Finance 520-1

GENERAL SEMINAR IN
Finance

Basic Time Series
Analysis

SPRING 2007

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.

**Homeworks**

The following homeworks
invite students to explore time series concepts more deeply. In addition, the
homeworks provide practical, data applications.

Homework
#1: exercises designed to clarify the proof of the Wold decomposition
theorem

Homework
#2: questions designed to familiarize students with VARs and the inverse Fourier
transform of the spectrum. Also, the question asks students to use VARs to
study the dynamic properties of aggregate quantities and rates of return, and
to do a particular test of the expectations hypothesis of the long term
interest rate.

Homework
#3: exercise for students to gain experience with the use of frequency
domain tools. This involves the (King-Rebelo) derivation of the
Hodrick-Prescott filter in the frequency domain; the derivation of a seasonal
adjustment procedure using projection methods; and the derivation of the
classic Sims result concerning the plim of a miss-specified distributed lag
regression. Substantively, we re-examine the expectations hypothesis of the
long term interest rate from the perspective of the frequency domain; derive
the classic (initially counterintuitive) result that optimal seasonal
adjustment results in dips at the seasonal frequencies; show that the HP filter
is a ‘high pass filter’; explore conditions under which the sum of
distributed lag coefficients is well-estimated.

Homework
#4: this is designed to take students through

Homework #5 will expose
students to ideas in:

Boivin and Giannoni, DSGE Models
in a Data-Rich Environment

Bekaert,
Cho, and Moreno, New-Keynesian Macroeconomics and the Term Structure, 2006.

Data
for homework #5

Homework #5: The first two questions in this
three-part homework have two purposes: to let students see once again the power
of the Kalman filter by exposing them to some promising recent research that
uses it. The homework summarizes the recent work by Bernanke, Boivin, Eliasz
and Giannone which exploits the fact that in practice there are many different
variables which measure the same basic economic concept (e.g., there are
several measures of hours worked, there are several measures of inflation, output, etc.). The homework also shows
how to evaluate the term structure implications of fully specified, dynamic
equilibrium models, along the lines of recent work by Bekaert, Cho and

Homework
#6: the first part of this homework is designed to show students another
example of how useful the tools of GMM can be for deriving the asymptotic
sampling distribution of an estimator. The example used is the skewness
statistic when the underlying data are iid. This statistic is of substantive
interest and the exercise also allows the student to discover a surprising
result – the sampling distribution of the skewness statistic is very
sensitive to whether the mean of the underlying process is estimated or known.
The next step is to apply GMM to testing for skewness in the data used in
homework #2. Recognizing that asymptotic sampling theory may be a bad
approximation in finite samples, students are asked to execute a suitably
constructed

Homework
#7: This homework analyzes
historical time series on Indian per capita GDP going back to 1884, to evaluate
the Niall Ferguson hypothesis that

**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 and Sargent)

·
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 (

6.
Kalman filter
(sections 13.1, 13.2, 13.6)

·
State Space,
Observer System

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

7.
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)

8.
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

9.
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)

10.
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

11.
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
– we skipped this!

i.
Defined

ii.
The proper
handling of cointegration in VARs

12.
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