrictions for the whole family of phase diagrams. We see that these are bounded due to limitations for and obtained in Appendix D (conditions 43 and 44). These limitations require all phase diagrams be started from points above the dashed line, Figure 6. As such only data above the dashed line can be described by a GARCH-double-normal model (which is the case for the empirical data collected for the securities we have considered here).

##### 5.1. Time Windows

In Figure 6, we see three-parameter diagrams for three different double Gaussian distributions. Parameters for these diagrams are given in Table 2.

Figure 6 demonstrates how altering the parameters and of the GARCH-double-normal model enables us to capture different time windows of the empirical data. The data used in Figure 6 is for the Bank of America time series from 6 October 2000 to 6 October 2018. We truncate the time series into different lengths. We start with of the overall length and increment by up to the whole length of the time series. In other words, the first, most left point, corresponds to 43 days of data (from 06/10/2000 to 04/12/2000), the second point corresponds to moments obtained for 86 days of data (from 06/10/2000 to 31/01/2001), and so on. For the leftmost phase diagram, we use a double Gaussian distribution with . This allows us to fit , and for the time window of duration in the interval, days. When fitting higher order moments for longer time windows, we need to use double Gaussian distributions with parameters summarised in Table 2. It is not possible to gain a GARCH process with a double Gaussian distribution to capture all of the empirical data’s higher order standardised moments. We denote this behaviour as the local ability to model higher order moments of financial time series by the GARCH-double-normal model. Figure 6 only uses the data from the Bank of America; however, this behaviour is seen throughout the empirical data we have studied. In order to capture the empirical data, we must first decide on the time window we wish to model and then ascertain a suitable distribution that will capture this window.

##### 5.2. Time Dependence of GARCH-Double-Normal Parameters

Once we have fixed the time window we wish to analyse, we can study what happens when the window with this fixed duration shifts in time. This can be done by attributing to the higher order moments a time moment, , corresponding to the middle point (the median) of the time window, see equation (1). This can be seen in Figure 7, where we detail the schematic of a fixed time window moving in time for a long time series.

If we fix the double Gaussian distribution (in essence, select certain and ), we can gain the set of GARCH parameters, , and that describes the particular time median. If we change the time window we look at by moving its time median, then the GARCH parameters , , and also change. Below, we observe that the GARCH parameters , , and significantly vary with the moving time window, highlighting the nonstationarity of our modelling.

Given equations (8) and (22), we are able to define trajectories in (, ) space for a fixed value of and . Unlike the GARCH-normal methods, we now have the trajectories which change when varies. These can be seen below:

Now for each desired dataset we can use the trajectories in the same manner as we have done with the GARCH-normal model. We can plot along the trajectories of (, ) using the running parameter , overlaying this with the empirical value (see Figure 8).

From the above method, we can recover the value of that allows the fitting of , , and , where is the median of the running window, enabling us to create Figure 9. This is done for several banks: Lloyds Bank, Barclays Bank, and Bank of America, and a commodity, Gold ETFs. We seek to find a fingerprint of the companies’ GARCH parameters through the financial crisis. It is evident from Figure 9 that the banking sector has a unique behaviour in response to the crisis. We see an initial flat signal, but when the crisis period occurs we see an increase in the parameter value followed by a very dramatic reduction. This behaviour is mirrored in the commodity, Gold. We propose to use this specific behaviour exhibited by the banking companies as an indicator for future banking crisis periods. In Figure 9, we highlight an interval of the value which reflects the error interval of the parameter. This was recovered using the same method as described in Appendix A, the interval reflects the maximum and minimum values, a similar method to that used in Figure 1.

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#### 6. Likelihood Estimations for and

Here, we consider a likelihood estimation for the parameters of a GARCH model using higher order moments. We focus on the procedure of fitting when the empirical data is located outside of the “GARCHable” region of the GARCH-normal (1,1) process shown in Figure 5(b). As we can see from below, the Maximum Likelihood Estimators (MLE) and the methods outlined in this study are the same when we are inside the “GARCHable” region. However, when the empirical points are outside of this region, each of the methods presented above does not allow us to estimate the GARCH parameters. In this section, we will analyse if and how a likelihood estimate can resolve the issue.

