Non-stationary GARCH modelling for fitting higher order moments of financial series within moving time windows
FFitting higher order moments of empirical financial series withGARCH models
Luke De Clerk
Department of Physics, Loughborough University, Leicestershire, LE11 3TU, United Kingdom
Sergey Savel’ev
Department of Physics, Loughborough University, Leicestershire, LE11 3TU, United Kingdom
Abstract
Here we have analysed a GARCH(1,1) model with the aim to fit higher order moments for differentcompanies’ stock prices. When we assume a gaussian conditional distribution, we fail to captureany empirical data. We show instead that a double gaussian conditional probability better capturesthe higher order moments of the data. To demonstrate this point, we construct regions (phasediagrams) in higher order moment space, where a GARCH(1,1) model can be used to fit the higherorder moments and compare this with empirical data from different sectors of the economy. Wefound that, the ability of the GARCH model to fit higher order moments is dictated by the timewindow our data spans. Primarily, if we have a time series, using a GARCH(1,1) model with adouble gaussian conditional probability (a GARCH-double-normal model) we cannot necessarily fitthe statistical moments of the time series. Highlighting, that the GARCH-double-normal modelonly allows fitting of specific lengths of time series. This is indicated by the migration of thecompanies’ data out of the region of the phase diagram where GARCH is able to fit these higherorder moments. In order to overcome the non-stationarity of our modelling, we assume that one ofthe parameters of the GARCH model, α , has a time dependence. Keywords:
GARCH, Phase Diagrams, Double Gaussian, Empirical Data
JEL Classification: C10, C40, C50, C80
1. Introduction
Modelling of financial time series is a very extensive area of research. In general, there arelarge over simplifications of a financial time series, so there has been research into the modellingof time series with a plethora of different mathematical tools. The main issues are; the largetails present in most financial time series, the heteroskedasticity of volatility and the conditionalsecond order moments of price change, volatility; demonstrating the presence of a further stochasticdynamic in addition to those of price change. These stochastic processes evolve in the same systembut have different time scales; a fast changing stochastic process (the price) and a slow changingstochastic process (the volatility). This motivates the creation of the Autoregressive ConditionalHeteroskedasticity models (ARCH) by Engle [1] and later generalised (GARCH) by Bollersev [2].
Preprint submitted to Elsevier February 24, 2021 a r X i v : . [ ec on . E M ] F e b he autoregressive processes allow a stochastic model to predict price change probability of a giventime series. The level of return at a certain instance is described by a probability distribution(usually gaussian). Where the standard deviation of the process varies with time and is defined byboth the standard deviation and level of return at the previous time instance(s). However, GARCHis not limited to simply financial systems but to any system where this two scale stochastic processis seen, for instance the study by Kumar et al. on atmospheric cycles [3] or a study on pathogengrowth by Ali in [4].GARCH has been thoroughly researched. Whilst it captures the heteroskedasticity of volatility(something standard time series models do not) and the two time scales of price dynamics. Theoriginal model proposed by Bollerslev, [2], does not capture the leveraging effect of volatility andit does not accommodate for the jumps seen in volatility. When we derive expressions for higherorder moments, it is not possible to solve past the sixth order moment analytically for the smallestGARCH model. For longer-time history models, for example, GARCH(2,2), generating higher ordermoment expressions is a difficult task. Extensive research has been undertaken to adapt the originalBollerlsev GARCH model to fit the different empirical observations of time series. For example,there is the A-GARCH (asymmetric GARCH) model to account for the asymmetrical effect ofvolatility to price change, [5] and also the GJR-GARCH model to account for this asymmetry,[6, 7, 8]. One particularly notable improvement to account for the asymmetry is the E-GARCHmodel of Nelson, [9]. We can also hypothesise a GARCH model that has more than one GARCHterm and so looks at the development of a higher-dimensional system, such models are the BEKK-GARCH and DCC-GARCH models, [10, 11]; they have shown promise alongside the use of aCAPM (capital asset price model). Moreover, the increase in computational power in the early1990s allowed for the collection of trade (tick) data. With this increased data collection, one needsto consider how to model the type of volatility seen on short time scales, and so realised volatilitymeasures were created and incorporated in to GARCH frameworks, [12, 13]. In the majority ofmodelling, one assumes a stationarity of time series, however, this is not necessarily the case andas such the fractionally integrated GARCH (FIGARCH) models were proposed, [14, 15, 16]. Onefinal development that motivates the work in this