Empirical commodity storage model: the challenge of matching data and theory

1. Introduction

The importance of proper empirical estimations of key parameters in agricultural commodity markets is evident in the face of the international concerns over price volatility for major food commodities.

The commodity storage model, as originally described by Gustafson (1958) and discussed in for example Scheinkman and Schechtman (1983), Williams and Wright (1991), Deaton and Laroque (1992, 1995, 1996), Carter, Rausser and Smith (2011) and Wright (2011), recognises the role of storage and provides a basis for rationalising many of the observed qualitative features of the behaviour of prices of storable commodities.

The discrete time annual storage model assumes that in each year price is formed after the realisation of a stochastic harvest, when decisions on how much to store out of the available supply are made. Price series that are appropriate for testing such a model are therefore annual price series. The evidence on the empirical validity of the annual competitive storage model is still mixed. Based on a pseudo maximum likelihood (PML) econometric procedure, Deaton and Laroque (1995, 1996) reject the practical relevance of storage arbitrage in explaining annual prices. They conclude that the price serial correlations implied by the annual models they estimate are significantly lower than those measured on the series of price indices they use. Cafiero et al. (2011, 2015) present more positive evidence for the role of storage arbitrage. Cafiero et al. (2011) estimate the storage model using the same data, model specification and PML econometric approach as Deaton and Laroque (1995, 1996), but using a much finer grid to approximate the equilibrium price function. Based on their econometric estimations, they find that, contrary to Deaton and Laroque's claim, the competitive storage model generates the high degree of price autocorrelation for five of the 12 commodities considered by Deaton and Laroque (1995, 1996), and for seven commodities when they add a marginal storage cost parameter in the model.

The series of real annual prices used in (for example) Deaton and Laroque (1992, 1995, 1996), Cafiero et al. (2011, 2015) and Cafiero, Bobenrieth and Bobenrieth (2011) have been formed by taking the simple average of prices over the calendar year, that is, from January through December, with no explicit recognition of the fact that the actual span of the marketing season may not coincide with the calendar year, therefore smoothing the most prominent feature of the price series in the storage model: its price spikes.¹Cafiero and Wright (2006) discuss this spurious smoothing problem.² This data issue is particularly delicate in this literature, in which a main focus of the discussion is in the ability of the storage model to explain observed price correlation.

The question of what annual price to use best deal with the spurious smoothing of price spikes leaves room for various choices. An annual price data set constructed as the average of daily prices over the marketing year is a candidate. Alternatively, the use of a single month per year (as in Roberts and Schlenker, 2013a, 2013b) avoids the complications of inter-seasonal anticipation of information.

In this article, we illustrate this issue using the case of US maize prices. (Maize is a major agricultural commodity, considered in the influential papers of Deaton and Laroque, 1992, 1995, 1996, and in Cafiero et al. 2011). In the northern hemisphere, where most of the maize produced is obtained, harvesting occurs from September through to November (FAO, 2006, Table 2, p. 5). We form an index of annual real prices in four different ways: as an average of market day prices over the calendar year, over the marketing year (from September through to the following August), over a quarter (a single quarter per year) and over a month (a single month per year). The first-order serial correlation of the price series data we use is actually highest when the price index is constructed by averaging the daily prices over the calendar year.

We focus on the price of maize in the United States because there is a long series of prices for US maize, consistently referring to the same commercial grade (US No. 2). This price series is widely considered as the traditional representative price for maize produced in the United States, and it is also accepted to be the world's most representative price (FAO, 2006, p. 4).

In our estimations we implement the maximum likelihood (ML) approach of Cafiero et al. (2015). This estimation procedure allows for the estimation of the structural parameters of the storage model using only price data. Cafiero et al. (2015) show that while their ML estimator imposes no additional assumptions on the model, it has small sample properties significantly superior to those of the PML estimator of Deaton and Laroque (1995, 1996).

2. The model

In this section we present the model. Although the results of this section are well known, we present them here in order for the article to be self-contained.

We model a simple competitive commodity market in which storers are risk neutral, face a constant discount rate $r>0$ and have no other costs of storage.³ Supply shocks, $ωt,$ are i.i.d. The state variable is the total available supply at time t, defined as $zt≡ωt+(1−d)xt−1,$ where $xt−1$ is storage at time $t−1,$ and $0≤d<1$ is the physical deterioration rate of stocks. Price is formed as $pt=F(ct),$ where consumption at time $t$ is given by $ct≡zt−xt.$ The inverse consumption demand, $F:R→R$⁠, is continuous, strictly decreasing, with $EF(ωt)>0$⁠, where $E$ denotes the expectation taken with respect to the random variable $ωt$⁠.⁴

where $Et$ denotes expectation conditional on information at time $t.$

Existence and uniqueness of the SREE, $f$⁠, as well as some of its properties are given by the following Theorem.

$f$is strictly decreasing whenever it is strictly positive. The equilibrium level of inventories is strictly increasing for$z>F−1(p∗)$.

Proof of the Theorem: Deaton and Laroque (1992), Theorem 1.

