Download the Jupyter Notebook for this section: post_estimation.ipynb
PostEstimation Tutorial¶
[1]:
%matplotlib inline
import pyblp
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
pyblp.options.digits = 2
pyblp.options.verbose = False
pyblp.__version__
[1]:
'0.11.0'
This tutorial covers several features of pyblp
which are available after estimation including:
Calculating elasticities and diversion ratios.
Calculating marginal costs and markups.
Computing the effects of mergers: prices, shares, and HHI.
Using a parametric bootstrap to estimate standard errors.
Estimating optimal instruments.
Problem Results¶
As in the fake cereal tutorial, we’ll first solve the fake cereal problem from Nevo (2000). We load the fake data and estimate the model as in the previous tutorial. We output the setup of the model to confirm we have correctly configured the Problem
[2]:
product_data = pd.read_csv(pyblp.data.NEVO_PRODUCTS_LOCATION)
agent_data = pd.read_csv(pyblp.data.NEVO_AGENTS_LOCATION)
product_formulations = (
pyblp.Formulation('0 + prices', absorb='C(product_ids)'),
pyblp.Formulation('1 + prices + sugar + mushy')
)
agent_formulation = pyblp.Formulation('0 + income + income_squared + age + child')
problem = pyblp.Problem(product_formulations, product_data, agent_formulation, agent_data)
problem
[2]:
Dimensions:
=================================================
T N F I K1 K2 D MD ED
        
94 2256 5 1880 1 4 4 20 1
=================================================
Formulations:
===================================================================
Column Indices: 0 1 2 3
    
X1: Linear Characteristics prices
X2: Nonlinear Characteristics 1 prices sugar mushy
d: Demographics income income_squared age child
===================================================================
We’ll solve the problem in the same way as before. The Problem.solve method returns a ProblemResults class, which displays basic estimation results. The results that are displayed are simply formatted information extracted from various class attributes such as ProblemResults.sigma and ProblemResults.sigma_se.
[3]:
initial_sigma = np.diag([0.3302, 2.4526, 0.0163, 0.2441])
initial_pi = [
[ 5.4819, 0, 0.2037, 0 ],
[15.8935, 1.2000, 0, 2.6342],
[0.2506, 0, 0.0511, 0 ],
[ 1.2650, 0, 0.8091, 0 ]
]
results = problem.solve(
initial_sigma,
initial_pi,
optimization=pyblp.Optimization('bfgs', {'gtol': 1e5}),
method='1s'
)
results
[3]:
Problem Results Summary:
=======================================================================================================
GMM Objective Gradient Hessian Hessian Clipped Weighting Matrix Covariance Matrix
Step Value Norm Min Eigenvalue Max Eigenvalue Shares Condition Number Condition Number
       
1 +4.6E+00 +6.9E06 +2.7E05 +1.6E+04 0 +6.9E+07 +8.4E+08
=======================================================================================================
Cumulative Statistics:
===========================================================================
Computation Optimizer Optimization Objective Fixed Point Contraction
Time Converged Iterations Evaluations Iterations Evaluations
     
00:01:36 Yes 51 57 45483 141195
===========================================================================
Nonlinear Coefficient Estimates (Robust SEs in Parentheses):
=====================================================================================================================
Sigma: 1 prices sugar mushy  Pi: income income_squared age child
          
1 +5.6E01  1 +2.3E+00 +0.0E+00 +1.3E+00 +0.0E+00
(+1.6E01)  (+1.2E+00) (+6.3E01)

prices +0.0E+00 +3.3E+00  prices +5.9E+02 3.0E+01 +0.0E+00 +1.1E+01
(+1.3E+00)  (+2.7E+02) (+1.4E+01) (+4.1E+00)

sugar +0.0E+00 +0.0E+00 5.8E03  sugar 3.8E01 +0.0E+00 +5.2E02 +0.0E+00
(+1.4E02)  (+1.2E01) (+2.6E02)

mushy +0.0E+00 +0.0E+00 +0.0E+00 +9.3E02  mushy +7.5E01 +0.0E+00 1.4E+00 +0.0E+00
(+1.9E01)  (+8.0E01) (+6.7E01)
=====================================================================================================================
Beta Estimates (Robust SEs in Parentheses):
==========
prices

