risks
Article Least Quartic Regression Criterion to Evaluate Systematic Risk in the Presence of Co-Skewness and Co-Kurtosis
Giuseppe Arbia, Riccardo Bramante and Silvia Facchinetti *
Department of Statistical Sciences, Università Cattolica del Sacro Cuore, Largo Gemelli 1, 20123 Milan, Italy; [email protected] (G.A.); [email protected] (R.B.) * Correspondence: [email protected]
Received: 23 July 2020; Accepted: 2 September 2020; Published: 8 September 2020
Abstract: This article proposes a new method for the estimation of the parameters of a simple linear regression model which is based on the minimization of a quartic loss function. The aim is to extend the traditional methodology, based on the normality assumption, to also take into account higher moments and to provide a measure for situations where the phenomenon is characterized by strong non-Gaussian distribution like outliers, multimodality, skewness and kurtosis. Although the proposed method is very general, along with the description of the methodology, we examine its application to finance. In fact, in this field, the contribution of the co-moments in explaining the return-generating process is of paramount importance when evaluating the systematic risk of an asset within the framework of the Capital Asset Pricing Model. We also illustrate a Monte Carlo test of significance on the estimated slope parameter and an application of the method based on the top 300 market capitalization components of the STOXX® Europe 600. A comparison between the slope coefficients evaluated using the ordinary Least Squares (LS) approach and the new Least Quartic (LQ) technique shows that the perception of market risk exposure is best captured by the proposed estimator during market turmoil, and it seems to anticipate the market risk increase typical of these periods. Moreover, by analyzing the out-of-sample risk-adjusted returns we show that the proposed method outperforms the ordinary LS estimator in terms of the most common performance indices. Finally, a bootstrap analysis suggests that significantly different Sharpe ratios between LS and LQ yields and Value at Risk estimates can be considered more accurate in the LQ framework. This study adds insights into market analysis and helps in identifying more precisely potentially risky assets whose extreme behavior is strongly dependent on market behavior.
Keywords: co-skewness; co-kurtosis; leastquarticcriterion; systematicriskevaluation; portfoliooptimization
1. Introduction Traditional linear regression models based on the normality assumption neglect any role in the higher moments of the underlying distribution. This approach is not justified in many situations where the phenomenon is characterized by strong non-normality like outliers, multimodality, skewness and kurtosis. Although this situation may occur in many circumstances, quantitative finance is a field where the consequences of non-normality are particularly relevant and may affect dramatically the investors’ decisions. Consider, for instance, the Capital Asset Pricing Model (henceforth CAPM; see (Sharpe 1964; Lintner 1965)), a well-known equilibrium model, which assumes that investors construct their portfolio on the basis of a trade-off between the expected return and the variance of the returns of the market portfolio. In the traditional CAPM framework, a regression slope reflects the exposure of an asset to systematic risk, indicating how fluctuations in the returns are related to movements
Risks 2020, 8, 95; doi:10.3390/risks8030095 www.mdpi.com/journal/risks Risks 2020, 8, 95 2 of 14 of the market as a whole. In its original formulation based on Least Squares estimation, CAPM is restricted to the first two moments of the empirical distribution of historical returns; however, due to the complexity of the corresponding generating process, this simple model often fails in detecting risk premia. Many papers in the financial literature heavily criticize such an approach emphasizing the role of co-moments to account for non-normal extreme events in investors’ decisions. Such an approach leads to a series of modifications that incorporate the consideration of higher moments of the distribution of returns (Barone-Adesi et al. 2004). A completely different approach to modeling extreme financial events is based on the notion of the copula (Trivedi and Zimmer 2005). For an application to financial data, see (Nikoloulopoulos et al. 2012). Our findings contribute to the literature in the following way. We propose a new linear regression estimation criterion, the Least Quartic criterion, which is based on a quartic loss function in place of the usual quadratic specification considered in the ordinary LS. It represents an extension of the ordinary LS criterion that, in non-Gaussian situations, provides slope coefficient estimations that outperform the ordinary LS in terms of out-of-sample risk-adjusted performance. This allows us to show how higher moments can be formally taken into account redefining the estimation criterion even in the presence of co-skewness and co-kurtosis, as it occurs in many practical circumstances. Along with the detailed description of the methodology, we examine in particular its application to finance. In particular, within the CAPM framework, we show that using the LQ criterion, it is possible to provide a measure of each asset’s systematic risk which accounts for non-normality by incorporating higher-order moments. This finding adds insights into market analysis and helps in identifying more precisely potentially risky assets whose extreme behavior is strongly dependent on market behavior. The article is organized as follows. Section2 is devoted to a literature review on the role of higher moments in financial risk evaluation, useful to provide the reader with the framework within which the results of the proposed methodology could be interpreted. The methodology of our proposal is reported in Section3. Specifically, in Section 3.1 we formally introduce various definitions of co-skewness and co-kurtosis and we clarify their statistical nature with particular reference to the analysis of systematic risk. Section 3.2 is devoted to our proposal: a regression estimator which involves the minimization of the fourth power of the regression errors. In Section4, we illustrate an application of the proposed methodology using some real data referring to the returns of the top 300 market capitalization components of the STOXX® Europe 600. Specifically, an empirical comparison between the linear and quartic estimators is pointed out. Finally, Section5 concludes.
2. The Role of Higher Moments in Financial Risk Measurement The traditional theory of CAPM uses essentially a Least Square linear regression strategy to measure the relationship between returns of an asset in relation to the market, thus providing a measure of the so-called systematic risk. In this respect, only the first two moments of the joint distribution between the asset and the market are relevant in the analysis (Sharpe 1964; Lintner 1965; Mossin 1966). In the last decades, many papers have recognized the shortcomings associated with such an approach (starting from the criticism contained, e.g., in (Fama and French 1992)), and extended it so as to incorporate considerations linked to the higher-order moments of the generating probability distributions. In particular, some authors (e.g., (Ranaldo and Favre 2005; Jondeau and Rockinger 2006)) agree on the fact that the use of a pricing model limited to the first two moments may be misleading and may wrongly indicate insufficient compensation for the investment. They thus indicate that a higher-moment approach is more appropriate to detect non-linear relationships between assets and portfolio returns while accommodating for the specific risk–return payoffs. The financial literature is very rich in contributions that include considerations related to the third moment. For instance, papers like Kraus and Litzenberger(1976, 1983), Barone-Adesi and Talwar(1983), Barone-Adesi(1985), Sears and Wei(1988), Rubinstein(1994) and Harvey and Siddique(2000) propose a three-moment CAPM which includes the third-order moment. More recently papers like Barberis and Huang(2008), Brunnermeier et al.(2007), Mitton and Vorkink(2007), Boyer et al.(2010), Green and Hwang(2012) Risks 2020, 8, 95 3 of 14 and Byun and Kim(2016) empirically document that the skewness of individual assets may have an influence on portfolio decisions. Moreover, the paper of Jondeau et al.(2019) investigates the ability of the average asymmetry in individual stock returns prediction. Rather surprisingly, comparatively less attention has been paid to the role of the fourth moment. In this respect, Krämer and Runde(2000) show evidence of leptokurtic stock returns, while, more recently, Conrad et al.(2013) discuss the role of (risk-neutral) skewness and kurtosis showing that they are strongly related to future returns. Co-moments of the asset’s distribution with the market portfolio have also been recognized as influencing investors’ expected returns. In this respect, Hwang and Satchell(1999) , Harvey and Siddique(1999, 2000) and Bakshi et al.(2003) introduce the idea of co-moments analysis in financial market risk evaluation, while Beaulieu et al.(2008) propose exact inference methods for asset pricing models with unobservable risk-free rates and co-skewness, and Dittmar(2002) tests explicitly the influence of co-skewness and co-kurtosis on investors’ decisions, showing that systematic kurtosis is better than systematic skewness in explaining market returns. Fang and Lai(1997) and Christie-David and Chaudhry(2001) describe a model where the excess returns are expressed as a function of the covariance, of the co-skewness and of the co-kurtosis between the returns of a risky asset and those of the investor’s portfolio. They show that volatility is an insufficient measure of risk for risk-averse agents and that systematic skewness and systematic kurtosis increase the explanatory power of the return generating process of future markets. Their empirical results agree on the fact that investors are averse both to variance and kurtosis in their portfolio requiring higher excess rates as compensation. The papers by Fang and Lai(1997) and Christie-David and Chaudhry(2001) have the merit of introducing, in a formal way, the consideration of co-skewness and co-kurtosis in a CAPM framework, showing their empirical relevance in explaining risk premia. In this paper, we follow the same approach showing how higher moments considerations can be formally taken into account, redefining the estimation criterion of the fundamental systematic risk regression. Further references to the use of higher moments in analyzing future markets may be found in Levy(1969) , Badrinath and Chatterjee(1988) , Hwang and Satchell(1999), Fernandez-Perez et al.(2018) and Liu et al.(2020) , among others. Other approaches related to higher-order considerations in the evaluation of risk may be found in the extreme value approach suggested by McNeil and Frey(2000), in the idea of bivariate Value at Risk (VaR) by Arbia(2003) and in the tail VaR introduced by Barg ès et al.(2009). A totally different approach to model tail financial events is based on the notion of the copula; see papers like Nikoloulopoulos et al.(2012), Dissmann et al.(2013), So and So and Yeung(2014) , Kim et al.(2013) and Cortese(2019) for recent examples in the financial literature. For other recent approaches to modeling anomalous tails of financial events see, Zhang and Huang(2006) , McElroy and Jach(2012) and So and Chan(2014) among others. Furthermore, it should be remarked that a large part of the financial econometrics literature (e.g., (Brooks et al. 2005; Dubauskas and Teresiene˙ 2005)) has considered models based on conditional higher moments which have essentially extended the ARCH formulation of Engle(1982).
