Monia Ferchichi
email: monia.ferchichi@lasalle-beauvais.fr

Loïc Sauvée

Modelling the Choice of the Organizational Form in the European Bioethanol Industry

Keywords: Bioethanol industry, organizational choice, procurement, transaction costs, scale

With increasing interest in renewable energy from agriculture, including bioethanol for environmental benefits, rural growth and development, and energy security, there is a need for a better understanding of the economic organization of this emerging industry. Study of the organization of the bioethanol industry represents an under-researched area and a new application of transaction cost theory to an emerging industry in Europe.

Refinement of the theory can also result from challenging applications. This paper provides an application of transaction-cost economics theory to the existing European bioethanol industry in challenging the empirical convention of excluding production cost variables from transaction-cost analysis. Utilizing survey data from 41 bioethanol firms using sugar beets and/or cereals, we study the relationship between transaction's frequency, physical asset specificity, site specificity and scale in explaining firms' decisions to insure procurement of inputs either externally or through contracts. Consistent with transaction cost theory, the resolution of both dichotomous logit and probit models shows that both physical asset specificity and scale are good predictors of organizational forms. Given this evidence, this paper reconsiders the impact of scale and transaction costs on the choice of organizational form.

Introduction

Recent years have seen a growing interest in the development of renewable energy industries for environmental benefits, rural growth and development, and energy security. While the technical aspects of bioethanol processing have been extensively studied, less is known about the economic organization and governance of the bioethanol industry. How should bioethanol firms procure their inputs: from the spot market or through contracts with independent biomass producers? What are the characteristics of purchase and supply contracts in bioethanol?

To begin addressing some of these questions we focus on a fundamental alternative facing any firm [START_REF] Coase | The Nature of the Firm[END_REF], the "make-or-buy decision" [START_REF] Klein | The Make-or-Buy Decision: Lessons from Empirical Studies[END_REF]. Using the transaction cost economics framework developed by [START_REF] Williamson | The Economic Institutions of Capitalism[END_REF][START_REF] Williamson | Comparative Economic Organization: The Analysis of Discrete Structural Alternatives[END_REF][START_REF] Williamson | The Mechanisms of Governance[END_REF][START_REF] Williamson | The Economics of Governance" The Richard T. Ely lecture[END_REF], we examine the vertical structure of the current bioethanol industry with survey data from 41 European bioethanol producers. Our analysis focuses not only on the frequency of the transaction and asset specificity-the main variable of interest in the empirical transaction cost literature [START_REF] Klein | The Make-or-Buy Decision: Lessons from Empirical Studies[END_REF]-but also on the economies of scale related to he technology of production. While transaction cost economics (TCE) allows production costs to affect the choice of organizational form-indeed, in the integrated Riordan and [START_REF] Williamson | The Economic Institutions of Capitalism[END_REF] model, production costs and transaction costs are determined jointly-in practice, most of the empirical literature has taken production costs as given and focused on transaction costs. Or, as [START_REF] Langlois | Capabilities and Governance: the Rebirth of Production in the Theory of Economic Organization[END_REF] describe the problem, the TCE literature tends to assume that knowledge about production is easily acquired and costless, while market transactions are fraught with hazards brought about by information and agency costs.

Our results suggest that the frequency of inputs' procurement is not a statistically significant determinant of external organization in bioethanol industry. Instead, characteristics of the production process, such as scale of the bioethanol firms, appear to be more important. Moreover, several firms in our sample both purchase their inputs from the spot market and through contracts with farmers and/or agricultural cooperatives, a practice difficult to explain within the usual framework of TCE. Of course, our results may be specific to bioethanol and may not be generalized to vertical relationships in other industries. Nevertheless, the findings suggest that the standard TCE framework may require adaptation to account for complex arrangements such as simultaneous external and hybrid procurement. In this article we focus on scale in empirical analysis as a TCE variable and offer some evidence to support the hypothesized relationship in TCE between the choice of organizational form and scale.

Literature

Biomass and bioethanol

Biomass and bioethanol researchers have rarely considered organizational issues or applied an adequate organizational theory to this nascent industry3 . Some authors indirectly discuss organization when considering non-technical barriers in biomass production. [START_REF] Roos | Critical Factors to Bioenergy Implementation[END_REF] and [START_REF] Costello | Institutional Opportunities and Constraints to Biomass Development[END_REF] develop broad-based frameworks for considering organizational issues. They identify critical factors in the choice of organization including the degree of integration, the scale of operation, the degree of competition, the institutional environment such as national and local policy including public infrastructure availability, and the perceptual beliefs of key actors. Rösch and Kaltschmitt (1999) identify similar topics adding insurance issues and efficiency of knowledge and information flows along the supply chain. [START_REF] Lunnan | Agriculture-Based Biomass Energy Supply -A Survey of Economic Issues[END_REF] takes an in-depth view of the institutional environment created by bioenergy policy, and especially how bioenergy policy and more general agricultural policy can be coordinated.

More closely related to the organization of exchange, [START_REF] Downing | Development of New Generation Cooperatives in Agriculture for Renewable Energy Research, Development, and Demonstration Projects[END_REF] describe the role of agricultural cooperatives in research, financing, and exchange mechanisms in the agro-bioenergy industry. They do not, however, compare cooperatives to contracting and spot market procurement as suggested by organizational theory. [START_REF] Overend | Biomass Power Industry: Assessment of Key Players and Approaches for DOE and Industry Interaction[END_REF] describes the main features of a general biomass industry and recommends optimal exchange structures. Contrary to Downing et al. on the choice of organizational form, Overend recommends spot markets and short term contracting. Recently, [START_REF] Gallagher | Scale, Organization, and Profitability of Ethanol Processing[END_REF] analysed the appropriate scale, organization and implied profitability of a representative U.S. ethanol processing firms. [START_REF] Altman | Scale and Transaction Costs in the U.S. Biopower Industry[END_REF] studied the organization of the biopower industry through an application of transaction cost theory. However, as we will discuss below, in our sample only 6 out of 41 firms procure inputs on the spot market while the others purchase their inputs via contracts. Why do so few firms use the spot market, and what explains the choice of contracting? We turn to the theoretical framework of TCE for answers.

Transaction cost economics

Central to TCE is the discriminating alignment hypothesis, which states that the choice of organizational form depends on the characteristics of the transaction (Williamson 1996, p.371). Economic agents behave in such a way that transactions, which vary in degrees of asset specificity, uncertainty and frequency, are aligned with organizational forms, which can be considered efficient if no feasible alternative can be implemented with net gains. The central problem in Williamson's framework is the bilateral dependency that results from an increase in asset specificity or relationship-specific investment. Asset specificity describes the condition under which the value of assets depends on a particular exchange relationship. Parties that invest in relationship-specific assets risk loosing some of the rents accruing to those assets if their trading partners take advantage of unanticipated changes in circumstances to renegotiate the terms of the exchange relationship in their favor. To protect those investments, parties will craft governance structures such as detailed long-term contracts with adequate adjustment provisions or vertical integration (internal procurement).

Riordan and [START_REF] Williamson | The Economic Institutions of Capitalism[END_REF] formulation seeks to integrate TCE with neoclassical production theory. Extending the basic TCE model to include production costs, [START_REF] Williamson | The Economic Institutions of Capitalism[END_REF] and Riordan and [START_REF] Williamson | The Economic Institutions of Capitalism[END_REF] argue that markets have a production cost advantage over internal organization because the market can realize economies of scale and scope from aggregation of demand (Williamson 1985, p.92). Internal organization, since it only supplies the firm itself, cannot achieve the same benefits of scale achieved by a market. Thus the greater the potential for realizing external economies of scale, the less likely hybrid and internal organization will be observed 4 . This issue can be depicted as a cost-minimization problem. Market organization and internal organization are assumed to have different transaction costs5 . When asset specificity is low, market transaction costs are lower than the costs of hybrid form and of course internal organization (because of administration costs), but as asset specificity rises, the costs of market transactions increase more rapidly than the costs of hybrid form and internal organization as well, such that at some threshold level of asset specificity, internal procurement is the least costly alternative.

These implications are also explained graphically in [START_REF] Williamson | Comparative Economic Organization: The Analysis of Discrete Structural Alternatives[END_REF] where M denotes market governance costs, H hierarchy (or internal organization) and X is used to indicate governance costs of hybrid forms, such as long-term contracting. Adding hybrid simply implies: M(0) < X(0 )< H(0) and M' > X' > H' > 0. . The lower envelope curve is the locus of minimum governance costs. The organization forms that correspond to those points will be most efficient. If additional curves were added for multiple organizational structures, including different contractual arrangements (short term, long term, formal, informal), firm organization (joint ventures, strategic alliances, cooperatives), and even government, the resulting lower boundary would be a concave envelope of least cost organizational forms.

The benefit of this version of the model is that comparative statics analysis can be easily conducted. Shift parameters include technological change, policy and uncertainty. For instance, if policy is implemented that discourages hierarchy (perhaps to restrict monopoly power), this would cause an upward shift in H(k). The change in policy would increase 2 k and make hierarchy less likely compared to the hybrid form. However, the range of market optimality would remain unaffected. This analysis suggests that the greater the level of asset specificity, the more likely that firms will rely on hybrid or hierarchical forms of procurement. Moreover, given the high fixed-cost, low-variable-cost nature of bioethanol production, we expect substantial scale economies to exist, suggesting that firms needing to procure large quantities of inputs will tend to rely on spot-market procurement rather than internal or hybrid procurement. Therefore, given the existence of economies of scale, as scale of a bioethanol plant increases, external procurement should be more likely and hybrid procurement less likely.

Empirical research in transaction cost economics

Previous empirical research in a variety of industry settings has tended to support the basic predictions of TCE, particularly regarding the relationship between asset specificity and vertical integration [START_REF] Shelanski | Empirical Research in Transaction Cost Economics[END_REF][START_REF] Masten | Case Studies in Contracting and Organizations[END_REF][START_REF] Rindfleisch | Transaction Cost Analysis: Past, Present, and Future Applications[END_REF][START_REF] Klein | The Make-or-Buy Decision: Lessons from Empirical Studies[END_REF][START_REF] Macher | Transaction Cost Economics: An Assessment of Empirical Research in the Social Sciences[END_REF]. In the vein of TCE, research works on hybrid forms have shown that the underlying logic of such alternative modes of organization are far from being elucidated [START_REF] Ménard | The Economics of Hybrid Organizations[END_REF][START_REF] Ménard | A New Institutional Approach to Organization[END_REF].

