1. Use the money market and FX diagrams to answer the following questions about the
relationship between the British pound () and the U.S. dollar ($). The exchange rate
is in U.S. dollars per British pound, E$/. We want to consider how a change in the
U.S

B. An Illustrated Example
1
The Cobweb Model
Consider the following model for the supply and
demand of a product say wheat:
st b pt -1 t
d t a pt
0
0
d t st
Suppliers make their production decisions based
on previously observed price, but supply is
af

More on Cointegration
Allan Brunner
Macroeconometrics
Johns Hopkins University
Overview
The Engle-Granger Approach Using
Interest Rate Data
The Johansen Approach
An Illustration of the Approach
2
I. A Look at Interest Rates, using the
Engle-Granger App

More on VAR Models
Allan Brunner
Macroeconometrics
Johns Hopkins University
Overview
Estimation of VAR Models
Approaches to Identifying VAR Models
Using the VAR Toolkit
2
I. Estimating VAR Models
3
Estimating the Reduced-Form Model
yt a10
eyt
a11 a

V. Analysis Using a
Structural VAR Model
1
Structural Vector Moving-Average Models
Suppose that we have estimated and identified a
structural VAR model. The next step is to study its
properties.
It will be useful to convert the structural VAR into a
str

Introduction to ARMA Models
Allan Brunner
Macroeconometrics
Johns Hopkins University
Overview
Autoregressive Moving-Average Models
Review Stationarity Conditions for ARMA
Models
Tools for Identifying and Estimating ARMA
Models
Autocorrelation function

Introduction to Cointegration
Allan Brunner
Macroeconometrics
Johns Hopkins University
Overview
Motivation
Cointegration and Common Trends
Cointegration and Error Correction
The Engle-Granger Methodology
2
I. Motivation
3
Predictions from Growth Theor

A More General ECM
Consider the following model:
rSt 10 S (rLt-1 - rSt -1 ) 11 rSt -1 12rLt-1 e St
rLt 20 - L (rLt-1 - rSt -1 ) 21rSt -1 22rLt-1 e Lt
This version of the model allows for more general
short-term dynamics, but maintains the same LR
behavi

Models with Time-Varying Parameters
Allan Brunner
Macroeconometrics
Johns Hopkins University
Overview
Introduction
ARCH and GARCH Models
ARCH-M Models
Regime Switching Models
Other Nonlinear Models
2
I. Introduction
3
Time-Invariant Parameters
So fa

III. Dickey-Fuller Tests
1
How Do We Distinguish Between Stationary
and Non-Stationary Variables?
In the simplest case, we are trying to distinguish
between the following two representations:
yt a 0
y t -1 t
y t a 0 a 1 y t -1 t
This suggests a simple

IV. The Engle-Granger Methodology
1
Step 1
Suppose we have two variables y and z.
The first step is to use the ADF test to see if
both y and z are I(1) variables.
Unless both are I(1) variables, the two
variables cannot be cointegrated.
If we have thr

More on Trends in Univariate Models
Allan Brunner
Macroeconometrics
Johns Hopkins University
Overview
Quick Review
Two Problems with UR Tests
Low Power
Fooled by Structural Breaks
Univariate Decompositions
2
Quick Review
p
y t a 0 a 2 t yt -1
y
i
t

IV. Estimation and Identification
of VARs
1
VAR Analysis
1. Estimate the reduced-form VAR model
2. Place enough restrictions on the model to
identify a structural VAR model
3. Use analytical tools to study the dynamic
properties of the VAR model:
Impulse

II. Stationarity
1
Different Definitions of Stationarity
In this course, we will consider a time-series to be
stationary if its covariance stationary if its
mean and all autocovariances are unaffected by a
change of time origin:
E ( yt ) E ( y t s )
E[(

C. Solving Homogeneous
Difference Equations
1
The Case of a 2nd Order DE
The homogeneous equation is
y t - a 1 y t -1 - a 2 y t -2 0
Given our previous findings, we can guess
(correctly) that the homogeneous solution has the
form
h
t
yt A
Plugging thi

II. Identification Schemes
1
The AS-AD Model
(initial equilibrium)
2
Effects of a AD Shock
3
Effects of a AS Shock
4
What Do We Expect the IRFs
to Look Like?
5
Various Types of Identification Schemes
Short-run restrictions shock 1 does not affect
variabl

III. Pitfalls of Estimating ARMA Models
1
Digression: The Lag Operator
Ly t yt 1
2
L y t yt 2
y t a 0 a 1 yt 1 t
y t a 0 a1 Ly t t
(1 a1 L) y t a 0 t
2
AR and MA Models Can Look Similar
Suppose that the true model is an AR(1)
y t a 0 a 1 yt 1 t
An MA(q)

