Identification Through Copulas

Recently I attended a talk (see Zimmer and Trivedi below) where a paper referenced work by Han and Vytlacil that used copulas to estimate probit models with dummy endogenous regressors. The seminar offered an extension to other types of models. However, here I wanted to summarize the approach more generally. You can find the referenced working paper below for more details, which I am told is forthcoming in the Journal of Econometrics.

Copula functions can be used to simulate a dependence structure independently from the marginal distributions.

Based on Sklar's theorem the multivariate distribution F can be represented by copula C as follows:

F(x₁…x_p) = C{ F₁(x₁),…, F_p(x_p); θ}

The parameter θ represents the dependence between the two distributions F1 and F2. Now let's set up the framework for what we are trying to model.

Suppose we want to predict some outcome Y. Let

Y = f(x,D)

where x is a vector of controls and D is a treatment indicator. We are interested in estimating the coefficient on D as our measure of the treatment effect. However, suppose that there is selection bias, such that those that choose to engage in the program indicated by D are more likely to have higher levels of Y regardless of treatment. (more on selection bias and unobserved heterogeneity and endogeneity).

We can model selection as follows:

D = g(x,z)

where x is a vector of controls and z is an instrument, correlated with the probability of D, but uncorrelated with selection. We can jointly model the outcome and selection functions using copulas where:

P(Y, D|x,z) = C{ F(.), G(.); θ} 

As it turns out, the term θ captures the dependence between outcome and selection allowing for unbiased estimation of treatment effects associated with D. Han and Vytlacil extend the results to cases without instruments.

References:

Han, S. and E. Vytlacil (2015). Identification in a generalization of bivariate probit models with dummy endogenous regressors.Working paper, University of Texas at Austin.

A Note on IdentiÖcation of Discrete Bivariate Copulas. Pravin K. Trivedi and David M. Zimmer August 5, 2016

Copula Based Regression

Recently I ran across an article in the Casualty Actuarial Society's publication Variance that discussed copula based regression. 

From the abstract: 

In this paper, we present copula regression as an alternative to OLS and GLM. The major advantage of a copula regression is that there are no restrictions on the probability distributions that can be used. In this paper, we will present the formulas and algorithms necessary for conducting a copula regression analysis using the normal copula. However, the ideas presented here can be used with any copula function that can incorporate multiple variables with varying degrees of association.

In the paper they outline a 3 step process for accomplishing this:

1) Assume a model for the joint distribution of all the variables (response and covariates)
2) Estimate the parameters of the model (the parameters for the selected marginal distributions and the parameters of the copula)
3) Compute the predicted values of Y given a set of covariates by using the conditional mean of Y given the covariates.

I'd also like to point out this interesting virtual course on copula regression from Dr. Edward Frees at the University of Wisconsin: https://sites.google.com/a/wisc.edu/jed-frees/multivariate-regression-using-copulas  
 
I have not had a chance to view these materials in detail, but absolutely think it could be valuable to anyone wanting to learn more about these methods.


Additional Reading:

Modeling Dependence with Copulas and R
Copula Based Agricultural Risk Models
Intro to Copulas using SAS 
Copulas, R, and the Financial Crisis

References: 
Parsa, Rahul A, and Stuart A. Klugman, "Copula Regression," Variance 5:1, 2011, pp. 45-54.
Link: http://www.variancejournal.org/issues/?fa=article&abstrID=6831
Link: http://www.variancejournal.org/issues/05-01/45.pdf
Presentation: http://www.casact.org/research/dare/documents/P1-Parsa_1.pdf

Modeling Dependence with Copulas and Quantmod in R

A while back I produced a few posts related to copulas. In A Basic Intro to Copulas I played around with some examples using SAS. This gave me some idea about how you could fit copulas based on parameters estimated from real data. And then I did some exploratory work using the copula package in R. In Copulas Functions, R and the Financial Crisis I mentioned some properties of copulas and how they were used in some of the risk modeling related to mortgage backed securities that went bad during the financial crisis. I used R to make some copula based plots....but it just did not click with me how I could use the copula functions in R to do anything concrete.

However, recently I ran across a post Modelling Dependence with Copulas in R, in which the author walks through a number of motivating examples giving some really powerful intuition about how copulas work, as well as a concrete application using stock data.

In this post (or really just the R code below) I modified the code somewhat to pull the data from yahoo finance using the quantmod package in R. Then I just pulled out the key segments of code that estimated the copulas, and concretely used parameters derived from the actual stock data to simulate returns for two stocks, Yahoo and Cree, Inc. Except I used Monsanto and ADM.

One thing discussed in the post that I found very interesting was related to the VineCopula application. This was used to help decide which copula specification might make the most sense for a given data structure.

"In the first example above I chose a normal copula model without much thinking, however, when applying these models to real data one should carefully think at what could suit the data better. For instance, many copulas are better suited for modelling asymetrical correlation other emphasize tail correlation a lot, and so on. My guess for stock returns is that a t-copula should be fine, however a guess is certainly not enough. Fortunately, the VineCopula package offers a great function which tells us what copula we should use. Essentially the VineCopula library allows us to perform copula selection using BIC and AIC through the function BiCopSelect()"

Below is the last plot from my code, of simulated returns vs actual daily returns for Monsanto and ADM for the last 3 years. I'll have to echo the above, I didn't use the VineCopula package to select a copula, but my intent here was just to look at some example code and a practical application of copulas using R with real data.

I won't share the previous plots etc. but refer you to the original post where they walk through the code almost line by line and demonstrate really well how this works. My code is below, but they have their entire program for their orignal post at github.





R Code: