Estimating Custom Maximum Likelihood Models in Python (and Matlab)
In this post I show various ways of estimating "generic" maximum likelihood models in python. For each, we'll recover standard errors.
We will implement a simple ordinary least squares model like this
\begin{equation} \mathbf{y = x\beta +\epsilon} \end{equation}where \(\epsilon\) is assumed distributed i.i.d. normal with mean 0 and variance \(\sigma^2\). In our simple model, there is only a constant and one slope coefficient (\(\beta = \begin{bmatrix} \beta_0 & \beta_1 \end{bmatrix}\)).
For this model, we would probably never bother going to the trouble of manually implementing maximum likelihood estimators as we show in this post. However, for more complicated models for which there is no established package or command, there are benefits to knowing how to build your own likelihood function and use it for estimation. It is also worthwhile noting that most of the methods shown here don't use analytical gradiants or hessians, so are likely (1) to have longer execution times and (2) to be less precise than methods where known analytical gradiants and hessians are built into the estimation method. I might explore those issues in a later post.
tl;dr: There are numerous ways to estimate custom maximum likelihood models in Python, and what I find is:
 For the most features, I recommend using the
Genericlikelihoodmodel
class from Statsmodels even if it is the least intuitive way for programmers familiar with Matlab. If you are comfortable with object oriented programming you should definitely go this route.  For fastest run times and computationally expensive problems Matlab will most likely be significantly even with lots of code optimizations.
Python packages we'll use for this post:
import pymc3 as pm3
import numpy as np
import numdifftools as ndt
import pandas as pd
from scipy.stats import norm
import statsmodels.api as sm
from statsmodels.base.model import GenericLikelihoodModel
from scipy.optimize import minimize
First we generate data for this model here and add a column for the model constant:
N = 10000
x = 10 + 2*np.random.randn(N)
y = 5 + x + np.random.randn(N)
df = pd.DataFrame({'y':y, 'x':x})
df['constant'] = 1
df.head()
x y constant 0 4.857731 10.623466 1 1 11.285848 16.136891 1 2 8.454859 14.482064 1 3 8.673113 15.358859 1 4 10.655889 14.795186 1
OLS Estimation
Since this is such a simple and universally used model, there are numerous packages available for estimating it. For this problem, you would undoubtedly want to use one of these existing packages. For example, statsmodels
has an OLS
method. We use it here for benchmarking purposes for comparing our maximum likelihood estimation of the same model below.
sm.OLS(df.y,df[['constant','x']]).fit().summary()
OLS Regression Results ============================================================================== Dep. Variable: y Rsquared: 0.803 Model: OLS Adj. Rsquared: 0.803 Method: Least Squares Fstatistic: 4.086e+04 Date: Wed, 08 Mar 2017 Prob (Fstatistic): 0.00 Time: 11:19:36 LogLikelihood: 14124. No. Observations: 10000 AIC: 2.825e+04 Df Residuals: 9998 BIC: 2.827e+04 Df Model: 1 Covariance Type: nonrobust ============================================================================== coef std err t P>t [95.0% Conf. Int.]  constant 4.9956 0.050 99.532 0.000 4.897 5.094 x 0.9982 0.005 202.135 0.000 0.989 1.008 ============================================================================== Omnibus: 1.709 DurbinWatson: 2.000 Prob(Omnibus): 0.426 JarqueBera (JB): 1.731 Skew: 0.022 Prob(JB): 0.421 Kurtosis: 2.953 Cond. No. 51.8 ============================================================================== Warnings: [1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
Matlab
Matlab is a tool used by many econometricians for estimating generic likelihood models so I include it here for comparison purposes. The file neg_loglike.m
defines the loglikelihood function (negative loglikelihood):
function [ll] = neg_loglike(theta,Y,X)
mu = X*theta(1:2);
ll = 1*sum(log(normpdf(Y,mu,theta(3))));
end
When this code is run in Matlab:
data = csvread('/tmp/data.csv');
X = data(:,[3 1]);
Y = data(:,2);
% Optimization Settings
opt = optimset('GradObj','off',...
'Hessian','off',...
'LargeScale','off',...
'Display','off',...
'MaxIter', 100000,...
'MaxFunEvals', 1000000, ...
'TolFun', 1.000e006,...
'TolX', 1.000e006,...
'FunValCheck','off',...
'DerivativeCheck','off',...
'Diagnostics','off',...
'Algorithm','trustregionreflective');
% starting values for estimation
theta_start=.5*ones(size(X,2)+1,1);
[theta,fval,flag,out,grad,hess] = fminunc(@(theta) neg_loglike(theta,Y,X), ...
theta_start, opt);
se = sqrt(diag(inv(hess)));
ans = [theta se]
We get the following parameter estimates:
Parameter Standard Error Intercept 4.99560 0.050186 Slope 0.99823 0.004938 Sigma 0.99344 0.007027
Python Handcoded using Scipy
Of all the python methods, this one is most similar to Matlab above. Defining the loglikelihood (in this case as in matlab a negative loglikelihood since there is no maximize function):
def neg_loglike(theta):
mu = theta[0] + theta[1]*x
return 1*norm(mu, theta[2]).logpdf(y).sum()
We'll optimize the loglikelihood over our parameters using minimize
from python's scipy.optimize
package:
theta_start = np.array([1,1,1])
res = minimize(neg_loglike, theta_start, method = 'NelderMead',
options={'disp': True})
Optimization terminated successfully. Current function value: 14123.593389 Iterations: 109 Function evaluations: 193
The default method, BFGS
sometimes fails to converge, so I usually use NelderMead
. Unfortunately, NelderMead
doesn't return an estimated Hessian, so we need to calculate it ourselves using the numdifftools
package:
Hfun = ndt.Hessian(neg_loglike, full_output=True)
hessian_ndt, info = Hfun(res['x'])
se = np.sqrt(np.diag(np.linalg.inv(hessian_ndt)))
results = pd.DataFrame({'parameters':res['x'],'std err':se})
results.index=['Intercept','Slope','Sigma']
results.head()
Extending Statsmodels
This method is perhaps the best way to proceed, but unless you are wellversed in object oriented programming is likely to be confusing. In my view it is worth the trouble to setup your custom maximum likelihood problem like this, by leveraging the machinery of statsmodels for working with our custom likelihood function. Since we have \(K+1\) parameters, proceed carefully, since by default statsmodels
assumes the number of parameters is equal to the column dimension of your independent variable. First, define the loglikelihood function (note this is not the negative loglikelihood):
def _ll_ols(y, X, beta, sigma):
mu = X.dot(beta)
return norm(mu,sigma).logpdf(y).sum()
Then, using the loglikelihood define our custom likelihood class (I'll call it MyOLS
). Note that there are two key parts to the code below:
 The function
nloglikeobs
, is only acting as a "traffic cop" and spits the parameters into \(\beta\) and \(\sigma\) coefficients and calls the likelihood function_ll_ols
above.  The
fit
function is where we inform statsmodels that our model has \(K+1\) variables. We'll provide a name for the additional variable (\(\sigma\)) and provide default starting values.
class MyOLS(GenericLikelihoodModel):
def __init__(self, endog, exog, **kwds):
super(MyOLS, self).__init__(endog, exog, **kwds)
def nloglikeobs(self, params):
sigma = params[1]
beta = params[:1]
ll = _ll_ols(self.endog, self.exog, beta, sigma)
return ll
def fit(self, start_params=None, maxiter=10000, maxfun=5000, **kwds):
# we have one additional parameter and we need to add it for summary
self.exog_names.append('sigma')
if start_params == None:
# Reasonable starting values
start_params = np.append(np.zeros(self.exog.shape[1]), .5)
return super(MyOLS, self).fit(start_params=start_params,
maxiter=maxiter, maxfun=maxfun,
**kwds)
Once the model structure is setup, we can use fit()
for estimating the model. statsmodels
handles all of the optimizations and hessian calculations for us.
