Estimating Short and Long-Run Demand Elasticities: A Primer with Energy-Sector Applications.

• Author: Cuddington, John T.

INTRODUCTION

Many empirical exercises estimating demand and supply functions are concerned with estimating dynamic effects of price and income changes over time. (1) Researchers are typically interested in estimating both short-run (SR) and long-run (LR) elasticities, along with their standard errors. Energy demand analysis offers many applications; see Dahl (1993) for a comprehensive survey of energy elasticity estimates. For example, consider a public utility requesting a rate increase from the public service commission. The utility and regulators want to know how a proposed price hike will impact demand in the SR and the LR. Searching the literature on energy demand elasticity estimates, one finds that authors often fail to provide standard errors for either their short or long-run elasticity estimates, or both. (2,3) Thus, it is hard to know whether the LR elasticities are statistically different from their SR counterparts. Moreover, it is difficult to determine whether elasticity estimates across studies are really statistically different from each other.

Section I of this paper first reviews a number of commonly used dynamic demand specifications and highlights the implausible a priori restrictions that they place on short and long-run elasticities. We emphasize that these restrictions are easily avoided, and are indeed testable, when a general-to-specific modeling approach is employed. Section II discusses estimation issues, including a simple way to get standard errors as well as point estimates for both short and long-run elasticities. Section III provides an empirical application--estimating residential demand for electricity in Minnesota. Section IV concludes.

1. ALTERNATIVE DYNAMIC DEMAND SPECIFICATIONS AND IMPLICATIONS FOR SR AND LR ELASTICITIES

   Modern time-series econometricians emphasize the merits of beginning with a hypothesized data generating process (DGP) for all variables in the data sample being analyzed. In the case of sectoral supply and demand analysis, the DGP will typically involve a system of potentially simultaneous equations. As Andrew Harvey (1990, p. 2) has stressed: "Econometric models typically consist of sets of equations which incorporate feedback effects from one variable to another. Treating the estimation of a single equation from such a system as an exercise in multiple regression will, in general, lead to estimators with poor statistical properties."

   In many cases, however, a system of equations can be reduced to a single equation. The assumptions needed to reduce the empirical analysis to a single-equation exercise (or so-called partial system) with no loss of information regarding the parameters of interest are often testable within a systems framework. Even when there is some loss of information, a limited information approach may have merits. According to Juselius (2006, p. 198): "Note, however, that in order to know whether we can estimate from a partial system we need first to estimate the full system and test in that system. But if we need to estimate the full system, why would we bother to discuss estimation in a partial system? Two reasons come to mind: (1) by conditioning on weakly exogenous variables, one can often achieve a partial system which has more stable parameters than the full system and (2) it is sometimes very likely a priori that weak exogeneity holds. In particular when the number of potentially relevant variables to include in the VAR model is large it can be useful to impose weak exogeneity restrictions from the outset."

   Studies of energy demand elasticities have often used a single-equation DGP by assuming "that the particular market conditions of electrical and natural gas energy favor single equation analyses free from any endogeneity problem" (Balestra 1967; Uri 1975; Bohi 1981). The most common justification given is that the supply of electricity and natural gas may be considered perfectly elastic because supply is rarely, if ever, interrupted, and construction of pipeline and transmission and distribution lines are made with the purpose of satisfying not only immediate but also future consumption. As a result, most of the time there is excess capacity (Balestra 1967). Most studies implicitly assume that all regressors are (weakly) exogenous, so that these estimation approaches yield asymptotically valid statistical inference.

   Suppose the empirical task at hand is to estimate a demand function for residential electricity demand (q) using time series data. For expositional simplicity, demand is assumed to depend only on own real price (p), the real price of substitutes (ps), and real income (y). (4,5)

   [q.sub.t] = [[beta].sub.0] + [[beta].sub.p][p.sub.t] + [[beta].sub.s]p[s.sub.t] + [[beta].sub.y][y.sub.t] + et (1.1)

   where all variables are in natural logs. Typically, this equation will have serially correlated errors, which is taken as prima facie evidence that dynamic considerations are important when modeling demand for many commodities. Here, four popular approaches for modeling the dynamics in order to estimate SR and LR price and income elasticities are considered. The first three are, in fact, nested as special cases of the general autoregressive distributed lag (ADL) model. After discussing these approaches, we emphasize the merits of a general-to-specific methodology.

