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Random Coefficients Logit Tutorial with the Fake Cereal Data

[1]:
import pyblp
import numpy as np
import pandas as pd

pyblp.options.digits = 2
pyblp.options.verbose = False
pyblp.__version__
[1]:
'0.8.0'

In this tutorial, we’ll use data from Nevo (2000) to solve the paper’s fake cereal problem. Locations of CSV files that contain the data are in the data module.

Theory of Random Coefficients Logit

The random coefficients model extends the plain logit model by allowing for correlated tastes for different product characteristics. In this model (indirect) utility is given by

(1)\[u_{jti} = \alpha_i p_{jt} + x_{jt} \beta_i^x + \xi_{jt} + \epsilon_{jti}\]

The main addition is that \(\beta_i = (\alpha_i, \beta_i^x)\) have individual specific subscripts \(i\).

Conditional on \(\beta_i\), the individual marketshares follow the same logit form as before. But now we must integrate over heterogeneous individuals to get the aggregate marketshares:

(2)\[s_{jt}(\alpha, \beta, \theta) = \int \frac{\exp(\alpha_i p_{jt} + x_{jt} \beta_i^x + \xi_{jt})}{1 + \sum_k \exp(\alpha_i p_{jt} + x_{kt} \beta_i^x + \xi_{kt})} f(\alpha_i, \beta_i \mid \theta).\]

In general, this integral needs to be calculated numerically. This also means that we can’t directly linearize the model. It is common to re-parametrize the model to separate the aspects of mean utility that all individuals agree on, \(\delta_{jt} = \alpha p_{jt} + x_{jt} \beta^x + \xi_{jt}\), from the individual specific heterogeneity, \(\mu_{jti}(\theta)\). This gives us

(3)\[s_{jt}(\delta_{jt}, \theta) = \int \frac{\exp(\delta_{jt} + \mu_{jti})}{1 + \sum_k \exp(\delta_{kt} + \mu_{kti})} f(\mu_{it} | \theta).\]

Given a guess of \(\theta\) we can solve the system of nonlinear equations for the vector \(\delta\) which equates observed and predicted marketshares \(s_{jt} = s_{jt}(\delta, \theta)\). Now we can perform a linear IV GMM regression of the form

(4)\[\delta_{jt}(\theta) = \alpha p_{jt} + x_{jt} \beta^x + \xi_{jt}.\]

The moments are constructed by interacting the predicted residuals \(\hat{\xi}_{jt}(\theta)\) with instruments \(z_{jt}\) to form

(5)\[\bar{g}(\theta) =\frac{1}{N} \sum_{j,t} z_{jt}' \hat{\xi}_{jt}(\theta).\]

Application of Random Coefficients Logit with the Fake Cereal Data

To include random coefficients we need to add a Formulation for the nonlinear characteristics \(X_2\).

Just like in the logit case we have the same reserved field names in product_data:

  • market_ids are the unique market identifiers which we subscript \(t\).

  • shares specifies the marketshares which need to be between zero and one, and within a market ID, \(\sum_{j} s_{jt} < 1\).

  • prices are prices \(p_{jt}\). These have some special properties and are always treated as endogenous.

  • demand_instruments0, demand_instruments1, and so on are numbered demand instruments. These represent only the excluded instruments. The exogenous regressors in \(X_1\) (of which \(X_2\) is typically a subset) will be automatically added to the set of instruments.

We proceed with the following steps:

  1. Load the product data which at a minimum consists of market_ids, shares, prices, and at least a single column of demand instruments, demand_instruments0.

  2. Define a Formulation for the \(X_1\) (linear) demand model.

    • This and all other formulas are similar to R and patsy formulas.

    • It includes a constant by default. To exclude the constant, specify either a 0 or a -1.

    • To efficiently include fixed effects, use the absorb option and specify which categorical variables you would like to absorb.

    • Some model reduction may happen automatically. The constant will be excluded if you include fixed effects and some precautions are taken against collinearity. However, you will have to make sure that differently-named variables are not collinear.

  3. Define a Formulation for the \(X_2\) (nonlinear) demand model.

    • Include only the variables over which we want random coefficients.