3. Methodology In what follows, we present the methodology of our approach. First, we introduce some formal definitions of co-skewness and co-kurtosis that will be used later while discussing an augmented version of the CAPM which takes into account higher moments of the distribution of returns. Then we describe our proposal, the Least Quartic criterion, a new regression interpolation criterion that takes into account higher-order moments characteristics in non-normal situations.
3.1. Co-Moments Let us consider two random variables X and Y. The generic bivariate moment of order r, s r s centered around the mean is defined as µr,s = E(x y ), where x = [X E(X)] and y = [Y E(Y)]. − − Risks 2020, 8, 95 4 of 14
Risks 2020, 8, x FOR PEER REVIEW 4 of 14 Similarly, the standardized bivariate moment of order r, s centered around the mean is defined as , 𝜆 = µr,s , with 𝜎 = 𝜇 and analogously for 𝜎 . Starting from these definitions, the co-skewness λ , r,s = r s , with σx = √µ , 2,0 and analogously for σy . Starting from these definitions, the co-skewness of σ xσ y ofa bivariatea bivariate distribution distribution is defined is defined formally formally as the as mixedthe mixed moments moments of orders of ordersr and r sandsuch s such that r that+ s =𝑟+3, 𝑠=3that is, thatµ1,2 isand 𝜇 , µ and2,1. 𝜇 In , a. In bivariate a bivariate normal normal distribution distributionµ1,2 𝜇 , = =𝜇µ2,1 , ==00,, see (Kendall (Kendall et et al. al. 1983;1983; Kotz et al. 20002000);); however,however, in in general general non-normal non-normal distributions, distributions, we we have, have, conversely, conversely,µ1,2 𝜇 , ∈ℜandandµ2,1 µ1,2 , ∈ < 𝜇 , .∈ℜ The. The corresponding corresponding standardized standardized moment moment is defined is defined as λ1,2 as= 𝜆 , 2 =and similarly and similarly for λ2,1. for Figure 𝜆 , 1. ∈ < σxσy Figureillustrates 1 illustrates graphically graphically the behavior the behavior of positive of positive and negative and negative co-skewness co-skewness showing showing the two the density two densityfunctions functions and the and associated the associated scatter diagrams. scatter diagrams.
(a) (b) (c)
Figure 1. VariousVarious types types of of co-skewness: co-skewness: (a)( apositive) positive left-tail left-tail co-skewness; co-skewness; (b) (positiveb) positive right-tail right-tail co- skewness;co-skewness; (c) (negativec) negative co-skewness. co-skewness. Scatter Scatter diagrams diagrams (upper (upper pane); pane); probability probability density density functions (lower panel).
