Edgeworth Price Cycles in Gasoline: Evidence from the United States

Abstract

Studies of gasoline prices in multiple countries have found sequences of a sharp price increase followed by gradual decreases. This pattern is linked to Maskin and Tirole (Econometrica 56:571–599, 1988) duopoly pricing game and labeled Edgeworth price cycles. We examine data on average daily MSA-level retail gasoline prices for 350 MSAs in the US from 1996–2010. We confirm the finding of others and show that a relatively small number of US MSAs in contiguous upper Midwestern states evidence price cycling. However, our lengthy data set allows us to see that these MSAs began cycling in 2000. Thus, we can examine prices in cycling and non-cycling MSAs before and after cycling and find that prices are lower in MSAs that began cycling.

Appendix: A Markov-switching Model for Identifying Edgeworth Price Cycles

We employ a Markov switching model that is based on Neftçi (1984). Let $p_t$ be the retail price in a given MSA during week $t$, which is assumed to follow a mean-zero linearly regular stationary process. Define $\{I_t \}$ as a second-order (“two-state”) Markov switching process:

$$\begin{aligned} I_t =\left\{ {\begin{array}{l} +1\quad \text{ if} \Delta _t p_{t}^{(\ell )} >0 \\ -1\quad \text{ if} \Delta _t p_t^{(\ell )} \le 0, \\ \end{array}} \right. \end{aligned}$$

(2)

where $\Delta _t$ denotes the first-difference operator.²⁰ The associated transition probabilities, denoted $\lambda _{ij}$ for $i,j=\{0,1\}$, are given by:

$$\begin{aligned} \left. {\begin{array}{l} \lambda _{11} =\Pr (I_t =+1|I_{t-1} =+1,I_{t-2} =+1) \\ \lambda _{00} =\Pr (I_t =-1|I_{t-1} =-1,I_{t-2} =-1) \\ \lambda _{10} =\Pr (I_t =+1|I_{t-1} =+1,I_{t-2} =-1) \\ \lambda _{01} =\Pr (I_t =-1|I_{t-1} =-1,I_{t-2} =+1) \\ \end{array}} \right\} . \end{aligned}$$

(3)

If a MSA’s retail or wholesale gasoline price series exhibits sharp increases and gradual decreases, as is suggested by the Maskin and Tirole (1988), then $\{I_t \}$ remains in state $-1$ longer than it remains in state $+1$. In this case, the retail price cycle is said to be asymmetric and would imply$\lambda _{00} >\lambda _{11}.$ If, on the other hand, the series is symmetric over the cycle, then $\lambda _{00} =\lambda _{11}$.

Our objective is to obtain estimates of the transition probabilities given in Eq. (3). Let $s_T$ denote a realization of $\{I_t \}$. The log-likelihood function is then given by:

$$\begin{aligned} L\left( {s_T ,\lambda _{11} ,\lambda _{00} ,\lambda _{10} ,\lambda _{01} ,\pi _0 } \right)&= \ln \pi _0 +\phi _{11} \ln \lambda _{11} +\psi _{11} \ln (1-\lambda _{11} )\nonumber \\&+\phi _{00} \ln \lambda _{00} +\psi _{00} \ln (1-\lambda _{00} ) \\&+\phi _{10} \ln \lambda _{10} +\psi _{10} \ln (1-\lambda _{10} )\nonumber \\&+\phi _{01} \ln \lambda _{01} +\psi _{01} \ln (1-\lambda _{01} ).\nonumber \end{aligned}$$

(4)

The variable $\pi _0$ is the initial condition (i.e., the probability of observing the initial two states), while the variables $\phi _{11} ,\ldots ,\psi _{01}$ represent the number of observed occurrences of the transitions.