6.3E+01
(+1.5E+01)
==========
Additional postestimation outputs can be computed with ProblemResults methods.
Elasticities and Diversion Ratios¶
We can estimate elasticities, \(\varepsilon\), and diversion ratios, \(\mathscr{D}\), with ProblemResults.compute_elasticities and ProblemResults.compute_diversion_ratios.
As a reminder, elasticities in each market are
Diversion ratios are
Following Conlon and Mortimer (2018), we report the diversion to the outside good \(D_{j0}\) on the diagonal instead of \(D_{jj}=1\).
[4]:
elasticities = results.compute_elasticities()
diversions = results.compute_diversion_ratios()
Postestimation outputs are computed for each market and stacked. We’ll use matplotlib functions to display the matrices associated with a single market.
[5]:
single_market = product_data['market_ids'] == 'C01Q1'
plt.colorbar(plt.matshow(elasticities[single_market]));
[6]:
plt.colorbar(plt.matshow(diversions[single_market]));
The diagonal of the first image consists of own elasticities and the diagonal of the second image consists of diversion ratios to the outside good. As one might expect, own price elasticities are large and negative while crossprice elasticities are positive but much smaller.
Elasticities and diversion ratios can be computed with respect to variables other than prices
with the name
argument of ProblemResults.compute_elasticities and ProblemResults.compute_diversion_ratios. Additionally,
ProblemResults.compute_long_run_diversion_ratios can be used to used to understand substitution when products are eliminated from the choice set.
The convenience methods ProblemResults.extract_diagonals and ProblemResults.extract_diagonal_means can be used to extract information about own elasticities of demand from elasticity matrices.
[7]:
means = results.extract_diagonal_means(elasticities)
An alternative to summarizing full elasticity matrices is to use ProblemResults.compute_aggregate_elasticities to estimate aggregate elasticities of demand, \(E\), in each market, which reflect the change in total sales under a proportional sales tax of some factor.
[8]:
aggregates = results.compute_aggregate_elasticities(factor=0.1)
Since demand for an entire product category is generally less elastic than the average elasticity of individual products, mean own elasticities are generally larger in magnitude than aggregate elasticities.
[9]:
plt.hist(
[means.flatten(), aggregates.flatten()],
color=['red', 'blue'],
bins=50
);
plt.legend(['Mean Own Elasticities', 'Aggregate Elasticities']);
Marginal Costs and Markups¶
To compute marginal costs, \(c\), the product_data
passed to Problem must have had a firm_ids
field. Since we included firm IDs when configuring the problem, we can use ProblemResults.compute_costs.
[10]:
costs = results.compute_costs()
plt.hist(costs, bins=50);
plt.legend(["Marginal Costs"]);
Other methods that compute supplyside outputs often compute marginal costs themselves. For example, ProblemResults.compute_markups will compute marginal costs when estimating markups, \(\mathscr{M}\), but computation can be sped up if we just use our precomputed values.
[11]:
markups = results.compute_markups(costs=costs)
plt.hist(markups, bins=50);
plt.legend(["Markups"]);
Mergers¶
Before computing postmerger outputs, we’ll supplement our premerger markups with some other outputs. We’ll compute HerfindahlHirschman Indices, \(\text{HHI}\), with ProblemResults.compute_hhi; populationnormalized gross expected profits, \(\pi\), with ProblemResults.compute_profits; and populationnormalized consumer surpluses, \(\text{CS}\), with ProblemResults.compute_consumer_surpluses.
[12]:
hhi = results.compute_hhi()
profits = results.compute_profits(costs=costs)
cs = results.compute_consumer_surpluses()
To compute postmerger outputs, we’ll create a new set of firm IDs that represent a merger of firms 2
and 1
.
[13]:
product_data['merger_ids'] = product_data['firm_ids'].replace(2, 1)
We can use ProblemResults.compute_approximate_prices or ProblemResults.compute_prices to estimate postmerger prices. The first method, which is discussed, for example, in Nevo (1997), assumes that shares and their price derivatives are unaffected by the merger. The second method does not make these assumptions and iterates over the \(\zeta\)markup equation from Morrow and Skerlos (2011) to solve the full system of \(J_t\) equations and \(J_t\) unknowns in each market \(t\). We’ll use the latter, since it is fast enough for this example problem.
