# Estimating the Competitive Storage Model with Stochastic Trends in Commodity Prices

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## Abstract

**:**

## 1. Introduction

## 2. Storage Model

#### 2.1. State-Space Formulation with a Stochastic Trend

#### 2.2. Stochastic Trends and Storage Decisions

## 3. Statistical Inference

#### 3.1. Preliminaries and Prior Selection

#### 3.2. Bayesian Inference Using Particle Markov Chain Monte Carlo

#### 3.3. State Prediction for Diagnostics and Marginal Likelihood

## 4. Ability to Isolate the Trend and Storage Model Component

## 5. Empirical Application

#### 5.1. Estimation Results for the Storage SSM with a Stochastic Trend

#### 5.2. Model Comparisons

#### 5.2.1. Alternative Models

#### 5.2.2. Marginal Likelihood Model Comparisons and Diagnostics Checks

#### 5.2.3. Estimates for Annual Storage Costs and Price Elasticity of Demand

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Numerical Solution of the Price Function

- Select an initial guess, e.g., ${\mathcal{S}}_{1}=\{{\hat{x}}_{1}^{*},{\hat{x}}_{1}^{**},{s}_{1}(x)\}=\{0,C,{s}_{1}(x)\}$, where ${s}_{1}(x)$ is the linear function such that $s(0)=0$, $s(C)=C$. Set $n=1$;
- Update the left kink point ${\hat{x}}_{n+1}^{*}$ according to:$${\hat{x}}_{n+1}^{*}=D\left(\beta \int {f}_{{\mathcal{S}}_{n}}(z)\varphi (z)dz\right);$$
- Update the right kink point ${\hat{x}}_{n+1}^{**}$ according to:$${\hat{x}}_{n+1}^{**}=D\left(\beta \int {f}_{{\mathcal{S}}_{n}}((1-\delta )C+z)\varphi (z)dz\right)+C.;$$
- Update the grid $\{{x}_{n+1}^{(j)}\}$ to be on $[{\hat{x}}_{n+1}^{*},{\hat{x}}_{n+1}^{**}];$
- For each grid point j, find the update ${s}_{n+1}({x}_{n+1}^{(j)})$ as the solution in s to:$$s={x}_{n+1}^{(j)}-D\left(\beta \int {f}_{{\mathcal{S}}_{n}}((1-\delta )s+z)\varphi (z)dz\right),$$
- Until convergence, set $n\leftarrow n+1$ and go back to Step 2.

## Appendix B. Computational Details for Statistical Inference

#### Appendix B.1. Particle Filter

- For period $t=1$ (initialization): Sample ${x}_{1}^{k}\sim {\pi}_{\theta}({x}_{1})=\mathcal{U}({x}_{1}|-2,C+2)$ for $k=1,\dots ,N$, and set the corresponding (normalized) IS weights to ${W}_{1}^{k}=1/N$. For initialization, set ${\overline{x}}_{1}^{k}={x}_{1}^{k}$;
- For periods $t=2,\dots ,T$: Sample ${x}_{t}^{k}\sim {\pi}_{\theta}({x}_{t}|{\overline{x}}_{t-1}^{k})=\mathcal{N}\left({x}_{t}|(1-\delta )\sigma ({\overline{x}}_{t-1}^{k}),1\right)$ for $k=1,\dots ,N$, and set ${x}_{1:t}^{k}=({x}_{t}^{k},{\overline{x}}_{1:t-1}^{k})$. Compute the IS weights as:$$\begin{array}{c}\hfill {w}_{t}^{k}={W}_{t-1}^{k}{\pi}_{\theta}({p}_{t}|{p}_{t-1},{x}_{t-1:t}^{k}),\end{array}$$

#### Appendix B.2. Marginal Likelihood

#### Appendix B.3. Residuals for the Deterministic Trend Models

## Appendix C. Additional Results

C | Natgas | Coffee | Cotton | Aluminum |
---|---|---|---|---|

10 | 164.13 | 420.07 | 522.80 | 545.36 |

15 | 163.89 | 425.93 | 530.93 | 549.77 |

20 | 163.68 | 429.32 | 536.81 | 553.16 |

25 | 163.57 | 431.10 | 539.89 | 555.41 |

**Table A2.**Estimates for the storage model parameters under the storage model with deterministic trends.

Natgas | Coffee | Cotton | Aluminum | |||
---|---|---|---|---|---|---|

$\delta $ | Linear | Post. mean | 0.0110 | 0.0041 | 0.0009 | 0.0046 |

Post. std. | 0.0051 | 0.0049 | 0.0007 | 0.0026 | ||

RCS3 | Post. mean | 0.0073 | 0.0023 | 0.0009 | 0.0037 | |

Post. std. | 0.0040 | 0.0014 | 0.0007 | 0.0021 | ||

RCS7 | Post. mean | 0.0087 | 0.0055 | 0.0034 | 0.0015 | |

Post. std. | 0.0080 | 0.0021 | 0.0007 | 0.0010 | ||

b | Linear | Post. mean | 4.85 | 4.18 | 3.56 | 2.09 |

Post. std. | 0.200 | 0.282 | 0.153 | 0.106 | ||

RCS3 | Post. mean | 3.18 | 4.37 | 3.62 | 3.25 | |

Post. std. | 0.164 | 0.190 | 0.154 | 0.172 | ||

RCS7 | Post. mean | 5.99 | 4.04 | 3.64 | 2.11 | |

Post. std. | 0.920 | 0.316 | 0.163 | 0.095 |

## Notes

1 | The knots for the RCS3 specification are located at the 25%, 50%, and 75% quantiles of the time index and for the RCS7 at the 12.5%, 25%, 37.5%, 50%, 67.5%, 75%, and 87.5% quantiles. |