paper is the conversion from discrete time modelsto continuous time models leading to the creation of the continuous-GARCH (coGARCH) models,[17]. With these developments of the GARCH model, there is an increase in complexity of analyticalsolutions to the moment equations. Therefore, we wish to seek how effective the original GARCHmodel is at fitting higher order moments of empirical financial data series for different sectors ofthe economy.GARCH is a stochastic model, therefore it enables one to fit time series for perceived stochasticphenomena, for example financial time series for stock prices. Three independent stochastic quan-tities of a particular time series can, in principle, be fitted by three GARCH parameters. If onecannot fit a GARCH model it is not enough to simply say that the time series we have in questionis not stochastic. There are a multitude of factors that could affect the accuracy of the GARCHmodel that can be fitted to a given time series, as presented in this paper. In particular, we showthat there is a limit to the time window that can be modelled by a GARCH process.The question is, therefore, how to fit prices for a time series. This is a difficult process asselecting values of the GARCH parameters depends on what we wish to achieve. One can set atask to fit a certain set of statistical data moments, for example (cid:104) x (cid:105) , (cid:104) x (cid:105) and (cid:104) x (cid:105) by the threeparameters of the GARCH(1,1) model. Alternatively, we can choose another set (cid:104) x (cid:105) , (cid:104) x (cid:105) and (cid:104) x (cid:105) . Obviously, we will have different parameter values for the two cases. Moreover, we can askif we can or cannot fit three empirically estimated moments of a chosen stock price series by three2ARCH(1,1) parameters. Here we will show, that it is often not possible. And by the use of phasediagrams in ( (cid:104) x n (cid:105) , (cid:104) x m (cid:105) ) space, we shall see where GARCH(1,1) can fit the corresponding moments.The region where the parameters of the GARCH model can fit empirical moments shall be referredto throughout as the GARCH existence region or the ‘GARCHable’ region.If we evaluate the time series and concludes that GARCH(1,1) parameters cannot fit empiricalmoments, then we can judge that the time series might no longer be purely stationary in natureor a significant modification of the GARCH(1,1) model is needed. Most time series are not purelystationary. As in the case of stocks and shares, the global economic climate/factors are a majorstressor when determining the price of a given stock. Such a complex dependence of factors leadsto a very fluid economic environment. With the changing of the economic climate, the time seriesof certain stock prices also changes in response to this. We can draw an analogy to a phase change.When we have a very large economic change, the ‘state’ of the time series changes in responseto this. In the case of a small economic change, we can deem the time series of this company togradual evolve its ‘state’. So in both cases, the GARCH model fitting the empirical moments wouldalso change. Moreover, we may not be able to find a GARCH model fitting long time series, dueto this change of state. So it may be plausible that we have a limit to the time horizon GARCHcan be used for. However, with this changing state in the stock price, one would expect GARCHmodels with different sets of parameters to fit the empirical moments for each time window of anempirical series. This presents the GARCH model with parameters evolving in time. Within thistime dependence we can see the response of these parameters to the economic environment. Theinformation about the changing of states of time series can identify changing economic factors andtrends, including crisis periods [18]. Our final motivation for our work is to allow an individual toevaluate how best to model a series they are interested in to gain the most accurate parameters, ifpossible, of a GARCH(1,1) model for their particular needs.The estimation data in this paper is for the period of 6th October 2005 to 6th October 2011 formost data sets presented, unless otherwise stated. This can be divided in to a pre-crisis, post-crisisand crisis period. This division is extremely valuable in deducing the statistics that are evidentwithin an economic crisis. This will be reflected in the results we gain from evaluating certain ordermoments in the years from 2005 to 2011.The paper is organised as follows, in section 2, we initially analyse the sixth order moment forseveral companies and discuss the economic environments. In section 3, we discuss the methodswe will be using and how we have created the phase diagrams. Section 4 presents our findings onGARCH models with a gaussian conditional probability (we will proceed calling these GARCH-normal models) for time series fitting, whilst also showing their failure to describe higher ordermoments of financial time series. In section 5, we discuss GARCH models with a double gaussianconditional probability (we shall proceed calling these the GARCH-double-normal models) to ac-count for this shortfall. We also show how with the assumption of time dependent parameters thedata we analyse is non-stationary. Section 6 concludes our paper.
