Method of Simulated Moments (MSM)#

This tutorial shows you how to do a Method of Simulated Moments estimation in estimagic. The Method of Simulated Moments (MSM) is a nonlinear estimation principle that is very useful for fitting complicated models to the data. The only ingredient required is a function that simulates the model outcomes you observe in some empirical dataset.

In the tutorial here, we will use a simple linear regression model. This is the same model which we use in the tutorial on maximum likelihood estimation.

Throughout the tutorial, we only talk about MSM estimation. However, the more general case of indirect inference estimation works exactly the same way.

Steps of MSM estimation#

Load (simulate) empirical data
Define a function to calculate estimation moments on the data
Calculate the covariance matrix of the empirical moments (with get_moments_cov)
Define a function to simulate moments from the model
Estimate the model, calculate standard errors, do sensitivity analysis (with estimate_msm)

Example: Estimate the parameters of a regression model#

The model we consider here is a simple regression model with only one explanatory variable (plus a constant). The goal is to estimate the slope coefficients and the error variance from a simulated data set.

The estimation mechanics are exactly the same for more complicated models. A model is always defined by a function that can take parameters (here: the mean, variance and lower_cutoff and upper_cutoff) and returns a number of simulated moments (mean, variance, soft_min and soft_max of simulated exam points).

Model:#

\[ y = \beta_0 + \beta_1 x + \epsilon, \text{ where } \epsilon \sim N(0, \sigma^2)\]

We aim to estimate \(\beta_0, \beta_1, \sigma^2\).

import estimagic as em
import numpy as np
import pandas as pd

rng = np.random.default_rng(seed=0)

1. Simulate data#

def simulate_data(params, n_draws, rng):
    x = rng.normal(0, 1, size=n_draws)
    e = rng.normal(0, params.loc["sd", "value"], size=n_draws)
    y = params.loc["intercept", "value"] + params.loc["slope", "value"] * x + e
    return pd.DataFrame({"y": y, "x": x})

true_params = pd.DataFrame(
    data=[[2, -np.inf], [-1, -np.inf], [1, 1e-10]],
    columns=["value", "lower_bound"],
    index=["intercept", "slope", "sd"],
)

data = simulate_data(true_params, n_draws=100, rng=rng)

2. Calculate Moments#

def calculate_moments(sample):
    moments = {
        "y_mean": sample["y"].mean(),
        "x_mean": sample["x"].mean(),
        "yx_mean": (sample["y"] * sample["x"]).mean(),
        "y_sqrd_mean": (sample["y"] ** 2).mean(),
        "x_sqrd_mean": (sample["x"] ** 2).mean(),
    }
    return pd.Series(moments)

empirical_moments = calculate_moments(data)
empirical_moments

y_mean         1.868333
x_mean         0.081097
yx_mean       -0.723189
y_sqrd_mean    5.227749
x_sqrd_mean    0.932272
dtype: float64

3. Calculate the covariance matrix of empirical moments#

The covariance matrix of the empirical moments (moments_cov) is needed for three things:

to calculate the weighting matrix
to calculate standard errors
to calculate sensitivity measures

We will calculate moments_cov via a bootstrap. Depending on your problem, there can be other ways to calculate the covariance matrix.

moments_cov = em.get_moments_cov(
    data, calculate_moments, bootstrap_kwargs={"n_draws": 5_000, "seed": 0}
)

moments_cov

	y_mean	x_mean	yx_mean	y_sqrd_mean	x_sqrd_mean
y_mean	0.017775	-0.009000	-0.015085	0.060904	-0.001010
x_mean	-0.009000	0.009407	0.016998	-0.032881	0.001586
yx_mean	-0.015085	0.016998	0.052317	-0.081053	-0.010686
y_sqrd_mean	0.060904	-0.032881	-0.081053	0.254822	0.008873
x_sqrd_mean	-0.001010	0.001586	-0.010686	0.008873	0.012473

get_moments_cov mainly just calls estimagic’s bootstrap function. See our bootstrap_tutorial for background information.

4. Define a function to calculate simulated moments#

In a real world application, this is the step that would take most of the time. However, in our very simple example, all the work is already done by numpy.

def simulate_moments(params, n_draws=10_000, seed=0):
    rng = np.random.default_rng(seed)
    sim_data = simulate_data(params, n_draws, rng)
    sim_moments = calculate_moments(sim_data)
    return sim_moments

simulate_moments(true_params)

y_mean         1.996739
x_mean         0.006312
yx_mean       -0.997919
y_sqrd_mean    5.999877
x_sqrd_mean    0.996197
dtype: float64

5. Estimate the model parameters#

Estimating a model consists of the following steps:

Building a criterion function that measures a distance between simulated and empirical moments
Minimizing this criterion function
Calculating the Jacobian of the model
Calculating standard errors, confidence intervals and p-values
Calculating sensitivity measures

This can all be done in one go with the estimate_msm function. This function has sensible default values, so you only need a minimum number of inputs. However, you can configure almost any aspect of the workflow via optional arguments. If you need even more control, you can call the lower level functions, which the now famliliarestimate_msm function is built on, directly.

start_params = true_params.assign(value=[100, 100, 100])

res = em.estimate_msm(
    simulate_moments,
    empirical_moments,
    moments_cov,
    start_params,
    optimize_options="scipy_lbfgsb",
)

res.summary()

	value	standard_error	ci_lower	ci_upper	p_value	free	stars
intercept	1.869534	0.134356	1.606201	2.132867	5.151662e-44	True	***
slope	-0.721958	0.231564	-1.175815	-0.268102	1.822359e-03	True	***
sd	1.099789	0.137635	0.830029	1.369549	1.342806e-15	True	***

What’s in the result?#

MomentsResult objects provide attributes and methods to calculate standard errors, confidence intervals and p-values. For all three, several methods are available. You can even calculate cluster robust standard errors.

A few examples are:

res.params

	value	lower_bound
intercept	1.869534	-inf
slope	-0.721958	-inf
sd	1.099789	1.000000e-10

res.cov(method="robust")

	intercept	slope	sd
intercept	0.018052	-0.015855	-0.013072
slope	-0.015855	0.053622	0.024682
sd	-0.013072	0.024682	0.018943

res.se()

	value	lower_bound
intercept	0.134356	-inf
slope	0.231564	-inf
sd	0.137635	1.000000e-10

How to visualize sensitivity measures?#

from estimagic import lollipop_plot

sensitivity_data = res.sensitivity(kind="bias").abs().T

fig = lollipop_plot(sensitivity_data)

fig = fig.update_layout(height=500, width=900)
fig.show(renderer="png")

../../_images/3de84dbdbf8531af3609089df2d306f66125bd49a81d69be8da45b69e162a3c3.png