Estimation and decomposition of food price inflation risk

Abstract

Ensuring aggregate food price stability requires a forward-looking assessment of the risk that unexpected deviations in individual food items’ inflation lead to large shocks in the aggregate food price inflation. To do so, we propose using a multivariate GARCH framework in combination with the Euler method to (1) estimate the conditional standard deviation and quantiles of the food price inflation shocks and (2) attribute the total risk to the underlying food items. For the FAO food price index, we find that even though meat inflation systematically has the highest weight in the aggregate index, cereal inflation is the main contributor to the total food price inflation risk over the period 1990–2018. The use of time series models and the Cornish-Fisher expansion make the risk characterization forward-looking and a potentially helpful tool for risk management.

Appendices

Appendix

Portfolio representation of inflation when the reference weights $w_0$ are not constant

When collecting data for empirical analysis, we observe that unlike the FAO food price indices that have constant reference weights over time, other food price indices such as the European Harmonised Index of Consumer Prices (HICP) have time-varying piecewise constant $w^{i}_{0}$, as the reference weights are updated annually.

In this case, we recommend to proceed as follows. First, we assume that the modeler knows at time $t-1$ a constant $k_{t-1}^i$ such that

$$\begin{aligned} w_{0,t}^i = k_{t-1}^i w^{i}_{0,t-1} \end{aligned}$$

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with $k_{t-1}^i = 1$ when there is no base weight change. This is a realistic assumption for a short prediction horizon such as one month; statisticians typically have an accurate view on the next period, meaning $k_{t-1}^i$ can be reasonably determined. From the above assumption, we then have

$$\begin{aligned} \begin{aligned} \frac{P_{c,t} - P_{c,t-1} }{P_{c,t-1}}&= \frac{ \sum \limits _{i=1}^{n} w^{i}_{0,t} P^{i}_{t} - \sum \limits _{i=1}^{n} w^{i}_{0,t-1} P^{i}_{t-1} }{P_{c,t-1}} \\&= \frac{ \sum \limits _{i=1}^{n} k_{t-1}^i w^{i}_{0,t-1} P^{i}_{t} - \sum \limits _{i=1}^{n} w^{i}_{0,t-1} P^{i}_{t-1} }{P_{c,t-1}} \\&= \sum _{i=1}^{n} \frac{w^{i}_{0,t-1} (k_{t-1}^i P^{i}_{t}- P^{i}_{t-1})}{P_{c,t-1}} \\&= \sum _{i=1}^{n} \bigg (\frac{w^{i}_{0,t-1} P^{i}_{t-1}}{P_{c,t-1}}\bigg ) \bigg (\frac{k_{t-1}^i P^{i}_{t}- P^{i}_{t-1}}{P^i_{t-1}} \bigg ). \\ \end{aligned} \end{aligned}$$

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As such, we obtain the relation between index inflation ${\varPi }_{c,t}$ and the component inflation $\pi ^i_t$:

$$\begin{aligned} {\varPi }_{c,t} =\sum _{i=1}^n {\tilde{w}}_{t}^{i} \, \pi _{t}^{i} \end{aligned}$$

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with ${\tilde{w}}_{t}^{i} = k_{t-1}^i w^{i}_{t}$, and ${\varPi }_{c,t}$, $\pi ^{i}_t$, and $w^i_{t}$ are as defined in Equations (2) and (3). From here, we can apply the same formulae in Sect. 2.2 to construct the multivariate model and risk measures like the case where the reference weight $w_{0}^i$ is constant.