2. Raw Data Analysis
In order to determine the behaviour of the moments of financial time series, we first highlightthe time dependence of the sixth order moment for a group of companies and a government bond(gilt) through the previous financial crisis of 2008. To do this we use the daily closing price of eachtrading day over yearly periods for 8 years, 2002-2010. We then use the below equation to calculatethe nth order moment: 3 x n (cid:105) ( t ) = 1 N (cid:88) t − N δt<τ 3. Stochastic Model In this section we focus upon a GARCH-normal(1,1) model. One can see from Bollersev’s work[2] that for such a model; x t is a random variable with zero mean and possesses the variance, σ t . Wedefine x t ≡ ζ t σ t . Here ζ t is a random process with standard deviation equal to one. Depending onthe system we wish to model, the variable ζ t can be described by different probability distributions,see for example, [8, 17, 20, 21, 22]. However, as mentioned by Mantegna et al. in [23], it is oftenchosen to be gaussian. We shall also simulate a gaussian random variable, hence we denote thismodel the GARCH-normal model. The GARCH(1,1) class of stochastic processes are defined viathe relation: σ t = α + α x t − + β σ t − (5)If we know the exact probability density, p ( x ), of a process, we can write the definition ofmoments by, E [ x m ] = (cid:82) ∞−∞ P ( x ) x m dx . However, we do not know the analytical expression for theprobability distribution of the GARCH process. To resolve this problem Bollerslev, [2], proposedrecurrence relations for moments of the GARCH-normal(1,1) model: E ( x mt ) = a m (cid:104)(cid:80) m − n =0 a − n ( E ( x nt )) α m − n (cid:0) mm − n (cid:1) µ ( α , β , n ) (cid:105) [1 − µ ( α , β , m )] (6)5here, µ ( α , β , m ) = m (cid:88) j =0 (cid:18) mj (cid:19) a i α j β m − j , a j = j (cid:89) i =1 (2 i − 1) (7)Therefore, we can derive equations for the variance, fourth order and sixth order standardisedmoments: σ = α − α − β (8)Γ = E ( x t ) E ( x t ) = 3 + 6 α − α − α β − β (9)Γ = E ( x t )( E ( x t )) = 15(1 − α − β ) (1 + α + β )1 − α − β + α β − α − β )( β +2 α β +3 α )1 − α − α β − β )1 − α − α β − α β − β (10)The relation (6) and (7) are defined for all moments if we fix the three GARCH parameters, α , α and β . For a moment to exist it is clear that µ ( α , β ) < 1. Therefore, we can solve µ ( α , β )for β = β ( α ). In doing so, we can create figure 2. In this figure, we see the different curves of β = β ( n )1 ( α ) where n takes the value; 2, 4, 6, 8, 10, and 12. When β <β ( n )1 ( α ) the correspondingmoments have finite value, for β >β ( n )1 ( α ), these moments diverge. Since the particular line β = β ( n )1 ( α ) separates the region of parameters where the sixth moment exists and where it doesnot, we can interpret this as a phase diagram in parameter space, [2]. Below, we extend this idea.However, the inverse problem to estimate the three GARCH parameters, if three moments areknown, is much more complicated and reduces to a set of transcendental equations which can behard to solve. In figure 2 we present a filled area that shows the region of existence of the sixth ordermoment. The red circle in this figure represents parameter values that allow for the existence ofthe second, fourth and sixth moment but not the eighth. While for the black square in figure 2 onlythe second and fourth moments are finite. We can daw the conclusion, that GARCH-normal(1,1)cannot model time series if the empirical fourth order standardised moment value, Γ > 6, as seenby the work undertaken in Appendix B. We therefore, need to ask how is it possible to accesshigher values of Γ for empirical data where a GARCH-normal model is no longer sufficient. 