    • Do not absorb or include fixed effects.

    • It will include a random coefficient on the constant (to capture inside good vs. outside good preference) unless you specify not to with a 0 or a -1.

  4. Define an Integration configuration to solve the market share integral from several available options:

    • Monte Carlo integration (pseudo-random draws).

    • Product rule quadrature.

    • Sparse grid quadrature.

  5. Combine Formulation classes, product_data, and the Integration configuration to construct a Problem.

  6. Use the Problem.solve method to estimate paramters.

    • It is required to specify an initial guess of the nonlinear parameters. This serves two primary purposes: speeding up estimation and indicating to the solver through initial values of zero which parameters are restricted to be always zero.

Specification of Random Taste Parameters

It is common to assume that \(f(\beta_i \mid \theta)\) follows a multivariate normal distribution, and to break it up into three parts:

  1. A mean \(K_1 \times 1\) taste which all individuals agree on, \(\beta\).

  2. A \(K_2 \times K_2\) covariance matrix, \(\Sigma\) .

  3. Any \(K_2 \times D\) interactions, \(\Pi\), with observed \(D \times 1\) demographic data, \(d_i\).

Together this gives us that

(6)\[\beta_i \sim N(\beta + \Pi d_i, \Sigma).\]

Problem.solve takes an initial guess \(\Sigma_0\) of \(\Sigma\). It guarantees that \(\hat{\Sigma}\) (the estimated parameters) will have the same sparsity structure as \(\Sigma_0\). So any zero element of \(\Sigma\) is restricted to be zero in the solution \(\hat{\Sigma}\). For example, a popular restriction is that \(\Sigma\) is diagonal, this can be achieved by passing a diagonal matrix as \(\Sigma_0\).

As is common with multivariate normal distributions, \(\Sigma\) is not estimated directly. Rather, its matrix square (Cholesky) root \(L\) is estimated where \(LL' = \Sigma\).

Loading the Data

The product_data argument of Problem should be a structured array-like object with fields that store data. Product data can be a structured NumPy array, a pandas DataFrame, or other similar objects.

[2]:
product_data = pd.read_csv(pyblp.data.NEVO_PRODUCTS_LOCATION)
product_data.head()
[2]:
market_ids city_ids quarter product_ids firm_ids brand_ids shares prices sugar mushy ... demand_instruments10 demand_instruments11 demand_instruments12 demand_instruments13 demand_instruments14 demand_instruments15 demand_instruments16 demand_instruments17 demand_instruments18 demand_instruments19
0 C01Q1 1 1 F1B04 1 4 0.012417 0.072088 2 1 ... 2.116358 -0.154708 -0.005796 0.014538 0.126244 0.067345 0.068423 0.034800 0.126346 0.035484
1 C01Q1 1 1 F1B06 1 6 0.007809 0.114178 18 1 ... -7.374091 -0.576412 0.012991 0.076143 0.029736 0.087867 0.110501 0.087784 0.049872 0.072579
2 C01Q1 1 1 F1B07 1 7 0.012995 0.132391 4 1 ... 2.187872 -0.207346 0.003509 0.091781 0.163773 0.111881 0.108226 0.086439 0.122347 0.101842
3 C01Q1 1 1 F1B09 1 9 0.005770 0.130344 3 0 ... 2.704576 0.040748 -0.003724 0.094732 0.135274 0.088090 0.101767 0.101777 0.110741 0.104332
4 C01Q1 1 1 F1B11 1 11 0.017934 0.154823 12 0 ... 1.261242 0.034836 -0.000568 0.102451 0.130640 0.084818 0.101075 0.125169 0.133464 0.121111

5 rows × 30 columns

The product data contains market_ids, product_ids, firm_ids, shares, prices, a number of other firm IDs and product characteristics, and some pre-computed excluded demand_instruments0, demand_instruments1, and so on. The product_ids will be incorporated as fixed effects.

For more information about the instruments and the example data as a whole, refer to the data module.