2 | The smoothed mean $E({k}_{t}|{p}_{1:T})={p}_{t}-E(logf({x}_{t})|{p}_{1:T})$ was computed using the particle smoothing algorithm, which adds to the BPF, as outlined in Appendix B.1, a backward sampling step (Doucet and Johansen 2009, Section 5). |

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**Figure 1.**Filtered price components for simulated data. Upper panel: time series plot of the simulated log price $log{p}_{t}$ (blue line), the actual stochastic trend component ${k}_{t}$ (green line), and its estimated filtered mean $E({k}_{t}|{p}_{1:t})$ (red line). Lower panel: time series plot of the actual storage model component $logf({x}_{t})$ (green line) and its estimated filtered mean $E(logf({x}_{t})|{p}_{1:t})$ (red line). The gray shaded areas indicate the 95% credible intervals under the filtering densities for ${k}_{t}$ and $logf({x}_{t})$, and the black horizontal lines in the lower panel mark the boundaries of the storage regimes. The prices were simulated using parameters set at $(v,\delta ,b)=(0.097,\phantom{\rule{0.166667em}{0ex}}0.011,\phantom{\rule{0.166667em}{0ex}}0.42)$ and $C=10$. The posterior mean of the parameters obtained by fitting the model to the simulated price data are $(\hat{v},\hat{\delta},\hat{b})=(0.101,\phantom{\rule{0.166667em}{0ex}}0.013,\phantom{\rule{0.166667em}{0ex}}0.436)$.

**Figure 2.**Commodity prices and filtered price components. Upper panels: time series plot of the log price $log{p}_{t}$ (blue line) and the estimated filtered mean of the stochastic trend component $E({k}_{t}|{p}_{1:t})$ (red line). Lower panels: time series plot of the estimated filtered mean of the storage model component $E(logf({x}_{t})|{p}_{1:t})$ (red line). The gray shaded areas indicate the 95% credible intervals under the filtering densities for ${k}_{t}$ and $logf({x}_{t})$, and the black horizontal lines in the lower panels mark the boundaries of the storage regimes. The parameters are set to their posterior mean as given in Table 1.

**Figure 3.**Fit stochastic and deterministic trends. Smoothed stochastic trend (purple solid line), linear trend (red dashed line), RCS3 trend (blue plus sign), and RCS7 trend (green square).

**Table 1.**MCMC posterior analysis of the storage SSM with a stochastic trend. The reported numbers are the posterior mean, posterior standard deviation, and Effective Sample Size (ESS) for the parameters. The results are based on 12,000 PMMH iterations, discarding the first 2000 burn-in iterations.

Natgas | Coffee | Cotton | Aluminum | ||
---|---|---|---|---|---|

v | Post. mean | 0.0972 | 0.0574 | 0.0443 | 0.0422 |

Post. std. | 0.0083 | 0.0034 | 0.0023 | 0.0022 | |

ESS | 634 | 627 | 914 | 636 | |

$\delta $ | Post. mean | 0.0112 | 0.0025 | 0.0019 | 0.0014 |

Post. std. | 0.0048 | 0.0014 | 0.0010 | 0.0009 | |

ESS | 580 | 544 | 708 | 770 | |

b | Post. mean | 0.4196 | 1.4847 | 1.2969 | 1.0283 |

Post. std. | 0.2594 | 0.4242 | 0.3080 | 0.3058 | |

ESS | 515 | 741 | 1188 | 786 | |

C | 10 | 25 | 25 | 25 |

**Table 2.**Log-marginal likelihood values of the storage SSM and the alternative models. The log-Bayes factor of the storage SSM relative to the alternative models is in parentheses, and the marginal likelihood preferred values for C are in brackets.

Natgas | Coffee | Cotton | Aluminum | |
---|---|---|---|---|

Storage SSM | 164.13 | 431.10 | 539.89 | 555.41 |

[10] | [25] | [25] | [25] | |

LGLL SSM | 146.76 | 404.09 | 510.28 | 540.88 |

(17.37) | (27.01) | (29.61) | (14.53) | |

Linear trend | 138.36 | 386.70 | 507.51 | 546.59 |

[20] | [25] | [25] | [20] | |

(25.77) | (44.40) | (32.38) | (8.82) | |

RCS3 trend | 144.68 | 405.50 | 518.41 | 531.56 |

[15] | [25] | [25] | [25] | |

(19.45) | (25.60) | (21.48) | (23.85) | |

RCS7 trend | 137.58 | 401.78 | 519.11 | 540.74 |

[25] | [25] | [25] | [20] | |

(26.55) | (29.32) | (20.78) | (14.67) |

**Table 3.**Diagnostics on the PIT and Pearson residuals. Skewness, kurtosis, and p-value of the Jarque–Bera test (JB) for the PIT residuals. Lag-1 autocorrelation (${\rho}_{1}$) and p-value of the Ljung–Box test (LB) for the Pearson residuals and their squared values, including 12 lags.