If we follow the method of error estimation described in appendix A, we need to shift a central point of the studied time window. For each central point of the studied time window we estimate the , . Considering an ensemble of these moments for different central points as a dataset, we estimate the mean values, , , the standard deviations , and the mutual Pearson correlation coefficient , of these time windows. The statistics of the studied data points of the fourth and sixth have not been investigated as of yet. In the absence of such a study, we assume that the data’s distribution can be approximated by a bi-variant Gaussian function. To fit this distribution to the data, we can maximise the corresponding cost function, which is in line with the common use of the Maximum Likelihood Estimation (MLE). As such, we can follow the MLE methods, [21, 23], and derive a cost function that needs to be optimised:

Moreover, if we take the derivative of this cost function with respect to and and setting the derivatives equal to zero, we will recover equations (14) (section 4.2), which have solutions inside of the “GARCHable” region. Note, if we assume that the fourth- and sixth-order standardised moments are distributed according to a Gaussian law, the minimum of the cost function (26) will correspond to the maximum probability density. Furthermore, based on the cost function we can define a distance as:

By defining, , , and . Since we consider a GARCH process, we have to assume, and described by equations (9) and (10), respectively. We have used a similar algorithm to that defined in the phase diagram section to find the minimum distances between the “GARCHable” region and the empirical data. The values and , which minimise the distance , could be considered as the best likelihood estimation of the parameters. We have investigated the dependence of the minimum distance on the size of empirical data collection windows for Gold ETFs and Barclays Bank (see Figure 10). The distances exceed 1 indicating the low probability of the system described by the GARCH process in agreement with our conclusions above.

When we study the empirical data we discover the empirical values of the fourth- and sixth-order standardised moments are highly correlated, , see insert of Figure 10. This is to be expected since both higher order standardised moments are affected by rare-events within our time series. As such, if the number of rare-events increases so does the value of both higher order standardised moments. Due to this high level of dependence (correlation) between our random variables ( and ), we can evaluate the probability of fitting our GARCH model to empirical data using higher order standardised moments by just one of the random variables. If we choose to use as our random variable and assuming this variable is to follow the distribution: , where and are defined as above. When we calculate such values, we see the values of the probability being low , highlighting our conclusion that fitting empirical data by a GARCH-normal (1,1) model is very unlikely when we wish to fit higher order standardised moments.

#### 7. Analysis of COVID-19 Time Window

The study focusses primarily upon the 2008 financial crash; however, in recent times, the world has undergone a much more profound economic and social shock, the COVID-19 pandemic. It is reasonable to assume, the pandemic would create a similar, if not greater, impact on economic and financial systems, given the nature of the crisis period. Therefore, we carry out the analysis, namely, deriving the time evolution of the GARCH parameter, and the time evolution of the sixth-order standardised moment for the empirical time series of several stock instruments within the COVID-19 pandemic time period.

In Figure 11, we show the sixth-order standardised moments for Lloyds Bank, Barclays Bank and the Bank of America for the period 2019–2021. In each panel, we show the error interval using the same method described previously. It can be seen that Barclays Bank and the Bank of America show quintessential features of their sixth-order standardised moment evolution that is indicative of a crisis period: a sharp increase, followed by a sharp decrease around the start of the pandemic. Whereas, Lloyds Bank sees a steadier increase followed by a sharp decrease further into 2020, this could be attributed to the U.K.’s handling of the pandemic, as the UK did not go into lockdown until March 2020.

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We also investigate the evolution of the GARCH parameter, in the same time period. In Figure 12, we see the evolution of for Lloyds Bank, Barclays Bank and the Bank of America. It can be seen that in all empirical time series analysed, we recover the signal of the crisis period: a sharp increase followed by a sharp decrease, around the start of the pandemic. A strong indication of a crisis period affects the instruments.

