Monte Carlo Studies

Monte Carlo methods have been used for centuries, but only in the past several decades has the technique gained the status of a full-fledged numerical method capable of addressing the most complex applications. The Monte Carlo method may be thought of as similar to a political poll, where a carefully selected statistical sample is used to predict the behavior or characteristics of a large group.

Enrico Fermi in the 1930's used Monte Carlo in the calculation of neutron diffusion, and later designed the Fermiac, a Monte Carlo mechanical device used in the calculation of criticality (The point at which a nuclear reaction is self-sustaining) in nuclear reactors.

In the 1940's, a formal foundation for the Monte Carlo method was developed by von Neumann, who established the mathematical basis for probability density functions (PDFs), inverse cumulative distribution functions (CDFs), and pseudorandom number generators. The work was done in collaboration with Stanislaw Ulam, who realized the importance of the digital computer in the implementation of the approach.

Before digital computers were available to the labs, "computer" was a job title. Parallel computing was done by rows and columns of mathematicians. The applications, which arose mostly from the Manhattan Project, included design of shielding for reactors.

Uses of Monte Carlo methods have been many and varied since that time. In the late 1950's and 1960's, the method was tested in a variety of engineering fields. At that time, even simple problems were compute-bound. Many complex problems remained intractable through the seventies. With the advent of high-speed supercomputers, the field has received increased attention, particularly with parallel algorithms which have much higher execution rates.

In econometrics, the general idea behind a Monte Carlo study is to (1) model the data-generating process, (2) generate several sets of artificial data, (3) employ these data and an estimator to create several estimates, and (4) use these estimates to gauge the sampling distribution properties of that estimator.

A useful reference is the paper:

Design and analysis of Monte Carlo experiments
Written by Kleijnen, J.P.C. (Tilburg University, Center for Economic Research)

Mean Square Error (MSE) and Variance

The difference between the variance of an estimator and its MSE is that the variance measures the dispersion of the estimator around its mean whereas the MSE measures its dispersion around the true value of the parameter being estimated. for unbiased estimators they are identical.

Biased estimator with smaller variances than unbiased estimators are easy to find. The MSE estimator has not been as popular as the best unbiased estimator because of the mathematical difficulties in its derivation. Furthermore, when it can be derived its formula often involves unknown coefficients (the value of beta), making its application impossible. Monte Carlo studies have shown that approximating the estimator by using OLS estimates of the unknown parameters can sometimes circumvent this problem (a little confused here, using approximated OLS estimates to substitute the real beta?)

Note: Weighted Square(d) Error Criterion can be a very interested topic to explore!
Peter Kennedy: When the weights are equal, the criterion is the popular mean square error (MSE) criterion. It happens that the expected value of a loss function consisting of the square of the difference between beta and its estimate (i.e. the square of the estimation error) is the same as the sum of the variance and the squared bias. 
Please refer to following derivation:

OLS: It is not the case that the OLS estimator is the minimum mean square error estimator in the Classic Linear Regression model. Even among linear estimators, it is possible that a substantial reduction in variance can be obtained by adopting a slightly biased estimator.

Toronto Milk Producers

The Toronto Sunday World, Mar 23, 1914.

A meeting of the Toronto Milk and Cream Producers' Association will be held at the Labor Temple on Thursday, commencing at 2 p.m. The meeting is called to discuss and decide on prices of milk and cream for the ensuing season, May to October, and any other business in the interest of the association. In the evening a banquet will be held at the Grand Union Hotel.

The Story of Maximum Likelihood

The theory of maximum likelihood is very beautiful indeed: a conceptually simple approach to an amazingly broad collection of problems. This theory provides a simple recipe that purports to lead to the optimum solution for all parametric problems and beyond, and not only promises an optimum estimate, but also a simple all-purpose assessment of its accuracy. And all this comes with no need for the specification of a priori probabilities, and no complicated derivation of distributions. Furthermore, it is capable of being automated in modern computers and extended to any number of dimensions. Maximum-likelihood estimation was recommended, analyzed and vastly popularized by R. A. Fisher between 1912 and 1922 (although it had been used earlier by Gauss, Laplace, Thiele, and F. Y. Edgeworth). Reviews of the development of maximum likelihood have been provided by a number of authors.


When we analyze an analysis of variance or linear regression, typically we estimate parameters for the model using the principle of least squares. The idea of least squares is that we choose parameter estimates that minimize the average squared difference between observed and predicted values. That is, we maximize the fit of the model to the data by choosing the model that is closest, on average, to the data.

For many other procedures such as logistic, Poisson, and proportional hazards regression, least squares usually cannot be used as an estimation method. Instead, most often we turn to the method of maximum likelihood. In maximum likelihood estimation, we search over all possible sets of parameter values for a specified model to find the set of values for which the observed sample was most likely. That is, we find the set of parameter values that, given a model, were most likely to have given us the data that we have in hand.

By way of analogy, imagine that you are in a jury for a civil trial. Four things are presented to you in the course of the trial: 1) charges that specify the purpose of the trial, 2) prosecution's version of the truth, 3) defendant's version of the truth, and 4) evidence. Your task on the jury is to decide, in the context of the specified charges and given the evidence presented, which of the two versions of the truth most likely occurred. You are asked to choose which version of the truth was most likely to have resulted in the evidence that was observed and presented.

Analogously, in statistical analysis with maximum likelihood, we are given: 1) a specified conceptual, mathematical, and statistical model, 2) one set of values for the parameters of the model, 3) another set of values for the parameters of the model, and 4) observed data. We want to find the set of values for the parameters of the model that are most likely to have resulted in the data that were actually observed. (We do this by searching over all possible sets of values for the parameters, not just two sets.)

In analysis of variance or linear regression, we measure the fit of the model to the data using the regression sum of squares. With maximum likelihood, the likelihood measures the fit of the model to the data, Therefore, we want to choose parameter values that maximize the likelihood. In analysis of variance or linear regression if we want to compare the fit of two models, we form the ratio of two mean squares to yield an F-test . With maximum likelihood, we do this by forming the ratio of two likelihoods to yield a chi-square test.

Asymptotic Properties

Since econometricians quite often must work with small samples, depending estimators on the basis of their asymptotic properties is legitimate only if it is the case that estimators with desirable asymptotic properties have more desirable small-sample properties than do estimators without desirable asymptotic properties.

Rejection From Rice

We regret having to inform you that Rice University cannot offer you admission for graduate study.

You can be assured that your application received very careful consideration.

Our decision is based on high standards of selectivity and on the constraints of space, and faculty.

For these reasons, we must limit the number of admissions in all departments.

The other members of the departmental graduate committee join me in wishing you success in your

future endeavors.

Yours sincerely,

Simon Grant, Director

Economics Graduate Program