The Role of coefficient drivers of time-varying coefficients in estimating the total effects of a regressor on the dependent variable of an equation

Swamy, Paravastu Ananta Venkata Bhattanatha; Chang, I.-Lok; Von zur Mühlen, Peter; Achameesing, Amit Article The Role of coefficient drivers of time-varying coefficients in estimating the total effects of a regressor on the dependent variable of an equation Journal of Risk and Financial Management Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Swamy, Paravastu Ananta Venkata Bhattanatha; Chang, I.-Lok; Von zur Mühlen, Peter; Achameesing, Amit (2022) : The Role of coefficient drivers of time-varying coefficients in estimating the total effects of a regressor on the dependent variable of an equation, Journal of Risk and Financial Management, ISSN 1911-8074, MDPI, Basel, Vol. 15, Iss. 8, pp. 1-14, https://doi.org/10.3390/jrfm15080331 This Version is available at: https://hdl.handle.net/10419/274853 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Citation: Swamy, Paravastu Ananta Venkata Bhattanatha, I-Lok Chang, Peter von zur Muehlen, and Amit Achameesing. 2022. The Role of Coefficient Drivers of Time-Varying Coefficients in Estimating the Total Effects of a Regressor on the Dependent Variable of an Equation. Journal of Risk and Financial Management 15: 331. https:// doi.org/10.3390/jrfm15080331 Academic Editors: S. Ejaz Ahmed and Nataša Šarlija Received: 10 June 2022 Accepted: 6 July 2022 Published: 27 July 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Journal of Risk and Financial Management Article The Role of Coefficient Drivers of Time-Varying Coefficients in Estimating the Total Effects of a Regressor on the Dependent Variable of an Equation Paravastu Ananta Venkata Bhattanatha Swamy 1, I-Lok Chang 2, Peter von zur Muehlen 1,* and Amit Achameesing 3 1Federal Reserve Board, 20th & Constitution Avenue, Washington, DC 20551, USA; [email protected] 2Department of Mathematics and Statistics, American University, 4400 Massachusetts Avenue, Washington, DC 20016, USA; [email protected] 3Business School, University of Stellenbosch, Carl Cronje Dr, Bellville, Cape Town 7530, South Africa; [email protected] *Correspondence: [email protected] Abstract: Typically, the explanatory variables included in a regression model, in conjunction with the omitted relevant regressors implied by the usual error term, have both direct and indirect effects on the dependent variable. Attempts to obtain their separate estimates have been plagued with simultaneity issues. To circumvent these problems, this paper defines their sum as “total effects”, develops a time-varying coefficients methodology for their estimation without simultaneity bias, and applies these techniques to estimate the total effects of commercial bank credit per-capita on real GDP per-capita in Mauritius. An innovation is the introduction of extraneous variables that act as “coefficient drivers” chosen on the basis of best predictive performance, as measured by the smallest value of Theil’s U-statistic we were able to locate in the estimation. Keywords: total effects; bank credit; economic growth; direct effects; indirect effects; threshold regression; real GDP; coefficient drivers; Theil’s U statistic 1. Introduction As is—or should be—known from Pratt and Schlaifer (1984), every regressor included in a regression equation has both direct and indirect effects on its dependent variable. In contrast to traditional econometric practice, which side-steps the issue of indirect effects, we shall follow Pratt and Schlaifer (1984) and account for such direct and indirect effects by estimating their sum as “total effects.” Since it is unlikely in most economic settings that the total effects of a given regressor are constant, we generalize the proposed model, by allowing all of its coefficients to be time-varying, necessitating the use of “modified generalized least squares.” 