Tutorial¶
We now illustrate the basic capabilities of the grmpy
package.
We start by outlining some basic functional form assumptions before introducing to alternative models that can be used to
estimate the marginal treatment effect (MTE).
We then turn to some simple use cases.
Assumptions¶
The grmpy
package implements the normal linear-in-parameters version of the generalized Roy model. Both potential outcomes and the choice \((Y_1, Y_0, D)\) are a linear function of the individual’s observables \((X, Z)\) and random components \((U_1, U_0, V)\).
We collect all unobservables determining treatment choice in \(V = U_C - (U_1 - U_0)\). Individuals decide to select into latent indicator variable \(D^{*}\) is positive. Depending on their decision, we either observe \(Y_1\) or \(Y_0\).
Parametric Normal Model¶
The parametric model imposes the assumption of joint normality of the unobservables \((U_1, U_0, V) \sim \mathcal{N}(0, \Sigma)\) with mean zero and covariance matrix \(\Sigma\).
Semiparametric Model¶
The semiparametric approach invokes no assumption on the distribution of the unobservables. It requires a weaker condition \((X,Z) \indep {U_1, U_0, V}\)
Under this assumption, the MTE is:
additively separable in \(X\) and \(U_D\), which means that the shape of the MTE is independent of \(X\), and
identified over the common support of \(P(Z)\), unconditional on \(X\).
The assumption of common support is crucial for the application of LIV and needs to be carefully evaluated every time. It is defined as the region where the support of \(P(Z)\) given \(D=1\) and the support of \(P(Z)\) given :math:`D=0 overlap.
Model Specification¶
You can specify the details of the model in an initialization file (example). This file contains several blocks:
SIMULATION
The SIMULATION block contains some basic information about the simulation request.
Key |
Value |
Interpretation |
---|---|---|
agents |
int |
number of individuals |
seed |
int |
seed for the specific simulation |
source |
str |
specified name for the simulation output files |
ESTIMATION
Depending on the model, different input parameters are required.
PARAMETRIC MODEL
Key |
Value |
Interpretation |
---|---|---|
semipar |
False |
choose the parametric normal model |
agents |
int |
number of individuals (for the comparison file) |
file |
str |
name of the estimation specific init file |
optimizer |
str |
optimizer used for the estimation process |
start |
str |
flag for the start values |
maxiter |
int |
maximum numbers of iterations |
dependent |
str |
indicates the dependent variable |
indicator |
str |
label of the treatment indicator variable |
output_file |
str |
name for the estimation output file |
comparison |
int |
flag for enabling the comparison file creation |
SEMIPARAMETRIC MODEL
Key |
Value |
Interpretation |
---|---|---|
semipar |
True |
choose the semiparametric model |
show_output |
bool |
If True, intermediate outputs of the LIV estimation are displayed |
dependent |
str |
indicates the dependent variable |
indicator |
str |
label of the treatment indicator variable |
file |
str |
name of the estimation specific init file |
logit |
bool |
If false: probit. Probability model for the decision equation |
nbins |
int |
Number of histogram bins used to determine common support (default is 25) |
trim_support |
bool |
Trim the data outside the common support (default is True) |
bandwidth |
float |
Bandwidth for the locally quadratic regression |
gridsize |
int |
Number of evaluation points for the locally quadratic regression (default is 400) |
reestimate_p |
bool |
Re-estimate \(P(Z)\) after trimming (default is False), not recommended |
rbandwidth |
int |
Bandwidth for the double residual regression (default is 0.05) |
derivative |
int |
Derivative of the locally quadratic regression (default is 1) |
degree |
int |
Degree of the local polynomial (default is 2) |
ps_range |
list |
Start and end point of the range of \(p = u_D\) over which the MTE shall be plotted |
In most empirical applications, bandwidth choices between 0.2 and 0.4 are appropriate for the locally quadratic regression. [11] find that a gridsize of 400 is a good default for graphical analysis. For data sets with less than 400 observations, we recommend a gridsize equivalent to the maximum number of observations that remain after trimming the common support. If the data set of size N is large enough, a gridsize of 400 should be considered as the minimal number of evaluation points. Since grmpy’s algorithm is fast enough, gridsize can be easily increased to N evaluation points.
The “rbandwidth”, which is 0.05 by default, specifies the bandwidth for the LOWESS (Locally Weighted Scatterplot Smoothing) regression of \(X\), \(X \ \times \ p\), and \(Y\) on \(\widehat{P}(Z)\). If the sample size is small (N < 400), the user may need to increase “rbandwidth” to 0.1. Otherwise grmpy will throw an error.
