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**PENN STATE 2024 LECTURES ON NUMERICAL METHODS **

**KEN JUDD **

## Overview

I will be visiting Penn State March 11 – March 22, 2023. My lectures will be

March 12, 14, 19 and 21

6pm-7:30pm

113A Chamber Building.

Here is an overview of my planned lectures:**Tuesday, March 12:** Discussion of unconstrained optimization methods applied to maximum likelihood and moment estimators:

A systematic approach of how to proceed when you need to find an optimum and don’t have a good initial guess.

The nested fixed point method (NFXP)

The surrogate method of optimization

Parallelization**Thursday, March 14: **Structural estimation, NFXP and MPEC:

Discussion of efficient methods for NFXP.

Introduce mathematical programming with equilibrium constraints (MPEC) and compare the two approaches.**Tuesday, March 19:** BLP models: The advantages of MPEC (Dube, Fox and Su) and the importance of quadrature rules (Skrainka and Judd).

Estimation of the discount factor in the Zurcher bus model. It is greater than 1 (Reich and Mueller)**Thursday, March 21:** Solving and estimating static games.

Carlos Rangel will be at Penn State March 11-15. He will be available to describe the programming details behind the first two lectures.

Below are the topics I will cover. The links below take you to the papers related to the topics. I certainly cannot cover all of them in detail. I have included links presentations by Che-Lin Su and Ben Skrainka which do present the details.

**These topic descriptions are incomplete. I will add slides and code soon. I am also flexible. I can adjust my lectures to suit student interest.**

**1: Unconstrained Optimization**

Here is a basic lecture describing unconstrained optimization methods such as maximum likelihood methods and moment estimators. This will include the Nested Fixed Point method (which computational mathematicians call “implicit programming”).

Software for computing the objective: write your own code, or use state-of-the-art methods to solve equations

Optimization methods: Search, derivative-free, gradient, Newton

Surrogate methods: polynomial approximation, sparse grids

Parallelism: all methods require the evaluation of the objective at many parameter vectors, implying that parallelism can be easily exploited

**2: Structural Estimation: NFXP vs MPEC vs Surrogate**

We first compare nested methods to basic constrained optimization.

**Nested Methods and a Superior Alternative**

**Evolution of Su-Judd **

The SJ papers are versions of the Su-Judd paper on MPEC, a small piece of which was published in Econometrica as a note. Here are the four versions of Su-Judd, leading to the 2012 Econometrica paper.

**2008 SJ Original submission to ECTA**

**2010 SJ First revision for ECTA**

**Che-Lin Su Presentations**

**Surrogate Method**

Markus Trunschke, Gregor Reich and I have applied the surrogate method to the Zurcher bus model.

**Rust comment:**

In the 2014 and 2015 versions, Rust et al. compares NFXP and MPEC. They argue that NFXP is more stable and faster than MPEC. However, Che-Lin Su showed that the comparisons were not valid.

**Rust Iskhakov Schjerning June 2014**

**3: BLP**

**3a: Model description**

The standard description of BLP has always confused me. Carlos Rangel and I have worked on a cleaner presentation that makes clear the computational details.

**3b: NFXP vs MPEC**

Nevo’s A Research Assistant’s Guide to Random Coefficients Discrete Choice Models of Demand describes the BLP model and a commonly used computational method. Che-Lin and I looked at Nevo’s code and found some obvious weaknesses. In particular, the stopping rules were too loose. I had Che-Lin use MPEC to solve the Nevo problem. He found a better value for the objective and was happy because he was trained as a computational mathematician. I told him that was not enough, and that he should check to see if Nevo’s estimate was statistically different than his result. It was. I decided my job was done. Che-Lin went on to write a paper with Dube and Fox.

AMPL code for MPEC applied to BLP.

Dubé, Fox, & Su (2012) compares Nevo to an MPEC approach

**3c: BLP: quadrature vs MC**

We first examine the performance of simulation methods for a simple portfolio problem. It shows that using Monte Carlo integration to approximate the objective in an optimization problem is unreliable.

Portfolio problem PSU 2019 nb.pdf

Skrainka-Judd (2012) shows that these problems become serious for estimating BLP models, but also shows that numerical quadrature formulas are far more accurate and reliable. Essentially, Monte Carlos methods have a tendence to underestimate standard errors. Using Monte Carlo integration helps with publishing papers because (1) you are more likely to get identification, and (2) you can try different seeds for your RNG and use the one that gives you results the editor will like. Editors will likely detect “regression fishing” but I doubt that they are as good at detecting “seed fishing”.

**4:** **Infinite- versus Finite-Horizon DP**

The Rust bus model assumes that Zurcher will live forever, whereas his horizon was certainly finite. Therefore, the infinite-horizon assumption is a misspecification. Does that matter? Also, Rust has claimed that the discount factor is difficult to estimate even though it is a very important parameter.

A paper by Mueller and Reich has addressed both questions. They show that the discount rate likely exceeds 1. Contrary to standard beliefs in economics, this does not make solving the DP problem difficult. They use methods from the 1970’s to solve DP problems with arbitrary discount factors as part of an MPEC estimation approach. This is made easy by the use of homotopy solution methods. Furthermore, they show that while assuming an infinite horizon is wrong, it has little impact on the estimates because the DP solution is the limit of finite-horizon DP specifications.

They also show that there is a break in the estimated (real) discount factor at the same time that Volcker took over monetary policy and drove inflation down and increased real interest rates.

**5: Static games**

We begin with a simple example of using MPEC to estimate a dynamic game. Che-Lin Su examined different methods for estimating static games:

Estimating discrete-choice games of incomplete information: Simple static examples by Che-Lin Su

**Office Hours:**

I will also be available for office visits. I ask that you first send me something written that describes the computational aspects of what you are working on or want to work on. Do not assume that I have read the papers related to your research. You should be able to explain the mathematical and computational structure of your problems in a way that I can understand without knowing published economic papers (which are often poorly written).

Basically, the part of your research I can help with is the computational methods you use. Feel free to send me your writeup NOW so that I will be acquainted with your work when we meet.

Looking forward to seeing you soon.