The choice of organizational form is usually modelled as a function of asset specificity and other explanatory variables. Cross-sectional analyses often utilize a logit or probit model to deal with the qualitative and discontinuous nature of the dependent variable. Some contractual attributes, such as prices, length of contracts, or other measurable contract provisions, can be modelled as continuous variables, although the presence of a contract provision is typically measured as a qualitative variable.

Studies that use scale or size as an explanatory variable tend not to support the TCE hypothesized relationship between the choice of organizational form and scale. [START_REF] Wiggins | Oil Field Unitization: Commercial Failure in the Presence of Imperfect Information[END_REF] find that, contrary to TCE theory, that firm size is positively related to vertical integration in oil field organization. In agriculture, the use of contracts and vertical integration are found to be positively correlated with farm size [START_REF] James | Markets, Contracts, or Integration? The Adoption, Diffusion, and Evolution of Organizational Form[END_REF]. [START_REF] Wilson | Adaptation to Uncertainty and Small Numbers Exchange: The New England Fresh Fish Market[END_REF] uses the size of the fishing operation as an indicator of trust. Anecdotally, larger fishermen tend to have long term reciprocal relationships with buyers. Again size is found to be positively correlated with internal organization. These studies contradict the hypothesis made by TCE theory that scale is negatively related to internal organization. From the empirical transaction cost literature, support for scale as a transaction cost variable is much weaker than support for asset specificity.

Empirical results

Data

Our data comes from a survey of bioethanol firms and experts of the sector. We identified 41 active companies in Europe which are producing ethanol from sugar beets and/or cereals.

In the bioethanol industry, key assets include the plant and storage, collection, and transportation equipment. The degree of asset specificity of these assets varies. For example, a bioethanol operation that utilizes wheat that can be easily redeployed to use barley or other cereals would be considered, for given levels of supplier concentration, to have a low degree of asset specificity6 . Other systems that are not as flexible with respect to biomass quantity and quality and are not as redeployable would have higher degrees of asset specificity, ceteris paribus. Thus the theory would suggest that the types of organizational arrangements should vary with the type of equipment employed by the plant7 .

There are two organizational alternatives. The first one is external procurement. This system involves independent firms purchasing sugar beets or cereals from independent producers. This category covers spot markets. The second organizational choice is the hybrid one by which firms can procure their inputs through contracts with farmers and/or with cooperatives. We note that the vertical integration or internal procurement is excluded in this case because bioethanol plants are not able to procure neither sugar beets nor cereals internally and are, therefore, obliged to purchase their inputs from farmers or cooperatives through spot markets or contracting.

These organizational choices are of interest when the choice of scale is also taken into consideration.

Of our 41 bioethanol firms, 35 rely on hybrid procurement by contracting. These companies produce ethanol especially from sugar beets through contracts with the farmers and cereals through contracts with farmers and/or cooperatives. The remaining 6 firms procure all their inputs externally in using spot markets. These companies are Spanish and German ones that have chosen to enter the bioethanol production.

As in most forms of fuel production, bioethanol plants rarely rely on spot market purchases for their inputs. Several empirical TCE studies compare hybrid (contract) procurement and internal procurement [START_REF] Joskow | Vertical Integration and Long-Term Contracts: The Case of Coal Burning Electric Generating Plants[END_REF][START_REF] Joskow | Contract Duration and Relation-Specific Investments: Empirical Evidence from Coal Markets[END_REF][START_REF] Joskow | The Performance of Long-Term Contracts: Further Evidence from the Coal Markets[END_REF]. In this paper we compare hybrid procurement and external procurement through spot markets.

The frequency of the transaction is an important issue to bioethanol firms since it affects their organisational choice according to our questionnaire with experts of the sector. In our sample, 15 firms have a low frequency of inputs procurement. These firms produce ethanol from sugar beets and the procurement of such a product occurs once a year during the harvest period (October-December). The remaining 26 firms have a high frequency of purchasing cereals in order to saturate the production capacity during the year. Table 1 summarizes statistics on our frequency variable. [START_REF] Williamson | The Economic Institutions of Capitalism[END_REF] calls "site specificity", is measured as the average input hauling distance in increments of 0-25, 25-50, and over 50 miles. Following [START_REF] Joskow | Vertical Integration and Long-Term Contracts: The Case of Coal Burning Electric Generating Plants[END_REF][START_REF] Joskow | Contract Duration and Relation-Specific Investments: Empirical Evidence from Coal Markets[END_REF][START_REF] Joskow | The Performance of Long-Term Contracts: Further Evidence from the Coal Markets[END_REF] work on coal-fired plants, low average hauling distances are expected to be indicators of high site specificity. The logic here is that firms that procure inputs from a greater distance are less restricted in space. Those firms that procure all their inputs near their plant often do so because sources at any greater distance are not feasible. In fact, it is likely that the initial location decision of these bioethanol plants was based on proximity to their primary source. The flexibility of the bioethanol technology with respect to the use of sugar beets or cereals is an important issue to bioethanol firms. We use the degree of flexibility as an ordinal indicator of physical asset specificity. If the plant can easily switch between primary sources (cereals/sugar beets), for given levels of supplier concentration, the degree of physical asset specificity of the firm would be low. The asset is easily redeployable and the value of alternative uses of the bioethanol plant is high. If it is difficult and costly to convert the firm to the use of cereals instead of sugar beets then, controlling for supplier concentration, the asset is not as redeployable and the degree of asset specificity would be high. The value of the plant in alternative uses is relatively low.

In our survey, firms were asked to rate the flexibility of their plant as either: inflexible (their bioethanol plant can not be converted to use another input); moderately flexible (their plant could use mainly cereals or sugar beets after minor adjustment and delays); highly flexible (their plant could easily be converted to use mainly cereals or sugar beets without adjustments and delay). Table 3 summarizes the responses to this question. 8 Besides physical asset specificity and site specificity, the transaction cost literature (following [START_REF] Williamson | The Mechanisms of Governance[END_REF] also considers human asset specificity (transaction-specific knowledge or human capital), brand-name capital, "dedicated assets" (substantial, general-purpose investments that would not have been made outside a particular transaction, the commitment of which is necessary to serve a large customer), and temporal specificity (assets which must be used in a particular sequence). Physical asset specificity and site specificity are the most obvious transaction cost variables relevant to bioethanol production. Temporal specificity is discussed in footnote 11 below. Scale of bioethanol plants is measured by the level of inputs the plant uses per year. The scale of the plants could also be measured in terms of scale or production capacity in metric tons which is a common scale measure for bioethanol plants. In our sample the production capacity ranges from a low of 2000 tons per year to a high of 240000 tons per year. Table 4 summarizes statistics on our scale variable. Table 5 summarizes the explanatory variables used in this analysis and the expected signs based on the transaction cost economics results and our surveys with both ethanol plants leaders and experts of the sector.

Table 5. Explanatory Variables and Expected Sign

Explanatory variable Expected sign Frequency of inputs procurement

Increase in the probability of observing external procurement Average hauling distance Increase in the probability of observing external procurement Flexibility to inputs use Increase in the probability of observing external procurement Scale Increase in the probability of observing external procurement

Analytical methods

Because our dependant variable is categorical, we use both dichotomous logit and probit models since the organizational choice of bioethanol firms can be either an external procurement through spot markets or a hybrid form through contracts with farmers and/or cooperatives. In models of this class, the dependant variable can take on only two values, which it is convenient to denote by 0 and 1.

The logit and probit models, together with their multi-response and multivariate generalizations, are now widely used in applied econometric work. Such models are typically estimated by maximum likelihood methods which require the numerical maximization of a log likelihood function [START_REF] Davidson | Convenient Specification Tests for Logit and Probit Models[END_REF]. Since this is usually much more expensive than, say, calculating ordinary least squares estimates for a linear regression model, investigators often display a natural reluctance to test the specification of the model as thoroughly as would normally be done in the regression case. There is thus a clear need for specification tests of logit and probit models which are easy to understand and inexpensive to compute.

In this context, it seems natural to investigate the use of Lagrange multiplier, or score tests, because they require only estimates under the null hypothesis, and they can often be computed by means of artificial linear regressions. The literature on LM tests for logit and probit models is, however, remarkably limited. The recent survey of qualitative response models by [START_REF] Amemiya | Qualitative response models: A survey[END_REF] does not mention LM tests at all, and the survey of LM tests by Engle (1982) describes only one such test for logit and probit models, which appears to be new.

The probability that i y 9 , the i th observation on the dependent variable is equal to 1 is given by )

( β i x F
The distribution function F is an increasing function of i x which has the properties that 0

) ( = -∞ F and 1 ) ( = +∞ F . i
x is a possibly nonlinear function, which depends on i X a row vector of exogenous variables, and β , a column vector of parameters to be estimated.

The only difference between the logit and probit models is that they employ different functions for F [START_REF] Gourieroux | Econométrie des Variables Qualitatives[END_REF]. In the case of the logit model,

β β i x i i i i e x F x y ob p - + = = = = 1 1 ) ( ) / 1 ( Pr , N i ,..., 1 = ∀ (1)
where F denotes the logistic function.

In the case of the probit model,

dz e x F x y ob p z x i i i i i 2 2 2 1 ) ( ) / 1 ( Pr - ∞ - = = = = β π β N i ,..., 1 = ∀ (2)
where F denotes the cumulative distribution function of the standard normal distribution.

The logistic and cumulative normal differ little, and only at the tails [START_REF] Amemiya | Qualitative response models: A survey[END_REF]. Hence, unless the sample size is very large, the empirical results obtained from the two will be very close. Note that, for both (1) and ( 2

), ) ( 1 ) ( z F z F - = -
. Provided the functions have this symmetry property, everything we say below will apply to them as well.

In specific form, there are two choices of organizational form for the 41 European bioethanol plants: either spot markets or contracts. Thus 41 = N and ) / 1 ( Pr

i i i x y ob p = =
is the probability of observing the external organization. Thus the likelihood function for these dichotomous models associated with the observation i y can be expressed as

[ ] = = - + = 0 : 1 : ) ( 1 log ) ( log ) , ( i i y i i y i i x F x F y LogL β β β (5)
In the linear case, this function is globally concave for both the logit and probit models. Thus ML estimates may be found in a straightforward fashion by maximizing it.