3. Econometric procedure

We estimate the model described in Section 2 assuming a linear inverse demand function, $F(c)=a+bc$⁠, with $b<0$⁠, and normal harvests. The discount rate $r$ is set at 5 per cent. We follow the approach of Deaton and Laroque (1992, 1995, 1996) in using only price data. We use the ML procedure introduced by Cafiero et al. (2015). We now provide a general overview of the estimation procedure; a detailed discussion is available in Cafiero et al. (2015).

The first iteration uses a guess $f<1>sp$ on the right hand side of Equation (4). Conditional on $f<1>sp$⁠, we evaluate $f<2>sp$ on an equally spaced grid of 1,000 points over a range of available supply z from −5 to 45. Iterations continue until the maximum difference between $f<n+1>sp$ and $f<n>sp$ evaluated at each grid point is less than the preset tolerance of 10⁻¹³, in absolute value.

We first use a grid-search routine to locate a candidate maximum for the log of the likelihood function, and then use a gradient-based constrained maximisation algorithm to search for a maximum in the neighbourhood of the candidate. To approximate the solution function f and the derivatives needed to calculate $Jt$ we use the MATLAB^® Spline Toolbox™. To maximise the function (Equation (3)) we first use the MATLAB^® routine fminsearch, to locate a preliminary maximiser, and then the routine fmincon, both included in the Optimization Toolbox™. The inner-loop tolerance is fixed at 10⁻¹³, while the outer-loop tolerances are fixed at 10⁻⁴ and 10⁻⁶ for fminsearch and fmincon, respectively. A grid of 64 vectors distributed uniformly on the set ${a=[0.3;3]×b=[−7;−0.3]×d=[0;0.3]}$ is fixed as the set of initial conditions for each sample. We checked that our parameter estimates are robust to the use of two alternative algorithms: fminunc and ktrlink from KNITRO^® optimisation package on MATLAB^®.

4. Data used in the econometric estimation

We use the series of maize prices obtained from Global Financial Data described as ‘Corn (US), No. 2, yellow, Chicago Board of Trade’ from January 1949 to December 2012. From the daily prices we first form monthly averages, which we divide by the January 1977–December 1979 average, consistent with the description in Pfaffenzeller, Newbold and Rayner (2007), to form a series of nominal monthly price indices. We next deflate the nominal values by dividing them by the corresponding United States Monthly Consumer Price Index reported by the US Bureau of Labor Statistics.

The deflated monthly price index (plotted in Figure 1) exhibits a downward trend over the sample period. We detrend the price index assuming a log-linear trend.⁷ The resulting series is plotted in Figure 2. In our estimations using quarterly and monthly data we take the months and quarters included in the September–December period. Calendar year averages, marketing year averages and the December prices are plotted in Figure 3.⁸

5. Results

We estimate the annual storage model using eight different annual price indices, formed by averaging prices over the calendar year, the marketing year, quarters and single months. The estimated parameters are reported in Table 1 along with the value of the maximised likelihood, and the implied threshold price, $p∗$⁠.

To evaluate the models' fit, we follow the method presented in Cafiero et al. (2011), using the estimated parameters to generate a series of 300,000 prices, and then extract from it all possible consecutive subsamples of the same length as the observed data. On each extracted subsample we measure various moments thus generating simulated distributions of implied mean, median, coefficient of variation, first- and second-order autocorrelation, skewness and kurtosis. We then identify, in each of the simulated distributions, the percentiles corresponding to the values of the corresponding moments observed in the detrended price data. Table 2 shows the percentiles. The moments measured on the price series lie, in all cases considered, within symmetric 90 percent central confidence regions.

6. What data to use: does it matter?

The slope of the consumption demand $b$ is a key parameter of the storage model, related to the sensitivity of the market price to a negative supply shock. As shown in Table 1, different ways to form the index of annual prices for the series of detrended real US maize imply large differences in $b$⁠. The slope of the consumption demand estimated using calendar year averages is $−2.71$⁠, lower in absolute value than the slope estimated using marketing year averages, $−2.86$⁠, and lower in absolute value than the slopes obtained using either quarterly or monthly data.

The different values for $b$ imply different values for the threshold price $p∗$⁠. To illustrate the implications of these differences in $p∗$⁠, we divide each price series by its corresponding $p∗$⁠, thus providing normalised price series that measure the relative distance of prices to each corresponding threshold price (Figure 4). It is striking that the two series with the steepest slope parameters (the October–December quarter and the November month series) exhibit empirical histograms of normalised prices quite distinct from the other series, with most of their probability mass well below 1 (the value of normalised price that corresponds to the threshold price $p∗$⁠).

In Figure 5 we present the histograms of stocks implied by our parameter estimates, for each price in the sample (divided by the maximum stock level for each series), for each of the price samples. In symmetry with the price histograms, both the October–December quarter and the November month normalised stocks series exhibit empirical histograms quite distinct from the histograms of the other series, with more of their probability mass near one (their maximum level of stocks, given our normalisation).