   Approach 1: Estimate the LR Demand Function with an AR(1) Error Process

   [q.sub.t] = [[beta].sub.0] + [[beta].sub.p][p.sub.t] + [[beta].sub.s]p[s.sub.t] + [[beta].sub.y][y.sub.t]+ [e.sub.t] (1.2)

   where

   [[epsilon].sub.t]= [rho][[epsilon].sub.t-1] + [u.sub.t]

   In this specification, the error term [[epsilon].sub.t] can be interpreted as the deviation of quantity demanded from the LR demand equation. The speed of adjustment toward the LR equilibrium is given by 1 -p.

   To estimate the AR(1) model in (1.2), a generalized least squares (GLS) estimator is typically used. Alternatively, the long-run relationship is quasi-differenced to yield the following regression:

   [mathematical expression not reproducible] (1.3)

   This equation is then estimated using non-linear least squares (NLS) regression to obtain estimates of the LR price, cross-price, and income elasticities ([[beta].sub.p],[[beta].sub.s],[[beta].sub.y]) and other structural parameters

   ([[beta].sub.0],[rho]).6

   Approach 2: Estimate a Partial Adjustment Model (PAM) (7)

   In the partial adjustment model (PAM), the long-run level of demand q (*) is:

   [mathematical expression not reproducible] (1.4)

   A partial adjustment mechanism describes how actual quantity [q.sub.t] adjusts gradually towards q* with speed of adjustment k where 0

   [mathematical expression not reproducible] (1.5)

   Substituting (1.4) into (1.5) produces an equation that is nonlinear in the five structural parameters ([[beta].sub.0],[[beta].sub.p],[[beta].sub.s],[[beta].sub.y],k):

   [mathematical expression not reproducible] (1.6)

   Again, this specification is easily estimated using NLS regression, yielding both parameter estimates and their associated standard errors. With the PAM, however, authors typically just apply OLS regression involving the regressors in the (just-identified) equation above, and then reverse engineer the long-run elasticities. Getting their associated standard errors is tricky, however, so they are often not calculated or not reported.

   Note that both the AR(1) and PAM specifications include the contemporaneous price pt on the right-hand side, suggesting a need to use instrumental variables (IV) estimation to avoid the possibility of endogeneity bias.

   Approach 3: Estimate an Error Correction Model (ECM)

   A single-equation error-correction model (as opposed to a vector error-correction system) is similar to the PAM except that the long-run demand q* enters with a one-period lag:

   [mathematical expression not reproducible] (1.7)

   Rewriting (1.7) with the log-level of q rather than the log-difference as the dependent variable for comparability to the previous specifications yields:

   [mathematical expression not reproducible] (1.8)

   The PAM and the ECM are quite similar: [p.sub.t], [ps.sub.t], and yt enter the PAM specification, whereas the one-period lags, [p.sub.t-1], [ps.sub.t-1], and [y.sub.t-1], are included in the ECM. [lambda] provides information about the speed of adjustment in both models.

   The absence of the contemporaneous price among the regressors in the ECM is a restrictive a priori assumption when estimating demand or supply equations for most commodities. Even though we may presume that the very-short-run price elasticity is low, forcing it to be zero seems questionable with quarterly or annual frequency data, at least. Omitting contemporaneous price from the demand equation might seem to legitimize the use of OLS or non-linear LS rather than IV estimation, but it may merely replace the criticism of simultaneity bias with that of omitted variable bias. Moreover, consistency of the OLS estimator in demand equations typically requires the assumption of weak exogeneity of the regressors. (8)

   The "simple" ECM in (1.7) assumes that the error term is serially uncorrelated. After discussing the ADL model below, more general ECMs with lagged differences of the regressors will be considered.

   Approach 4: Estimate an Autoregressive Distributed Lag (ADL) Model

   The autoregressive distributed lag or ADL(L,R,V,S) model regresses quantity demanded on L lags of itself, R lags of prices, V lags of cross prices, and S lags of income:

   [mathematical expression not reproducible] (1.9)

   Sometimes, ADL models include contemporaneous values of the additional regressors, as shown above. Other times, only lagged values are included. We'll allow contemporaneous (not just lagged) values of the explanatory variables to enter the demand equation for reasons just discussed above. To make the lag intervals explicit, we might label an ADL model as ADL(1-L,0-R,0-V,0-S), this implies that lags 1 to L, 0 to R, 0 to V, and 0 to S of q, p, ps, and y...