We shall denote the gradient of ( 5) with respect to β by the row vector

) (β G Its expression is done by [ ] ' 0 : ' 1 : ) ( 1 ) ( ) ( ) ( ) ( i y i i i i y i i i x x F x f x x F x f G i i = = - - = β β β β β (6)
β y L = (7) [ ] [ ] 0 ) ( ) ( 1 ) ( ) ( ) ( ˆ) , ( log ' 41 1 = = - - = ∂ ∂ = β β β β β β β G x x F x F x f x F y y L i i i i i i i (8) where ) (β G
denotes the gradient associated to the log likelihood ) , ( log

β y L ∂ evaluated at
the point β ˆ. In the case of the logit model this gradient is done by the expression

[ ] 0 ) ( ) ( ' 41 1 = - = = i i i i L x x F y G β β (9)
For the probit model, this gradient is expressed as

[ ] [ ] 0 ) ( 1 ) ( ) ( ) ( ) ( ' 41 1 = - - = = i i i i i i i P x x F x F x F x F y G β β β β β (10)
Explanatory variables: frequency, average hauling distances and flexibility are coded as single categorical variables with two or three categories, while scale is a continuous explanatory variable (the tons of bioethanol per year).

In both dichotomous logit and probit models the p-values are valid, making significance tests meaningful, but the signs and magnitudes of the coefficients have no direct interpretation [START_REF] Alban | Econométrie des Variables Qualitatives[END_REF]. The partial marginal effects for continuous variables can be calculated as:

[ ] [ ] [ ] j i j i i j i i i i j i i j i i x f x x F x x x x F x x F x p β β β β β β β β β ) ( ) ( ) ( ) ( ) ( ) ( ) ( = ∂ ∂ = ∂ ∂ ⋅ ∂ ∂ = ∂ ∂ = ∂ ∂ (11)
where [ ] j i

x denotes the j th explanatory variable for the observation i.

For the logit model, this derivative is expressed as

[ ] j x x j i i i i e e x p β β β 2 ) 1 ( + = ∂ ∂ (12)
For the probit model, the partial marginal effects can be calculated as

[ ] j i j i i x x p β β π - = ∂ ∂ 2 ) ( 2 1 exp 2 1 (13)
The marginal effects of limited explanatory variables are calculated as the difference between probabilities [START_REF] Altman | Scale and Transaction Costs in the U.S. Biopower Industry[END_REF].

Potentially complicating issues include endogeneity (due to simultaneity of organizational form and asset specificity), unobserved variables (such as transaction costs/only the choice or organizational form is observed), and causality problems (between asset specificity and organizational form for instance). These issues are addressed in the literature (see [START_REF] Saussier | Transaction Costs and Contract Incompleteness: the Case of Électricité de France[END_REF][START_REF] Joskow | Contract Duration and Relation-Specific Investments: Empirical Evidence from Coal Markets[END_REF]. In general these studies tend not to change the support of the theory but rather reinforce the validity of the tests.

Given the small data set (41 observations), and various limited explanatory variables (scale is the only continuous variable), we focus on testing the basic theory with these data and interpreting the marginal effects. Addressing endogeneity and related issues requires more and better data.

Results

In these models the probability of the choice of organizational form (CHOICE) is regressed against four key explanatory variables10 : frequency of the procurement (FREQ), flexibility with respect to inputs use (FLEX) as a measure of physical asset specificity, average hauling distance (DIST) as a measure of site specificity, scale of the plant (SCALE) measured in tons of bioethanol per year. Results of the logit model are reported in table 6 and those of the probit model in table 7. All the coefficients have the expected signs for both logit and probit models, though only one of the asset specificity variables, FLEX, is statistically significant at the 90% level for the logit model and at 95% level for the probit model. (The other, DIST, has a p-value of 0.3165, which may be due to the small sample size). SCALE is also statistically significant at the 90% level for the logit model and at 95% level for the probit model. In these models, external procurement (1) is the comparison group. The model as a whole is statistically significant at the 95% level. The marginal effects are also consistent with TCE. The marginal effects of the two asset specificity variables, flexibility with respect to inputs use and average hauling distance, have the expected signs. Also higher the frequency of inputs procurement and larger the scale of the plant, more important will be the probability of external procurement. This is also consistent with the theory and the surveys that we have made with bioethanol plants leaders and experts of the sector. When the frequency of inputs procurement becomes high and the average hauling distance increases from 0-25 miles to 50+ miles the probability of observing external procurement increases by 4% for both logit and probit models. In other words, in order to satisfy plant needs of inputs and to realize the expected bioethanol production, the procurement of raw materials through spot markets seems to be the most convenient organizational choice and a change from high to low site specificity is associated with an increase in the probability of external organization. A higher level of plant flexibility with respect to inputs increases the probability of observing external procurement by 10% for the logit model and 5% for the probit model which means that a change from high to low physical specificity is associated with an increase in the probability of external procurement and a decrease of the hybrid one. Scale also has the correct sign. A change in scale from 2000 to 240000 tons per year increases the probability of external organization by 6% in the logit model and 7% in the probit model.

Conclusions

The bioethanol plants face important organizational and strategic challenges, but there is so far little literature applying organizational economics to this industry. Application of transaction-cost theory has the dual benefit of shedding valuable light on the challenges facing the growing market for biomass, and at the same time advancing theory by testing its hypotheses in a new setting.

To address these issues we apply TCE to the European bioethanol plants with a special focus on the effect of scale on the choice of organizational form. A survey of bioethanol firms produced information on several important variables such as the frequency of the procurement, physical asset specificity, spatial asset specificity and scale of bioethanol production. These variables are regressed in both dichotomous logit and probit models providing similar results, against the choice of market organizational form. Contrary to preliminary empirical evidence in the transaction cost literature, larger scale of operation is associated with greater reliance on external procurement, which provides some support for the transaction cost theory predicted relationship between organization and scale.

Given this evidence both transaction costs and economies of scale are reconsidered in the choice of organizational form and scale of bioethanol plants. Transaction costs and economies of scale are demonstrated to be two key trade-offs in organizational decisions.

  Figure 1 : Governance costs as a function of asset specificity

9

  The value of i y depends on certain variables and unknown parameters.

  the log likelihood function made by the i th observation. Therefore this function can be expressed as

  β ˆ must of course satisfy the first-order conditions

Table 1 . Frequency of Transaction in the European Bioethanol Firms

 1

			Frequency	Percent	Valid Percent	Cumulative Percent
	Valid	Low	15	36,6	36,6	36,6
		High	26	63,4	63,4	100,0
	Total		41	100,0	100,0
		Spatial asset specificity, or what

Table 2 . Frequency of Average Hauling Distances

 2 Table 2 summaries this variable 8 .

			Frequency	Percent	Valid Percent	Cumulative Percent
	Valid	0-25 miles	15	36,6	36,6	36,6
		25-50 miles	2	4,9	4,9	41,5
		Over 50	24	58,5	58,5	100,0
	Total		41	100,0	100,0

Table 3 . Flexibility of European Bioethanol Plants

 3

			Frequency	Percent	Valid Percent	Cumulative Percent
	Valid	Inflexible	30	73,2	73,2	73,2
		Moderately	3	7,3	7,3	80,5
		Highly flexible 8	19,5	19,5	100,0
	Total		41	100,0	100,0

Table 4 . Summary Statistics-Scale

 4

	Variable	Observations	Mean (Tons/year)	Standard Deviation (Tons/year)	Minimum (Tons/year)	Maximum (Tons/year)
	Scale	41	10952,553	70130,558	2000	240000

Table 6 . Regression results : Logit model

 6

		Estimate	Std. Error	z value	Pr(>|z|)
	(INTERCEPT)	-2.819e+01	3.419e+03	-0.008	0.9934
	FREQ	1.399e+01	3.419e+03	0.004	0.9967
	DIST	2.213e+00	2.318e+00	0.955	0.3397
	FLEX	3.551e+00	1.820e+00	1.951	0.0511 .
	SCALE	4.216e-05	2.187e-05	1.928	0.0539 .
	Signif. Codes : 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
	(Dispersion parameter for binomial family taken to be 1)
	Null deviance: 34.137 on 40 degrees of freedom
	Residual deviance: 12.751 on 36 degrees of freedom
	AIC: 22.751
	Number of Fisher Scoring iterations: 19

Table 7 . Regression results : Probit model

 7

		Estimate	Std. Error	z value	Pr(>|z|)
	(INTERCEPT)	-1.155e+01	7.184e+02	-0.016	0.9872
	FREQ	3.248e+00	7.184e+02	0.005	0.9964
	DIST	1.342e+00	1.340e+00	1.002	0.3165
	FLEX	2.004e+00	9.186e-01	2.181	0.0292
	SCALE	2.478e-05	1.160e-05	2.137	0.0326
	Signif. Codes : 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
	(Dispersion parameter for binomial family taken to be 1)
	Null deviance: 34.137 on 40 degrees of freedom
	Residual deviance: 12.715 on 36 degrees of freedom
	AIC: 22.715
	Number of Fisher Scoring iterations: 19

  Table 8 reports these marginal effects.

Table 8 . Marginal effects of logit and probit models

 8

	Variable	Logit	Probit
	FREQ	0.0402	0.0431
	DIST	0.0417	0.0424
	FLEX	0.1036	0.0526
	SCALE	0.0590	0.0697

loic.sauvee@lasalle-beauvais.fr Professor, head of department STIM, Institut Polytechnique LaSalle Beauvais, France

For example,[START_REF] Klass | Biomass for Renewable Energy, Fuels, and Chemicals[END_REF] identifies storage and shipping strategies for wood biomass and[START_REF] Van Loo | Handbook on Biomass Combustion and Cofiring[END_REF] discuss how organization has solved technical issues in some cases in Europe.

The idea is that the firms minimize total costs (production and transaction costs) in their choice of organizational form. The TCE empirical literature, however, tends to focus on asset specificity and uncertainty to the neglect of production costs.

It is recognized that there are other transaction costs in addition to those associated with the procurement of inputs. This model focuses on the procurement costs in the biomass transaction. Other costs will influence firm behavior as well. This model focuses only on the sugar beets/cereals procurement decision.

Supplier-market concentration must be taken into account because assets that are specialized to a particular use may not be relationship-specific assets, in the TCE sense, if there is a thick market for biomass. For given levels of supplier-market concentration, the degree to which assets can be put to different uses can be a good proxy for asset specificity.

We assume here that supplier concentration does not vary systematically across plant types. Unfortunately we do not have measures of supplier density in our data, and there are insufficient degrees of freedom to include county or state dummy variables.

Several other variables were considered for this analysis including the number of suppliers, plant dispatch type as a temporal asset specificity variable, uncertainty related to the inputs procurement and technology type. These variables are not included in the final model either because they are statistically insignificant or insufficient data.


domain: qfin stat

text:
Victor Elvira

Samir Orujov
email: author.samir.orujov@univ-ubs.fr

Audrey Poterie

Farid Rajabov

Francois Septier

VS-LTGARCHX: A Flexible Subset Selection Approach for Estimation of log-TGARCHX Models and Its Application to BTC Markets

Keywords: GARCHX, log-GARCH, Variable Selection, Bitcoin Volatility

The log-TGARCHX model is less restrictive in terms of inclusion of exogenous variables and asymmetry lags compared to the GARCHX model. However, adding less (more) covariates than necessary may lead to underfitting (overfitting), respectively. In this context, we propose a new algorithm, called VS-LTGARCHX, which incorporates a variable selection procedure into the log-TGARCHX estimation process. Furthermore, the VS-LTGARCHX algorithm is applied to extremely volatile BTC markets using 42 conditioning variables. Interestingly, our results show that the VS-LTGARCHX models outperform the specified benchmark models in one-step-ahead forecasting.