1 Recognizing that available data for the variables included in our model do not contain sufficient information about the indirect effects of the regressor of our model, we shall utilize additional information over and above the information already contained in the specified variables of the model by introducing so-called “coefficient drivers” without knowing whether this additional information is relevant or not. These coefficient drivers are variables not actually included in the set of regressors but having an influence on how the coefficients associated with regressors impact the dependent variable over time. As a consequence, the coefficients themselves become functions of coefficient drivers not otherwise in a model but, nevertheless, playing important roles in how the dependent variable responds to its regressors over time. The choice of such variables is inductive and should be guided by empiricism which we advocate using Theil’s U-statistic, J. Risk Financial Manag. 2022,15, 331. https://doi.org/10.3390/jrfm15080331 https://www.mdpi.com/journal/jrfm J. Risk Financial Manag. 2022,15, 331 2 of 14 a measure of predictive performance that is invariant to scaling, rather than a Neyman– Pearson test criterion, to improve the accuracy of results. In all this, we were motivated by a desire to obtain results that are as precise and empirically relevant as feasible. The remaining part of this paper is divided into six sections. Section 2provides the motivation for the model to be estimated and gives an economic background. In Section 3, we develop a model with time-varying coefficients. The novelty of this model is that the coefficient on the regressor included in a regression equation measures the regressor’s total effect on the dependent variable. Section 4gives some implications of the model developed in Section 3for the relationship between economic growth and financial development. Section 5is concerned with the estimation of the total effect of commercial bank credit (CBC) on real gross domestic (RGDP) for the period 1970–2019 in Mauritius. Section 6offers a detailed rationale for our choice of, and need for, estimating a model with time-varying coefficients. Section 7concludes. 2. The Economic Background Early economists, such as Bagehot (1873) and Schumpeter (1912), suggested that finance leads to economic development. More recent theory on finance and endogenous growth likewise suggest that more finance can have a positive effect on economic growth; Greenwood and Jovanovic (1990); Pagano (1993); King and Levine (1993); Berthelemy and Varoudakis (1996). However, the empirical literature has found mixed evidence of the effects of finance on growth. A comprehensive review by Levine (2005) found that more finance tends to be beneficial to the economy, so that countries with a smaller share of credit to GDP should attempt to increase it to promote investment and growth. In general, the empirical literature is ambiguous in its conclusions, suggesting a diminishing-returns non-linear relationship between “financial deepening” and economic growth, in that too much finance might be harmful for growth (see Deidda and Fattouh 2002;Huang and Lin 2009;Arcand et al. 2012,2015;Cecchetti and Kharroubi 2012,2013;Law and Singh 2014). In the case of Mauritius, empirical studies on finance and growth have generally found a positive link between GDP (or investment or economic growth) and different quantitative measures of financial development (FD), such as the ratio of liquid liabilities of banks to GDP, and private sector credit (see Jouan 2005;Jankee 2006;Seetanah 2008; Nowbutsing et al. 2010;Muyambiri and Odhiambo 2018). None of these studies considered the possibility of the time-variability of the total effects of finance on growth. To pursue this possibility, this paper applies a time-varying coefficient (TVC) model to explore how the relationship between financial development and economic growth in Mauritius may have changed over time, possibly as a consequence of changes in economic policies and