Interfuel Substitution: Evidence from the Markov Switching Minflex Laurent Demand System with BEKK Errors.

• Author: Serletis, Apostolos

1. INTRODUCTION

   In a series of recent papers, Serletis et al. (2010) and Jadidzadeh and Serletis (2016), and Hossain and Serletis (2017) use state-of-the-art advances in microeconometrics to investigate interfuel substitution (energy elasticities) and the demand for energy. They take the dual approach to demand system generation developed by Diewert (1974), and pioneered by Berndt and Wood (1975), Fuss (1977), and Pindyck (1979) in the context of interfactor and interfuel substitution. This approach allows estimation in a systems context, assuming a flexible functional form for the aggregator function, and the computation of the relevant elasticity measures, like the income elasticities, the own- and cross-price elasticities, and the Allen and Morishima elasticities of substitution, consistent with theoretical regularity (in terms of curvature, positivity, and monotonicity of the aggregator function). There are also other papers on the interfuel substitution, including Bachmeier and Griffin (2006), Hyland and Haller (2018), and Steinbuks (2012).

   Until recently, all of the studies in the interfuel substitution literature had assumed homoskedastic errors. The exemption is Hossain and Serletis (2017), who build on Serletis and Isakin (2017) and Serletis and Xu (2019) and introduce recent state-of-the-art advances in financial econometrics to the empirical energy demand literature. In particular, they relax the homoskedasticity assumption and instead assume that the covariance matrix of the errors of the demand system is time varying, thus improving the flexibility of the demand system to capture important features of the data. Hossain and Serletis (2017) use the Normalized Quadratic (NQ) expenditure function, and over nearly 100 years (from 1919 to 2012) of data, to investigate the demand for energy and the degree of substitutability among fossil fuels in the United States. In doing so, they generate inference consistent with neoclassical microeconomic theory and the data generating process.

   In this article, we follow Hossain and Serletis (2017). However, instead of using the Normalized Quadratic demand system used in Hossain and Serletis (2017), in this paper we use the minflex Laurent demand system, introduced by Barnett (1983) and Bamett and Lee (1985), to investigate the degree of substitutability among crude oil, natural gas, and coal in the United States. It is to be noted that both the Normalized Quadratic and minflex Laurent models are flexible functional forms, but the minflex Laurent model uses the Laurent series expansion, which is a generalization of the Taylor series expansion, to approximate the unknown aggregator function. Moreover, as noted by Barnett and Lee (1985), the minflex Laurent model is well suited for use with time series data like ours, because its regularity region expands as real income increases.

   We also advance the methodology by relaxing the assumption of constant parameters in the aggregator function and thus the resulting demand system. We are motivated by the fact that consumer preferences seem to change quite dramatically in response to shocks that hit the economy. In particular, commodity price shocks, large-scale events (such as wars and financial crises), changes in government policy, and technological and institutional changes can induce significant shifts in consumer tastes and preferences. For example, Guiso et al. (2017) find that qualitative and quantitative measures of risk aversion increase substantially after financial crises, and that fear is a potential mechanism that influences financial decisions, whether by increasing the curvature of the utility function or the salience of bad outcomes. The change in behavior may be permanent, typically known as a 'structural break,' temporary, or shift to another style of behavior, usually called a 'regime switch.' Thus, what is required is a framework that is sufficiently flexible to allow for different types of behavior at different times and that also utilizes all of the available observations on the series to be used for estimating the demand system. Two classes of data-driven models that allow this to occur have been used so far in empirical demand analysis: the time-varying coefficient model and the Markov switching model. The former assumes that consumer preferences change in every period (which might not be the case in the real world), whereas the latter assumes a switching mechanism from one state of preferences to another that is controlled by an unobserved variable governed by a Markov process. In this paper, we take the Markov-switching approach, associated with Hamilton (1989).

   The rest of the paper is organized as follows. Section 2 sketches out related neoclassical microeconomic theory and applied consumption analysis. Section 3 discusses the minflex Laurent demand system and Section 4 the modelling of the changing parameters and the conditional covariance matrix of the minflex Laurent demand system. Section 5 discusses related econometric issues, Section 6 presents the data and the empirical results based on the Markov switching minflex Laurent demand system with BEKK errors, and Section 7 addresses robustness issues. The final section concludes the paper.

2. NEOCLASSICAL DEMAND THEORY

   Let's consider an economy with identical households with the following utility function

   u = u(o,g,c) (1)

   where o is the quantity demanded of oil, g is the quantity demanded of natural gas, and c is the quantity demanded of coal. It is to be noted that crude oil is not consumed directly, but is used as a factor of production in the refining industry (in the production of gasoline, diesel, heating oil, and jet fuel), and competes with natural gas, which also competes with coal in producing electricity and in the manufacturing of chemicals and metals.

   The representative household's optimization problem can then be written as

   [mathematical expression not reproducible] (2)

   where x=(o,g,c) and p = ([p.sub.o],[p.sub.g],[p.sub.c]) is the corresponding vector of prices. The solution of the first-order conditions is the Marshallian demand functions

   x = x(p,y). (3)

   The demand system (3) can also be represented in budget share form s, where s= ([S.sub.1],...,[S.sub.K]), and [w.sub.j] = [p.sub.j][x.sub.j](p,y)/ y is the expenditure share of good j. Since the Marshallian demand functions are homogenous of degree zero inp and y, we could write the demand system in budget share form as

   [s.sub.j]=[s.sub.j](v) (4)

   where v = ([v.sub.j],...,[v.sub.k]) with [v.sub.j] denoting the income normalized price, [p.sub.j] / y.

   An effective method to derive the demand system in budget share form is to apply Diewert's (1974) modified version of Roy's identity to the indirect utility function, h(p,y)--see Barnett and Serletis (2008) for more details.

3. THE MINFLEX LAURENT MODEL

   We use the flexible functional forms method for the approximation of aggregator functions. The advantage of this method is that the corresponding demand system will adequately approximate systems resulting from a broad class of aggregator functions. The representative agent's optimization problem (2) is equivalent to

   [mathematical expression not reproducible] (5)

   where v is the vector of income normalized prices, with the i th element being [p.sub.i] l/ y. We define the indirect utility function corresponding to the optimization problem (5) as

   [mathematical expression not reproducible]. (6)

   The indirect utility function, /(v), is the unknown aggregator function, and we use the minflex Laurent model to approximate the reciprocal of I(v)

   [mathematical expression not reproducible] (7)

   where k denotes the number of goods (in our case, k = 3), [v.sub.i] denotes the income normalized price ([p.sub.i] / y), c is a constant, and [delta] =([[delta].sub.l],...,[[delta].sub.k])', and [d.sub.ij]. and [h.sub.ij] are all parameters. The application of Roy's identity to (7) yields the share equations of the ML demand system (for i = 1,...,k )

   [mathematical expression not reproducible] (8)

   Since the share equations are homogeneous of degree zero in the parameters, we follow Barnett and Lee (1985) and impose the normalization

   [mathematical expression not reproducible] (9)

   and the restrictions

   [d.sub.ij]-[d.sub.ij], [h.sub.ij]= [h.sub.ij], [d.sub.ij][h.sub.ij]=0, i [not equal to] j. (io)

   These restrictions render the ML model minimal, in the sense that imposition of any further prior restrictions compromises the model's flexibility property.

   Finally, the...