Neftçi argues that it is necessary to estimate $\pi _0$ when the number of observations contained in the series is small and when the initial state may contain useful information (e.g., when the process $I_t$ does not in fact start at $t=1$, which is usually the case). Neftçi develops a methodology for deriving the limiting probabilities of the initial conditions in terms of the transition probabilities.²¹ If, however, the number of observations that are available in the sample is large (i.e., in an asymptotic sense), the initial state may be treated as a nuisance parameter (Billingsley 1961). Since the number of MSA-specific daily price observations that are available in our dataset covers (1996–2010), ignoring the influence of the initial condition is reasonable.²² With $\pi _0 =0$, the maximum likelihood estimates (MLEs) of the four unknown parameters $\Lambda =[\lambda _{00} ,\lambda _{11} ,\lambda _{10} ,\lambda _{01} {]}^{\prime }$ are obtained by setting the four score equations of the log-likelihood function equal to zero and solving the parameters in terms of the transition counts.²³ The general form of the score equations is given by:

$$\begin{aligned} \frac{\partial L}{\partial \lambda _{ij} }=\frac{\phi _{ij} }{\lambda _{ij} }-\frac{\psi _{ij} }{1-\lambda _{ij} }=0. \end{aligned}$$

(5)

Solving Eq. (5) in terms of $\lambda _{ij}$ gives:

$$\begin{aligned} \left. {\begin{array}{l} \frac{\phi _{ij} }{\lambda _{ij} }=\frac{\psi _{ij} }{1-\lambda _{ij} } \\ \Rightarrow \phi _{ij} (1-\lambda _{ij} )=\psi _{ij} \lambda _{ij} \\ \Rightarrow \lambda _{ij} (\phi _{ij} +\psi _{ij} )=\phi _{ij} \\ \Rightarrow \lambda _{ij} =\frac{\phi _{ij} }{\phi _{ij} +\psi _{ij} }=\hat{{\lambda }}_{ij} \\ \end{array}} \right\} , \end{aligned}$$

(6)

where $\hat{{\lambda }}_{ij}$ denotes the (approximate) MLE of $\lambda _{ij}$. McQueen and Thorley (1991) show that the asymptotic variance of $\hat{{\lambda }}_{ij}$ is given by

$$\begin{aligned} \sigma ^{2}(\hat{{\lambda }}_{ij} )=\frac{\hat{{\lambda }}_{ij} (1-\hat{{\lambda }}_{ij} )}{\phi _{ij} +\psi _{ij} }. \end{aligned}$$

(7)

Testing for the presence of Edgeworth price cycles (asymmetry) in gasoline prices involves testing the null hypothesis $H_0 :\lambda _{00} =\lambda _{11}$ against the (two-sided) alternative $H_1 :\lambda _{00} \ne \lambda _{11}$.

1.1 Hypothesis Testing

Neftçi demonstrates how the test for asymmetry can be evaluated using the estimate of the transition probabilities to construct a confidence region (ellipsoid), the center of which corresponds to the MLEs of $\lambda _{11}$ and $\lambda _{00}$. All points within the confidence ellipsoid represent the true value of the latter estimate for a given confidence level.²⁴ However, Sichel (1989, p. 1259) demonstrates that this procedure “has low power and is sensitive to noise”. Specifically, he shows that Neftçi’s test may fail to identify asymmetry that is actually present, and instead applies an asymptotic $t$-test that appears to give higher power.

McQueen and Thorley (1991) test the symmetry hypothesis in their data by considering asymptotic Lagrange Multiplier, Likelihood Ratio, and Wald tests (all of which are approximately equal for large sample sizes). They note that: “The choice of test statistics is normally a matter of computational convenience” (p. 256). Again, the length of our time series data suggests that we can rely on the direct analytical solutions for the MLEs and (asymptotic) variances of the Markov transition probabilities. This fact motivates the use of the Wald test since it uses the MLEs and asymptotic variance estimates of the unconstrained log-likelihood function, which correspond to the “unrestricted” estimates obtained by appealing to Eqs. (6) and (7). The computed value the Wald test under $H_0$ is given by:

$$\begin{aligned} \frac{(\hat{{\lambda }}_{00} -\hat{{\lambda }}_{11} )^{2}}{\sigma (\hat{{\lambda }}_{11} )-\sigma (\hat{{\lambda }}_{00} )}\sim \chi _{df=1}^2. \end{aligned}$$

(8)

This test statistic is used to determine whether there is a statistically significant Edgeworth price cycling effect within a given MSA over the sample period.