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--- the_complete_and_consistent_data_base_coco_for_the_national_scale [2020/02/13 08:04] – [COCO2: Data Preparation] matsz
+++ the_complete_and_consistent_data_base_coco_for_the_national_scale [2020/02/13 09:26] – [COCO2: Estimation procedure] matsz
@@ Line 552: / Line 552: @@
 ====COCO2: Estimation procedure====
+**Include file //‘coco2_def.gms’//**
+The approach to determine consumer prices is to distribute food expenditure on groups with consumption quantities given from COCO1 results such that endogenous consumer prices link endogenous expenditure with exogenous quantities. Deviations of estimated expenditure and consumer prices from their supports is penalised in an entropy framework. Estimation is done year by year, starting with the most recent year where hard data are usually available to a greater extent than for the oldest years in the database. Including consumer price changes (always relative to the previously solved year) serves to stabilise the results to some extent such that the objective does not only have supports for the consumer prices, but also for their changes. The entropy problem is solved by maximizing:
+\begin{align}
+\begin{split}
+max_t &- \sum_{m,j,k} CPS_{m,j,2}*HCOM_{m,j,k}/1000/TOFO_{m,t}*\\
+&PE_{m,j,k}*LOG(PE_{m,j,k}/PQ_k)\\
+&-\sum_{m,j,k} CPS_{m,j,2}*HCOM_{m,j,k}/1000/TOFO_{m,t}*\\
+&PED_{m,j,k}*LOG(PED_{m,j,k}/PQ_k)\\
+&-\sum_{m,FOPOS,k} EXS_{m,FOPOS,2}/TOFO_{m,t}*\\
+&PEX_{m,FOPOS,k}*LOG(PEX_{m,FOPOS,k}/PQ_k)\\
+&-\sum_{m,j,k} PFAC_{m,k}*LOG(PFAC_{m,,k}/PQ_k)*1000\\
+\end{split}
+\end{align}
+where //m// represents the region, //j// the food item with consumer price, FOPOS the food group, //t// stands for the current estimation year, t_1 for the year estimated before and k for the number of support points (=3).
+Parameters are
+| \(HCOM_{m,j,t}\) |Human consumption, result from COCO1|
+| \(UVAD_{m,j,t\_1}\)	|Consumer price from last simulation of year t+1|
+|\(CPS_{m,j,k}\)	|Support points for consumer prices |
+|\(DCPS_{m,j,k}\)	|Support points for consumer price changes|
+|\(EXS_{m,FOPOS,k}\)	|Support points for group expenditures|
+|\(TOFACS_{m,k}\)	|Support points for total food expenditure slack|
+|\(PQ_k\)		|A priori probabilities for support points|
+|\(TOFO_{m,t}\)	|Total food expenditure and entropy variables|
+|\(PE_{m,j,t}\)	|Probability of support points for consumer prices|
+|\(PED_{m,j,t}\)	|Probability of support points for consumer price changes|
+|\(CP_{m,j}\)	|Consumer prices|
+|\(DCP_{m,j}\)	|Consumer price changes|
+|\(PEX_{m,FOPOS,t}\)	|Probability of support points for group expenditure|
+|\(PFAC_{m,k}\)	|Probability of support points for food expenditure slack|
+|\(EX_{mFOPOS}\)	|Group expenditures|
+|\(TOFAC_m\)	|Food expenditure slack|
+Constraints are as follows:
+Summing up probabilities for support points
+\begin{equation}
+\sum_{k\forall_{m,j}(CP.L_{m,j}\ge 0\wedge HCOM_{m,j,i}\ge 0)} PE_{m,j,k}=1
+\end{equation}
+\begin{equation}
+\sum_{k\forall_{m,j}(DCPS_{m,j}\ge 0\wedge HCOM_{m,j,i}\ge 0)} PE_{m,j,k}=1
+\end{equation}
+\begin{equation}
+\sum_{k\forall_{m,j}(EX.L_{m,FOPOS}\ge 0)} PE_{m,FOPOS,k}=1
+\end{equation}
+\begin{equation}
+\sum_{k\forall_{m}(TOFAC.LO_m\ge TOFAC.UP_m)} PFAC_{m,k}=1
+\end{equation}
+Define consumer price changes from support points
+\begin{equation}
+DCP_{m,j} = \sum_{k\forall_{m,j}(CP.L_{m,j}\ge 0\wedge HCOM_{m,j,i}\ge 0 \wedge DCPS_{m,j,2}\ge 0)} PED_{m,j,k}*DCPS_{m,j,k}