4. GARCH-Normal Models If we want to fit the second, fourth and sixth moments, the values of the parameters mustbe below the divergence curve; β < β (6)1 ( α ) which does not cover all parameter space for theexistence of the fourth ( β < β (4)1 ) and the second ( β < β (2)1 ) order moments. This can result insome values of the fourth and second moment, or fourth order standardised moment and the secondorder moment being unreachable for GARCH modelling.For the fourth and sixth order moment we can obtain the divergence line explicitly, and soderive: β (4)1 = (cid:113) − α − α (11)6 igure 2: The ‘Bollerslev Phase Diagram’, [2], showing the divergence of moments in a GARCH-normal(1,1) model.The highlighted area shows the existence region for the sixth order moment. The red circle presents an example of α , β values that allow for the sixth, fourth and second order moments to exist, whilst the black square shows anexample of α , β values that allow for only the fourth and second order moment to exist. (6)1 = ( − α + (cid:112) α − α + 1 + 1) − α − α ( − α + (cid:112) α − α + 1 + 1) (12)For higher order moments, the divergence lines are defined by high order algebraic equations, whichcannot be solved analytically. Here we will consider the situation of when we need to fit only the second and fourth moments,or equivalently, fitting the variance (cid:104) x (cid:105) and fourth order standardised moment Γ . Since theGARCH-normal(1,1) model has three parameters we can conclude that we can express two GARCHparameters, for instance, α and β , as a function of the third parameters α . In doing so, we derive: β = 1 − α σ emp − (cid:118)(cid:117)(cid:117)(cid:116) α σ emp − α ( σ emp ) ,emp − + 2 (13) α = (cid:118)(cid:117)(cid:117)(cid:116) α σ emp − α ( σ emp ) ,emp − + 2 (14)It is clear from these equations that for any value of Γ ,emp > σ emp > 0, we can find afamily of one-parametric GARCH models. So, we obtain the parametric curves; ( α ( α ) , β ( α )),in ( α , β ) space. Such curves represent the ‘company trajectories’ with already fixed (empirical)variance, σ emp and empirical fourth order standardised moments, Γ ,emp .In figure 3, we see an extension of figure 2 for a banking stock, a commodity, a pharmaceuticaland a mining company, respectively. The dotted lines represent the parameters of the GARCH-normal model for the given company. They allow one to see the stability of the time series, inessence, what statistical moment can exist for the GARCH description of the empirical data. Itis evident, for the longest time period (18 years) that apart from the gold ETF, trajectories of allother companies lie above the divergence line of the sixth order moment. Implying, the values ofthe fourth and second order empirical statistical moments do not allow for any higher moments tobe fitted via a GARCH-normal model.If one decreases the time window of data collection, for example a year, 6th October 2017 to6th October 2018, or even six months, 6th April 2018 to 6th October 2018, then we can see themigration of the company trajectory to deeper inside the stability region in the ( α , β ) plane,where higher moments are finite (see figures 3a, 3b, 3c, 3d). We have also examined the timelengths of nine months, fifteen months and three years. In these figures (figures 3a, 3b, 3c, 3d), it isclear that the Lloyds Bank and Rio Tinto 6 month time series allows the largest number of higherorder moments to exist. Apart from Lloyds Bank and Rio Tinto all other time series present onlywith the sixth moment as the highest finite moment. However, in general, it is clear to see that theshorter a time series we take, the more moments exist for a GARCH-normal(1,1) model.As we traverse a company’s trajectories in ( α , β ) space we can work out the value of the sixthorder standardised moment generated from the GARCH-normal(1,1) model for the specific α and β values. In table 1, we see the minimum and maximum of Γ generated, when the sixth ordermoment exists. We can see in general Γ does not vary significantly along the company trajectory.8 a) (b)(c) (d) Figure 3: The stability phase diagram for the GARCH-normal(1,1) moments with an overlap of companytrajectories. Panel (a) shows the trajectory for Lloyds Bank, where the shortest time window allows up tothe eighth order moment