Skew (ξ_{t}) | Kurt (ξ_{t}) | JB (ξ_{t}) | ρ_{1} (η_{t}) | LB_{12} (η_{t}) | LB_{12} (${\mathit{\eta}}_{\mathit{t}}^{\mathbf{2}}$) | |
---|---|---|---|---|---|---|

Storage SSM | ||||||

Natgas | 0.053 | 3.069 | 0.915 | 0.075 | 0.027 | 0.297 |

Coffee | 0.452 | 4.050 | <0.001 | 0.224 | <0.001 | 0.452 |

Cotton | −0.024 | 3.669 | 0.035 | 0.452 | <0.001 | <0.001 |

Aluminum | −0.090 | 3.105 | 0.723 | 0.245 | <0.001 | <0.001 |

LGLL SSM | ||||||

Natgas | 0.033 | 4.298 | <0.001 | 0.084 | 0.055 | 0.452 |

Coffee | 0.801 | 7.679 | <0.001 | 0.257 | <0.001 | <0.001 |

Cotton | −0.230 | 6.325 | <0.001 | 0.502 | <0.001 | <0.001 |

Aluminum | −0.381 | 4.652 | <0.001 | 0.268 | <0.001 | <0.001 |

Linear trend | ||||||

Natgas | 0.181 | 5.295 | <0.001 | 0.023 | 0.206 | 0.276 |

Coffee | −0.680 | 5.736 | <0.001 | 0.246 | <0.001 | 0.095 |

Cotton | −0.073 | 5.938 | <0.001 | 0.481 | <0.001 | <0.001 |

Aluminum | 0.516 | 6.389 | <0.001 | 0.247 | <0.001 | <0.001 |

RCS3 trend | ||||||

Natgas | −0.109 | 4.022 | 0.003 | 0.014 | 0.002 | 0.031 |

Coffee | −0.498 | 4.681 | <0.001 | 0.182 | <0.001 | 0.004 |

Cotton | 0.143 | 4.832 | <0.001 | 0.449 | <0.001 | <0.001 |

Aluminum | 0.391 | 5.246 | <0.001 | 0.261 | <0.001 | <0.001 |

RCS7 trend | ||||||

Natgas | −0.093 | 4.343 | <0.001 | −0.010 | <0.001 | 0.004 |

Coffee | −0.666 | 5.939 | <0.001 | 0.195 | <0.001 | 0.002 |

Cotton | 0.131 | 4.319 | <0.001 | 0.462 | <0.001 | <0.001 |

Aluminum | −0.039 | 3.332 | 0.419 | 0.198 | <0.001 | <0.001 |

**Table 4.**Estimates for the annual storage costs (net of interest costs) in percent of the average price and price elasticities of demand. Numbers in bold indicate the figures of the deterministic trend model that is preferred by the marginal likelihood (see Table 2).

Natgas | Coffee | Cotton | Aluminum | |||||
---|---|---|---|---|---|---|---|---|

Costs | Elast. | Costs | Elast. | Costs | Elast. | Costs | Elast. | |

Storage SSM | 12.6 | −1.03 | 2.9 | −0.07 | 2.2 | −0.07 | 1.7 | −0.10 |

linear trend | 12.5 | −0.02 | 4.8 | −0.02 | 1.1 | −0.02 | 5.4 | −0.05 |

RCS3 trend | 8.4 | −0.04 | 2.7 | −0.02 | 1.1 | −0.02 | 4.4 | −0.02 |

RCS7 trend | 10.0 | −0.01 | 6.4 | −0.02 | 4.0 | −0.02 | 1.8 | −0.04 |

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## Share and Cite

**MDPI and ACS Style**

Osmundsen, K.K.; Kleppe, T.S.; Liesenfeld, R.; Oglend, A. Estimating the Competitive Storage Model with Stochastic Trends in Commodity Prices. *Econometrics* **2021**, *9*, 40.
https://doi.org/10.3390/econometrics9040040

**AMA Style**

Osmundsen KK, Kleppe TS, Liesenfeld R, Oglend A. Estimating the Competitive Storage Model with Stochastic Trends in Commodity Prices. *Econometrics*. 2021; 9(4):40.
https://doi.org/10.3390/econometrics9040040

**Chicago/Turabian Style**

Osmundsen, Kjartan Kloster, Tore Selland Kleppe, Roman Liesenfeld, and Atle Oglend. 2021. "Estimating the Competitive Storage Model with Stochastic Trends in Commodity Prices" *Econometrics* 9, no. 4: 40.
https://doi.org/10.3390/econometrics9040040