In finance, finance, for the systematic co-skewness of of a a joint distribution, a a slightly different different version of µ 2,1 , the standardized momentmoment isis usually usually considered considered which which is definedis defined by by the the term termµ (Christie-David (Christie-David and 3,0 Chaudhry 2001; Fang and Lai 1997; Harvey and Siddique 1999, 2000). Always referring , to the financial and Chaudhry 2001; Fang and Lai 1997; Harvey and Siddique 1999, 2000). Always referring to the interpretation, from Figure1 it can be argued that, in general, investors will tend to avoid positive financial interpretation, from Figure 1 it can be argued that, in general, investors will tend to avoid right-tail co-skewness (involving cases of big losses when the market also experiences big losses) and positive right-tail co-skewness (involving cases of big losses when the market also experiences big to favor instead left-tail positive co-skewness (involving cases of big revenues when the market also losses) and to favor instead left-tail positive co-skewness (involving cases of big revenues when the experiences big revenues). Prudential investors will prefer negative co-skewness to mitigate the risk of market also experiences big revenues). Prudential investors will prefer negative co-skewness to losses when the markets perform badly. mitigate the risk of losses when the markets perform badly. In a similar way, the co-kurtosis of a bivariate distribution is defined by the mixed moments of In a similar way, the co-kurtosis of a bivariate distribution is defined by the mixed moments of orders r and s such that r + s = 4. Co-kurtosis can, therefore, assume three different manifestations orders r and s such that 𝑟+𝑠=4. Co-kurtosis can, therefore, assume three different manifestations defined respectively by the mixed moments µ1,3, µ3,1 and µ2,2 whose behavior is likely to be correlated defined respectively by the mixed moments 𝜇 , , 𝜇 , and 𝜇 , whose behavior is likely to be in empirical situations. In a bivariate normal distribution with the correlation coefficient ρx,y, we have correlated in empirical2 2 situations. In a bivariate normal distribution with the correlation coefficient that µ1,3 = 3ρx,yσxσy and analogously for µ3,1, see (Kendall et al. 1983; Kotz et al. 2000). In non-Gaussian 𝜌 , , we have that 𝜇 , =3𝜌 , 𝜎 𝜎 and analogously for 𝜇 , , see (Kendall et al. 1983; Kotz et al. 2000). distributions, we have, instead, µ1,3 and µ3,1 . The corresponding standardized moment In non-Gaussian distributiµ1,3 ons, we ∈have, < instead,∈ <𝜇 , ∈ℜ and 𝜇 , ∈ℜ. The corresponding is defined as λ1,3 = 3 and similarly for λ 3,1 , . A measure of excess co-kurtosis (with respect to σxσy standardized moment is defined as 𝜆 , = and similarly for 𝜆 , . A measure of excess co- 2 2 the bivariate normal distribution) is intuitively provided by the expression κ1,3 = µ1,3 3ρx,yσxσy kurtosis (with respect to the bivariate normal distribution) is intuitively provided by the −expression and similarly for κ 3,1 . Analogously, we can look at co-kurtosis through the mixed moment µ2,2. 𝜅 , =𝜇 , −3𝜌 , 𝜎 𝜎 and similarly for 𝜅 , . Analogously, we can2 look 2 at2 co-kurtosis through the In a bivariate normal distribution, we have that µ2,2 = 1 + 2ρx,y σxσy, see (Kendall et al. 1983; mixed moment 𝜇 . In a bivariate normal distribution, we have that 𝜇 = 1+2𝜌 𝜎 𝜎 , see Kotz et al. 2000). Conversely, , in non-Gaussian distributions, we have µ , +. The corresponding , 2,2 (Kendall et al. 1983; Kotz et al. 2000). Conversely,µ2,2 in non-Gaussian distributions,∈ < we have 𝜇 , ∈ℜ . standardized moment is defined as λ2,2 = 2 3 and the relative measure with respect to the bivariate σxσy , The corresponding standardized moment is defined as 𝜆 , = and the relative measure with 2 2 2 normal distribution is provided by the quantity κ2,2 = µ2,2 1 + 2ρ σ σ . In the financial literature, respect to the bivariate normal distribution is provided by −the quantityx,y 𝜅x y =𝜇 − 1+2𝜌 𝜎 𝜎 . the systematic co-kurtosis of a risky asset has been often defined as , a modified , version , of the In the financial literature, the systematicµ3,1 co-kurtosis of a risky asset has been often defined as a standardized (3,1) moment given by (Christie-David and Chaudhry 2001; Fang and Lai 1997) while modified version of the standardizedµ4,0 (3,1) moment given by , (Christie-David and Chaudhry the joint moment of order (2,2) has been completely neglected. The , range of possible situations that 2001;may ariseFang here and is Lai more 1997) complex while thanthe joint those moment considered of order for co-skewness. (2,2) has been Figure completely2 illustrates neglected. graphically The range of possible situations that may arise here is more complex than those considered for co- skewness. Figure 2 illustrates graphically only some of the possible cases where positive and negative
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Risks 2020, 8, x FOR PEER REVIEW 5 of 14 only some of the possible cases where positive and negative co-kurtosis may emerge in empirical circumstances.co-kurtosis may Again, emerge finance in empirical provides circumstances. interesting Again substantive, finance interpretations provides interesting of the co-kurtosissubstantive parameters.interpretations In fact, of the looking co-kurtosis at Figure parameters.2, one may In argue fact, lookingthat, generally at Figure speaking, 2, one investors may argue prefer that, negativegenerally co-kurtosis speaking, thatinvestors reduces prefer the chances negative of co big-kurtosis losses when that marketsreduces performthe chances badly. of Thisbig losses conclusion when ismarkets in accordance perform with badly. the speculationsThis conclusion of Fang is in andaccordance Lai(1997 with) and theChristie-David speculations of and Fang Chaudhry and Lai( 2001(1997)) onand investors’ Christie- behaviorDavid and in Chaud the presencehry (2001) of kurtosis. on investors’ behavior in the presence of kurtosis.