to exist. Panel (b) shows the trajectory of Gold ETFs, where the shortest timewindow allows to the sixth moment to exist. Panel (c) shows the same for GSK, which also allows up tothe sixth moment to exist and lastly, panel (d) shows the same for Rio Tinto, which allows up to the eighthorder moment to exist. Table 1: The minimum and maximum values of Γ along the company’s trajectories in ( α , β ) space. Let us consider the algorithms we can use to fit empirical values of (cid:104) x (cid:105) , (cid:104) x (cid:105) and (cid:104) x (cid:105) which canbe reformulated as: variance, σ emp , as well as the fourth and sixth order standardised moments,Γ ,emp and Γ ,emp respectively. In the first approach, we present α and β , as a function of α ,that is α ( α , σ emp , Γ ,emp ) and β ( α , σ emp , Γ ,emp ), from equations (13) and (14), then numericallysolve the equation: Γ ( α ( α , σ emp , Γ ,emp ) , β ( α , σ emp , Γ ,emp ) , α ) = Γ ,emp (15)to find the value of α . This method is inspired by the trajectory analysis we use in the previoussection. We search for α by traversing the trajectory and trying to fit the empirical sixth orderstandardised moment. However, if Γ is lower than the minimum or larger than the maximum ofpossible Γ stated in table 1, this equation cannot be solved. Indicating that the GARCH-normalmodel with such a value of Γ does not exist.In the second approach to fit empirical values of x , x and x , we first fit the empirical fourthand sixth order standardised moments using the fact that Γ ( α , β ) and Γ ( α , β ) do not dependon α , see equations (9) and (10). Therefore, we can reduce the problem to two equations:Γ ( α , β ) = Γ ,emp , Γ ( α , β ) = Γ ,emp (16)allowing us to evaluate values of α , β and reserve α to the fitting of variance; α = σ emp (1 − α − β ). The set of equations (16) can be further reduced to one equation by eliminating β usingthe first equation of the set namely: β = (cid:115) − α − α Γ ,emp − − α (17)and substituting it to the second equation of (16). This enables us to write the one-variable equation;Γ ( β ( α , Γ ,emp ) , α ) = Γ ,emp . Note, that we similarly can exclude α , resulting in equations for β . The equations (16) can only be solved for some region in standardised moment space, (Γ ,Γ ), in figure 4, we see two highlighted points. The first point (1 . , 8) is inside the ‘GARCHable’10 igure 4: The phase diagram for (ln(Γ ), ln(Γ )) space. The filled yellow region shows the ‘GARCHable’ region,whilst the rest of the space is where the certain values of Γ and Γ cannot be fitted via a GARCH-normal(1,1)model. The two highlighted points (1 . , 8) and (2 . , 8) show the values of Γ and Γ that can be fitted by a GARCHmodel and those that cannot, respectively. The other data points in the space, represent the empirical data for severalcompanies, truncated to 10-90 day windows, incremented in 10 day periods. It is shown here that no empirical datacan be fitted via a GARCH-normal(1,1) model. region. This is the region of phase space where the respective values of the fourth and sixth orderstandardised moments can be fitted via a GARCH-normal model. The second point (2 . , , Γ ) space, figure 4. Tosee the effect of the length of the time window, the interval of time we take the data over, on theability of GARCH to fit empirical moments, we divide data in to different economic periods. Infigure 5, we show the different regions we wish to evaluate, we start with taking six month periods;6th April to the 6th October in the years 2005, 2008, 2011, 2014 and 2017. We break these sixmonth time periods in to 10, 20, 30, 40, 50, 60, 70, 80 and 90 day windows, an example of thiscan be seen for the several stocks in figure 4. We then overlap these data points on top of the‘GARCHable’ region detailed above.We never see the empirical data inside of the GARCH-normal(1,1) phase region for the timeperiods analysed. Therefore, we can say that a GARCH-normal(1,1) model is unable to fit higherorder moments of the time series analysed. 