[14]:
changed_prices = results.compute_prices(
firm_ids=product_data['merger_ids'],
costs=costs
)
We’ll compute postmerger shares with ProblemResults.compute_shares.
[15]:
changed_shares = results.compute_shares(changed_prices)
Postmerger prices and shares are used to compute other postmerger outputs. For example, \(\text{HHI}\) increases.
[16]:
changed_hhi = results.compute_hhi(
firm_ids=product_data['merger_ids'],
shares=changed_shares
)
plt.hist(changed_hhi  hhi, bins=50);
plt.legend(["HHI Changes"]);
Markups, \(\mathscr{M}\), and profits, \(\pi\), generally increase as well.
[17]:
changed_markups = results.compute_markups(changed_prices, costs)
plt.hist(changed_markups  markups, bins=50);
plt.legend(["Markup Changes"]);
[18]:
changed_profits = results.compute_profits(changed_prices, changed_shares, costs)
plt.hist(changed_profits  profits, bins=50);
plt.legend(["Profit Changes"]);
On the other hand, consumer surpluses, \(\text{CS}\), generally decrease.
[19]:
changed_cs = results.compute_consumer_surpluses(changed_prices)
plt.hist(changed_cs  cs, bins=50);
plt.legend(["Consumer Surplus Changes"]);
Bootstrapping Results¶
Postestimation outputs can be informative, but they don’t mean much without a sense sampletosample variability. One way to estimate confidence intervals for postestimation outputs is with a standard bootstrap procedure:
Construct a large number of bootstrap samples by sampling with replacement from the original product data.
Initialize and solve a Problem for each bootstrap sample.
Compute the desired postestimation output for each bootstrapped ProblemResults and from the resulting empirical distribution, construct boostrap confidence intervals.
Although appealing because of its simplicity, the computational resources required for this procedure are often prohibitively expensive. Furthermore, human oversight of the optimization routine is often required to determine whether the routine ran into any problems and if it successfully converged. Human oversight of estimation for each bootstrapped problem is usually not feasible.
A more reasonable alternative is a parametric bootstrap procedure:
Construct a large number of draws from the estimated joint distribution of parameters.
Compute the implied mean utility, \(\delta\), and shares, \(s\), for each draw. If a supply side was estimated, also computed the implied marginal costs, \(c\), and prices, \(p\).
Compute the desired postestimation output under each of these parametric bootstrap samples. Again, from the resulting empirical distribution, construct boostrap confidence intervals.
Compared to the standard bootstrap procedure, the parametric bootstrap requires far fewer computational resources, and is simple enough to not require human oversight of each bootstrap iteration. The primary complication to this procedure is that when supply is estimated, equilibrium prices and shares need to be computed for each parametric bootstrap sample by iterating over the \(\zeta\)markup equation from Morrow and Skerlos (2011). Although nontrivial, this fixed point iteration problem is much less demanding than the full optimization routine required to solve the BLP problem from the start.
An empirical distribution of results computed according to this parametric bootstrap procedure can be created with the ProblemResults.bootstrap method, which returns a BootstrappedResults class that can be used just like ProblemResults to compute various postestimation outputs. The difference is that BootstrappedResults methods return arrays with an extra first dimension, along which bootstrapped results are stacked.
We’ll construct 90% parametric bootstrap confidence intervals for estimated mean own elasticities in each market of the fake cereal problem. Usually, bootstrapped confidence intervals should be based on thousands of draws, but we’ll only use a few for the sake of speed in this example.
[20]:
bootstrapped_results = results.bootstrap(draws=100, seed=0)
bootstrapped_results
[20]:
Bootstrapped Results Summary:
======================
Computation Bootstrap
Time Draws
 
