Wouter J. den Haan - CFM-LSE Numerical Summer School

Essentials

Instructors:

• Wouter J. den Haan, two days
• Petr Sedlacek, three days

Prerequisites!!!: This course is not for everybody. It is important that you have a solid knowledge of macroeconomic models taught at the graduate level before you start a course on numerically solving these models. So make sure to read the course prerequisites carefully. If you don't have the right background yet, then it is better to wait a year.

• Required knowledge about dynamic models:
  • Know what an Euler and a Bellman equation are; being able to derive them.
  • Know what state variables are.
  • Understanding of key economic mechanisms in representative-agent models such as consumption smoothing and precautionary savings.
• You should also be able to do some Matlab programming.

Course outline:

Monday - Solving and analyzing your first dynamic model

• Overview: This morning we teach you how to use Dynare to solve dynamic stochastic macroeconomic models. We also teach you how to incorporate Dynare programs into bigger Matlab programs so that you can, for example, loop quickly over different parameter values. We also give you the information to calculate standard errors for business cycle statistics like the ratio of two standard deviations of HP-filtered series. With knowledge of both the theoretical and the empirical tool, you will be able to build a theory, calculate its key properties, and confront these properties with the analogues in the data.
• Topics:
  • State variables
  • Policy rules (i.e. the recursive solution to dynamic models)
  • Impulse response functions
  • Perturbation analaysis
  • Certainty equivalence
  • Dynare
  • Using the homotophy idea to get good initial values for the steady state (often the hardest part of running Dynare)
  • Parameter values and properties of basic neoclassical model
  • Stylized facts
• Applications & exercises: In the afternoon, you are asked to write a program to investigate how one can generate sufficient volatility in the unemployment rate in a simple matching model (i.e. how to solve the Shimer puzzle).
• Other: Dynare uses perturbation analysis to solve dynamic models. This year we do not teach the underlying theory in class, but we provide you with the notes to understand it.

Tuesday - Key tools from the numerical approximation literature and projection methods

• Overview: In the morning, we teach you numerical integration and function approximation. These tools are used inside and oustide economics. Within economics they are not only used to solve models, but they are also widely used in, for example, econometrics. Numerical integration makes it possible to approximate the conditional expectation with just a few lines of code. When you are familiar with numerical integration and function analysis, then projection methods are quite easy to understand. In contrast to the approximation technique underlying Dynare, projection methods are global approximation methods. It is a bit more involved to program a projection method, but for some models it is a much better choice.
• Topics:
  • Numerical integration (Gaussian quadrature)
  • Function approximation (Splines & Polynomials)
  • Projection methods
  • Endogenous grid points
  • Fixed point iteration
  • Time iteration
• Applications & exercises: In the afternoon, you are asked to solve a simple model in which the risk premium varies over the business cycle.This is difficult to do for perturbation analysis since uncertainty affects the solution in only a limited way. In fact, one needs at least third-order perturbation to have any time-variation in the risk premium. In contrast, this is easy when using projection methods.

Wednesday - Topics

• Overview:Perturbation is cheap to implement, but less/not appropriate when nonlinearities are important, when there is substantial volatility (e.g. idiosyncratic risk), or when there are occasionally binding constraints. Projection methods can in principle deal with all these complexities, but is more expensive to implement and very costly if there are many state variables. We will teach you the Parameterized Expectations Algorithm (PEA) which is a projection method but is less computer-intensive than classic projection methods. We discuss the version of PEA proposed by Den Haan and Marcet, but also the recent version of Maliar, Maliar, & Judd which contains a simple but powerful improvement. After the discussion of Value Function Iteration you will be familiar with most of the available tools to solve dynamic models. We will discuss the advantages and disadvantages of the different methods. As part of this discussion we will talk about accuracy tests, occasionally binding constraints, and penalty functions. There is one topic that has to be discussed in any course on solving dynamic models and that is stability and uniqueness of the solution. We will teach you what the Blanchard-Kahn conditions have to say about this. Explaining the Blanchard-Kahn conditions automatically teaches you about sun spots or self-fullfilling expectations an exciting part of economics.
• Topics:
  • Parameterized Expectations Algorithm
  • Value Function Iteration
  • Accuracy tests: Euler errors, Dynamic Euler equation test, DHM statistic
  • Occasionally binding constraints and penalty functions
  • Blanchard-Kahn conditions
  • Sun spots and self-fulfilling expectations
• Applications & exercises: In the afternoon we use PEA to look at another asset pricing model and check for the possiblity of sun spots.

Thursday - Kalman filter & full information methods

• Overview: In the morning, we teach you possibly the most powerful tool in economics, namely the Kalman filter with which you can estimate unobserved components. This tool also plays a key role in the estimation of dynamic models with full information methods like Maximum Likelihood and Bayesian estimation. Full information methods estimate the model use the complete specification including specifications of the distribution of all the exogenous shocks. We will teach you how to calculate the likelihood of a model given the data. More importantly, we discuss some tricky issues you will run into in practice, namely the singularity problems which will restrict the number of observables you can use.
• Topics:
  • Kalman filter
  • State space form
  • Maximum Likelihood
  • Avoiding the singularity problem
• Applications & exercises: In the afternoon, you are asked to estimate a time series model of the labor market and document how matching efficiency dropped during the Great Recession in the United States.

Friday - Bayesian estimation

• Overview: With the tools learned on Thursday, Bayesian estimation is actually not that difficult, since Bayesian estimation is like Maximum Likelihood with a bit more information (the prior). One challenge of Bayesian estimation is the evaluation of the posterior. To be able to do this, we teach you Markov Chain Monte Carlo (MCMC) techniques. We also discuss the advantages and disadvantages of this method relative to its alternatives, such as Maximum Likelihood, Generalized Method of Moments (GMM), and Simulated Method of Moments (SMM).
• Topics:
  • Bayesian estimation
  • MCMC
  • Metropolis Hastings
  • Maximum Likelihood, GMM, SMM
• Applications & exercises: In the afternoon you are asked to estimate a dynamic model, extract the unobserved components, and discuss which shock was most important for which recession

LSE Economics