Setting Up and Solving the Problem Without Demographics

Formulations, product data, and an integration configuration are collectively used to initialize a Problem. Once initialized, Problem.solve runs the estimation routine. The arguments to Problem.solve configure how estimation is performed. For example, optimization and iteration arguments configure the optimization and iteration routines that are used by the outer and inner loops of estimation.

We’ll specify Formulation configurations for \(X_1\), the linear characteristics, and \(X_2\), the nonlinear characteristics.

  • The formulation for \(X_1\) consists of prices and fixed effects constructed from product_ids, which we will absorb using absorb argument of Formulation.

  • If we were interested in reporting estimates for each fixed effect, we could replace the formulation for \(X_1\) with Formulation('prices + C(product_ids)').

  • Because sugar, mushy, and the constant are collinear with product_ids, we can include them in \(X_2\) but not in \(X_1\).

[3]:
X1_formulation = pyblp.Formulation('0 + prices', absorb='C(product_ids)')
X2_formulation = pyblp.Formulation('1 + prices + sugar + mushy')
product_formulations = (X1_formulation, X2_formulation)
product_formulations
[3]:
(prices + Absorb[C(product_ids)], 1 + prices + sugar + mushy)

We also need to specify an Integration configuration. We consider two alternatives:

  1. Monte Carlo draws: we simulate 50 individuals from a random normal distribution.

  2. Product rules: we construct nodes and weights according to a product rule that exactly integrates polynomials of degree \(5 \times 2 - 1 = 9\) or less.

[4]:
mc_integration = pyblp.Integration('monte_carlo', size=50, seed=0)
mc_integration
[4]:
Configured to construct nodes and weights with Monte Carlo simulation.
[5]:
pr_integration = pyblp.Integration('product', size=5)
pr_integration
[5]:
Configured to construct nodes and weights according to the level-5 Gauss-Hermite product rule.
[6]:
mc_problem = pyblp.Problem(product_formulations, product_data, integration=mc_integration)
mc_problem
[6]:
Dimensions:
============================================
 T    N     F    I     K1    K2    MD    ED
---  ----  ---  ----  ----  ----  ----  ----
94   2256   5   4700   1     4     20    1
============================================

Formulations:
===========================================================
       Column Indices:           0       1       2      3
-----------------------------  ------  ------  -----  -----
 X1: Linear Characteristics    prices
X2: Nonlinear Characteristics    1     prices  sugar  mushy
===========================================================
[7]:
pr_problem = pyblp.Problem(product_formulations, product_data, integration=pr_integration)
pr_problem
[7]:
Dimensions:
=============================================
 T    N     F     I     K1    K2    MD    ED
---  ----  ---  -----  ----  ----  ----  ----
94   2256   5   58750   1     4     20    1
=============================================

Formulations:
===========================================================
       Column Indices:           0       1       2      3
-----------------------------  ------  ------  -----  -----
 X1: Linear Characteristics    prices
X2: Nonlinear Characteristics    1     prices  sugar  mushy
===========================================================

As an illustration of how to configure the optimization routine, we’ll use a simpler, non-default Optimization configuration that doesn’t support parameter bounds.

[8]:
bfgs = pyblp.Optimization('bfgs', {'gtol': 1e-10})
bfgs
[8]:
Configured to optimize using the BFGS algorithm implemented in SciPy with analytic gradients and options {gtol: +1.0E-10}.

We estimate three versions of the model:

  1. An unrestricted covariance matrix for random tastes using Monte Carlo integration.

  2. An unrestricted covariance matrix for random tastes using the product rule.

  3. A restricted diagonal matrix for random tastes using Monte Carlo Integration.

Notice that the only thing that changes when we estimate the restricted covariance is our initial guess of \(\Sigma_0\). The upper diagonal in this initial guess is ignored because we are optimizing over the lower-triangular Cholesky root of \(\Sigma\).