structural economic changes in the country since independence in 1968. In contrast to existing fixed and variable coefficient models, 2 which ignore the indirect effects of the regressor on the dependent variable, the TVC model of Swamy and von zur Muehlen (2020) measures the total effects of bank credit on RGDP from 1970 to 2019. In this paper, we focus on total bank credit as a measure of financial development, because the transaction activities of a commercial bank are different from those of other financial intermediaries, such as an insurance company, in that the former, when transacting with the latter, discharges its payment obligations to the latter by issuing deposits, whereas when agents belonging to the latter group transact with each other, they do so by transferring existing deposits. When an insurance company lends to a household, it pays by transferring money it holds with a bank (an asset to the insurance company), thereby leaving the total stock of money unaffected. In contrast, when the bank extends a loan to a household, it discharges its obligation to pay by crediting the household’s account, thereby increasing the total stock of money (See Werner 2005). The theoretical background of and interest in the potential role of bank credit in promoting GDP is the literature inaugurated by Werner (1992), who argued that in order for GDP to expand, more money is needed to settle those transactions, implying that when banks create money, credit, and new purchasing power, they contribute to GDP not J. Risk Financial Manag. 2022,15, 331 3 of 14 merely sectorally but, more importantly, to the expansion of GDP as a whole. Werner (2012) argued that it is the portion of bank credit allocated to GDP-type spending as opposed to financial transactions that drives GDP. If this conjecture is correct, we should expect a diminishing effect of total bank credit on GDP over time if bank credit in Mauritius underwent shifts from real to financial spending. In this paper, we focus on the effects of bank credit on RGDP. 3. A Model with Time-Varying Coefficients In this section, we describe a relationship between per-capita RGDP and per-capita CBC utilizing time-varying coefficients and carefully selected coefficient drivers, particularly to improve predictive performance, where, importantly, these time-varying coefficients are to be taken as random variables. We assert that the total effect of per-capita x1t= per-capita CBC on per-capita yt = RGDP per-capita can be cast in terms of two relationships, as follows: First, yt is related to x1t plus an unspecified set of excluded relevant variables, denoted Wt , via the following relation with time-varying coefficients, based on the methodology introduced by Swamy and Tinsley (1980), yt=α0t+α1tx1t+Wt, (1) where W t contains the effects of excluded relevant variables. Here Wt is written as a scalar and not potentially a vector, as done by Pratt and Schlaifer (1984). Second, recognizing that Equation (1) suffers from a simultaneity problem caused by the correlation of x1t with Wt ,Pratt and Schlaifer (1984) proposed augmenting (1) with the stochastic relationship, Wt=λ0t+λ1tx1t, (2) where λ0tis a random error term and i.i.d. (0, ω). Our contribution is (1) the introduction of time-varying coefficients that (2) are potentially driven by variables not otherwise part of the model. Substituting the right-hand side of Equation (2) for Wtin Equation (1), gives yt=α0t+α1tx1t+(λ0t+λ1tx1t)=(α0t+λ0t) + (α1t+λ1t)x1t, (3) where λ0t , which is i.i.d. (0, ω ), is the error term of Equation (3), coefficient α1t is the direct and the term λ1t represents the indirect effect of x1t on yt . 