Finance 520-1

GENERAL SEMINAR IN
Finance

Basic Time Series
Analysis

SPRING 2008

The purpose of the course is
to introduce, primarily at an intuitive level, the basic concepts of time series
analysis, as applied in finance and economics. Topics will include:

·
Basic concepts
and models: ergodicity, stationarity,
ARMA models, Vector Autoregressions, ARCH/GARCH

·
Foundational
results for time series models: Wold and Spectral
Decomposition Theorems

·
Filtering tools: Kalman Filter and other filters such as the band pass
filter and the Hodrick-Prescott filter

·
Statistical
Inference:

o Classical asymptotic theory: laws of large numbers;
central limit theorems; sampling results for Generalized Method of Moments and
Maximum Likelihood.

o Bayesian inference: priors and posteriors; the
marginal likelihood as a way to assess model fit; Monte Carlo Markov Chain and
Gibbs sampling; use of Bayesian methods to achieve parameter parsimony in
forecasting models, and for estimating structural models.

·
Nonstationarity in economic time series: unit roots, determination of
cointegration rank, deterministic trends.

·
Identification of
impulse response functions.

The textbook is Hamilton,
Time Series Analysis. Chapter 11 in Sargent’s
Macroeconomic Theory is also very useful.

Midterm and final from last
year:

The course meets 2:00-3:15
Mondays, and 4:00-5:45 on Wednesdays, in room 4214, Leverone.

Grades will be based on a
midterm (30%), homeworks (30%) and a final (40%).

The midterm will be
Wednesday, April 30.

There was no class on March
31 and there will also be no class on April 28. These will be made up by taking
two Friday homework sessions.

TA for the
course: Jingling Guan, office hours, Tuesday 4-5pm, Leverone
448.

My office
hours: Tuesday, 10-11am.

Homework #1, due April 16.

Homework #2,
due April 23 (data for homework).

Homework #3, due April 30.

Homework #4, due May 12 (the due date for this homework
has been pushed back).

Homework #5, due May 16

Homework #6, due May 22.

Homework #7, due June 1.

**Course Outline**

The first half of the course
studies Time Series models and their properties. The second part discusses
statistical inference about those models.

1.
Stochastic
Processes (chapter 3)

·
Stationarity, ergodicity

· White noise, AR, MA, ARMA
models

· Yule Walker equations

2. Linear Projections
(section 4.1, 4.2)

·
Necessity and
sufficiency of orthogonality conditions

·
Recursive
property of projections

·
Law of Iterated
Projections

·
Projections
versus regressions

3.
Wold decomposition theorem (page 109; Sargent)

4.
Vector autoregressions (sections 10.1-10.3)

5.
Spectral analysis
(Chapter 6, Sargent and Christiano-Fitzgerald, Technical
Appendix 1, page 71, and also.)

·
Lag operator
notation

·
Autocovariance generating function (section 3.6)

·
Complex numbers,
key integral property of complex numbers, inverse Fourier transform

·
Filters and band
pass filter

·
Spectral density

·
Spectral
decomposition theorem motivated according to band-pass filter approach in
Sargent (

1.
Kalman filter (sections 13.1, 13.2, 13.6)

·
State Space,
Observer System

·
Derivation of the
Kalman filter: ‘forward pass’,
‘backward pass’

2.
Maximum
likelihood estimation (chapter 5)

·
Gaussian maximum
likelihood

·
Conditional
versus unconditional maximum likelihood

·
Identification of
moving averages

·
Using the Kalman filter to build the Gaussian likelihood (section
13.4)

3.
Introduction to
generalized method of moments (section 14.1)

·
Basic ideas from
asymptotic sampling theory convergence in probability, mean square and
distribution; law of large numbers; mean value theorem (chapter 7)

·
Basic setup of
GMM: the ‘GMM orthogonality conditions’

·
Asymptotic
sampling distribution of a sample mean and spectral density at frequency zero
of GMM orthogonality condition

·
Optimal weighting
matrix

·
Sampling
properties of GMM estimator, hypothesis testing

4.
GMM and Maximum
Likelihood (chapter 14)

·
Preliminary:
martingale difference sequence

i.
definition

ii.
ARCH processes
(chapter 21)

iii.
Some asymptotic
results for m.d.s.

·
Demonstration
that ML is a special case of GMM

i.
Using GMM and
structure of Gaussian density to derive asymptotic sampling distribution of
Gaussian ML

ii.
Using GMM without
applying structure of the likelihood function to derive asymptotic sampling
distribution of quasi-ML

·
Asymptotic
efficiency of Gaussian ML (Cramer-Rao lower bound)
(this discussion is not in the book, but a good background discussion can be
found in an econometric text such as, e.g., Theil)

·
Robustness of GMM
versus efficiency of ML: example of estimating ARCH processes (chapter 21)

5.
Bayesian
inference (chapter 12)

·
Bayes rule, priors, posteriors

·
Estimating a mean
using iid observations with known variance, priors
and ‘padding data’

·
Markov Chain,
Monte Carlo Methods

i.
Approximating the
posterior distribution

ii.
Approximating the
marginal distribution and using it to evaluate model fit

6.
Nonstationarity (chapter 18, 19)

·
Trend stationarity

i.
Experiment:
impact on long-term forecast of a shock

ii.
Breakdown of
standard covariance stationary sampling theory

1. some preliminary results concerning sums

2. superconsistency of OLS slope estimator

·
Unit roots

i.
Experiment:
impact on long-term forecast of a shock

·
Distinction
between unit roots and trend stationarity based on
spectral density at frequency zero

·
Cointegration.

i.
Defined

ii.
The proper
handling of cointegration in VARs

7.
Very brief review
of chapter 22 discussion of time series analysis with changes in regime

·
One approach to
distinguishing whether recent change in volatility of macroeconomic and finance
data reflects change in the transmission mechanism or heteroscedasticity
in fundamental disturbances

·
Much data
(especially for emerging market economies) ‘looks’ like it reflects
periodic, large regime changes, suggesting the importance of chapter 22 methods