11 igure 5: The timeline of time series windows that we investigate within the paper. We investigate periods ofeconomic turmoil, pre-crisis, crisis and post-crisis periods, as well as relatively stable periods. We also highlight herethe truncation of 3 years, 15 months, 1 year, 9 months and 6 months. 5. GARCH-Double-Normal Models It is clear from this initial analysis of the GARCH-normal model that gaussian conditionaldistributions do not allow a fitting for Γ with a finite value of Γ . This problem can potentially beresolved if we replace the conditional gaussian distribution with a distribution allowing to have abetter flexibility in higher order GARCH standardised moments. Therefore, we seek a distributionwith a larger fourth order standardised moment value in the region where Γ exists. We introducethe double gaussian distribution [24], where p ( x ): p ( x ) = ae − x σ + be − x σ (18)In addition to an obvious normalisation condition: a + b = 1 (19)we also have constraints on the 2nd moment: E [ x ] = aσ + bσ = 1 (20)due to the requirement that the conditional distribution for a GARCH process should have varianceequal to one. We can introduce two more variables (higher order moments; 4th and 6th momentsof the conditional distribution) which fully define all parameters in equation (18): E [ x ] = aσ + bσ = µ = η E [ x ] = aσ + bσ = µ = η 15 (22)The parameterisation (21)-(22) of the double Gaussian distribution allows us to generalise Boller-slev’s equation (9). The 2nd order moment is not affected and is still determined by equation (8),while the 4th and 6th order standardised moments for GARCH with double-gaussian distributioncan be written as: Γ = η (1 − α − β ) (1 + α + β )1 − α − β )1 − η α − α β − β (23)Γ = η (1 − α − β ) (1 + α + β )1 − α − β + 3(1 + α + β )( η α +2 α β + β )1 − η α − α β − β ))1 − η α − η α β − α β − β (24)12sing the methods described prior, based on the existence of solutions of the set of equations(16) and equations (23) and (24), we create a family of phase diagrams parameterised by η and η . To understand which empirical values are achievable using GARCH-double-normal, we need tounderstand restrictions for the whole family of phase diagrams. We see that these are bounded dueto limitations for η and η obtained in Appendix A (conditions A.9 and A.10). These limitationsrequire all phase diagrams be started from points above or on the dashed line, figure 6. Only dataabove the dashed line can be described by a GARCH-double-normal model (which is the case forthe empirical data collected for the securities we have considered here). Figure 6: The phase diagrams for (Γ , Γ ) space for GARCH-double-normal(1,1) models corresponding to differentparameters given in table 2. From equations (21) and (22), we gain the condition for the distribution to be doublegaussian, seen by the dotted line. We then overlay the empirical data for Bank of America, truncated from 1% to100% of the length of the time series, incremented in one percent steps. To highlight the ability of the GARCH-double-normal model to fit higher order moments for specific lengths of time windows we present three ‘GARCHable’regions. Each has a different time window that it can fit, shown by the letter, associated with table 2. In figure 6, we see three phase diagrams for three different double gaussian distributions. Pa-rameters for these phase diagrams are given in table 2.Figure 6 demonstrates how altering the parameters η and η of the GARCH-double-normalmodel enables us to capture different lengths of time windows of the empirical data. The dataused in figure 6 is for the Bank of America time series for 6th October 2000 to 6th October 2018.We truncate the time series in to different lengths. We start with 1% of the overall length andincrement by 1% up to the whole length of the time series. In other words, the first, most left pointcorresponds to 40 days of data, (from 06/10/2000 to 01/12/2000), the second point corresponds tomoments obtained for 80 