+\end{equation}
+Of course consumer prices changes are also related to the last simulation result (which is for T+1 due to backward looping)
+\begin{equation}
+DCP_{m,j} =UVAD_{m,j,t\_1}-CP_{m,j}
+\end{equation}
+Define consumer prices from support points and probabilities
+\begin{equation}
+CP_{m,j} = \sum_{k\forall_{m,j}(CP.L_{m,j}\ge 0\wedge HCOM_{m,j,i}\ge 0)} PE_{m,j,k}*CPS_{m,j,k}
+\end{equation}
+Define group expenditure from support points and probabilities
+\begin{equation}
+EX_{m,FOPOS} = \sum_{k\forall_{m,j}(EX_{m,FOPOS}\ge 0)} PEX_{m,FOPOS,k}*EXS_{m,FOPOS,k}
+\end{equation}
+Define total expenditure slack from support points and probabilities
+\begin{equation}
+TOFAC_m=\sum_{k\forall_{m}(TOFAC.LO_m\ge TOFAC.UP_m)} PFAC_{m,k}*TOFACS_m
+\end{equation}
+Exhaustion of food expenditure may be relaxed with a slack factor different from one. However, this “last resort” to achieve feasibility in the expenditure allocation problem is limited to years and countries with precarious data and subject to strong penalties.
+\begin{equation}
+\sum_{FOPOS} EX_{m,FOPOS}=TOFO_{m,t}*TOFAC_{m,k}
+\end{equation}
+Consistency of group expenditure
+\begin{equation}
+EX_{m,FOPOS}=\sum_{j\forall_{m,FOPOS}(j\in FOPOS\wedge HCOM_{m,j} \ge 0)}CP_{m,j}*HCOM_{m,j}/1000
+\end{equation}
+For most countries the exhaustion of total expenditure is the only evident hard constraint (and even this is relaxed in problem cases). However, as the penalties for group expenditure are set high, and furthermore as the range of expenditure supports defines additional implicit hard constraints, the problem may turn out infeasible (typically solved by additional leeway). To meet the expenditure constraints the solver would tend to concentrate deviations from supports on the most important expenditure items while setting the less important items close to their supports. A more balanced distribution of deviations from supports was achieved in practice by weighting all contributons to the overall objective (except the last one for the total expenditure slack) with expected expenditure shares. The weights may be interpreted as expected expenditure shares because supports are specified in a symmetric way such that the central, second (of three) supports, which is used in the objective function, is equal to the expectation.
+**Include file //‘coco2_solve.gms’//**
+The initialisation, solving, reporting and storage is organised in the next include files with a few elements worth mentioning
+  * The initialisation tries to ensure positive consumer margins by the assignments of expected values and by specifying bounds on estimated consumer prices. The reference point for these margins is an average of EU and national prices that reflects the importance of domestic sales vs. imports.
+  * Bounds and spread of supports around expected consumer prices are set high for items without ILO style prices (say “table olives” TABO) or where the fit of available price information is questionable (e.g. cabbage prices for “OVEG”).
+  * A checking parameter (“p_checks”) permits to check the iniitalisation in case of infeasibilites. The most frequent case observed in the last years is that lower bounds on oils expenditure become binding, suggesting the need for some systematic mismatch of price and expenditure information for this group.
+====COCO2: Final completions====