Introduction

The generalized autoregressive conditional heteroskedasticity model (GARCH) due to [START_REF] Bollerslev | Generalized autoregressive conditional heteroskedasticity[END_REF] is a popular class of econometric models to capture the dynamics of conditional variance (volatility) of time series. A natural extension of the GARCH model, called GARCHX, uses exogenous variables to model the conditional variance of the time series. In fact, additional conditioning variables can significantly improve volatility modeling, as correctly stated by [START_REF] Sucarrat | garchx: Flexible and Robust GARCH-X Modelling[END_REF]. Nevertheless, in GARCHX models, a positivity restriction on the intercept parameter and non-negativity constraints on all other parameters and all exogenous variables should be imposed. The latter restrictions are to ensure the positivity of conditional volatility, but limit the use of GARCHX models [START_REF] Sucarrat | garchx: Flexible and Robust GARCH-X Modelling[END_REF]. Furthermore, if we are interested in invertibility and stationarity, then additional conditions should be imposed on the autoregressive and moving average parameters of the conditional variance equation [START_REF] Hamilton | Time series analysis[END_REF]. This restrictive nature of most members of the GARCH family models is relaxed in the exponential GARCH model with exogenous variables (EGARCHX) [START_REF] Nelson | Conditional Heteroskedasticity in Asset Returns: A New Approach[END_REF] and logarithmic GARCH model with exogenous variables (log-GARCHX). The log-GARCHX model is indeed the restricted form of logarithmic GARCH model with exogenous covariates and threshold variables (log-TGARCHX), as we emphasize herein further. Therefore, the terms log-GARCHX and log-TGARCHX are sometimes used interchangeably throughout the paper. In the last two members of the GARCH family (EGARCHX and log-TGARCHX), fitted volatility is guaranteed to be positive, and non-negativity constraints on parameters and covariates are lifted. However, the advantage of log-GARCHX over EGARCHX is that unconditional variance in the EGARCH formulation may not exist if the errors are too fat-tailed [START_REF] Sucarrat | The log-GARCH model via ARMA representations[END_REF]. Moreover, probabilistic properties of EGARCH and log-GARCH models have been studied, and it was shown that the consistency and asymptotic normality of the quasi-maximum likelihood estimation (QMLE) of log-GARCH models hold under conditions milder than those of EGARCH. Furthermore, log-GARCH models may capture richer asymmetry dynamics and its shock transmission mechanism may fit the reality better compared to EGARCH and GARCH [START_REF] Francq | GARCH models without positivity constraints: Exponential or log GARCH[END_REF].

Arguably, one of the most attractive features of the log-TGARCHX model is that it accepts the ARMA representation and therefore the parameter estimation can be carried out in any standard statistical software using least squares and quasi-maximal likelihood [START_REF] Francq | QML INFERENCE FOR VOLATILITY MODELS WITH COVARIATES[END_REF][START_REF] Sucarrat | The log-GARCH model via ARMA representations[END_REF]. Moreover, the latter makes the standard asymptotic theory about parameter estimates valid. Under these circumstances, hypothesis testing about the statistical significance of exogenous covariates and ARCH/GARCH parameters is standard. Conversely, because of the respective positivity and non-negativity constraints imposed on parameters in generic GARCHX models, the conventional asymptotic theory fails, and non-standard methods should be applied. These questions have been discussed in detail in [START_REF] Francq | QML INFERENCE FOR VOLATILITY MODELS WITH COVARIATES[END_REF].

The contributions of this work are twofold. First, we propose an algorithm called VS-LTGARCHX to incorporate the variable selection procedure into the log-TGARCHX estimation process. The algorithm is very flexible in the sense that any variable selection procedure can be employed within its framework. However, in this paper, we use three widely used variable selection methods that will be elaborated on in Section 3, separately within the VS-LTGARCHX algorithm to select the best subset of regressors while estimating log-TGARCHX models. We show that enhancing log-TGARCHX models with suitable variable selection procedures generates useful results for conditional volatility modeling in terms of prediction and interpretation. Second, we apply the proposed VS-LTGARCHX algorithm to time series of Bitcoin (BTC), one of the most popular decentralized cryptocurrencies operating on the blockchain (distributed ledger) platform. In fact, BTC has drawn public attention since its inception because of the novel underlying technology that enables the fast transfer of funds internationally without third-party involvement. However, the log-TGARCHX model has not yet been used to study the volatility of BTC, despite the former's higher flexibility compared to many other GARCH family models. We contribute to the existing body of literature by filling in the latter gap.

The paper is structured as follows. In Section 2, we describe and contrast the GARCHX and log-TGARCHX models and emphasize the advantages of the latter over the former in terms of the inclusion of exogenous variables and model estimation. In Section 3, we present a novel procedure to select the subset of variables from a much larger subset of variables to be used within the log-TGARCHX model. In Section 4, we apply our procedure to the BTC market and show that, indeed, the one-step-ahead volatility predictions can be improved using the suggested procedure, and in Section 5 we conclude.

On generalized autoregressive conditional heteroskedasticity based models

This section briefly introduces the GARCHX and the log-TGARCHX, and we discuss why the latter may be superior to the former, particularly in the presence of multiple exogenous variables. The last subsection provides the ARMA representation of log-TGARCHX models, which helps to simplify the parameter estimation and inference procedures.

The GARCHX model

Assume a covariance stationary process given by

yt = µt + ϵt, ϵt = σtηt, σt > 0, ηt ∼ iid (0, 1) , (1)
where {yt} t≥0 is the time series process (e.g. financial return), µt is the conditional mean function known a priori, and the error term ϵt is a random variable with mean zero and conditional variance σ 2 t . The term ηt is an independently and identically distributed innovation process with zero mean and unit variance, as indicated by iid (0, 1).

In Eq. ( 1), the conditional variance of yt can evolve over time, while the unconditional variance is assumed to be fixed (which is implied by the stationarity of yt). When the conditional variance of the errors (and thus that of the yt) is fixed, they are referred to as homoscedastic errors. Conversely, if the conditional variance of ϵt is varying, we call them heteroskedastic errors. The presence of heteroskedasticity may be undesirable in the standard linear regression framework, where the sole purpose of the researcher is to model the conditional mean of a time series process. Indeed, it may distort the standard error estimates of the model parameters, and thus lead to misleading statistical inference about those parameters. Heteroskedasticity, its potential challenges for statistical inference about linear regression parameters, and remedies are addressed extensively in most of the introductory and intermediate econometrics textbooks. More information about this point can be found in [START_REF] Brooks | Introductory Econometrics for Finance[END_REF]; [START_REF] Wooldridge | Introductory Econometrics: A Modern Approach[END_REF]; [START_REF] Greene | Econometric Analysis[END_REF].

In some cases, modeling the conditional variance directly may be more crucial for some entities than modeling the mean. For example, risk-averse investors may be more interested in the variance of returns on their investment and value exposed to risk than in the mean of them because the variance is a measure of the risk. In fact, changes in variance have crucial implications for financial markets, since investors usually require higher compensation for exposing themselves to higher risks [START_REF] Hamilton | Time series analysis[END_REF]. The latter creates a need for an explicit modeling of the conditional variance of returns in financial markets.

The GARCH model of [START_REF] Bollerslev | Generalized autoregressive conditional heteroskedasticity[END_REF] is a parametric model for the direct modeling of conditional variance. This model is the generalized version of the autoregressive conditional heteroskedasticity model (ARCH) due to [START_REF] Engle | Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation[END_REF]. Both models assume linear dynamics for the conditional variance of the ϵt process in Eq. (1). Formally, a GARCH(p1,p2) model is defined by

σ 2 t = α0 + p 1 i=1 αiϵ 2 t-i + p 2 j=1 αp 1 +j σ 2 t-j , α0 > 0, αi ≥ 0, ∀i ∈ {1, . . . , p1 + p2}, (2)
where α0 is the volatility intercept and p1 ∈ N and p2 ∈ N are ARCH (squared lags of ϵt) and GARCH (squared lags of σt) orders respectively. Note that if p2 = 0, the GARCH(p1,p2) model is the ARCH(p1) model.

Furthermore, the natural extension of the GARCH model in which exogenous variables can be added is called the GARCHX model [START_REF] Sucarrat | garchx: Flexible and Robust GARCH-X Modelling[END_REF]. Then, a GARCHX(p1,p2,p3) is defined by

σ 2 t =α0 + p 1 i=1 αiϵ 2 t-i + p 2 j=1 αp 1 +j σ 2 t-j + p 3 k=1 β k x k,t , α0 > 0, αi ≥ 0 ∀i ∈ {1, . . . , p1 + p2}, β k ≥ 0 and x k,t ≥ 0 ∀k = {1, . . . , p3}, (3)
where {x k,t } p 3 k=1 are p3 stationary exogenous covariates and {β k } p 3 k=1 are the unknown associated parameters to be estimated. Other parameters in Eq. ( 3) are GARCHX(p1,p2,p3) parameters defined in Eq. ( 2). The inclusion of exogenous covariates into the GARCH equation comes with the additional cost of non-negativity restrictions on both the exogenous variables {x k,t } p 3 k=1 and their associated parameters {β k } p 3 k=1 in order to guarantee the positivity of the conditional variance. Furthermore, inference procedures under nullity are nonstandard [START_REF] Sucarrat | garchx: Flexible and Robust GARCH-X Modelling[END_REF] and there is a need for alternative asymptotics, as in [START_REF] Pedersen | TESTING GARCH-X TYPE MODELS[END_REF]. This restrictive nature of GARCHX models is relaxed to a great extent by log-TGARCHX models, which we demonstrate in the next subsection.