(a) (b) (c)
FigureFigure 2.2. Various types of co co-kurtosis:-kurtosis: (a (a) )positive positive lepto lepto co co-kurtosis;-kurtosis; (b ()b positive) positive plati plati co- co-kurtosis;kurtosis; (c) (cnegative) negative co- co-kurtosis.kurtosis. Scatter Scatter diagrams diagrams (upper (upper panel); panel); probability probability density density functions functions (lower (lower panel). panel).
3.2.3.2. TheThe LeastLeast QuarticQuartic CriterionCriterion AA newnew regressionregression interpolationinterpolation criterion criterion that that takes takes into into account account higher-order higher-order moments moments characteristicscharacteristics inin non-normalnon-normal situationssituations isis nownow introduced.introduced. AlternativeAlternative estimationestimation approachesapproaches thatthat accountaccount for for the the presence presence of non-normality of non-normality and outliers and outliers in regression in regression may be found may in be the found contributions in the ofcontributions Theil(1950) andof Theil Sen( (1950)1968) amongand Sen others. (1968) among others. TheThe LQLQ criterioncriterion consistsconsists ofof aa simplesimple optimizationoptimization procedureprocedure thatthat providesprovides aa closedclosed formform forfor thethe slopeslope regressionregression coecoeffifficientcient estimator, like the the ordinary ordinary LS LS.. In In particular, particular, it it represents represents an an extension extension of ofthe the ordinary ordinary LS LS criterion criterion.. Indeed, Indeed, in in the the Gaussian Gaussian distribution distribution case case,, the the estimators reduce to to the the ordinaryordinary LSLS solution.solution. InIn non-Gaussiannon-Gaussian situations,situations, thethe proposedproposed procedureprocedure providesprovides estimatorsestimators thatthat outperformoutperform thethe ordinaryordinary LSLSin interms terms of of out-of-sample out-of-sample risk-adjusted risk-adjusted performance. performance. AccordingAccording to to CAPM, CAPM let, let us us consider consider the the simple simple linear linear regression regression model model where where the i-th the asset 푖-th return, asset forreturni = ,1, for 2, ...푖 =, N1,2 is, … modeled, 푁, is modeled by by ri,t = bi rM,t + εi,t (1) 푟푖,푡 = 푏푖· ∙ 푟푀,푡 + 휀푖,푡 (1) where b is the CAPM slope coefficient, while r and rM t are the historical returns of the i-th asset and where i푏 is the CAPM slope coefficient, whilei,t 푟 and, 푟 are the historical returns of the 푖-th the market,푖 respectively, at a given point in time t푖,푡 T expressed푀,푡 in terms of deviations from their asset and the market, respectively, at a given point∈ in time 푡 ∈ 푇 expressed in terms of deviations respective expected values. Let us assume that rM is non-stochastic and that the error term εi obeys from their respective expected values. Let us assume that 푟 is non-stochastic and that the error term some non-Gaussian distribution characterized by excess kurtosis.푀 휀 obeys some non-Gaussian distribution characterized by excess kurtosis. 