3 This indirect effect arises because x1t affects Wt as in (2), and Wt affects yt as in (1). The sum of these direct and indirect effects, (α1t+λ1t), is the total effect of x1ton ytalluded to in the Introduction. Pratt and Schlaifer (1984) claim that while the direct effect α1t and the indirect effect λ1t are non-unique, their sum ( α1t + λ1t ), called the total effect, is unique. To prove that the total effect is unique, we need to show in how many ways the total effect can be non-unique and how we can avoid all these ways. As such, in order to avoid issues in proving the uniqueness of total effects, we chose to write the effects of excluded relevant regressors in terms of a scalar Wt , compared with Pratt and Schlaifer (1984) 4 , who write this as the product of two vectors.5 Note that Equation (3) is free from simultaneity problems because x1t is independent of λ0t . The case for estimating total effects is further strengthened when one considers two principal defects of models such as the widely used Kalman filter (Durbin and Koopman 2001): their lack of an i.i.d. error term and their inability to measure the indirect effects of regressors. To proceed, re-write the relationship between ytand x1tas yt=γ0t+γ1tx1t, (4) where γ0t= (α0t+λ0t) and γ1t= (α1t+λ1t) = the total effect of x1ton yt. J. Risk Financial Manag. 2022,15, 331 4 of 14 In vector form yt=(1x1t)(γ0tγ1t)0=x0 tγt where x0 t = (1 x1t ) is a 1 × 2 vector, γt = ( γ0tγ1t)0 a 2 × 1 vector, and from now on all vectors are denoted by bold symbols. The sample information is (ytxt), t= 1, 2, . . . , T. As is evident from (1), the information contained in data on yt and x1t is adequate to estimate direct effects with precision, but it may not be enough to estimate indirect effects. Therefore, as promised in the Introduction, we now consider additional observable variables that hopefully contain information about λ1t . Since we do not know a priori what information any of these variables may contain, we use a heuristic approach of experimenting with various candidates that look promising from a theoretical point of view. We call them coefficient drivers because of the manner in which we shall use them. Consider two such variables, labelled zit and zjt , and posit the following two relationships: γ0t=π00 +π0izit +u0t, (5) γ1t=π10 +π1jzjt +u1t (6) Using appropriate matrix algebraic notation, Equations (5) and (6) can be combined into the following single equation: γt=Πzt+ut, where γt is defined in the equation below (4), Π = π00 π0i 0 π10 0π1j  is a 2 × 3 matrix with exclusion restrictions, zt= (1 zit zjt)0is a 3 ×1 vector, and ut=u0t u1t is a 2 ×1 vector. Note that the assumption of time-variability is key: Equations (5) and (6) would be impossible had we treated the coefficients of (4) as constant parameters. Equations (5) and (6) are new to our time-varying coefficients model. Later, we will check systematically what, if any, information is contained in these coefficient drivers. The coefficients of Equations (5) and (6) have further useful interpretations. The coefficient γ0t in Equation (5) is equal to ( α0t + λ0t ) where λ0t is random. The coefficient driver zit simply acts as an explanatory variable of γ0t . When (5) and (6) are inserted into (4), all the terms on the right-hand side of (6) get multiplied by x1t . Therefore, (i) π10 as the coefficient on x1t can absorb at least part of the direct-effect component of γ1t , and (ii) π1j becomes the coefficient on the interaction between zjt and x1t . Such a coefficient cannot absorb the direct-effect component of γ1t , but its estimate can indicate whether π1jzjt absorbs at least a part of the indirect-effect component of γ1t . Therefore, the estimate of the coefficient π1j reveals the strength or weakness of the relationship between the indirecteffect component of γ1t and zjt . If the intercept π10 of (6) does not completely absorb the direct-effect component, and if π1j zjt of the same equation does not completely absorb the indirect-effect component of γ1t , then the term u1t corrects the inaccuracies in both, if the equality sign of (6) holds. The sample information and the additional information can be combined by substituting the right-hand side of the equation γt = Πzt + ut , for γt in Equation (4). Doing so gives yt = x0 tΠzt + x0 tut = ( z0⊗x0 t ) vec(Π) + x0 tut where ⊗ denotes the Kronecker product and vec(Π)is the column stack of Π. Stacking the equations yt = = x0 tΠzt + x0 tut , t = 1, 2, . . . , T, gives y = Xzπ + ε , where y is a T × 1 vector of observations on yt , Xz is a T× 6 matrix of observations on ( z0⊗x0 t ), π= vec(Π) , ε = Dxu , ε is a T × 1 vector of errors, Dx is a T × 2T diagonal matrix with x0 1 , x0 2 , . . . , x0 T along the diagonal, and u is a 2T × 1 vector of the errors of equation which is below Equation (6) for t = 1, 2, . . . , T. In addition to (4)–(6), we assume that ut=(u0t , u1t)0=Φut−1+at, (7) J. Risk Financial Manag. 2022,15, 331 5 of 14 where Φ is diagonal with φ00 and φ11 as its diagonal elements, E at = 0 and Eata0 s = σ2 a∆aif t =s 0 if t 6=s . This assumption of diagonal Φ is required for convergence, because our estimation procedure of the model in (4)–(7) is an iterative procedure. If the variance–covariance matrix of the error vector u of Equation ε = Dxu is denoted by σ2 aΣu1, then the variance-covariance matrix of εcan be shown to be σ2 aDxΣu1D0 x. The exclusion restrictions imposed on Π can be written as r = Rπ + 0. Combining this equation with y = Xzπ + ε gives ye = Xzeπ + εe where ye = ( y r ) 0 , Xze = ( X0 zR0)0 , and εe= (ε0)0. The variance–covariance matrix of εeis singular. Since the coefficients of (5) and (6) are fixed parameters, they possess consistent estimators. We can find them by applying Paige’s (1979) numerically stable algorithm for the generalized least squares method to equation ye = Xzeπ + εe . We also find feasible generalized least squares estimators of π , see Swamy (1990). From these estimates, we derive the estimates of u using ε = Dxu ,asinSwamy (1990). Since the coefficients of Equation (4) are time-varying, they themselves do not possess consistent estimators. However, substituting the above estimates of π and u , and the data on zit and zjt on the right hand sides of (5) and (6), respectively, gives the estimates of γ0t and γ1t ,t= 1, . . . ,T. In other words, given data on zit and zjt , we find the estimates of π00 , π0i , π10 , and π1j from their consistent estimators obtained above and the derived estimates of u0t and u1t from ε = Dxu , to obtain the estimates of γ0t and γ1t from Equations (5) and (6), respectively. However, we do not know the statistical properties of these estimates. Since σ2 aΣu1 is unknown, we use its estimate in its place. Let ˆ Σu1 denote its estimate. The Cholesky factorization of ˆ Σu1 can be represented by FF0 . We denote the Cholesky factorization of DxΣu1D0 x as BB0 .Paige’s (1979) algorithm for performing the generalized least squares estimation can be directly applied to ye=Xze π+εe. After this estimation, we find the feasible version of the generalized least squares estimator of π ’s by replacing σ2 aΣu1 by ˆσ2 aˆ Σu1 . The vector u is replaced by its estimates. We substitute these feasible versions in place of the unknown coefficients and error terms in Equations (5) and (6), respectively. In conjunction with our data on coefficient drivers, the feasible estimates of π0s and u0s in (5) and (6) give the estimates of γ0t ’s and γ1t ’s. These estimates are also substituted into Equation (4). The consistency properties of the feasible generalized least squares estimators of fixed coefficients are known in the econometrics literature. The consistency of generalized least squares estimators of π0s are well defined but not of the time-varying coefficients. The only thing we can claim is that the estimates of time-varying coefficients are those implied by the consistent estimators of fixed coefficients. 