[9]:
results1 = mc_problem.solve(sigma=np.ones((4, 4)), optimization=bfgs)
results1
[9]:
Problem Results Summary:
==============================================================================================
GMM   Objective  Gradient      Hessian         Hessian     Weighting Matrix  Covariance Matrix
Step    Value      Norm    Min Eigenvalue  Max Eigenvalue  Condition Number  Condition Number
----  ---------  --------  --------------  --------------  ----------------  -----------------
 2    +6.1E-02   +3.8E-10     +2.3E-05        +2.7E+00         +5.6E+07          +1.8E+06
==============================================================================================

Cumulative Statistics:
================================================================
Computation  Optimization   Objective   Fixed Point  Contraction
   Time       Iterations   Evaluations  Iterations   Evaluations
-----------  ------------  -----------  -----------  -----------
 00:06:27        180           350        215888       675540
================================================================

Nonlinear Coefficient Estimates (Robust SEs in Parentheses):
======================================================
Sigma:      1         prices      sugar       mushy
------  ----------  ----------  ----------  ----------
  1      +1.6E-01
        (+5.2E+00)

prices   +3.7E+00    -8.9E+00
        (+4.2E+01)  (+1.8E+01)

sugar    -7.6E-02    +1.5E-01    +6.4E-02
        (+1.6E-01)  (+2.3E-01)  (+1.4E-01)

mushy    -1.9E-01    -6.5E-02    -7.8E-02    +4.8E-01
        (+2.1E+00)  (+1.4E+00)  (+1.8E+00)  (+2.9E+00)
======================================================

Beta Estimates (Robust SEs in Parentheses):
==========
  prices
----------
 -3.1E+01
(+5.6E+00)
==========
[10]:
results2 = pr_problem.solve(sigma=np.ones((4, 4)), optimization=bfgs)
results2
[10]:
Problem Results Summary:
==============================================================================================
GMM   Objective  Gradient      Hessian         Hessian     Weighting Matrix  Covariance Matrix
Step    Value      Norm    Min Eigenvalue  Max Eigenvalue  Condition Number  Condition Number
----  ---------  --------  --------------  --------------  ----------------  -----------------
 2    +7.0E-02   +5.6E-12     +7.0E-06        +2.3E+00         +5.3E+07          +1.4E+37
==============================================================================================

Cumulative Statistics:
================================================================
Computation  Optimization   Objective   Fixed Point  Contraction
   Time       Iterations   Evaluations  Iterations   Evaluations
-----------  ------------  -----------  -----------  -----------
 00:12:13        182           321        182808       572217
================================================================

Nonlinear Coefficient Estimates (Robust SEs in Parentheses):
======================================================
Sigma:      1         prices      sugar       mushy
------  ----------  ----------  ----------  ----------
  1      -7.4E-01
        (+2.0E+00)

prices   +1.3E+01    +1.0E-10
        (+6.1E+00)  (+8.2E-10)

sugar    -1.1E-01    -9.9E-12    -1.4E-11
        (+9.5E-02)  (+6.2E-08)  (+1.1E-07)

mushy    +1.5E-01    +1.7E-10    +3.1E-10    -5.4E-10
        (+6.8E-01)  (+1.1E-10)  (+8.4E-10)  (+8.6E-09)
======================================================

Beta Estimates (Robust SEs in Parentheses):
==========
  prices
----------
 -3.1E+01
(+3.9E+00)
==========
[11]:
results3 = mc_problem.solve(sigma=np.eye(4), optimization=bfgs)
results3
[11]:
Problem Results Summary:
==============================================================================================
GMM   Objective  Gradient      Hessian         Hessian     Weighting Matrix  Covariance Matrix
Step    Value      Norm    Min Eigenvalue  Max Eigenvalue  Condition Number  Condition Number
----  ---------  --------  --------------  --------------  ----------------  -----------------
 2    +7.9E-02   +3.5E-10     +4.8E-04        +2.7E+00         +5.8E+07          +4.8E+04
==============================================================================================

Cumulative Statistics:
================================================================
Computation  Optimization   Objective   Fixed Point  Contraction
   Time       Iterations   Evaluations  Iterations   Evaluations
-----------  ------------  -----------  -----------  -----------
 00:01:52         61           147         66429       209785
================================================================