days of data (from 06/10/2000 to 25/01/2001) and so on.13osition η η t min ( Days ) t max ( Days )Leftmost(Red) 5 41.7 171 (A) 600 (B)Centre(Blue) 7 81.7 943 (C) 1500 (D)Rightmost(Green) 12 240 1586 (E) 1587 Table 2: Parameters of the conditional double gaussian distributions used to construct ‘GARCHable’ regions in figure6. The table summarises the parameters of the distributions used to model the time windows (A) to (B), (C) to(D) and (E). These are the limits of the time horizon in days that the particular instance of the double gaussiandistribution can be used to fit the higher order moments of the empirical data of the Bank of America. For the leftmost phase diagram we use a double gaussian distribution with η = 5 , η = 41 . σ , Γ and Γ for the time window of duration in the interval, 171 ≤ t ≤ Once we have fixed the time window we wish to model, we can study what happens when thewindow with this fixed duration shifts in time. This can be done by attributing to the higher ordermoments a time moment, t , corresponding to the middle point (the median) of the time window wewish to model. If we fix the double gaussian distribution (in essence select certain η and η ), we cangain the set of GARCH parameters, α , α and β that describes the particular time median. If wechange the instance in time we look at by moving the time median of the window, then the GARCHparameters α , α and β also change. Below, we observe that the GARCH parameters α ( t ), α ( t )and β ( t ) significantly vary with time, highlighting the non-stationarity of our modelling.Given equations (8) and (23), we are able to define trajectories in ( α , β ) space for a fixedvalue of σ emp and Γ ,emp . Unlike the GARCH-normal methods, we now have the trajectories as afunction of η . These can be seen below: α = (cid:118)(cid:117)(cid:117)(cid:116) ( α σ emp − ( α σ emp ) )(Γ ,emp − η )(Γ ,emp − η )( η − − ( η − η ) (25) β = 1 − α σ emp − (cid:118)(cid:117)(cid:117)(cid:116) ( α σ emp − ( α σ emp ) )(Γ ,emp − η )(Γ ,emp − η )( η − − ( η − η ) (26)Now for each desired data set we can use the trajectories in the same manner as we have donewith the GARCH-normal model. We can plot Γ along the trajectories of ( α , β ) using the running14arameter α . Plotting Γ as a function of the running parameter, α , and overlaying this with theempirical value we gain figure 7. Figure 7: Γ as a function of the running parameter α . Here we show how we calculate the value of α for a particulartime window. The orange line is the value of Γ for the empirical time window we are modelling, whilst the blue lineshows Γ ( α ), equation (24) when α = α ( η , α , σ emp , Γ ,emp ) (equation (25)) and β = β ( η , α , σ emp , Γ ,emp )(equation (26)), for the GARCH-double-normal model. The intercept of the two lines shows the value of α whichallows us to model data for a certain median time and a certain time window within the GARCH-double-normal(1,1)model. From the above method we can recover the value of α ( t ) that allows the fitting of σ emp ( t ),Γ ,emp ( t ) and Γ ,emp ( t ), where t is the median of the running window, allowing us to create figure8. This is done for several banks: Lloyds Bank, Barclays Bank, Bank of America, HSBC andSantander, a commodity, Gold ETFs and a pharmaceutical company, GSK. We seek to find afingerprint of the companies’ GARCH parameters through the financial crisis. It is evident fromthis figure that the banking sector has a unique behaviour in response to the crisis. We see an initialfluctuating signal, but when the crisis period occurs we see a ‘cliff-edged’ drop in the parametervalue. This behaviour is not seen in the commodity or the pharmaceutical and so we infer thatthis is due to the response to the crisis period of 2008. It is our belief that this specific behaviourexhibited by the banking companies can be used as an indicator for future banking crisis periods. 6. 