6 4. Implications of the Model of above Section for the Relationship between Economic Growth and Financial Development Differencing both sides of each of Equations (4)–(6) gives ∆yt=∆γ0t+γ1tx1t−γ1,t−1x1,t−1+γ1,t−1x1t−γ1,t−1x1t =∆γ0t+(∆γ1t)x1t+γ1,t−1(∆x1t),(8) where ∆ is the difference operator, and ∆yt = yt−yt−1 , ∆γ0t = π0i(∆zit) + ∆u0t , ∆γ1t=π1j(∆zjt)+∆u1t, and the vector (∆u0t,∆u1t) is completely unknown. In the next section, we will be using the model in (4)–(7) to estimate the total effects of CBC per capita on RGDP per capita and, therefore, Equation (8) is nothing but an implication of our model. Dividing both sides of (8) by yt−1 gives a relationship between financial depth and economic growth, since CBC can be considered as a proxy for financial depth. Arcand et al. (2012) studied such a relationship and concluded that “there is a positive and robust correlation between financial depth and economic growth in countries with small and intermediate financial sectors, but . . . [they] also show that there is a threshold (which . . . [they] estimate to be at around 80–100% of GDP) above which finance starts having a negative effect on economic growth”. J. Risk Financial Manag. 2022,15, 331 6 of 14 The relationship between financial depth and economic growth we obtained above for Mauritius, using the model in (4)–(7), is more general than that of Arcand et al. (2012), and, notably, the total effects of CBC per capita on RGDP per capita we obtained for Mauritius are all positive throughout the sample period, 1970–2019, never turning negative. Since, with our sample and estimates, Arcand et al.’s (2012) threshold is never breached in Mauritius, we do not expect bank finance to have had a negative effect on economic growth during 1970–2019, at all. As a digression, we note that while on the surface, there may be some resemblance between the so-called hierarchical models and Swamy’s (1971) random coefficient model, such a similarity is superficial. Hierarchical models, being less general than the model given by (4)–(7), are not at all applicable to the kind of econometric work being considered here, because whatever methodological insights such modeling techniques might bring to the topic, they do not—nor can they—address the principal concern of this paper, which is to obtain consistent estimators of the total effects of the included regressors on the dependent variable when observations do not belong to different (hierarchical) clusters. We came to the preceding conclusion as follows: In Levy’s model, y ij is normally distributed with random mean µi and fixed variance σ2 y . The random mean can be written as µi = µ + bi , where bi is normally distributed with mean 0 and variance σ2 b . Combined, Levy’s model is yij = µ + bi + εij , where εij is normally distributed with mean 0 and variance σ2 y ,iindexes clusters and jindexes observations within each cluster. From this it follows that two different values of yij contain the same value of µ or bi and different values of εij , if the two values of yij belong to the same cluster and have the same value of µ , and different values of bi and εij otherwise. The fact that the distribution of bi has the property of countable additivity means that the probabilities implied by the distribution of bi and εij are frequentist, as are the probabilities implied by the distributions of random coefficients in Swamy’s (1971) random coefficient regression models. Levy (2012) estimates his hierarchical models using both maximum likelihood and Bayesian posterior distributions. It should be noted that these Bayes procedures employ frequentist probabilities but not subjective probabilities, as in Swamy’s (1971) random coefficient and Swamy and Tinsley’s (1980) stochastic coefficient regression models. To obtain subjective probabilities, Bayesian statisticians model their knowledge of each fixed parameter as random. Levy’s distributions of random parameters are not of this type because his distribution of the random variable µihas countable-additivity but not finite additivity properties. 