Nonlinear Coefficient Estimates (Robust SEs in Parentheses):
======================================================
Sigma:      1         prices      sugar       mushy
------  ----------  ----------  ----------  ----------
  1      +5.2E-02
        (+1.1E+00)

prices   +0.0E+00    -4.3E-01
                    (+8.0E+00)

sugar    +0.0E+00    +0.0E+00    +3.6E-02
                                (+5.8E-02)

mushy    +0.0E+00    +0.0E+00    +0.0E+00    +5.0E-01
                                            (+1.7E+00)
======================================================

Beta Estimates (Robust SEs in Parentheses):
==========
  prices
----------
 -3.0E+01
(+1.4E+00)
==========

We see that all three models give similar estimates of the price coefficient \(\hat{\alpha} \approx -30\). Note a few of the estimated terms on the diagonal of \(\Sigma\) are negative. Since the diagonal consists of standard deviations, negative values are unrealistic. When using another optimization routine that supports bounds (like the default L-BFGS-B routine), these diagonal elements are by default bounded from below by zero.

Adding Demographics to the Problem

To add demographic data we need to make two changes:

  1. We need to load agent_data, which for this cereal problem contains pre-computed Monte Carlo draws and demographics.

  2. We need to add an agent_formulation to the model.

The agent data has several reserved column names.

  • market_ids are the index that link the agent data to the market_ids in product data.

  • weights are the weights \(w_{it}\) attached to each agent. In each market, these should sum to one so that \(\sum_i w_{it} = 1\). It is often the case the \(w_{it} = 1 / I_t\) where \(I_t\) is the number of agents in market \(t\), so that each agent gets equal weight. Other possibilities include quadrature nodes and weights.

  • nodes0, nodes1, and so on are the nodes at which the unobserved agent tastes \(\mu_{jti}\) are evaluated. The nodes should be labeled from \(0, \ldots, K_2 - 1\) where \(K_2\) is the number of random coefficients.

  • Other fields are the realizations of the demographics \(d\) themselves.

[12]:
agent_data = pd.read_csv(pyblp.data.NEVO_AGENTS_LOCATION)
agent_data.head()
[12]:
market_ids city_ids quarter weights nodes0 nodes1 nodes2 nodes3 income income_squared age child
0 C01Q1 1 1 0.05 0.434101 -1.500838 -1.151079 0.161017 0.495123 8.331304 -0.230109 -0.230851
1 C01Q1 1 1 0.05 -0.726649 0.133182 -0.500750 0.129732 0.378762 6.121865 -2.532694 0.769149
2 C01Q1 1 1 0.05 -0.623061 -0.138241 0.797441 -0.795549 0.105015 1.030803 -0.006965 -0.230851
3 C01Q1 1 1 0.05 -0.041317 1.257136 -0.683054 0.259044 -1.485481 -25.583605 -0.827946 0.769149
4 C01Q1 1 1 0.05 -0.466691 0.226968 1.044424 0.092019 -0.316597 -6.517009 -0.230109 -0.230851

The agent formulation tells us which columns of demographic information to interact with \(X_2\).

[13]:
agent_formulation = pyblp.Formulation('0 + income + income_squared + age + child')
agent_formulation
[13]:
income + income_squared + age + child

This tells us to include demographic realizations for income, income_squared, age, and the presence of children, child, but to ignore other possible demographics when interacting demographics with \(X_2\). We should also generally exclude the constant from the demographic formula.

Now we configure the Problem to include the agent_formulation and agent_data, which follow the product_formulations and product_data.

When we display the class, it lists the demographic interactions in the table of formulations and reports \(D = 4\), the dimension of the demographic draws.

[14]:
nevo_problem = pyblp.Problem(
    product_formulations,
    product_data,
    agent_formulation,
    agent_data
)
nevo_problem
[14]:
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
===================================================================

The initialized problem can be solved with Problem.solve. We’ll use the same starting values as Nevo (2000). By passing a diagonal matrix as starting values for \(\Sigma\), we’re choosing to ignore covariance terms. Similarly, zeros in the starting values for \(\Pi\) mean that those parameters will be fixed at zero.

To replicate common estimates, we’ll use the non-default BFGS optimization routine, and we’ll configure Problem.solve to use 1-step GMM instead of 2-step GMM.