Note that the MTE identified by LIV consists of wo components: \(\overline{x}(\beta_1 - \beta_0)\) (which does not depend on \(P(Z) = p\)) and \(k(p)\) (which does depend on \(p\)). The latter is estimated nonparametrically. The key “p_range” in the initialization file specifies the interval over which \(k(p)\) is estimated. After the data outside the overlapping support are trimmed, the locally quadratic kernel estimator uses the remaining data to predict \(k(p)\) over the entire “p_range” specified by the user. If “p_range” is larger than the common support, grmpy extrapolates the values for the MTE outside this region. Technically speaking, interpretations of the MTE are only valid within the common support. In our empirical applications, we set “p_range” to \([0.005,0.995]\).
The other parameters in this section are set by default and, normally, do not need to be changed.
TREATED
The TREATED block specifies the number and order of the covariates determining the potential outcome in the treated state and the values for the coefficients \(\beta_1\). Note that the length of the list which determines the parameters has to be equal to the number of variables that are included in the order list.
Key |
Container |
Values |
Interpretation |
---|---|---|---|
params |
list |
float |
Parameters |
order |
list |
str |
Variable labels |
UNTREATED
The UNTREATED block specifies the covariates that a the potential outcome in the untreated state and the values for the coefficients \(\beta_0\).
Key |
Container |
Values |
Interpretation |
---|---|---|---|
params |
list |
float |
Parameters |
order |
list |
str |
Variable labels |
CHOICE
The CHOICE block specifies the number and order of the covariates determining the selection process and the values for the coefficients \(\gamma\).
Key |
Container |
Values |
Interpretation |
---|---|---|---|
params |
list |
float |
Parameters |
order |
list |
str |
Variable labels |
Further Specifications for the Parametric Model¶
DIST
The DIST block specifies the distribution of the unobservables.
Key |
Container |
Values |
Interpretation |
---|---|---|---|
params |
list |
float |
Upper triangular of the covariance matrix |
VARTYPES
The VARTYPES section enables users to specify optional characteristics to specific variables in their simulated data. Currently there is only the option to determine binary variables. For this purpose the user have to specify a key which reflects the corresponding variable label and assign a list to this label which contains the type (binary) as a string as well as a float (<0.9) that determines the probability for which the variable is one.
Key |
Container |
Values |
Interpretation |
---|---|---|---|
Variable label |
list |
string and float |
Type of variable + additional information |
SCIPY-BFGS
The SCIPY-BFGS block contains the specifications for the BFGS minimization algorithm. For more information see: SciPy documentation.
Key |
Value |
Interpretation |
---|---|---|
gtol |
float |
the value that has to be larger as the gradient norm before successful termination |
eps |
float |
value of step size (if jac is approximated) |
SCIPY-POWELL
The SCIPY-POWELL block contains the specifications for the POWELL minimization algorithm. For more information see: SciPy documentation.
Key |
Value |
Interpretation |
---|---|---|
xtol |
float |
relative error in solution values xopt that is acceptable for convergence |
ftol |
float |
relative error in fun(xopt) that is acceptable for convergence |
Examples¶
Parametric Normal Model¶
In the following chapter we explore the basic features of the grmpy
package. The resources for the tutorial are also available online.
So far the package provides the features to simulate a sample from the generalized Roy model and to estimate some parameters of interest for a provided sample as specified in your initialization file.
Simulation
First we will take a look on the simulation feature. For simulating a sample from the generalized Roy model you use the simulate()
function provided by the package. For simulating a sample of your choice you have to provide the path of your initialization file as an input to the function.
import grmpy
grmpy.simulate('tutorial.grmpy.yml')
This creates a number of output files that contain information about the resulting simulated sample.
data.grmpy.info, basic information about the simulated sample
data.grmpy.txt, simulated sample in a simple text file
data.grmpy.pkl, simulated sample as a pandas data frame
Estimation
The other feature of the package is the estimation of the parameters of interest. By default, the parametric model is chosen, in which case the parameter semipar in the ESTIMATION section of the initialization file is set to False. The start values and optimizer options need to be specified in the ESTIMATION section.
grmpy.fit('tutorial.grmpy.yml', semipar=False)
As in the simulation process this creates a number of output files that contain information about the estimation results.
est.grmpy.info, basic information of the estimation process
comparison.grmpy.txt, distributional characteristics of the input sample and the samples simulated from the start and result values of the estimation process
Local Instrumental Variables¶
If the user wishes to estimate the parameters of interest using the semiparametric LIV approach, semipar must be changed to True.
grmpy.fit('tutorial.semipar.yml', semipar=True)
If show_output is True, grmpy
plots the common support of the propensity score and shows some intermediate outputs of the estimation process.