5. Empirical Estimates and Lessons to Be Drawn We made an empirical application of the model given by Equations (4)–(7). In this application, with data for RGDP per capita and CBC per capita for Mauritius, as well as for various potential coefficient drivers (z it , zjt ) for the sample period 1970–2019, we obtained (i) estimates ˆγ0t and ˆγ1t of the time-varying coefficients γ0t and γ1t and in (4), respectively; (ii) estimates ˆπ00 , ˆπ0i , ˆπ10 , and ˆπ1j of the fixed coefficients in (5) and (6); (iii) estimates of the diagonal elements φ00 and φ00 of Φ that appears in the process (7); and (iv) an estimate of the variance–covariance matrix σ2 a∆a of the process (7). Except for ˆγ0t and ˆγ1t , these estimates are recorded in Tables 1and 2for different pairs of ( zit , zjt ). Evidently, the coefficients of (4) are not constant, unlike the coefficients of (5) and (6). Our experiments with a chosen set of coefficient drivers ( zit , zjt ), showed that some give better out-of-sample forecasts of ytthan the others. The results are shown below: J. Risk Financial Manag. 2022,15, 331 7 of 14 Table 1. Estimates of the Coefficients of Equations (5) and (6) and the Variance-Covariance of Process (7) When the Diagonal Parameter Matrix Φ of Process (7) is Restricted to be Zero *. Cases of Φ= 0 Coefficient Driver of (5) Estimates of Coefficinets of (5) Coefficient Driver of (6) Estimates of Coefficients of (6) AR Coefficients of Errors of Equation (7) Scalar of Cova-Riance Matrix of Equation (7) Covariance Matrix of Equation (7) Theil’s Measure of Forecast Accuracy Zit π00 π0iZjt π10 π1jφ00 φ11 ^ σ 2 a ^ ∆aU Statistic II GFCF 78,828.5877 5.785 CEXPI 2.148 −0.0304 0 0 1.61575374 52,958,174.6 −136.318188 0.47 (30.340) * (9.840) * (0.683) (−0.989) −136.318188 0.561897701×10−11 III GFCF 57,282.9121 26.144 OMT 9.128 −0.1667 0 0 5.12553742 ×1013 0.286771043 ×10−21 0.931448686 ×10−17 1..62 (77.329) * (11.461) * (1.742) (−3.417) * 0.931448686 ×10−17 0.302539840×10−12 IV GFCF 57971.6075 23.923 REEXR −28.067 0.1695 0 0 3.5351479 ×10−64671467.31 4,725,656.10 1.77 (79.507) * (10.886) * (−3.344) * (2.417) * 4725656.10 4,780,473.49 V PI 55,837.7511 28.594 CEXPI −19.113 0.1275 0 0 3.4874377 ×1080.114075712×10−15 0.220199723 ×10−11 1.02 (55.558) * (7.895) * (−4.958) * (2.865) * 0.220199723 ×10−11 0.425050323×10−7 VI PI 56,892.1517 40.422 OMT 10.159 −0.18522507 0 0 3.1687904 ×10−8333,057,160 359,975,297 1.53 (84.735) * (13.236) * (2.169) * (−4.221) * 359,975,297 389,068,994 VII PI 57,680.9252 36.655 REEXR −30.707 0.185 0 0 3.1374654 ×10−745,052,492.5 45,723,771.0 0.78 (84.488) * (12.039) * (−3.914) * (2.833) * 45,723,771.0 46,405,051.6 VIII PC 51,618.476 6.834 CEXPI −35.319 0.295 0 0 9.3242741 ×10−8158,748,760 −164,953,937 1.89 (49.150) * (7.326) * (−8.776) * (7.724) * −164, 953, 937 171, 401, 662 IX PC 57,387.614 7.647 OMT −14.894 0.0671 0 0 1.2701022 ×10−627,980,777.2 28,023,306.1 0.46 (48.140) * (5.522) * (−1.852) * (1.012) 28,023,306.1 28,065,899.7 XPC 57,106.0044 7.865 REEXR 2.25 −0.0857 0 0 619.238949 0.191039356×10−90.336970525 ×10−50.69 (48.442) * (5.546) * (0.196) (−0.845) 0.336970525 ×10−50.594375616×10−1 XI CEXPI 159,842.9 −512.65 PI 1.985 −0.000018 0 0 0.21070 311,100,912 −313,455,083 12.27 (10.948) * (−4.317) * (16.363) * (−7.119) * −313,455,083 315,827,069 XII OMT 9320.21 473.03 PI 9.492 −0.00016 0 0 0.32877 600,047,017 −631,200,283 8.91 (1.196) (6.265) * (8.089) * (−5.081) * −631,200,283 663,970,965 XIII REEXR 128,722.23 −612.732 PI 11.216 −0.0002 0 0 0.57717 ×10−7557,628,459 −584,797,650 10.96 (4.653) * (−2.564) * (7.795) * (−4.932) * −584,797,650 613,290,599 XIV CEXPI 33,091.9 161.689 GFCF 13.047 −0.00018 0 0 0.25716 ×10−70.103723501 ×1010 −0.113230521 ×1010 11.96 (4.417) * (3.314) * (9.17) * (−6.007) * −0.113230521 ×1010 0.527320203 XV OMT 9689.82 468.194 GFCF 9.806 −0.00013 0 0 0.60319 ×10−1831,699,126 −888,554,543 8.71 (1.261) (6.285 ) * (8.277) * (−5.291) * −888,554,543 871,140,258 XVI REEXR 127,993.07 −607.771 GFCF 11.581 −0.00016 0 0 0.39494 ×10−7769,980,308 −818,999,905 10.70 (4.691) * (−2.579 ) * (7.976) * (−5.133) * −818,999,905 871,140,258 XVII CEXPI 35,287.53 151.79 PC 11.413 −0.00004 0 0 0.31956 ×10−7973, 114, 924 −0.106389135 ×1010 18.91 (4.38) * (2.885) * (8.122) * (−4.919) * −0.106389135 ×1010 0.116313579 ×1010 XVIII OMT 8383.01 486.8 PC 8.581 −0.000029 0 0 0.26532 ×10−7789,283,708 −845,445,226 13.72 (1.015) (6.168) * (7.466) * (−4.387) * −845,445,226 905,602,920 XIX REEXR 130,269.95 −622.12 PC 10.17 −0.000036 0 0 0.45244 ×10−7750, 423, 836 −800, 984, 254 16.80 (4.505) * (−2.491) * (7.217) * (−4.284) * −800,984,254 854,951,221 J. Risk Financial Manag. 