BUG/TST: use correct expression for nested logit second choice probab…

…ilities

- Rewrite second choice probabilities as difference between probability with first choice eliminated and a first choice probability.
- Update choice probability tangents for nested case.
- Add nested logit second choice moments test.
- Add test of first and second choice probabilities, using the evd package from R to simulate nested logit deviates.

docs/background.rst

@@ -295,9 +295,9 @@ Micro moment parts based on second choice are averages over values :math:`v_{pij
 
 Its simulated analogue is
 
-.. math:: v_p = \frac{\sum_{t \in T} \sum_{i \in I_t} \sum_{j, k \in J_t \cup \{0\}} w_{it} s_{ijt} s_{ik(-j)t} w_{d_pijkt} v_{pijkt}}{\sum_{t \in T} \sum_{i \in I_t} \sum_{j, k \in J_t \cup \{0\}} w_{it} s_{ijt} s_{ik(-j)t} w_{d_pijkt}},
+.. math:: v_p = \frac{\sum_{t \in T} \sum_{i \in I_t} \sum_{j, k \in J_t \cup \{0\}} w_{it} s_{ijt} s_{ik(-j)t} w_{d_pijkt} v_{pijkt}}{\sum_{t \in T} \sum_{i \in I_t} \sum_{j, k \in J_t \cup \{0\}} w_{it} s_{ijt} s_{ik(-j)t} w_{d_pijkt}}.
 
-in which second choice probabilities are :math:`s_{ik(-j)t} = \frac{s_{ikt}}{1 - s_{ijt}}` if :math:`k \neq j` and zero if :math:`k = j`. Covariances are defined analogously.
+Covariances are defined analogously.
 
 
 Random Coefficients Nested Logit

docs/testing.rst

@@ -13,6 +13,7 @@ The full suite of tests also requires installation of the following software:
 
 - `Artleys Knitro <https://www.artelys.com/solvers/knitro/>`_ version 10.3 or newer: testing optimization routines.
 - `MATLAB <https://www.mathworks.com/products/matlab.html>`_: comparing sparse grids with those created by the function `nwspgr <http://www.sparse-grids.de/>`_ created by Florian Heiss and Viktor Winschel, which must be included in a directory on the MATLAB path.
+- `R <https://www.r-project.org/>`_: simulating nested logit errors created by the package `evd <https://cran.r-project.org/web/packages/evd/index.html>`_ created by Alec Stephenson, which must be installed.
 
 If software is not installed, its associated tests will be skipped. Additionally, some tests that require support for extended precision will be skipped if on the platform running the tests, ``numpy.longdouble`` has the same precision as ``numpy.float64``. This tends to be the case on Windows.
 

pyblp/markets/economy_results_market.py

@@ -443,8 +443,8 @@ def safely_compute_score_denominator_contributions(
         micro_chunks = self.generate_micro_chunks(probabilities, probabilities_tangent_mapping)
         for agent_indices, probabilities_chunk, probabilities_tangent_mapping_chunk in micro_chunks:
             _, denominator_mapping_chunk, _, tangent_mapping_chunk = self.compute_micro_dataset_contributions(
-                [dataset], self.delta, probabilities_chunk, probabilities_tangent_mapping_chunk, compute_jacobians=True,
-                agent_indices=agent_indices
+                [dataset], self.delta, probabilities_chunk, probabilities_tangent_mapping_chunk, xi_jacobian,
+                compute_jacobians=True, agent_indices=agent_indices
             )
             if dataset in denominator_mapping_chunk:
                 denominator += denominator_mapping_chunk[dataset]
@@ -474,8 +474,8 @@ def safely_compute_score_numerator_contributions(
         micro_chunks = self.generate_micro_chunks(probabilities, probabilities_tangent_mapping)
         for agent_indices, probabilities_chunk, probabilities_tangent_mapping_chunk in micro_chunks:
             weights_mapping_chunk, _, tangent_mapping_chunk, _ = self.compute_micro_dataset_contributions(
-                [dataset], self.delta, probabilities_chunk, probabilities_tangent_mapping_chunk, compute_jacobians=True,
-                agent_indices=agent_indices
+                [dataset], self.delta, probabilities_chunk, probabilities_tangent_mapping_chunk, xi_jacobian,
+                compute_jacobians=True, agent_indices=agent_indices
             )
             if dataset in weights_mapping_chunk:
                 weights_chunk = weights_mapping_chunk[dataset]