# pyblp.OptimalInstrumentResults.to_problem¶

OptimalInstrumentResults.to_problem(supply_shifter_formulation=None, demand_shifter_formulation=None)

Re-create the problem with estimated feasible optimal instruments.

The re-created problem will be exactly the same, except that instruments will be replaced with estimated feasible optimal instruments.

Note

Most of the explanation here is only important if a supply side was estimated.

The optimal excluded demand-side instruments consist of the following:

1. Estimated optimal demand-side instruments for $$\theta$$, $$Z_D^\text{opt}$$, excluding columns of instruments for any parameters on exogenous linear characteristics that were not concentrated out, but rather included in $$\theta$$ by Problem.solve().

2. Optimal instruments for any linear demand-side parameters on endogenous product characteristics, $$\alpha$$, which were concentrated out and hence not included in $$\theta$$. These optimal instruments are simply an integral of the endogenous product characteristics, $$X_1^\text{en}$$, over the joint density of $$\xi$$ and $$\omega$$. It is only possible to concentrate out $$\alpha$$ when there isn’t a supply side, so the approximation of these optimal instruments is simply $$X_1^\text{en}$$ evaluated at the constant vector of expected prices, $$E[p \mid Z]$$, specified in ProblemResults.compute_optimal_instruments().

3. If a supply side was estimated, any supply shifters, which are by default formulated by OptimalInstrumentResults.supply_shifter_formulation: all characteristics in $$X_3^\text{ex}$$ not in $$X_1^\text{ex}$$.

Similarly, if a supply side was estimated, the optimal excluded supply-side instruments consist of the following:

1. Estimated optimal supply-side instruments for $$\theta$$, $$Z_S^\text{opt}$$, excluding columns of instruments for any parameters on exogenous linear characteristics that were not concentrated out, but rather included in $$\theta$$ by Problem.solve().

2. Optimal instruments for any linear supply-side parameters on endogenous product characteristics, $$\gamma^\text{en}$$, which were concentrated out an hence not included in $$\theta$$. This is only relevant if shares were included in the formulation for $$X_3$$ in Problem. The corresponding optimal instruments are simply an integral of the endogenous product characteristics, $$X_3^\text{en}$$, over the joint density of $$\xi$$ and $$\omega$$. The approximation of these optimal instruments is simply $$X_3^\text{en}$$ evaluated at the market shares that arise under the constant vector of expected prices, $$E[p \mid Z]$$, specified in ProblemResults.compute_optimal_instruments().

1. If a supply side was estimated, any demand shifters, which are by default formulated by OptimalInstrumentResults.demand_shifter_formulation: all characteristics in $$X_1^\text{ex}$$ not in $$X_3^\text{ex}$$.

As usual, the excluded demand-side instruments will be supplemented with $$X_1^\text{ex}$$ and the excluded supply-side instruments will be supplemented with $$X_3^\text{ex}$$. The same fixed effects configured in Problem will be absorbed.

Warning

If a supply side was estimated, the addition of supply- and demand-shifters may create collinearity issues. Make sure to check that shifters and other product characteristics are not collinear.

Parameters
Returns

OptimalInstrumentProblem, which is a Problem updated to use the estimated optimal instruments.

Return type

OptimalInstrumentProblem

Examples