State Space Models and the Kalman Filter
Eric Zivot April 9, 2006
1
State Space Models
A state space model for an Ndimensional time series yt consists of a measurement equation relating the observed data to an m dimensional state vector t , and a Markovia
Econ 584 Lab 3 Spring 2006
Eric Zivot Due: Wednesday, May 17
1
Reading
1. Hamilton, J. (1993), Time Series Analysis, chapters 15 and 17 2. Hayashi, F. (2000), Econometrics, chapter 9. 3. Zivot, E. and J. Wang (2002), chapter 4 in Modeling Financial Time S
Econ 584 Lab 4
Eric Zivot Due: Friday June 2, 2006
1
Part I: Analytic Exercises
Question 1 Consider the VAR model yt = Ayt1 + t , t iid (0, ) where A= 1. Find the eigenvalues of A 2. Find the roots of the characteristic polynomial det(I2 Az) = 0. and show
Notes on Forecasting
Eric Zivot April 8, 2006
1
Forecasting
Let cfw_yt be a covariance stationary are ergodic process, e.g. an ARMA(p, q) process with Wold representation yt = +
X j=0
j tj , t WN(0, 2 )
(1)
= + t + 1 t1 + 2 t2 + and let It = cfw_yt , yt
University of Washington Department of Economics Econ 584 Lab #1 Fun with ARMA models Readings: (1) (2) Data: (1) Hamilton, chapters 1-3, 4 (pgs 72 - 87, 108 - 113), 5. Zivot and Wang, chapter 2.
Spring 2006 Eric Zivot
Real gross domestic product (quarter
Department of Economics University of Washington Economics 584 Computer Lab #2 Suggested Solutions
Eric Zivot Spring 2006
Empirical Exercises
Comparing forecasting models
Simulated values from the model
yt = 1.2 yt 1 0.4 yt 2 + t , t ~ iid N (0, (0.5) 2 )
Econ 584 Final Exam Spring 2006
Eric Zivot Exam is due Friday June 9 at 9:00 am in my office or in my mailbox. June 7, 2006
Question 1 1. Give state space representations of the form yt = Zt t + dt + t , t iid N(0, Ht ) t = Tt t-1 + ct + Rt t , t iid N(0,
1
Unit Root Tests
Consider the trend-cycle decomposition of a time series yt yt = T Dt + T St + Ct = T Dt + Zt The basic issue in unit root testing is to determine if T St = 0. Two classes of tests, called unit root tests, have been developed to answer th
Trend-Cycle Decompositions
Eric Zivot
April 22, 2005
1
Introduction
A convenient way of representing an economic time series yt is through the so-called
trend-cycle decomposition
yt = T Dt + Zt
(1)
where T Dt represents the deterministic trend and Zt repr