sm_ols_manual = MyOLS(df.y,df[['constant','x']]).fit()
print(sm_ols_manual.summary())
Optimization terminated successfully. Current function value: 1.412359 Iterations: 212 Function evaluations: 386 MyOLS Results ============================================================================== Dep. Variable: y LogLikelihood: 14124. Model: MyOLS AIC: 2.825e+04 Method: Maximum Likelihood BIC: 2.827e+04 Date: Wed, 08 Mar 2017 Time: 11:20:09 No. Observations: 10000 Df Residuals: 9998 Df Model: 1 ============================================================================== coef std err z P>z [95.0% Conf. Int.]  constant 4.9956 0.050 99.541 0.000 4.897 5.094 x 0.9982 0.005 202.154 0.000 0.989 1.008 sigma 0.9934 0.007 141.420 0.000 0.980 1.007 ==============================================================================
The great thing about using the statsmodels
infrastructure is that you have access to a lot of functionality once your model is setup and estimated. For example, suppose you want 95% confidence intervals, after estimation, we can use this command with the model object:
ci = sm_ols_manual.conf_int(alpha=.05)
resdf = pd.DataFrame(ci,columns=['lower','upper'])
resdf.index = ['constant','x','sigma']
resdf.head()
lower upper constant 4.897258 5.093985 x 0.988556 1.007912 sigma 0.979679 1.007215
The are lots of other methods available to your generic model instance, and it is worth the effort getting it running.
PyMC3
The Bayesian Statistics Package PyMC3
can also find the Hessian and Maximum Posterior values, which for Flat priors should give us something nearly identical to Maximum Likelihood. Note, the Hessian produced by PyMC3
using approx_hessian
is what you should use. The command find_hessian
doesn't yield a valid hessian matrix for any of the cases I have tried. One potential advantage of using PyMC3
is that the hessian could be calculated off of analytical gradiants and if this is the case would likely yield more accurate standard errors than any of the other methods presented in this post (including Matlab).^{1}
Setting the model up in PyMC3
with pm3.Model() as model:
b0 = pm3.Flat('constant',testval=5)
b1 = pm3.Flat('x',testval=1)
sigma = pm3.Flat('sigma',testval = 1)
mu = b0 + b1 * df.x
like = pm3.Normal('like',mu = mu, sd=sigma, observed=df.y)
Noting that with Flat priors the posterior is exactly proportional to the likelihood, we can use the find_MAP
function
with model:
# obtain MLE
beta_mle = pm3.find_MAP()
# obtain hessian at MAP
hessian = pm3.approx_hessian(beta_mle)
Optimization terminated successfully. Current function value: 14123.593388 Iterations: 7 Function evaluations: 11 Gradient evaluations: 11
The method pm3.approx_hessian
uses numdifftools
to calculate the standard errors evaluated at the parameter vector you specify. We evaluate it at the MAP estimates which coincides with MLE estimates for this model setup. Note, the only gotcha is that the pm3.approx_hessian
changes the order of parameters, so we need to match the standard errors we calculate from the hessian to the estimates carefully. In this case, they are in reverse order. Putting together we have
se_1 = np.sqrt(np.diag(np.linalg.inv(hessian)))
se = se_1[::1]
b = np.array([beta_mle['constant'], beta_mle['x'], beta_mle['sigma']])
results_pymc = pd.DataFrame({'parameters':b,'std err':se})
results_pymc.index=['Intercept','Slope','Sigma']
results_pymc.head()
parameters std err Intercept 4.995611 0.050186 Slope 0.998235 0.004938 Sigma 0.993442 0.007025
Comparison
Unsurprisingly given the simplistic nature of the model estimated here, all of the results are nearly identical. I did an informal time (timed single run) comparison and here are the results. Note that we are using PyMC3
in unintended ways and it wasn't built to optimize execution times for this type of problem.
Program  Execution Time (Seconds) 
Matlab with fminunc

.122 
Python with minimize

.9937 
Python statsmodels

.9848 
Python PyMC3

1.377 
Matlab is significantly faster for this problem.
Footnotes:
Without futher investigation, I can't tell if PyMC3 uses analytical gradiants for calculating the hessian.