[15]:
initial_sigma = np.diag([0.3302, 2.4526, 0.0163, 0.2441])
initial_pi = np.array([
  [ 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     ]
])
nevo_results = nevo_problem.solve(
    initial_sigma,
    initial_pi,
    optimization=bfgs,
    method='1s'
)
nevo_results
[15]:
Problem Results Summary:
==============================================================================================
GMM   Objective  Gradient      Hessian         Hessian     Weighting Matrix  Covariance Matrix
Step    Value      Norm    Min Eigenvalue  Max Eigenvalue  Condition Number  Condition Number
----  ---------  --------  --------------  --------------  ----------------  -----------------
 1    +2.0E-03   +5.4E-11     +1.3E-08        +7.3E+00         +6.9E+07          +8.4E+08
==============================================================================================

Cumulative Statistics:
================================================================
Computation  Optimization   Objective   Fixed Point  Contraction
   Time       Iterations   Evaluations  Iterations   Evaluations
-----------  ------------  -----------  -----------  -----------
 00:02:49        140           145        109732       341137
================================================================

Nonlinear Coefficient Estimates (Robust SEs in Parentheses):
=====================================================================================================================
Sigma:      1         prices      sugar       mushy     |   Pi:      income    income_squared     age        child
------  ----------  ----------  ----------  ----------  |  ------  ----------  --------------  ----------  ----------
  1      +5.6E-01                                       |    1      +2.3E+00      +0.0E+00      +1.3E+00    +0.0E+00
        (+1.6E-01)                                      |          (+1.2E+00)                  (+6.3E-01)
                                                        |
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    -5.8E-03               |  sugar    -3.8E-01      +0.0E+00      +5.2E-02    +0.0E+00
                                (+1.4E-02)              |          (+1.2E-01)                  (+2.6E-02)
                                                        |
mushy    +0.0E+00    +0.0E+00    +0.0E+00    +9.3E-02   |  mushy    +7.5E-01      +0.0E+00      -1.4E+00    +0.0E+00
                                            (+1.9E-01)  |          (+8.0E-01)                  (+6.7E-01)
=====================================================================================================================

Beta Estimates (Robust SEs in Parentheses):
==========
  prices
----------
 -6.3E+01
(+1.5E+01)
==========

Results are similar to those in the original paper with a (non-scaled) objective value of \(q(\hat{\theta}) = \text{8.96E-7} = 4.65 / N^2\) and a price coefficient of \(\hat{\alpha} = -62.7\).

Restricting Parameters

Because the interactions between price, income, and income_squared are potentially collinear, we might worry that \(\hat{\Pi}_{21} = 588\) and \(\hat{\Pi}_{22} = -30.2\) are pushing the price coefficient in opposite directions. Both are large in magnitude but statistically insignficant. One way of dealing with this is to restrict \(\Pi_{22} = 0\).

There are two ways we can do this:

  1. Change the initial \(\Pi_0\) values to make this term zero.

  2. Change the agent formula to drop income_squared.

First, we’ll change the initial \(\Pi_0\) values.

[16]:
restricted_pi = initial_pi.copy()
restricted_pi[1, 1] = 0
nevo_problem.solve(
    initial_sigma,
    restricted_pi,
    optimization=bfgs,
    method='1s'
)
[16]:
Problem Results Summary:
==============================================================================================
GMM   Objective  Gradient      Hessian         Hessian     Weighting Matrix  Covariance Matrix
Step    Value      Norm    Min Eigenvalue  Max Eigenvalue  Condition Number  Condition Number
----  ---------  --------  --------------  --------------  ----------------  -----------------
 1    +6.8E-03   +1.8E-09     +2.1E-05        +7.3E+00         +6.9E+07          +5.7E+05
==============================================================================================

Cumulative Statistics:
================================================================
Computation  Optimization   Objective   Fixed Point  Contraction
   Time       Iterations   Evaluations  Iterations   Evaluations
-----------  ------------  -----------  -----------  -----------
 00:03:50        112           208        155160       482159
================================================================