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In Figure 13, we show the time evolution for the sixth-order standardised moment and the GARCH parameter for the S & P 500 for the COVID-19 pandemic and the financial crash of 2007–2009. In panel (a), we can see that when the pandemic started, around 2020, we gain a large increase in the moment’s value. This is followed by a very sharp decrease in the value. Whilst in panel (b), we show the evolution of the GARCH parameter , we see a very similar behaviour compared to the banking stocks of the same period. A sharp increase is seen in the parameter value, followed by the now expected sharp drop around the start of the crisis period.

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In panel (c), we detail the behaviour of the sixth-order standardised moment for the index over the period 2004–2012. It can be seen that the sixth-order standardised moment is very noisy and as such, we are unable to determine any behaviour that could allude to the type of economic environment the index inhabits at this instance in time. This can be attributed to the structure of the index. The S&P 500 index is the 500 largest companies by market capitalisation, within the United States of America as ranked by Standard and Poor’s credit agency. Therefore, a plethora of different sectors will be represented in such an instrument. This means that sectors which would normally be at risk from financial crises, for example banks, will be mitigated by the other sectors within the index. Moreover, in panel (d), we show the evolution of the GARCH parameter, in time. For this parameter’s evolution, we see a similar response as has been seen within the COVID-19 pandemic, an increase in the value of the parameter followed by a decrease. However, the magnitude of increase and decrease is much smaller than for the COVID-19 period.

In the two crisis periods studied, we gain different responses for the index’s sixth-order standardised moment, but gain a similar behaviour in its GARCH parameter evolution , albeit some difference in the scaling of the parameter. In the COVID-19 pandemic, all sectors of the economy were affected and as such, the index’s sixth-order standardised moment reflects this. Whilst, the financial crisis of 2008 predominantly affected banking companies, so we see a reduced effect in the standardised moment. In addition, we can see that the COVID-19 peak is an order of magnitude higher than the peak gained in the banking crisis. Due to this observation, we can conclude that the GARCH parameter behaviour allows us to analyse the depth of the crisis, the larger the peak, the deeper the crisis penetrates into the economy for the analysis undertaken on this index.

#### 8. Conclusion

We use the time series of The Bank of America, Barclays Bank, Citi Bank, HSBC, Gold ETFs, GlaxoSmithKlein and Lloyds Bank, among others, to highlight the inability of a GARCH (1,1) model with the Gaussian conditional distribution to fit higher order moments of empirical time series.

In discovering this, we turn our attention towards different conditional distributions to try to capture the empirical data’s higher order moments. We show that with the use of a GARCH-double-normal model we can fit the empirical data’s higher order moments. However, through this enquiry, we still cannot capture the long run dynamics of the empirical data. We show that it is only possible to fit a model to empirical data within certain time horizons. To model a different time horizon we have to change the parameters of the double Gaussian distribution we use.

Fixing the distribution within certain time horizons to enable the fitting of higher order moments, highlights that the obtained GARCH-double-normal (1,1) model describes a nonstationary process. Therefore, if we wish to describe a long time series by a GARCH-double-normal model, we have to truncate it to smaller time windows. In doing so, we have to potentially fit GARCH-double-normal models with different parameters (, and ) to each time window. Therefore, we produce a time dependence of the GARCH model’s parameters, for example, . As such, we are able to build up a time signature of the parameter through the 2008 financial crash for several companies. We focus our attention on the banking sector to distinguish any shared behaviour in the evolution of , through this crisis period. The banking companies’ values of have a distinct behaviour from other sectors of the economy, giving hope of a standardised signal of these periods. It is seen through the banking sector’s empirical data that before the financial crash there is an increase in and during the financial period, the value of reduces extremely quickly. A behaviour that is found among banking companies but not other securities’ time series. Analysis undertaken upon the COVID-19 pandemic period reinforces this belief. This finding is potentially useful for either forecasting or predicting financial turbulence in economic periods.

