_ESTIMATING FOOD PRICE INFLATION_ _39_

**p** ˆ<sup>_d_</sup> mis<sup>byfillinginentriesfor</sup><sup>**p**</sup><sup>_d_</sup> mis<sup>basedontheinformationcontainedin</sup><sup>**p**ˆ</sup><sup>_d_</sup> obs<sup>.Sincethetruetargetsare</sup> unobserved, a direct criterion is difficult to establish but the objective can be summarized as minimizing the divergence _∥_ **p** ˆ<sup>_d_=1</sup><sup>_,...D_</sup> _−_ **_π_**<sup>_d_=1</sup><sup>_,...D_</sup> _∥_ , in turn estimated with an _L_ 1-norm based metric for the prediction function **p** ˆ<sup>_d_</sup> obs<sup>that</sup> that<sup>serves</sup> generates<sup>asaproxy</sup> **p** ˆ<sup>_d_</sup> mis<sup>for,validated</sup><sup>**_π_**</sup> obs<sup>_d_.bytheout-of-samplepredictionsitmakesfortheoutlier-filtered</sup>

# _B4. Intra-month estimates_

The price-level estimates are accompanied by intra-month price range estimates represented as an Open-High-Low-Close time series.

where _P_ is the imputed price series and E∆ _α_ is the expected change in the _α_ -percentile cases. The combined results can be plotted on a candle chart. The majority of price action can then be assumed to have occurred within the body of the candles, while the wicks indicate the average price of respectively the highest 50% of intra-month prices, and the average price of the lowest %50 of intra-month prices.

The first three quantities in equation B8 are estimated by modeling the time-varying distribution of the month-on-month inflation process as an autoregressive moving average process with fractionally integrated generalized autoregressive conditional heteroskedasticity (ARMA-fiGARCH) following Baillie et al. (1996). This is a time-varying density (B9) where _μt_ is a conditional mean process defined as an ARMA( _p, q_ ) process and the conditional variance is specified as a fractionally integrated GARCH process of order ( _p, d, q_ ) and _θ_ is a vector of remaining parameters of the distribution. The conditional variance is defined as follows. First, let the standard GARCH( _p, q_ ) process be defined as (B11) with _σt_<sup>2as the conditional variance,</sup><sup>_ω_an intercept, and</sup><sup>_L_the back-shift operator with</sup><sup>_α_(</sup><sup>_L_) = �</sup><sup>_q_</sup> _j_ =1<sup>_αjLj_</sup> and _β_ ( _L_ ) =<sup>�</sup><sup>_p_</sup> _j_ =1<sup>_βjLj_.ThismodelhasanARMArepresentationofthesquaredprocess:</sup> (B12) (1 _− L_ ) _φ_ ( _L_ ) _ε_<sup>2</sup> _t_<sup>= [1</sup><sup>_−α_(</sup><sup>_L_)</sup><sup>_−β_(</sup><sup>_L_)]</sup><sup>_ε_2</sup> _t_<sup>=</sup><sup>_ω_+ [1</sup><sup>_−β_(</sup><sup>_L_)](</sup><sup>_ε_2</sup> _t_<sup>_−σ_</sup> _t_<sup>2)</sup> with _φ_ ( _L_ ) =<sup>�</sup> _j_<sup>max(</sup><sup>_p,q_)</sup><sup>_−_1</sup> = _φj Lj_ . The fractionally integrated GARCH is obtained by replacing the back-shift operator (1 _− L_ ) with a truncated fractional difference operator Ignoring the approximation error due to the truncation, at _d_ = 0 the model equals the standard GARCH in which volatility shocks decay at an exponential rate. Similarly, when _d_ = 1, the AR polynomial of the GARCH has a unit root and the model equals the integrated GARCH in which shocks persist forever. When there are level shifts in volatility process, an integrated GARCH usually better describes the data