II. ARCH and GARCH Models
1
Heteroskedasticity
Homoskedasticity is the assumption of a
constant (time-invariant) variance.
However, we saw evidence of
heteroskasticity in the time series of daily
changes in stock prices.
If you are an investor, you wil

Solving Difference Equations
Allan Brunner
Macroeconometrics
Johns Hopkins University
Overview
An Alternative Methodology
Illustration: The Cobweb Model
Finding Homogeneous Solutions
Finding Particular Solutions
Method of Undetermined Coefficients
U

Introduction to VAR Models
Allan Brunner
Macroeconometrics
Johns Hopkins University
Overview
Approaches to Estimating Multivariate
Models
Sims (1980) Critique
Intro to VAR Models
Structural vs. Reduced-Form Models
Dynamics and Stability
Estimation a

II. Unit Roots and Regression Analysis
1
The Classic Regression Model
Consider the following regression:
y t a 0 a 1 zt e t
Does it matter whether y and z have deterministic
or stochastic trends?
Yes!
Inference in the classic regression model requires

II. Some Simulation Results
1
Why Simulation?
We could take any macroeconomic time
series and try to model it. But, we dont
know what ARMA(p,q) model generated
the data.
A more enlightening approach is to simulate
some data with a known DGP and try to
f

So, We Have Two Empirical Problems
First, we have a low powered test,
especially when the largest root is close
to 1.
Plus, the critical values depend on the
presence of a constant and a time trend.
How does one avoid utter failure?
1
Cambell and Perro

Introduction to
Trends in Univariate Models
Allan Brunner
Macroeconometrics
Johns Hopkins University
Overview
Univariate models with time-dependent
means i.e., non-stationary
Deterministic trends
Stochastic trends
Spurious Regressions
Various tests f

III. The Autocorrelation and Partial
Autocorrelation Functions
1
Autocorrelations are Normalizations of
Autocovariances
Autocovariances:
0
2
y
2
E[( yt ) ])
j E[( yt )( yt j )] for j 0
Autocorrelations:
0 0 /
2
y
j j /
2
y
1
for j 0
2
The Autocor

D. Finding Particular Solutions
1
Ingenuity and Perseverence
Depends heavily on the form of cfw_xt
Example: xt= a0
y t a 0 a 1 y t -1 a 2 y t -2 a 3 y t -3
Intuition tells us that with such a forcing process,
an unchanged value for y should solve the
e

II. Structural Breaks
1
UR Tests Are Affected by Structural Breaks
See Figure 4.10
Perron (1989) showed that the DF is very
susceptible to the presence of structural
breaks.
Stock market crash of 1929 (2008?)
Oil price shocks in 1970s.
2
Perrons Test

II. Sims Critique of Traditional
Macroeconometrics
1
Sims (1980) Critque
Traditional hypothesis tests and forecasts
were conducted using large-scale macro
models.
Each equation in the model was estimated
separately. The equations were then treated
as a

V. Forecasting (Lite)
1
Two Uses for Forecasting
Forecast! - use models to predict future
values
Also, forecasting is another tool for
evaluation models
Note: The text describes a number of tests
for evaluating models. None of these are
available in Ev

IV. A Model of the US PPI
1
US PPI in log levels
2
First Difference The Data
The data are clearly not stationary. The US
PPI trends upward, although with a
different rate of growth in different periods
We will discuss detrending methods in
Chapter 4
Fo

III. Introduction to VAR Models
1
Simple 1st-Order VAR
yt b10 - b12 zt 11 y t-1 12 zt-1 yt
zt b20 - b21 yt 21 yt-1 22 zt-1 zt
Note that both variables are treated endogenously
each appears on the RHS of the other equation.
This feature, along with unco

Point #2
Variables can only be cointegrated, if they have the
same order.
Example #1: M2 looks to be an I(2) variable. That is,
money growth appears to have a unit root. The CPI
has similar properties: Inflation appears to be I(1).
M2 and the CPI could

III. Cointegration and Error Correction
1
Example: Interest Rates
The term structure of interest rates suggests
that a long-term rate should be a function of
expected short-term rates.
What forces drive a wedge between these
two rates? Is it mostly vola

III. ARCH-M Models
1
Engle, Lilien, and Robins (1987)
ELR extended the ARCH framework to allow the
conditional mean to depend on the conditional
variance.
This approach is especially relevant to modeling asset
prices, where investors need to be compensa

II. Cointegration and Common Trends
1
Stock and Watson (1988)
Stock and Watson stressed that cointegration
between two or more variables implies that they
have a common trend.
Suppose we have two variables that are I(1).
y t yt e yt
z t zt e zt
Where t