_ANDR EE_<sup>_ ́_</sup> _DECEMBER - 2021_ _38_ beyond what seemed likely from initially observed data alone.

# _B2. Ensemble_

Let _Q_ be the quantity of interest, simple linear difference calculation using fixed-weight averages of _P_ . The ultimate goal of the multiple imputation strategy will be to obtain an estimate _Q_<sup>ˆ</sup> (B3)

**P** denoting the _true_ price index population. Since **P** is unknown, _Q_ is unknown. The amount of uncertainty in the estimate _Q_<sup>ˆ</sup> thus depends on what is known about **P** mis. Since we can only recreate it with uncertainty based on information in **P** obs, the idea is to summarize a distribution of _Q_ under varying estimates of **P** mis. In other words, the possible functions _Q_ given what has been observed in **P** obs have a posterior distribution _P_ ( _Q|_ **P** obs) which in turn can be decomposed into two parts

In this, _P_ ( _Q|_ **P** obs _,_ **P** mis) is the posterior distribution of inflation in the hypothetically complete price data and _P_ ( **P** mis _|_ **P** obs) is the posterior distribution of the missing price data given the observed price data. Suppose that _P_ ( **P** mis _|_ **P** obs) is used to draw various likely price data sets for **P** mis, denoted as **P**<sup> ̇</sup> mis. Then, associated inflation _Q_ can be calculated from ( **P**<sup> ̇</sup> mis _,_ **P** obs). By repeating this process multiple times, one can obtain the posterior distribution for _Q_ and equation B4 shows that _Q_ than equals the expectation over all draws:

## (B5)

which suggests that when _Q_<sup>ˆ(</sup><sup>_m_)</sup> is the estimated model using the _m_ -th imputation, then the combined model using all the imputations is equal to the ensemble estimate (B6) where the second equality is due to the linearity in the simple linear difference formulation of _Q_ .

# _B3. Performance against the unobserved prediction objective_

Let _p_<sup>_d_</sup> _it_<sup>beapossibleobservedpricequoteforfooditem</sup><sup>_d∈_(1</sup><sup>_, . . . , D_)thatmaybeobservedat</sup> location _i ∈_ (1 _, . . . , N_ ) and time _t ∈_ (1 _, . . . , T_ ). Let **p**<sup>_d_</sup> _∋ p_<sup>_d_</sup> _it_<sup>bethepossiblyincompletevectorofprices</sup> for food item _d_ generated by stacking all _N × T_ entries. _P_ = **p**<sup>1</sup> _, . . . ,_ **p**<sup>_D_</sup> is the _NT × D_ matrix that collects all the possible price points and consists of observed and missing parts, _P_ obs = ( **p**<sup>1</sup> obs<sup>_, . . . ,_</sup><sup>**p**</sup><sup>_D_</sup> obs<sup>)</sup> and _P_ mis = **p**<sup>1</sup> mis<sup>_, . . . ,_</sup><sup>**p**</sup><sup>_D_</sup> mis<sup>.Supposethetruepricesaregeneratedbysomeprocessthatisonlypartially</sup> observed and with error. In particular, for every individual price signal _d ∈_ (1 _, . . . , D_ ), the focus is on a _T_ -period sequence _{_ **_π_** _t_<sup>_d}T_</sup> _t_ =1<sup>that is a subset of the realized path of the stochastic sequence</sup><sup>**_π_**</sup><sup>_d_:=</sup><sup>_{_</sup><sup>**_π_**</sup> _t_<sup>_d}t∈_Z.</sup> Suppose that _{_ **_π_** _t}_<sup>_T_</sup> _t_ =1<sup>isunobserved,butthatthereisanobservedsequence</sup> In this equation, _M_<sup>_d_</sup> is a function that describes how price data of commodity _d_ is measured. It may produce measurement error in the form of additive outliers as well as data gaps. The important distinction is thus that true prices **_π_**<sup>_d_</sup> are assumed to exist but it is only possible to partially observe **p**<sup>_d_</sup> obs<sup>bysurveying,inadvertentlyintroducingerrorsandmissingentries.Thesequence</sup><sup>**p**</sup><sup>_d_= (</sup><sup>**p**</sup><sup>_d_</sup> obs<sup>_,_</sup><sup>**p**</sup> mis<sup>_d_)</sup> can thus be split in missing values **p**<sup>_d_</sup> mis<sup>andpossiblycontaminatedobservations</sup><sup>**p**</sup><sup>_d_</sup> obs<sup>.Thetwo-foldaim</sup> is to proxy **_π_** obs<sup>_d_withanestimate</sup><sup>**p**ˆ</sup><sup>_d_</sup> obs<sup>byfiltering</sup><sup>**p**</sup><sup>_d_</sup> obs<sup>fromoutliers,andproxy</sup><sup>**_π_**</sup> mis<sup>_d_byestimating</sup>