The log-TGARCHX model

The log-GARCHX model [START_REF] Sucarrat | The log-GARCH model via ARMA representations[END_REF] assumes logarithmic specification of the conditional variance process and includes both asymmetry error terms and stationary covariates. Formally, the log-GARCHX(p1,p2,p3,p4) is defined as follows:

ln σ 2 t = α0 + p 1 i=1 αi ln ϵ 2 t-i + p 2 j=1 αp 1 +j ln σ 2 t-j + p 3 k=1 β k x k,t + p 4 l=1 γ l I {ϵ t-l <0} ln ϵ 2 t-l , αi, β k , γ l ∈ R, ∀i ∈ {0, . . . , p1 + p2}, ∀k ∈ {1, . . . , p3}, ∀l ∈ {1, . . . , p4}, (4)
where α0 is log-volatility intercept which controls the level of volatility in a multiplicative way, (αi) i={1,...,p 1 } and (αp 1 +j ) j={1,...,p 2 } are real scalar sequences called ARCH-parameters and GARCH-parameters, respectively [START_REF] Sucarrat | The log-GARCH model via ARMA representations[END_REF]. Note that, when p3 = p4 = 0, the log-GARCHX(p1,p2,p3,p4) model is none other than the log-GARCH(p1, p2) model introduced by Pantula (1986), Geweke (1986) and [START_REF] Milhoj | A Conditional Variance Model for Daily Deviations of an Exchange Rate[END_REF] independently.

The log-GARCHX model is referred to as the asymmetric log-GARCHX model in the literature [START_REF] Sucarrat | The log-GARCH model via ARMA representations[END_REF]. However, we suggest calling the latter log-TGARCHX because it is equivalent to the TGARCHX model of [START_REF] Glosten | On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks[END_REF] except for the logarithmic transformation.

ARMA representation of log-TGARCHX model

The log-GARCH family of volatility models has several attractive features. One of them is the representability via non-linear ARMA process, which makes those models amenable to estimation using ARMA inference methods and almost with any statistical software. Due to the latter fact, the standard inference procedures remain applicable. Furthermore, the addition of exogenous, deterministic, and / or predetermined conditioning variables does not change the relationship between the ARMA coefficients and the log-GARCH coefficients under suitable conditions [START_REF] Sucarrat | Estimation and inference in univariate and multivariate log-GARCH-X models when the conditional density is unknown[END_REF][START_REF] Sucarrat | The log-GARCH model via ARMA representations[END_REF]. More precisely, the ARMAX representation for the log-TGARCHX model in Eq. ( 4) is as follows [START_REF] Sucarrat | The log-GARCH model via ARMA representations[END_REF]:

ln ϵt 2 = ϕ0 + p 1 i=1 ϕi ln ϵ 2 t-i + p 2 j=1 ϕp 1 +j ut-j + p 3 k=1 β k x k,t-1 + p 4 l=1 γ l I {ϵ t-l <0} ln ϵ 2 t-l + ut, (5)
with

ut = ln η 2 t -E ln η 2 t (6) ϕi =      α0 + 1 -p 2 j=1 αp 1 +j × E (ln η 2 t ) , i = 0 αi + αp 1 +i, 1 ≤ i ≤ p1 -αi, p1 + 1 ≤ i ≤ p2.
The parameter values (β l ) l={1,...,p 3 } for exogenous variables are the same in both the log-GARCHX specification and the corresponding ARMAX representation. The possibility of the ARMAX representation paves the way for a wide range of extensions of the log-GARCHfamily models. In this paper, we propose enhancing log-GARCHX models by including a variable selection procedure during model estimation. Although there are many variable selection methods, in this work we focus on three of them. However, other selection methods can also be easily adapted to the proposed framework. The following section describes our original methodology and the three variable selection methods.

3.

Enhancing log-TGARCHX model by adding variable selection algorithms As described by [START_REF] Sucarrat | The log-GARCH model via ARMA representations[END_REF], the log-TGARCHX model is estimated in two steps. The first step is to form the ARMAX (p1, p2, p3, p4) representation of the model as given in Eq. ( 5) and estimate the coefficients using the ordinary least squares (OLS) method. The second step consists of mapping the estimated coefficients to the coefficients of the model of interest, namely, log-TGARCHX specified in Eq. ( 4) using the maps in Eq. ( 6).

As an important contribution of this paper, we show that the log-TGARCHX model can be improved by selecting a relevant subset of exogenous variables and asymmetry lags, denoted by K⋆ ⊆ {1, . . . , p3} and L⋆ ⊆ {1, . . . , p4}, respectively. To that end, we propose a heuristic strategy that incorporates variable selection within the log-TGARCHX estimation procedure. More precisely, this approach describes i) how to select both the subset of exogenous variables, through the L⋆ index set, and that of the asymmetry terms through the K⋆ index set to be used in the log-TGARCHX model, and then ii) how to estimate parameters of the model. This new strategy that is made up of three steps is described below.

1.

Step 1. Fit an ARMA representation and get the residuals First, we fit the following ARMA(p1,p2) model:

ln ϵt 2 = ϕ0 + p 1 i=1 ϕi ln ϵ 2 t-i + p 2 j=1 ϕp 1 +j ut-j + ut. ( 7
)
The model is fitted by using maximum likelihood estimation together with the innovations algorithm [START_REF] Brockwell | Estimation for ARMA Models[END_REF].

Then, we compute the model residuals which are defined as:

ẑt = ln ϵt 2 -ln ϵt 2 , ( 8
)
where ln ϵt 2 denotes the adjusted value of ln ϵt 2 for model Eq. ( 7).

Step Perform the variable selection procedure for both exogenous variables and asymmetry orders

Next, variable selection is applied on the following model:

ẑt = a0 + p 3 k=1 b k I {ϵ t-k <0} ln ϵ 2 t-k + p 4 l=1 c l x l,t-1 + et, (9)
where a0, {b k } p 3 k=1 , {c l } p 4 l=1 are the model parameters to be estimated, I {ϵ t-k <0} ln ϵ 2 t-k is the k-th lagged asymmetry term evaluated at time t, x l,t-1 is the l-th stationary exogenous covariate evaluated at time t and et is the model error at time t. In this paper, we suggest applying one of the three variable selection methods introduced in Section 3.2.1. As a result of the variable selection process, we obtain the index sets L⋆ and K⋆ of size p3⋆ and p4⋆, respectively.

Step Fit the final log-TGARCHX model

Finally, we can estimate the log-TGARCHX(p1, p2, p3⋆, p4⋆) model:

ln σ 2 t = α0 + p 1 i=1 αi ln ϵ 2 t-i + p 2 j=1 αp 1 +j ln σ 2 t-j + k∈K * β k x k,t + l∈L * γ l I {ϵ t-l <0} ln ϵ 2 t-l . (10)
Parameters

{αi} p 1 +p 2 i=1
, {β k } k∈K ⋆ , {γ l } l∈L ⋆ of the final model are estimated using the two-step procedure described in Section 2.3.

In the next subsection, we shed light on the second step of the proposed methodology.

Variable selection algorithms

In this section, three variable selection strategies are introduced to select L⋆ and K⋆ (see Step 2 in Section 3.1). As described in the previous section, the variable selection procedure is performed on the linear model Eq. ( 9). The subset selection procedure is important, especially when the set of potential regressors is large or when the regressors are highly correlated. For instance, when the number of regressors is larger than the number of observations, it is well known that linear models may suffer from high variance, which is referred to as overfitting in the literature. Furthermore, parsimonious models are more easily interpretable than complex models with many regressors. Therefore, the selection of variables to identify the most influential variables can improve both predictive accuracy and interpretability [START_REF] Miller | Subset Selection in Regression[END_REF]. In fact, there are many variable selection methods in the related literature. Here, we have chosen three approaches: the least absolute shrinkage and selection operator (LASSO), adaptive best subset selection (ABESS), and Boruta.

Variable selection using LASSO

LASSO is a regression method introduced by [START_REF] Tibshirani | Regression Shrinkage and Selection via the Lasso[END_REF]. The method is still widely used to perform both variable selection and regularization. This is achieved by adding a penalty function of the model parameters. Consider the linear regression setting in Eq. ( 9). Then, the LASSO regression problem consists of estimating the model parameter vector Θ = a0, {b k } p 3 k=1 , {c l } p 4 l=1 that solves the following constrained OLS problem:

Θ lasso = arg min

Θ∈R (p 3 +p 4 +1) T t=1 zt -a0 - p 3 k=1 b k I {ϵ t-k <0} ln ϵ 2 t-k - p 4 l=1 c l x l,t-1 2 + λ( p 3 k=1 |b k | + p 4 l=1 |c l |) , (11)
where the objective function is nothing else than the sum of mean squared errors and λ > 0 is a regularization parameter. LASSO regression shrinks the magnitude of all model coefficients and also sets some of them to zero. Then, in Step 2 we use LASSO to perform the selection of variables in the model Eq. ( 9). The subsets of indices L⋆ and K⋆ obtained using LASSO are denoted by

K lasso ⋆ = {k| blasso k ̸ = 0, ∀k = 1, . . . , p3} L lasso ⋆ = {l|ĉ lasso l ̸ = 0, ∀l = 1, . . . , p4} (12)
corresponds to the sets of indices of non-zero parameters. Note that the value of the regularization parameter λ is usually unknown and therefore needs to be tuned. Several approaches have been suggested to adjust λ. Here, we adopt the algorithm outlined in Section 2.5 of [START_REF] Friedman | Regularization paths for generalized linear models via coordinate descent[END_REF]. The algorithm finds the minimum Lagrange multiplier value in the Lagrangian representation of the constrained optimization problem in the Eq. ( 11) which shrinks all slope coefficients to zero simultaneously, and that value is denoted by λmax. Then λmin is set to 0.001λmax and the grid is formed as a decreasing sequence of 100 values from λmax to λmin on the log scale.

Adaptive best subset selection (ABESS) algorithm

ABESS is a recently developed polynomial algorithm due to [START_REF] Zhu | A polynomial algorithm for best-subset selection problem[END_REF] for the vaiable selection. In simple linear regression setting e.g. Eq. ( 9), this algorithm chooses the subset of covariates by solving the following minimization problem:

Θ abess = arg min

Θ∈R (p 3 +p 4 +1) T t=1 zt -a0 - p 3 k=1 b k I {ϵ t-k <0} ln ϵ 2 t-k - p 4 l=1 c l x l,t-1 2 subject to ( p 3 k=1 ||b k ||0 + p 4 l=1 ||c l ||0) ≤ s, (13)
where s ∈ {1, . . . , p3 + p4} is a regularization parameter and || • ||0 is the ℓ0 norm.

Here, the Lagrangian of the problem is not continuous, and its solution may be computationally infeasible with large p. However, a recently developed splicing algorithm and special information criterion (SIC) have been proven to solve the minimization problem above in polynomial times and for an unknown sparsity parameter [START_REF] Zhu | A polynomial algorithm for best-subset selection problem[END_REF].