푖 To find the optimal interpolating line, in place of the familiar Least Squares criterion based on a To find the optimal interpolating line, in place of the familiar Least Squares criterion based on a quadratic loss function, let us define a quartic loss function l(bi) quadratic loss function, let us define a quartic loss function 푙(푏푖) X X 4 4 4 l(bi) = ε 4 = (ri,t bi rM,t) (2) 푙(푏푖) = ∑ 휀i푖,,t푡 = ∑(푟푖,푡 −− 푏·푖 ∙ 푟푀,푡) t t (2) 푡 푡 AnAn economic-theoreticaleconomic-theoretical motivation for the the choice choice of of aa least least quartic quartic criterion criterion may may be be found found in inthe the financial financial literature literature discussed discussed in Section in Section 2 and 2in and particular in particular in the empirical in the empiricalfindings of findingsFang and Lai (1997) and Christie-David and Chaudhry (2001) related to investors averseness to portfolio kurtosis. The quartic loss function can be seen as a particular case of the general multivariate loss function proposed by Alp and Demetrescu (2010) to model skewness, fat tails, non-ellipticity and tail
Risks 2020, 8, 95 6 of 14 of Fang and Lai(1997) and Christie-David and Chaudhry(2001) related to investors averseness to portfolio kurtosis. The quartic loss function can be seen as a particular case of the general multivariate lossRisks function2020, 8, x FOR proposed PEER REVIEW by Alp and Demetrescu(2010) to model skewness, fat tails, non-ellipticity6 andof 14 tail dependence of financial data. Higher-order loss functions (e.g., of order six) could be obviously considered,dependence butof thefinancial results data. are moreHigher-order difficult to loss interpret. functions (e.g., of order six) could be obviously considered,By expanding but the Equationresults are (2) more we have difficult to interpret. By expanding Equation (2) we have 4 3 2 l(bi) = µ4,0b 4µ3,1b + 6µ2,2b 4µ1,3bi + µ0,4 (3) i − i i − 𝑙 𝑏 ) =𝜇 , 𝑏 −4𝜇 , 𝑏 +6𝜇 , 𝑏 −4𝜇 , 𝑏 +𝜇 , (3) P 4 P 4 In the above expression, µ4,0 = t r and µ0,4 = t r represent the kurtosis of the market and In the above expression, 𝜇 , = ∑ 𝑟 , M,t and 𝜇 , = ∑ 𝑟 , i,t represent the kurtosis of the market and P 3 P 3 P 2 2 the i-th asset returns; whereas, µ3,1 = t r r i,t µ1,3 = t rM,tr and µ2,2 = t r r represent the the 𝑖 -th asset returns; whereas, 𝜇 , = ∑M ,𝑟t , 𝑟− , − 𝜇 , = ∑ 𝑟 , i,t𝑟 , and 𝜇 , = ∑M ,t𝑟 , i,t 𝑟 , represent measuresthe measures of co-kurtosis of co-kurtosis described described in Section in Section3.1. For 3.1. the sakeFor ofthe simplifying sake of simplifying the notation, the henceforth, notation, the subscript i for the co-moments involving ri is omitted. henceforth, the subscript i for the co-moments involving 𝑟 is omitted. Figure3 3 reports reports an an example example of of a quartica quartic polynomial polynomial which, which, in general, in general, identifies identifies a curve a curve with with two relativetwo relative minima minima and one and maximum. one maxi However,mum. However, in the specific in the case specific we are case examining, we are the examining, polynomial the to bepolynomial minimized to isbe subjectminimized to a seriesis subj ofect constraints to a series of deriving constraints from deri theving intrinsic from nature the intrinsic of the nature problem. of Inthe particular, problem. theIn particular, polynomial the parameters polynomial referring paramete to thers kurtosisreferring and to the the kurtosis co-kurtosis andµ 2,2theare co-kurtosis bound to be positive by definition. 𝜇 , are bound to be positive by definition.
Figure 3. The plot of a quartic polynomial. For For the the sake sake of of illustration illustration we we set set 𝜇µ , =1= 1;; 𝜇µ , =−4= 4;; 4,0 3,1 − µ𝜇2,2 , ==66;;µ 𝜇1,3 , ==22 and andµ 0,4𝜇 , = =11. .
In orderorder toto evaluateevaluate thethe minimum minimum of of the the loss loss function, function, let let us us now now set set to to zero zero the the first first derivative derivative of Equationof Equation (3). (3). We We obtain: obtain:
𝜕X X ∂ 𝑙 𝑏 ) =−4 𝑟3 −𝑏 ∙𝑟 𝑟 =0 3 l(b ) = 4 (r b rM t) r , M t = 0 , (r , b rM t) rM t = 0 ∂b i −𝜕𝑏 i,t − i· , , i,t − i· , , i t t