Although our estimates imply no stockouts in the sample data, the histograms for normalised prices and normalised implied stocks are coherent with the implied probabilities of stockouts in samples of the same size as the data, drawn from the simulated series of 300,000 observations: for the October–December quarter and the November month series, the implied probabilities of at least 1, 5 or 10 stockouts in the same sample periods used in our estimations are much lower than for the other price series (Table 3).

At the request of one referee, we calculate the price elasticity of consumption demand for maize implied by each of the data sets. Table 4 shows that our price elasticities are lower (in absolute value) than those implied by the estimates of Roberts and Schlenker (2013a, 2013b) for maize, comparable to the values of elasticities for maize implied by the estimates of Deaton and Laroque (1995, 1996),⁹ and Cafiero et al. (2011), within the range of values of elasticities of export demand for maize reported by Reimer, Zheng and Gehlhar (2012), and comparable to the elasticities of demand for aggregate calories from maize, rice and soybeans and wheat in Roberts and Schlenker (2013a, 2013b). Appendix A reports the procedure we use to calculate the elasticities, from the parameter estimates.

Bobenrieth, Wright and Zeng (2013) show that although quantity data might be unreliable, data on stocks-to-consumption can be a valuable complement to price, as warning of price spikes for maize, rice and wheat. Following their encouraging results, and at the suggestion of one referee to compare with data on quantities, we use our estimated model to predict stocks-to-use ratios (SURs), and compare them with the SURs constructed from maize marketing-year ending stocks and consumption from United States Department of Agriculture (USDA)/production, supply and distribution online (PSD) data, for the overlapping period 1961–2012.¹⁰ We adjust for essential stock following the procedure in Bobenrieth, Wright and Zeng (2013, pp. 5–6). In more detail, essential stocks are calculated as a fixed proportion of the consumption matching the minimum of observed SURs.¹¹ Figure 6 shows observed SURs and price-implied SURs, for calendar year averages and the December price series. It is encouraging that the dynamics of our predicted SURs follow the dynamics implied in PSD data on SURs. However, the goodness of fit is not homogeneous. Table 5 reports the root mean square error of the difference between the price-implied SURs and the observed SURs, for each of the price series considered. Price data constructed by taking the month of December offers the best fit. In contrast, model-implied SURs using calendar year price averages yield the worst fit.

7. Conclusions

We present the results of application of a ML estimator of the standard annual storage model, comparing the use of calendar and marketing year averages with quarterly and monthly averages (one quarter and one month per year, respectively), to form annual price indices. The results indicate serious differences in magnitudes of practical interest, including the location of the empirical distribution of prices relative to the cut-off price of zero stocks, the likelihood of stockouts and the fit to data on SURs.

This article explores the limits of econometric estimations of the standard commodity storage model, using annual price data. It is clear that calendar year averages are not appropriate to test the storage model, due to the averaging of two consecutive agricultural years. Although the use of marketing year averages, quarters or months can imply serious differences in magnitudes of policy interest, the theory of the storage model does not provide an answer to the question of what data set is best to represent annual prices. However, in terms of the ability of the estimated model to fit data on SURs, price data constructed by taking the month of December to represent the annual price offers the best fit. In contrast, model-implied SURs using calendar year averages yield the worst fit.

Acknowledgements

We thank, with the usual caveat, Brian Wright for helpful discussions. This study was supported by CONICYT/Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) Projects 1130257 and 1090017. Eugenio Bobenrieth's research for this article was done partially when he was a professor at Universidad de Concepción, Chile. Eugenio Bobenrieth acknowledges partial financial support from Project NS 100046 of the Iniciativa Científica Milenio of the Ministerio de Economía, Fomento y Turismo, Chile.

Conflict of interest

One of the authors, Carlo Cafiero, is a senior statistician and economist with the United Nations Food and Agriculture Organization.

References

Appendix A. Calculation of consumption demand elasticities

The estimated model is normalized at the mean and standard deviation of per capita maize production, assuming net supply shocks $N(0,1)$ and linear inverse consumption demand $F(c)=a+bc$⁠, where $c$ is interpreted as per capita consumption. For the calculation of consumption demand elasticities, we re-scale the distribution of maize production, setting its mean and standard deviation at $μ$ and $σ$⁠, respectively, and use the identification Proposition of Deaton and Laroque (1996, Proposition 1, p. 906) to correspondingly re-scale the consumption demand parameters. The re-scaled inverse consumption demand is $F(c)=a−bμσ+bσc$⁠. Therefore, the price elasticity of consumption demand, evaluated at mean real detrended price, is given by $11−a−bμσ/p¯$⁠, where $p¯$ denotes the mean of detrended real prices.

Maize production data from USDA/PSD and world population data from the US Census Bureau are available for the period 1961–2012.¹²Figure A.1 shows per capita production, which is detrended assuming linear trends, with three subsample periods: 1961–1970, 1971–2000 and 2001–2012, implying three distinct intercept and slope parameters. The values for the mean $μ$ and standard deviation $σ$ used in our calculation of consumption demand elasticities are obtained as the weighted averages of the intercepts and standard deviations of detrended per capita production, respectively, of each of these subsample periods. Figure A.2 shows detrended per capita production for the period 1961–2012.

Author notes