Although the ABESS algorithm is extremely fast compared to its competitors as shown in the latter reference, it seems to be more aggressive compared to LASSO in penalization as the form of the constraint function suggests. Therefore, when the set of the covariates upon which variable selection needs to be performed is relatively small, LASSO should be preferred. However, a comparison of the two is outside the scope of our study and needs further research.

Finally, one of the main advantages of ABESS is that the potential values of the tuning parameter s are finite, as the l0 norm penalty suggests. The selected variables by the ABESS algorithm are the ones which receive non-zero coefficients after the minimization of the latter equation. More precisely,

K abess ⋆ = {k| babess k ̸ = 0, k = 1, . . . , p3} L abess ⋆ = {l|ĉ abess l ̸ = 0, l = 1, . . . , p4}. (14)

Boruta algorithm

Boruta is an iterative feature selection algorithm based on the random forest algorithm [START_REF] Kursa | Feature Selection with the Boruta Package[END_REF]. At each iteration, the algorithm creates randomly shuffled copies of the original features. Then, a random forest is trained on the extended data that includes both the original features and their respective shuffled copies. To measure the importance of each feature, the algorithm computes a Z score on each feature and compares it to a threshold defined as the maximum Z score of all shuffled copies. Then, only variables for which importance is significantly higher than the threshold are kept in the model while the others are removed. This process is repeated until either all features are labeled as important/nonimportant or until the maximum number of iterations is reached. The detail of the algorithm is given in the original article by [START_REF] Kursa | Feature Selection with the Boruta Package[END_REF]. Ultimately, the two sets K Boruta ⋆ and L Boruta ⋆ correspond to the sets of indices of the retained exogenous variables and the asymmetric lags.

Application to BTC market

In this section, we apply the VS-LTGARCHX algorithm to the BTC market and study its performance compared to the log-GARCH (1,1) model and the full-blown log-TGARCHX model. The log-GARCH(1,1) model is a log-GARCH model with no exogenous variable, while the full-blown log-TGARCHX model is a log-GARCH model with the complete set of exogenous variables and asymmetry terms. These two extremes form the benchmark models with which to compete. The next subsection describes the BTC data used.

Data: data sources and data preprocessing

The sample period runs from December 18, 2017 to June 17, 2022 and includes 1643 observations in total collected at daily frequency. It is worth mentioning that the BTC price data were collected both in 5-minute and daily frequencies, which were taken from Binance exchange. The BTC price data at the daily frequency were used to obtain the BTC daily returns. Moreover, the BTC prices at 5-minute frequency were used to obtain the daily realized variance as described in [START_REF] Bergsli | Forecasting volatility of Bitcoin[END_REF]. Formally, let Rt, the return on the t-th day, be defined as

Rt = ln Pt -ln Pt-1, ( 15
)
where Pt is the close price of BTC on day t. Then, we define m equally spaced intraday returns between each consecutive two days t and t + 1. More precisely, the j-th intraday return for the the t-th day is denoted by R t+ j m and defined as:

R t+ j m = ln P t+ j m -ln P t+ j-1 m , for j = 0, . . . , m -1, ( 16
)
where ln P t+ j m (resp. ln P t+ j-1 m is the logarithm of the intraday price of BTC at time t + j m (resp. t + j-1 m ). Thus, the daily realized variance, also called volatility, for the t-th day can be defined as the sum of squared intraday returns following the previous reference:

V (d) t = m-1 j=0 R 2 t+ j m . ( 17
)
where the subscript d is used to emphasize the daily frequency of the realized variance.

For the purposes of this paper, we used BTC prices in 5-minute frequency to estimate the daily realized variance (volatility), which implies that m in Eq. ( 16) and Eq. ( 17) is set to 288 throughout our analysis.

Furthermore, we classify the exogenous variables into groups using a classification similar to, but not exactly the same as Chen et al. ( 2021). More precisely, the groups we defined are Blockchain technology, Public Opinion, Risks and Uncertainties, Financials, and Macroeconomic development. Also, the number of variables we use is more than in the latter reference. To be more precise, we consider 39 exogenous variables, 3 variables which are transformations of high-frequency returns (daily, weekly and monthly realized volatility) and a threshold variable, which constitute 42 variables in total. The threshold variable is nothing but a binary variable which receives the value of unity if the previous day return is negative and the value of zero if otherwise. The complete list of exogenous variables is given in the Appendix A. All variables were collected at daily frequency. However, unlike BTC prices, measurements on most exogenous variables are missing on weekends and public holidays. Then, to overcome inconsistency in frequency, the last observation carried forward (LOCF) method was used to fill the missing values.

In addition, we used only stationary exogenous variables in our study as shown in [START_REF] Sucarrat | The log-GARCH model via ARMA representations[END_REF]. Therefore, Augmented Dickey-Fuller (ADF) test due to [START_REF] Dickey | Distribution of the Estimators for Autoregressive Time Series With a Unit Root[END_REF] was applied to each exogenous variable to identify non-stationary ones. The nonstationary variables that take only positive values were exposed to the logarithmic difference transformation (d ln (Xt) = ln Xt -ln Xt-1) and the prefix dl was added to their names. In contrast, non-stationary variables with non-positive and positive values were subject to first differencing (Xt → Xt -Xt-1), and the prefix d was added to their names. Furthermore, we applied the test of stationarity [START_REF] Kwiatkowski | Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root[END_REF] to the data to confirm the results of the ADF test which is a standard practice (Brooks, 2008, p. 331). If the stationarity hypothesis is rejected by KPSS those series were subject to the first differencing and the prefix d was added to their names. In addition, all covariates were standardized to have zero mean and unit variance to prevent variables from having a different measurement scale that could affect the results of subset selection.

Finally, in the next subsection, we will describe in detail how the VS-LTGARCHX algorithm is applied to BTC market data and how the performance of different variants of the algorithm is evaluated.

Model performance and comparison to benchmark

Once the data preprocessing is performed, we divide the data set into train, test, and validation sets. The hyperparameters of the variable selection algorithms are then tuned. Ultimately, the final models are built and their respective performances are measured. The following subsections describe the methodology we have used to perform each of the latter steps.

Data split

First, the time series data set was divided into train, validation, and test sets. More precisely, the data partitioning consisted of dividing the sample period (from December 18, 2017 to June 17, 2022) into three sub-periods corresponding, respectively, to the train, validation and test sets.

The train set is used to build a prediction model (a learner ) with a fixed set of hyperparameters. The validation set is used to select the best combination of hyperparameters for the respective variable selection method based on some performance metrics. Ultimately, the test set is used to evaluate the performance of the final model, the one fitted with the best combination of hyperparameters [START_REF] Hastie | The Elements of Statistical Learning: Data Mining, Inference, and Prediction[END_REF].

Moreover, the benchmark models considered in this paper do not have any hyperparameters to tune. Therefore, the validation set is not needed, so it was merged with the train set for the benchmark models.

Finally, to respect the dynamic structure of the data, we use the recursive and rolling window scheme while partitioning the data set for all models considered [START_REF] Elliott | Economic Forecasting[END_REF].

Hyperparameter tuning

As has already been mentioned, the three variable selection methods, namely LASSO, ABESS, and Boruta, have hyperparameters that need to be tuned. To do that, we used two strategies: one for LASSO and ABESS and another for Boruta. In fact, a more efficient strategy can be used for Boruta, since it is a wrapper around the random forest algorithm [START_REF] Breiman | Bagging predictors[END_REF] that is heavily based on bootstrap aggregation (bagging).

To tune the hyperparameters of LASSO and ABESS, first, we fit the prediction models (learners) to the training set by fixing the hyperparameter value based on a predefined grid. Then, for each hyperparameter value, we obtain the prediction errors of the learners in the validation set. The value of the tuning parameter associated with the smallest validation root mean squared error (RMSE) is considered optimal.

The hyperparameter tuning procedure for the random forest behind the Boruta wrapper can be done more efficiently due to the bootstrap resampling applied to the initial training set [START_REF] Breiman | Random Forests[END_REF]. However, the use of the original bootstrap strategy would destroy the underlying temporal dependence in time series. Therefore, we utilize the moving block bootstrap mechanism to respect the dynamic nature of the data while conducting the bagging process. The idea is to divide the initial training set into N small blocks so that the temporal structure is preserved within each block, and then to perform a bootstrapping using the N blocks [START_REF] Kunsch | The Jackknife and the Bootstrap for General Stationary Observations[END_REF][START_REF]Moving blocks jackknife and bootstrap capture weak dependence[END_REF]. Unlike LASSO and ABESS which have only one hyperparameter to be tuned each, in Boruta there are multiple of them, which makes it costly to train the latter. Finally, we tune four hyperparameters; three of them are intrinsic to Boruta, namely, the number of trees, the minimum node size, the number of splitting variables, and one is due to the moving block bootstrap algorithm, namely, the block size (see Algorithm 2 in the Appendix C). Again, the set of hyperparameters associated with the smallest RMSE is considered optimal.

Evaluation of the predictive performance and comparison to the benchmark models

Once the optimal values of the hyperparameters are found, the variable selection is carried out by combining the training and the validation sets, and this is the end of Step 2 of the VS-LTGARCHX algorithm. The final step Step 3 consists of fitting the log-TGARCHX model as explained in detail in [START_REF] Escribano | Equation-by-equation estimation of multivariate periodic electricity price volatility[END_REF] and [START_REF] Sucarrat | The log-GARCH model via ARMA representations[END_REF] with the variables selected by the respective variable selection algorithms.

Moreover, we use the same portion of time series to build the final model for both benchmarks and VS-LTGARCHX variants, and use the same strategy for all models considered to generate one-step-ahead predictions. More precisely, to measure performance of each model, we generate one-step-ahead predictions from the test set (the same portion of time series for all models for fair comparison) using the rolling window scheme and finally calculate six different conventional metrics for each model based on their respective forecasts. The procedure is outlined in Algorithm 3.

Remark 1 The rolling window scheme is used instead of the recursive window scheme because it is known that the recursive window scheme may pose problems when evaluating nested models (Violante and Laurent, 2012, p. 466) and that the stationarity assumption of relative losses underlying the Model Confidence Set (MCS) test may not hold when using the expanding window scheme [START_REF] Hansen | The Model Confidence Set[END_REF].

Remark 2 We use the square root of daily realized volatility shown in Eq. (17) as a proxy for the true conditional standard deviation (σt) when calculating the performance metrics in accordance with the related literature (Violante and Laurent, 2012).

