pyblp.build_differentiation_instruments¶

pyblp.
build_differentiation_instruments
(formulation, product_data, version='local', interact=False)¶ Construct excluded differentiation instruments.
Differentiation instruments in the spirit of Gandhi and Houde (2017) are
(1)¶\[Z^\textit{Diff}(X) = [Z^\textit{Diff,Other}(X), Z^\textit{Diff,Rival}(X)],\]in which \(X\) is a matrix of product characteristics, \(Z^\textit{Diff,Other}(X)\) is a second matrix that consists of sums over functions of differences between nonrival goods, and \(Z^\textit{Diff,Rival}(X)\) is a third matrix that consists of sums over rival goods. Without optional interaction terms, all three matrices have the same dimensions.
Note
To construct simpler, firmagnostic instruments that are sums over functions of differences between all different goods, specify a constant column of firm IDs and keep only the first half of the instrument columns.
Let \(x_{jt\ell}\) be characteristic \(\ell\) in \(X\) for product \(j\) in market \(t\), which is produced by firm \(f\). That is, \(j \in \mathscr{J}_{ft}\). Then in the “local” version of \(Z^\textit{Diff}(X)\),
(2)¶\[\begin{split}Z_{jt\ell}^\textit{Diff,Other,Local}(X) = \sum_{k \in \mathscr{J}_{ft} \setminus \{j\}} 1(d_{jkt\ell} < \text{SD}_\ell), \\ Z_{jt\ell}^\textit{Diff,Rival,Local}(X) = \sum_{k \notin \mathscr{J}_{ft}} 1(d_{jkt\ell} < \text{SD}_\ell),\end{split}\]where \(d_{jkt\ell} = x_{kt\ell}  x_{jt\ell}\) is the difference between products \(j\) and \(k\) in terms of characteristic \(\ell\), \(\text{SD}_\ell\) is the standard deviation of these pairwise differences computed across all markets, and \(1(d_{jkt\ell} < \text{SD}_\ell)\) indicates that products \(j\) and \(k\) are close to each other in terms of characteristic \(\ell\).
The intuition behind this “local” version is that demand for products is often most influenced by a small number of other goods that are very similar. For the “quadratic” version of \(Z^\textit{Diff}(X)\), which uses a more continuous measure of the distance between goods,
(3)¶\[\begin{split}Z_{jtk}^\textit{Diff,Other,Quad}(X) = \sum_{k \in \mathscr{J}_{ft} \setminus\{j\}} d_{jkt\ell}^2, \\ Z_{jtk}^\textit{Diff,Rival,Quad}(X) = \sum_{k \notin \mathscr{J}_{ft}} d_{jkt\ell}^2.\end{split}\]With interaction terms, which reflect covariances between different characteristics, the summands for the “local” versions are \(1(d_{jkt\ell} < \text{SD}_\ell) \times d_{jkt\ell'}\) for all characteristics \(\ell'\), and the summands for the “quadratic” versions are \(d_{jkt\ell} \times d_{jkt\ell'}\) for all \(\ell' \geq \ell\).
Note
Usually, any supply or demand shifters are added to these excluded instruments, depending on whether they are meant to be used for demand or supplyside estimation.
 Parameters
formulation (Formulation) –
Formulation
configuration for \(X\), the matrix of product characteristics used to build excluded instruments. Variable names should correspond to fields inproduct_data
.product_data (structured arraylike) –
Each row corresponds to a product. Markets can have differing numbers of products. The following fields are required:
market_ids : (object)  IDs that associate products with markets.
firm_ids : (object)  IDs that associate products with firms.
Along with
market_ids
andfirm_ids
, the names of any additional fields can be used as variables informulation
.version (str, optional) –
The version of differentiation instruments to construct:
interact (bool, optional) – Whether to include interaction terms between different product characteristics, which can help capture covariances between product characteristics.
 Returns
Excluded differentiation instruments, \(Z^\textit{Diff}(X)\).
 Return type
ndarray
Examples