2022,15, 331 8 of 14 Table 2. Estimates of the Coefficients of Equations (5) and (6) and the Variance–Covariance of Process (7) When the Diagonal Parameter Matrix of Process (7) is NOT Restricted to be zero. Cases of Φ6=0 Coefficient Driver of (5) Estimates of Coefficinets of (5) Coefficient Driver of (6) Estimates of Coefficients of (6) AR Coefficients of Errors of Equation (7) Scalar of Cova-Riance Matrix of Equation (7) Covariance Matrix of Equation (7) Theil’s Measure of Forecast Accuracy Zit π00 π0iZjt π10 π1j φ00 φ11 ^ σ 2 a ^ ∆aU Statistic II GFCF 51,350.3509 0.1409 CEXPI −4.672 0.1343 0.995 0.957 6.28 ×10−79, 807,332.55 9, 860, 647.14 2.70 (35.847) * (0.040) (−0.6620) (3.673) * 9,860,647.14 9,914,251.55 III GFCF 55,539.7578 16.792 OMT 7.659 −0.0964 −0.995 0.696 16.5796 0.756565055×10−80.664381937×10−40.20 (50.23) * (4.535) * (1.144) (−1.537) 0.664381937×10−40.583430804 IV GFCF 52,279.2823 6.333 REEXR −4.47 0.0878 −0.995 0.95 1.00 ×10−67,670,048.55 7,740,947.45 0.34 (33.468) * (1.853) (−0.3850) (1.052) 7,740,947.45 7,812,501.72 V PI 51,332.4811 −0.5032 CEXPI −4.665 0.1357 −0.995 0.957 2.21 ×1012 0.80149479×10−20 0.152275360×10−15 2.72 (35.259) * (−0.1084) (−0.6496) (3.687) * 0.15227536×10−15 0.289306750×10−11 VI PI 52,898.0948 9.3061 OMT 8.824 0.0349 0.995 0.9447 3.03 ×10−8262,085,391 263,973,387 2.25 (33.973) * (1.829) (1.086) (0.599) 263,973,387 265,874,983 VII PI 56,582.0399 26.12 REEXR −15.468 0.0956 −0.995 0.5697 40.4588 0.151241664×10−80.203420888×10−43.99 (57.087) * (5.836) * (−1.428) (1.058) 0.203420888×10−40.273602238 VIII PC 67,119.6365 2.747 CEXPI 3.593 −0.0405 0.775 0.995 0.11762 66,448,679.4 8418.07436 0.10 (19.168) * (3.496) * (0.7648) (−1.3048) 8418.07436 1.06644672 IX PC 96,609.3714 0.8731 OMT 0.1205 0.0011 0.933 0.917 2.75 ×101926,617.228 −1.97251424 0.14 (9.121) * (2.863) * (0.226) (0.470) −1.97251424 0.112212652×10−3 XPC 77,794.8174 1.11636277 REEXR 0.0296 0.0016 0.995 0.945 1.25 ×1011,049,521.88 10.4809793 0.32 (2.178) * (2.3810) * (0.030) (0.275) 10.4809793 0.111900257×10−2 XI CEXPI 112,365.15 −264.24 PI 2.67 −0.000011 −0.239 0.995 0.489428 0.61054500 ×10−20 0.109457267 ×10−15 0.67 (12.517) * (−4.151) * (1.645) (−1.463) 0.10945727 ×10−15 0.196232765 ×10−11 XII OMT 55,627.7 244.122 PI 2.479 −0.000013 0.519 0.995 0.50171 48,659,198.5 5816.47734 0.65 (3.307) * (1.660) (2.247) * (−2.266) * 5816.47734 0.695272626 XIII REEXR 84,922.64 −104.828 PI 3.995 −0.000018 0.187 0.992 0.18590 0.49856292 ×10−27 0.10864136210−22 0.81 (3.56) * (−0.520) (1.856) (−1.610) 0.10864136 ×10−22 0.236739336 ×10−18 XIV CEXPI 97,632.09 −181.003 GFCF 3.59 −0.00001 −0.272 0.994 0.13742 222,179,723 −224,512,472 0.74 (11.94) * (−3.331) * (1.426) (−1.1103) −224, 512, 472 226, 869, 713 XV OMT 58,839.79 168.733 GFCF 3.024 −0.0000084 0.509 0.988 0.603 ×10−1827,247,425 −4847.13464 0.13 (4.792) * (1.570) (1.976) (−1.196) −4847.13464 0.417614474 XVI REEXR 82,544.39 −215.66 GFCF 10.601 −0.0000029 0.995 0.312 −0.22220 245,218,164 −247,154,805 0.09 (4.029) * (−1.206) (1.021) (−0.749) −247,154,805 249,106,741 XVII CEXPI 109,318.23 −244.646 PC 3.384 −0.0000052 −0.181 0.982 0.43383 189,498,520 −4896.63303 5.91 (12.841) * (−4.048) * (3.534) * (−2.430) * −4896.63303 0.126528772 XVIII OMT 76,300.71 30.931 PC 3.551 −0.0000054 0.337 0.974 0.22739 69,042,808.40 1611.51423 6.84 (4.867) * (0.223) (4.208) * (−2.530) * 1611.51423 0.376140277×10−1 XIX REEXR 78,978.33 −73.1 PC 5.113 −0.000007 0.146 0.995 0.15216 274,861,594 −13,457.1965 6.44 (3.571) * (−0.387) (1.580) (−1.686) −13457.1965 0.658863012 * 5% significance level is used in this table, t-ratios are given in parentheses, and they are significant if they are with asterisks.