Indeed, the judgment about predictive superiority of models relying solely on performance metrics may not be convincing due to underlying sampling issues. Therefore, we use the MCS procedure due to [START_REF] Hansen | The Model Confidence Set[END_REF], which involves hypotheses testing and model selection, to make inferences about the ranking of the models. When several models have been built, the MCS procedure helps in determining whether the performance of a given model in the set of tested models is statistically significant in comparison to the performance of other tested models, see [START_REF] Hansen | The Model Confidence Set[END_REF] for more details. Furthermore, in the MCS procedure, different aspects of the models can be evaluated by using user-specified loss functions [START_REF] Bernardi | The Model Confidence Set package for R[END_REF]. Here, we use two loss functions, namely, RMSE and QLIKE, within the MCS procedure due to the robustness of the latter [START_REF] Patton | Volatility forecast comparison using imperfect volatility proxies[END_REF]. However, other metrics such as mean error (ME), mean absolute error (MAE), mean absolute percentage error (MAPE), and mean percentage error (MPE) are also reported independently in the Appendix E.

The strategy used to assess the performance of the VS-LTGARCHX algorithm compared to the benchmark models is summarized in the Algorithm 3 displayed in the Appendix D.

Numerical results

Data preprocessing, data partition and parameter tuning

We collect daily data on the BTC market and 44 variables measured in daily frequency covering the period from December 18, 2017 to June 17, 2022. The missing values, if any, were filled using the last observation carried forward principle. The complete list of the variables is given in Table 4, 5 and 6 of the Appendix A. After the data are preprocessed and all variables are stationarized and standardized to have zero mean and unit variance, we partition the data set into train, validation and test sets in 60%-20%-20% chunks, initially (see Algorithm 1).

In the next subsections, we will shed light into the set of variables selected by LASSO, ABESS, and Boruta, respectively, and then compare the VS-LTGARCHX variants' predictive accuracy to those of the benchmark models.

Selected variables by VS-LTGARCHX algorithms

Before performing variable selection, the variable selection algorithms, namely LASSO, ABESS, and Boruta, must be calibrated. We follow the strategies introduced in Subsection 4.2.2. The hyperparameter tuning process for LASSO and ABESS is outlined in Algorithm 1, and that for random forest underlying the Boruta wrapper is summarized in Algorithm 2. The optimal values of the tuning parameters help us to choose a subset of variables from a larger set for each of the respective methods. The variables selected by various selection procedures are given in Table 1. ABESS selected the most parsimonious model as expected. Interestingly, the composition of the variables selected by LASSO and ABESS is almost the same except LASSO selects one extra variable, namely, the first lag of the logarithmic difference of BTC market capitalization (dl MKTCP). Moreover, the first lag of the difference in VIX index is selected by the three selection procedures. In fact, it has recently been found that the logarithmic difference of the VIX index has contemporaneous effects on the volatility of BTC [START_REF] López-Cabarcos | Bitcoin volatility, stock market and investor sentiment[END_REF]. In our study, we focus solely on lagged effects because contemporaneous feedback is not usually helpful for investor decision-making.

Another variable selected by the three procedures is the lagged difference of Google trends (d gtrends). Interestingly, Google trends lags were found to have predictive power over BTC returns [START_REF] Arratia | Do Google Trends forecast bitcoins? Stylized facts and statistical evidence[END_REF]. Moreover, several other recent studies have found the link of the Google trends variable with conditional volatility [START_REF] Bystrom | What Drives Bitcoin Volatility?[END_REF][START_REF] Aslanidis | The link between cryptocurrencies and Google Trends attention[END_REF][START_REF] Bakas | What drives volatility in Bitcoin market[END_REF].

Furthermore, the lag of the weekly realized volatility selected by the three selection procedures has already been shown to be effective in predicting the volatility of BTC [START_REF] Aharon | Bitcoin and the day-of-the-week effect[END_REF][START_REF] Bergsli | Forecasting volatility of Bitcoin[END_REF]. However, it is still an open question why the lag of daily and monthly realized volatilities were not selected by LASSO and ABESS as opposed to the latter references.

Interestingly, among the three variable selection methods, only Boruta selects a large set of technological variables. Indeed, we could not find a study to support this interesting finding, and the effect of BTC technology variables on its volatility should be further researched.

Predictive accuracy of VS-LTGARCHX algorithms

Before delving into the analysis of results and comparative statistics, one fact is worth noting to assert the importance of explanatory variables for conditional volatility forecasting. Figure 1 illustrates the forecasts generated from the two benchmark models. According to the latter, the full-blown log-TGARCHX model seems to perform better than log-GARCH(1,1) particularly during large volatility periods. Moreover, after the variable selection procedure is finished we generate one-step-ahead predictions from three variants of VS-LTGARHCX algorithm using the test set as shown in Algorithm 3. Figure 2 shows one-step-ahead forecasts of BTC conditional volatility from three different variants of the VS-LTGARCHX algorithm. As the figure reveals, the forecasts generated using the variants of the VS-LTGARCHX algorithm are more precise compared to the benchmark during relatively high volatility periods.

Furthermore, according to Table 2, the three variants of VS-LTGARCHX lead to better predictive accuracy than the log-GARCH (1,1) and log-TGRACHX benchmarks in terms of RMSE and QLIKE. Moreover, the ABESS and LASSO variants always outperform the Boruta alternative with regard to the latter criteria. 1: Plot of one-step-ahead forecasts from log-GARCH, log-TGARCHX models and realized standard deviation On the other hand, the results of the MCS procedure are summarized in Table 3. The intrinsic hypothesis test for MCS was carried out at the significance level 5%. The null hypothesis states that the model has predictive performance equivalent to all other models considered [START_REF] Bernardi | The Model Confidence Set package for R[END_REF]. Then, the p-value for each model represents the probability of observing a sample performance of that particular model at least as adverse to the null hypothesis as the current one. Therefore, it is natural for models with a p-value lower than the significance level (5%) to be removed from the set considered. The MCS procedure (hypothesis test+elimination) continues until no model can be removed from the set and the final set of surviving models is called the superior set models (SSM) [START_REF] Hansen | The Model Confidence Set[END_REF][START_REF] Bernardi | The Model Confidence Set package for R[END_REF].

Sep

As shown in Table 3, the MCS procedure has eliminated VS-LTGARCHX (Boruta) from SSM while using RMSE as a loss criterion. This means that the latter is inferior to VS-LTGARHCX (LASSO) and VS-LTGARHCX (ABESS). However, it does not mean that VS-LTGARCHX (Boruta) has the same predictive ability as log-GARCH (1,1) and log-TGARCHX models. In another experiment, we 2: Plot of forecasts from VS-LTGARCHX using LASSO, ABESS and Boruta variable selection procedures, respectively restricted the initial set of the models considered (M 0 ) to include only the models VS-LTGARCHX (Boruta), log-GARCH (1,1) and log-TGARCHX. Not surprisingly, VS-LTGARCHX (Boruta) was found to be superior to the other two models, using both MSE and QLIKE as loss criteria. This means that the VS-LTGARCHX algorithm outperformed the benchmarks in all of its variants. However, the Boruta variant is inferior to the LASSO and ABESS variants of VS-LTGARCHX.

As a robustness check, we report the extended list of performance metrics for the rolling window forecasting scheme in Table 7 of the Appendix E. The extended list of performance metrics includes not only QLIKE and RMSE, but also mean error (ME), mean absolute error (MAE), mean absolute percentage error (MAPE) and mean percentage error (MPE) (see Algorithm 3 in the Appendix D and [START_REF] Hyndman | Forecasting: principles and practice[END_REF] for details).

Furthermore, it is not surprising that the full-blown log-TGARCHX model outperforms all other models in terms of ME and MPE as shown in Table 7. Tibshirani correctly points out that : "...the OLS estimates often have low bias but large variance; prediction accuracy can sometimes be improved by shrinking or setting to 0 some coefficients. By doing so, we sacrifice a little bias to reduce the variance of the predicted values and hence may improve the overall prediction accuracy" (Tibshirani, 1996, p. 267). The unrestricted log-TGARCHX model (estimated using OLS) should have less bias compared to its restricted alternatives, as inferred from the latter quote. Therefore, ME and MPE which are nothing but the average and the weighted average of forecast errors, respectively, should have the smallest value for the unrestricted log-TGARCHX model. Moreover, as an additional robustness check, we follow [START_REF] Hansen | The Model Confidence Set[END_REF] and test our algorithm using the recursive forecasting scheme. An extended list of performance metrics for the recursive window forecasts ìn Table 8 of the Appendix E supports our conclusions in this subsection, thus showing the robustness of the proposed algorithms to varying environments.

Conclusion

Conditional volatility forecasting of a financial asset is important from the perspective of risk management, hedging, and portfolio management. Therefore, there are a plethora of models dealing with volatility forecasting, with the GARCH family of models being arguably the main approach. However, generic GARCH models are restrictive in nature with respect to the inclusion of exogenous variables that may contain important information about conditional volatility. Fortunately, log-GARCH models put very mild restrictions on inclusion of covariates in the conditional variance equation, and their estimation procedure is rather simple. However, this flexibility brings about a potential overfitting problem. To overcome this problem, we propose the VS-LTGARCHX algorithm to incorporate the variable selection procedure into the log-TGARCHX estimation process. The algorithm is very flexible in the sense that any variable selection procedure can be used within its framework. However, in this paper, we use LASSO, ABESS, and Boruta variable selection procedures separately within VS-LTGARCHX algorithm to select the subset of exogenous variables and asymmetry lags while estimating log-TGARCHX models. Furthermore, we apply the VS-LTGARCHX algorithm to extremely volatile BTC markets using 42 conditioning variables. Three different restricted versions of log-TGARHCX models obtained with the VS-LTGARCHX algorithm (using LASSO, ABESS, and Boruta selection procedures, respectively) outperform the benchmark log-GARCH (1,1) and log-TGARCHX (1,1) models in one-step ahead conditional volatility prediction based on different accuracy metrics. To assert the superiority of the VS-LTGARCHX variants, we use a formal procedure called MCS which indicates that the forecasts generated from the three variants of VS-LTGARCHX are superior to those from the benchmark models.

As a result, the main contributions of this paper are twofold. On the one hand, our study proposes a novel approach to model conditional volatility in the GARCH models domain which may be useful for academicians, investors, and policy-makers. On the other hand, we analyze volatility in the evolving and relatively young BTC market with 42 different exogenous variables. To the best of our knowledge, no study has been conducted to study BTC volatility with as many exogenous covariates as we do.

Ultimately, irrespective of the novelties of our paper, there are several limitations which paves the way for further research. First, the set of exogenous variables can be extended. Second, the selection procedures in this study cannot handle the simultaneous selection of the subset of ARCH, GARCH, asuymmetry terms and exogenous variables. Rather, our approach is based on the stepwise selection, which may be suboptimal due to possible correlation of variables. In addition, we have not considered ARCH/GARCH orders and asymmetry orders greater than 1, which presumably fettered further improvement of the predictive accuracy of VS-LTGARCHX. D. Evaluation of the predictive performance of VS-LTGARCHX variants and the benchmark models

A. Abbreviations

The following algorithm summarizes the strategy we follow to evaluate the predictive performance of the different variants of the VS-LTGARCHX algorithm.