#### Appendix

#### A. Error Bar Calculations

To calculate the error intervals for the plots in this study, we carried out a time kernel type calculation. For the time window we investigate, we displace the starting point, by a value between . Therefore, we move the starting point by where, days. We are able to work out the value of the moments for each displaced time window. For the range of values of the moment we gain, we can either take all of the values of the moments and plot them, as seen in Figure 5(a), creating error “clouds” or we can calculate the maximum and minimum values and get an error interval, as seen in Figure 1. The schematic for this method can be seen in Figure 14.

We also detail within the study a method of using two dimensional histograms to show the point density of the empirical data in the space versus the “GARCHable” region, see Figure 5(a). To do this, we displace the time window as described above and plot all of the moment values we gain from the displaced windows. Taking these and plotting a histogram of the empirical points we are able to show the number of points with certain moment values. To gain more points we move the time window more than the ten days highlighted above. In some cases we displace the window by 100 days.

#### B. Divergence Line Expressions

For the fourth- and sixth-order moment we can obtain the divergence line explicitly, and so derive:

For higher order moments, the divergence lines are defined by high order algebraic equations, which cannot be solved analytically.

#### C. Conditions for

For a general GARCH conditional probability distribution with variance equal to one, the equation for the sixth-order divergence line (the denominator of equation (10)) becomes:

Expanding in a series with respect to we derive:

Substituting this into our sixth-order divergence line we can equate coefficients up to the second order and so becomes:

If we now neglect orders higher than the second, we get the equation; . Substituting this into our equation for the fourth-order standardised moment, we obtain:

Considering the limit when we finally obtain:

#### D. Relations between the Parameters of the Double Gaussian Distribution and Its Higher Order Moments

The normalisation condition for the double Gaussian distribution described by equation (17) is . Substituting into equation (19), , we get:

Assuming , and substituting the equations for and into the fourth and sixth moment equations we derive:where and . Introducing the new variables, and , we can simplify the obtained equations:

Solving the above equations for and we finally obtain:

Since and must be positive, this gives us three conditions; , and . Due to the first condition, we can disregard the second as . Using relations between and and and we obtain:

We can then set-up equations for solving or :

Solving for , we can obtain relations for the parameters of the double Gaussian distribution:

And so,

Since, and must be both real and positive, this gives us the relation: . As such we get the following inequality:

Solving this inequality for , we get the condition: . Obviously, is always larger than , and so we always satisfy the condition shown in equation (D.11). As such, the parameters and have to only obey the conditions shown in equations (D.6) and (D.7).

#### E. Quarterly Time-Series Analysis

In this appendix, we show the time evolution for the sixth-order standardised moment for Lloyds Bank and the S&P 500 index, when we truncate the data in quarterly time periods (around 63 days). In Figure 15, it can be seen that compared with the analysis undertaken for the truncation of the sixth-month time period, the quarterly truncation causes very noisy signals. As such, we cannot discern any shared behaviour around any moment in time, particularly not the financial crash of 2008.

**(a)**

**(b)**

We present these findings as an illustrative exercise, taking such a short time window can cause the statistics of these periods to be inaccurate. If we are to assume a Gaussian distribution for the error of these statistics, then the standard error of the metrics scale is , where *n* is the length of time window we take, [34]. Given the sixth- month window is over 100 days, the error for this window is below one percent; however, when we take a window of half of this length, we double our error [30]. Therefore, the conclusions reached by the analysis of quarterly time periods should be studied with caution.

#### Data Availability

The authors wish to note the data that support the findings of this study are openly available for all empirical data from https://www.investing.com. Where it was possible, the data have been taken from the London Stock Exchange. The code is available upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

The authors would like to thank Professor Alistair Milne for his guidance and advice on this work. A preprint has previously been published [35]. The funding for the research was provided as part of the employment of the authors by Loughborough University.