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 the gaussian condi-tional distribution within a GARCH model to fit higher order moments of market time series.15 a) (b) Figure 8: α ( t ) for several companies between 2005 and 2011. We show here the evolution of α when we calculatethe parameter value in the shifting 6 month time window. In panel (a), we show the time evolution of α for BarclaysBank, Gold ETFs, GSK and Lloyds Bank, by the red, blue, yellow and purple lines respectively. Whilst in panel (b),we show the Bank of America’s evolution by the blue line, the evolution of HSBC by the red and Santander’s by theyellow line. In discovering this, we turn our attention towards different conditional distributions to try tocapture the empirical data. We show that with the use of a GARCH-double-normal model we canfit the empirical data’s higher order moment. However through this enquiry, we still cannot capturethe long run dynamics of the empirical data. We show that it is only possible to fit a model toempirical data within certain time horizons. To model a different time horizon we have to changethe parameters of the double gaussian distribution we use to model.Fixing the distribution within certain time horizons to enable the fitting of higher order mo-ments, highlights that the obtained GARCH-double-gaussian(1,1) model describes a non-stationaryprocess. 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 differentGARCH-double-normal models to each time window. Therefore, we produce a time dependence ofthe GARCH model’s parameters. We are able to build up a time signature of the α parameterthrough the 2008 financial crash for several companies. 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Relations between the parameters of the double gaussian distributionand its higher order moments The normalisation condition for the double gaussian distribution described by equation (18) is a + b = 1. Substituting a = 1 − b into the equation (20), E [ x ], we get: b = 1 − σ σ − σ (A.1)and, a = σ − σ − σ (A.2)Assuming σ < < σ , and substituting the equations for a and b into the fourth and sixthmoment equations we derive: µ = σ + σ − σ σ (A.3)and, µ = ( σ + σ ) − σ σ − σ σ ( σ + σ ) (A.4)18here, µ = η and µ = η . Introducing the new variables, X = σ + σ and Y = σ σ , we cansimplify the obtained equations: Y = X − µ (A.5) X ( µ − 1) + µ = µ (A.6)Solving the above equations for X and Y we finally obtain: X = µ − µ µ − Y = µ − µ µ − X and Y must be positive, this gives us three conditions; µ > µ > µ and µ > µ . Dueto the first one we can disregard the second as µ > µ . Using relations between µ and µ and η and η we obtain: η > η > η (A.10)We can then set-up equations for solving σ or σ : σ − Xσ + Y = 0 (A.11)and, σ = Yσ (A.12)Solving for σ we can obtain relations for the parameters of the double gaussian distribution: σ = 12 ( X + (cid:112) X − Y ) (A.13)and so, σ = 2 Y ( X + √ X − Y ) . (A.14)Since, σ and σ must be both real and positive, this gives us the relation; X > Y . As suchwe get the following inequality: µ − µ µ − µ − µ (3 − µ ) > µ , we get the condition; µ > − 1. Obviously, µ is always larger than − 1, and so we always satisfy the condition shown in equation (A.15.) As such, we have to onlyobey the conditions shown in equations (A.9) and (A.10).19 ppendix B. Conditions for Γ For a general GARCH conditional probability distribution with variance equal to one, the equa-tion for the sixth order divergence line (the denominator of equation (10)) becomes:1 − β − α β − η α β − η α = 0 (B.1)Expanding β in a series with respect to α we derive: β = 1 − Aα − Bα − Cα − ...... (B.2)Substituting this into our sixth order divergence line we can equate coefficients up to the secondorder and so β becomes: β = 1 − α − ( η + 1) α + O ( α ) (B.3)If we now neglect α orders higher than the second we get the equation; β = 1 − α − ( η + 1) α .Substituting this into our equation for the fourth order standardised moment, we obtain:Γ = η (1 − α − (1 − α − ( η + 1) α ))1 − η α − α (1 − α − ( η + 1) α ) − (1 − α − ( η + 1) α ) (B.4)Considering the limit when α → α → Γ = 2 η4