Algorithm 3 Evaluation of the Predictive Performance of VS-LTGARCHX variants and The Benchmark Models Inputs: Dt = {(rt, x * t-1 , σt)|t = 1, ..., T }, rt are log-returns, x * t-1 is deemed to be the selected variables from Algorithm 1 or 2 for VS-LTGARCHX variants, the full set of the variables at hand for the benchmark log-TGARCHX model, and an empty set for the benchmark log-GARCH(1,1). Fit the intended model using Dτ = {Dt|τ -⌊0.8T ⌋ + 1 ≤ t ≤ τ } 4:

Predict στ+1:

E(στ+1|Dτ ) = f (rτ , x * τ -1 ) 5:
Calculate the prediction errors, dτ = στ -στ 6: end for 7: Evaluate predictive performance using the following: Output: The table of predictive performance scores of the models considered.

M E = 1 #(S) τ ∈S dτ , RM SE = 1 #(S) τ ∈S

E. Study of robustness

The table below (Table 7) is an extended version of Table 2. In Table 7, together with RMSE and QLIKE metrics, we report mean error (ME), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE) and Mean Percentage Error (MPE) for rolling window forecasts (see Eq. ( 18) and [START_REF] Hyndman | Forecasting: principles and practice[END_REF] for details). Although we are aware of the potential problems posed by the recursive forecasting scheme when evaluating nested models, we follow [START_REF] Hansen | The Model Confidence Set[END_REF] and test our algorithm using the recursive scheme, also for the sake of robustness. Interestingly, our conclusions about the superiority of forecasts generated by the VS-LTGARCHX models over those generated by the benchmark models remain valid. Table 8 describes the performance metrics of the models in the test set (20% of the overall data). The training window is expanded by one observation at each forecast origin. The initial training set comprises 1314 (80% of the data) observations. The results of the MCS procedure are exactly the same as those in the rolling window forecasting scheme, and thus are omitted.

Competing interests

No competing interest is declared.

Author contributions statement

F.R. wrote a Python program to download data and reviewed the literature, the rest of the work was divided and completed equally by F.S., A.P., V.E and S.O.

Data availability statement

The data and code underlying this article are available in https://github.com/salahaddiniayyubi/Log-TGARCHX-Subset-Selection.git.

Acknowledgements

Samir Orujov gratefully acknowledges financial support from Total Energies and ADA University, Azerbaijan.

1:

  Data Split: Define set of indices S = {τ : ⌊0.8T ⌋ ≤ τ ≤ T } for the test data, with cardinality #(S) = ⌈0.2T ⌉ 2: for τ ∈ S do 3:

Table 1 .

 1 Variables Selected by ABESS, LASSO and Boruta, respectively

	# ABESS	LASSO	Boruta
	1	d VIX	dl MKTCP dl CPTRA
	2	d gtrends d VIX	dl ETRVU
	3	rv w	d gtrends	dl HRATE
	4		rv w	dl MWNTD
	5			dl MWTRV
	6			dl NADDU
	7			dl NTRAN
	8			dl NTRBL
	9			dl NTREP
	10			dl SPY
	11			dl DJI
	12			d CPTRV
	13			d ETRAV
	14			d TRFEE
	15			d TRVOU
	16			d VIX
	17			d gtrends
	18			rv d
	19			rv w

Table 2 .

 2 Predictive accuracy of models in the test set using rolling window

	Model RMSE QLIKE
	Log-GARCH	2.00	3.74
	Log-TGARCHX	1.85	3.82
	LASSO	1.60	3.64
	ABESS	1.59	3.64
	Boruta	1.66	3.69

Table 3 .

 3 MCS Superior Set Models (SSM) using MSE and QLIKE as loss criteria

	MCS (QLIKE)

Table 4 .

 4 Abbreviations for Blockchain Technology Variables

	# Abbreviation Expansion
	1	ATRCT	Bitcoin Median Transaction Confirmation Time in Minutes
	2	AVBLS	Bitcoin Average Block Size
	3	BLCHS	Bitcoin api.blockchain Size
	4	CPTRA	Bitcoin Cost Per Transaction
	5	CPTRV	Bitcoin Cost % of Transaction Volume
	6	DIFF	Bitcoin Difficulty
	7	ETRAV	Bitcoin Estimated Transaction Volume
	8	ETRVU	Bitcoin Estimated Transaction Volume USD
	9	HRATE	Bitcoin Hash Rate
	10 MIREV	Bitcoin Miners Revenue
	11 MKPRU	Bitcoin Market Price USD
	12 MKTCP	Bitcoin Market Capitalization
	13 MWNTD	Bitcoin My Wallet Number of Transaction Per Day
	14 MWNUS	Bitcoin My Wallet Number of Users
	15 MWTRV	Bitcoin My Wallet Transaction Volume
	16 NADDU	Bitcoin Number of Unique Bitcoin Addresses Used
	17 NTRAN	Bitcoin Number of Transactions
	18 NTRAT	Bitcoin Total Number of Transactions
	19 NTRBL	Bitcoin Number of Transaction per Block
	20 NTREP	Bitcoin Number of Transactions Excluding Popular Addresses
	21 TOTBC	Total Bitcoins
	22 TOUTV	Bitcoin Total Output Volume
	23 TRFEE	Bitcoin Total Transaction Fees
	24 TRFUS	Bitcoin Total Transaction Fees USD
	25 TRVOU	Bitcoin USD Exchange Trade Volume
	Source: www.nasdaq.com

Table 5 .

 5 Names and abbreviations of the exogenous variables within public opinion, risks and uncertainties, financials, and macroeconomic development classes. www.bitinfocharts.com 2 www.finance.yahoo.com 3 www.finance.yahoo.com 4 www.matteoiacoviello.com 5 www.fred.stlouisfed.org 6 www.economicpolicyuncertaintyinchina.weebly.com

	Public Opinion 1	Risks and Uncertainties	Financials 2		Macroeconomic
						Development 3
	Number of tweets (tweet)	Geopolitical risks (GPRD) 4	US Federal Funds Effective	Crude Oil (CL=F)
			Rate (FFER) 5
	Google trends (gtrends)	Daily China economic policy	Ripple-USD Exchange Rate	Nasdaq (NDAQ)
		uncertainty Index (CNEPU) 6	(XRP-USD)
			Bitcoin Futures price (BTC=F) S&P500 (SPY)
			Chinese	Yuan	to	dollar	DOW30 (DJI)
			exchange rate (CNYUSD=X)
						The Chicago Board Options
						Exchange	Volatility	Index
						(VIX)
						The Chicago Board Options
						Exchange Gold Volatility Index
						(GVZ)

1

Table 6 .

 6 Abbreviations for Realized Volatility Measures and Prefixes

	# Abbreviation Expansion
	1 rv d	Daily realized variance 1
	2 rv w	Weekly realized variance
	3 rv m	Monthly realized variance
	4 d	Prefix in variable names, denotes for

the first difference (d X := Xt -Xt-1) 5 dl Prefix in variable names, denotes the first logarithmic difference (dl X := ln Xt -ln Xt-1)

Table 7 .

 7 Predictive accuracy of models in the test set.

	ME RMSE MAE MPE MAPE QLIKE
	Log-GARCH 1.20	2.00	1.70 24.78	36.04	3.74
	Log-TGARCHX 0.42	1.85	1.39 3.02	36.66	3.82
	LASSO 0.65	1.60 1.23 15.47	29.66	3.64
	ABESS 0.67	1.59	1.24 16.11 29.53	3.64
	Boruta 0.47	1.66	1.25	9.94	31.61	3.69

Table 8 .

 8 Recursive window forecasts.

	ME RMSE MAE MPE MAPE QLIKE
	Log-GARCH 1.22	2.03	1.74 24.45	36.60	3.75
	Log-TGARCHX 0.07	1.78	1.31 7.91	40.34	3.93
	LASSO 0.59	1.62	1.24 14.07	30.29	3.65
	ABESS 0.61	1.61	1.24 14.66 30.02	3.65
	Boruta 0.45	1.64 1.22	9.65	31.15	3.68

BTC price data were taken in 5-minute frequencies from Binance exchange to calculate the daily realized volatily.

B. Time series CV for LASSO or ABESS

The following algorithm outlines the methodology we follow for data partitioning to perform cross-validation for LASSO and ABESS.

Algorithm 1 Time Series CV For Variable Selection With LASSO or ABESS

Input: Dt = {(zt, xt-1, σt|t = 1, ..., T } where σt is the sequence of the realized standard deviation, xt-1 are regressors including the asymmetries, zt are residuals within Eq. ( 9), Π is the parameter grid function BestSubsetSelector(Dt, Π)

Data Split: Training Sets (Ω k ) and Validation Sets (Υ k ), ∀k = 1, 2, ..., ⌊0.2N ⌋, where cardinalities #(Ω1) = ⌊0.6N ⌋ = τ1 and #(Υ1) = ⌈0.2N ⌉ = τ2. Then, the collection of train and validation sets are:

Define the set of parameters to be tuned with a search grid Π.

C. Time series CV for random forest (Boruta)

The following algorithm describes the data partitioning technique to perform time series cross-validation for the random forest (Boruta).

Algorithm 2 Time Series CV For Random Forest Behind The Boruta

Input: Dt = {(zt, xt-1, σt|t = 1, ..., T } where σt is the sequence of the realized standard deviation, xt-1 are regressors that also include asymmetries, zt are residuals within Eq. ( 9), number of trees (ntree), minimum node size (nmin), block size (ln), number of split variables (m), Π := (ntree, nmin, ln, m) ′ is four-dimensional parameter grid Data Split: Training Set T = {Dt|1 < t < ⌊0.8T ⌋}. Define the set of parameters to be tuned with a search grid Π. for π ∈ Π do for each b = 1 to ntree do Draw block bootstrap sample Z⋆ of size #(Z⋆) ≤ ⌊0.8T ⌋ with parameter ln. while node size n > nmin do Chose a random set of m variables among the p variables and apply the CART criterion on this subset. Cut on the best split. end while end for Make out-of-bag (OOB) predictions σt Calculate average OOB performance (RMSE) ∀π ∈ Π. end for Determine the optimal set of tuning parameters based on the lowest RMSE Apply Boruta with the optimal parameters Get The Best Subset of Covariates, x * Output: Selected subset of covariates (x * ) by Boruta.