00:00:24 100
======================
[21]:
bounds = np.percentile(
bootstrapped_results.extract_diagonal_means(
bootstrapped_results.compute_elasticities()
),
q=[10, 90],
axis=0
)
table = pd.DataFrame(index=problem.unique_market_ids, data={
'Lower Bound': bounds[0].flatten(),
'Mean Own Elasticity': aggregates.flatten(),
'Upper Bound': bounds[1].flatten()
})
table.round(2).head()
[21]:
Lower Bound  Mean Own Elasticity  Upper Bound  

C01Q1  4.31  0.81  3.88 
C01Q2  4.07  0.69  3.68 
C03Q1  3.71  0.55  3.20 
C03Q2  3.65  0.60  3.16 
C04Q1  3.31  0.68  2.97 
Optimal Instruments¶
Given a consistent estimate of \(\theta\), we may want to compute the optimal instruments of Chamberlain (1987) and use them to resolve the problem. Optimal instruments have been shown, for example, by Reynaert and Verboven (2014), to reduce bias, improve efficiency, and enhance stability of BLP estimates.
The ProblemResults.compute_optimal_instruments method computes the expected Jacobians that comprise the optimal instruments by integrating over the density of \(\xi\) (and \(\omega\) if a supply side was estimated). By default, the method approximates this integral by averaging over the Jacobian realizations computed under draws from the asymptotic normal distribution
of the error terms. Since this process is computationally expensive and often doesn’t make much of a difference, we’ll use method='approximate'
in this example to simply evaluate the Jacobians at the expected value of \(\xi\), zero.
[22]:
instrument_results = results.compute_optimal_instruments(method='approximate')
instrument_results
[22]:
Optimal Instrument Results Summary:
=======================
Computation Error Term
Time Draws
 
00:00:01 1
=======================
We can use the OptimalInstrumentResults.to_problem method to recreate the fake cereal problem with the estimated optimal excluded instruments.
[23]:
updated_problem = instrument_results.to_problem()
updated_problem
[23]:
Dimensions:
=================================================
T N F I K1 K2 D MD ED
        
94 2256 5 1880 1 4 4 14 1
=================================================
Formulations:
===================================================================
Column Indices: 0 1 2 3
    
X1: Linear Characteristics prices
X2: Nonlinear Characteristics 1 prices sugar mushy
d: Demographics income income_squared age child
===================================================================
We can solve this updated problem just like the original one. We’ll start at our consistent estimate of \(\theta\).
[24]:
updated_results = updated_problem.solve(
results.sigma,
results.pi,
optimization=pyblp.Optimization('bfgs', {'gtol': 1e5}),
method='1s'
)
updated_results
[24]:
Problem Results Summary:
=======================================================================================================
GMM Objective Gradient Hessian Hessian Clipped Weighting Matrix Covariance Matrix
Step Value Norm Min Eigenvalue Max Eigenvalue Shares Condition Number Condition Number
       
1 +8.0E14 +3.0E06 +1.6E04 +2.9E+04 0 +7.8E+07 +1.8E+08
=======================================================================================================
Cumulative Statistics:
===========================================================================
Computation Optimizer Optimization Objective Fixed Point Contraction
Time Converged Iterations Evaluations Iterations Evaluations
     
00:01:47 Yes 42 50 45102 139676
===========================================================================
Nonlinear Coefficient Estimates (Robust SEs in Parentheses):
=====================================================================================================================
Sigma: 1 prices sugar mushy  Pi: income income_squared age child
          
1 +2.1E01  1 +6.0E+00 +0.0E+00 +1.6E01 +0.0E+00
(+7.8E02)  (+5.2E01) (+2.0E01)

prices +0.0E+00 +3.0E+00  prices +9.8E+01 5.6E+00 +0.0E+00 +4.1E+00
(+6.5E01)  (+8.6E+01) (+4.5E+00) (+2.2E+00)

sugar +0.0E+00 +0.0E+00 +2.7E02  sugar 3.1E01 +0.0E+00 +4.9E02 +0.0E+00
(+7.2E03)  (+3.5E02) (+1.3E02)

mushy +0.0E+00 +0.0E+00 +0.0E+00 +3.0E01  mushy +9.7E01 +0.0E+00 5.4E01 +0.0E+00
(+1.0E01)  (+2.9E01) (+1.8E01)
=====================================================================================================================
Beta Estimates (Robust SEs in Parentheses):
==========
prices

3.1E+01
(+4.5E+00)
==========