Nonlinear Coefficient Estimates (Robust SEs in Parentheses):
=====================================================================================================================
Sigma:      1         prices      sugar       mushy     |   Pi:      income    income_squared     age        child
------  ----------  ----------  ----------  ----------  |  ------  ----------  --------------  ----------  ----------
  1      +3.7E-01                                       |    1      +3.1E+00      +0.0E+00      +1.2E+00    +0.0E+00
        (+1.2E-01)                                      |          (+1.1E+00)                  (+1.0E+00)
                                                        |
prices   +0.0E+00    +1.8E+00                           |  prices   +4.2E+00      +0.0E+00      +0.0E+00    +1.2E+01
                    (+9.2E-01)                          |          (+4.6E+00)                              (+5.2E+00)
                                                        |
sugar    +0.0E+00    +0.0E+00    -4.4E-03               |  sugar    -1.9E-01      +0.0E+00      +2.8E-02    +0.0E+00
                                (+1.2E-02)              |          (+3.5E-02)                  (+3.2E-02)
                                                        |
mushy    +0.0E+00    +0.0E+00    +0.0E+00    +8.6E-02   |  mushy    +1.5E+00      +0.0E+00      -1.5E+00    +0.0E+00
                                            (+1.9E-01)  |          (+6.5E-01)                  (+1.1E+00)
=====================================================================================================================

Beta Estimates (Robust SEs in Parentheses):
==========
  prices
----------
 -3.2E+01
(+2.3E+00)
==========

Now we’ll drop both income_squared and the corresponding column in \(\Pi_0\).

[17]:
restricted_formulation = pyblp.Formulation('0 + income  + age + child')
deleted_pi = np.delete(initial_pi, 1, axis=1)
restricted_problem = pyblp.Problem(
    product_formulations,
    product_data,
    restricted_formulation,
    agent_data
)
restricted_problem.solve(
    initial_sigma,
    deleted_pi,
    optimization=bfgs,
    method='1s'
)
[17]:
Problem Results Summary:
==============================================================================================
GMM   Objective  Gradient      Hessian         Hessian     Weighting Matrix  Covariance Matrix
Step    Value      Norm    Min Eigenvalue  Max Eigenvalue  Condition Number  Condition Number
----  ---------  --------  --------------  --------------  ----------------  -----------------
 1    +6.8E-03   +2.0E-09     +2.1E-05        +7.3E+00         +6.9E+07          +5.7E+05
==============================================================================================

Cumulative Statistics:
================================================================
Computation  Optimization   Objective   Fixed Point  Contraction
   Time       Iterations   Evaluations  Iterations   Evaluations
-----------  ------------  -----------  -----------  -----------
 00:03:19        111           157        117591       365555
================================================================

Nonlinear Coefficient Estimates (Robust SEs in Parentheses):
=====================================================================================================
Sigma:      1         prices      sugar       mushy     |   Pi:      income       age        child
------  ----------  ----------  ----------  ----------  |  ------  ----------  ----------  ----------
  1      +3.7E-01                                       |    1      +3.1E+00    +1.2E+00    +0.0E+00
        (+1.2E-01)                                      |          (+1.1E+00)  (+1.0E+00)
                                                        |
prices   +0.0E+00    +1.8E+00                           |  prices   +4.2E+00    +0.0E+00    +1.2E+01
                    (+9.2E-01)                          |          (+4.6E+00)              (+5.2E+00)
                                                        |
sugar    +0.0E+00    +0.0E+00    -4.4E-03               |  sugar    -1.9E-01    +2.8E-02    +0.0E+00
                                (+1.2E-02)              |          (+3.5E-02)  (+3.2E-02)
                                                        |
mushy    +0.0E+00    +0.0E+00    +0.0E+00    +8.6E-02   |  mushy    +1.5E+00    -1.5E+00    +0.0E+00
                                            (+1.9E-01)  |          (+6.5E-01)  (+1.1E+00)
=====================================================================================================

Beta Estimates (Robust SEs in Parentheses):
==========
  prices
----------
 -3.2E+01
(+2.3E+00)
==========

The parameter estimates and standard errors are identical for both approaches. Based on the number of fixed point iterations, there is some evidence that the solver took a slightly different path for each problem, but both restricted problems arrived at identical answers. The \(\hat{\Pi}_{12}\) interaction term is still insignificant.