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Advanced Seminar in
Macroeconomic Research I
Ruhr-University Bochum
Faculty of Management and Economics
Chair of International Economics
Joscha Beckmann
Lecture
• Joscha Beckmann
• Room GC 3/145; Email: [email protected]
• Office hours: By Appointment
Contact Details Lecturers
Advanced Seminar in Macroeconomic
Research I 2
Enrolment
• Due to the special character of the seminar, enrolment is
limited
• Please sign in on Moodle
• Registration via Flex Now is required
3
Dates
Friday, October 20, 10 am - 6 pm
Friday, November 3, 10 am - 4 pm
Friday, November 17, 10 am - 6 pm
Friday, December 8, 10 am - 4 pm
Thursday, January 25, 10 am - 2 pm
Friday, January 26, 10 am - 6 pm
Room: GC 02/120 and GC 03/42 only on January 25th
Exam: To be announced
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General Information
Aims and Scopes
Analyzing long-run and short-run dynamics on
financial markets
Introducing nonlinearities
Forecasting
Focus on exchange rates and interest rates
Empirical Estimation
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Research I
General Information
Lecture + Tutorial
Theoretical background
Introduction of empirical methods
Presentation of selected results
Implementation of empirical methods
Software: RATS. Details to be announced
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Research I
General Information
Course material on Moodle
Slides
References
Data and Code
Additional Material
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Research I
Course Outline
Chapter 1: Cointegration
1.1 Introduction and definitions
1.2 The Engle-Granger methodology
1.3 The Cointegrated VAR approach
1.3.1 Basics
1.3.2 Modelling Cycle
1.4 Case Study: The Exchange Rate Disconnect Puzzle
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Course Outline
Chapter 2: Modelling Nonlinearities
2.1 Theshold Models
2.2 Markov-Switching Models
2.3 Case Study: Interest Pass-Through in the EMU
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Course Outline
Chapter 3: Forecasting
3.1 Introduction and Definitions
3.2 Forecasting Evaluation
3.4 Case Study: Forecasting Exchange Rates
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General Information
References
Enders (2014): Applied Econometric Times
Series, Wiley.
Juselius (2006): The Cointegrated VAR Model:
Methodology and Applications, Oxford University
Press.
Selected empirical studies
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Introduction and Motivation
The influential study by Meese and Rogoff (1983), which
suggests that traditional exchange rate models are
unable to outperform a random walk in terms of
forecasting, has triggered different lines of research
All of them basically deal with the fragile relationship
between fundamentals and exchange rates. The so-
called exchange rate disconnect puzzle is widely viewed
as one of the most important questions in international
economics
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Research I
Introduction and Motivation
Consider exchange rates as a starting point
Fluctuations of nominal exchange rates higher compared
with the variations of fundamentals
Nonlinear relationships between exchange rates and
fundamentals? (De Grauwe and Vansteenkiste, 2007)
In-sample vs. out-of sample
Different fundamental exchange rate models
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Research I
Introduction and Motivation
Purchasing Power Parity
Purchasing power parity (PPP) serves as a condition of
equilibrium for good markets (Dornbusch, 1976a;
Frenkel, 1976; Bilson, 1978).
𝑠𝑡 = (𝑝𝑡 − 𝑝𝑡f)
𝑠𝑡 Nominal exchange rate expressed as the domestic
𝑝𝑡 and 𝑝𝑡𝑓
as logarithms of domestic and foreign price
levels.
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Introduction and Motivation
Empirical questions when analyzing exchange rates
Is there a long-run relationship between the nominal
exchange rate and fundamentals?
How can mean-reversion of real exchange rate be
analyzed ?
Are fundamental models able to forecast exchange
rates?
Standard regressions not suefficient to analyze such
issues15
Advanced Seminar in Macroeconomic
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Chapter 1: Cointegration
1.1 Introduction and definitions
1.2 The Engle-Granger methodology
1.3 The Cointegrated VAR approach
1.3.1 Basics
1.3.2 Modelling Cycle
1.4 Extensions: Nonlinear cointegration
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1.1.1 Stationary vs. Non-stationary series
• In a nutshell, a time series is stationary if it‘s mean and
all autocovariances are unaffected by a change of time
origin
• See Enders (2014) for a formal definition
• Main idea of ARIMA models: Achieve stationarity through
differencing
• A series is integrated of order d if it must be differenced d
times to achieve a stationary time series
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1.1.2 Stochastic vs. deterministic trends
• Consider the following Random Walk process
• Current value of 𝑌𝑡 fully depends on the past value and a
the white noise term 𝜀𝑡.
• Shocks no longer vanish over time. 𝑌𝑡 may also be
written as
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𝑌𝑡 = 𝑌𝑡−1 + 𝜀𝑡
𝑌𝑡 = 𝑌0 +
𝑖=1
𝑡
𝜀𝑡
1.1.2 Stochastic vs. deterministic trends
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Figure 4: A stationary time series
1.1.2 Stochastic vs. deterministic trends
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Figure 5: A random walk
• If a constant term is included, the following
representation arises:
• The path depends on a constant and a white noise
process
• This process is called a Random Walk with drift.
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𝑌𝑡 = 𝑌𝑡−1 + 𝜇0 + 𝜀𝑡
∆𝑌𝑡 = 𝜇0 + 𝜀𝑡
1.1.2 Stochastic vs. deterministic trends
• The representation
shows 𝑌𝑡 that does not converge to a constant value.
• 𝑌𝑡 follows two trends:
The deterministic trend 𝜇0𝑡 which increases linear over
time
The stochastic trend 𝑖=1𝑡 𝜀𝑡
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𝑌𝑡 = 𝑌0 + 𝜇0𝑡 +
𝑖=1
𝑡
𝜀𝑡
1.1.2 Stochastic vs. deterministic trends
• A series which only contains a deterministic trend is
called trend-stationary.
• Distinction between nonstationarity and trend
stationary is important
• Unit root tests can be used to distinguish between both
kind of series.
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1.1.2 Stochastic vs. deterministic trends
1.1.2 Stochastic vs. deterministic trends
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Figure 6: A trend-stationary process
1.1.2 Stochastic vs. deterministic trends
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Figure 7: A random walk with drift
1.1.2 Stochastic vs. deterministic trends
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Figure 8: Industrial Production, United States
1.1.2 Stochastic vs. deterministic trends
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Figure 9: Stock Prices, United States
1.1.3 Unit root testing
• Consider the following process 𝑌𝑡
which can be written as:
• 𝑌𝑡 is stationary if 𝜑 is statistically different from zero
• This Dickey Fuller test (Ho: 𝜑=0) follows a t-distribution
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𝑌𝑡 = 𝜇 + 𝜙𝑌𝑡−1 + 𝜀𝑡
Δ𝑦𝑡 = 𝜇 + 𝜑𝑦𝑡−1 + 𝜀𝑡
• The Augmented Dickey Fuller test includes
autoregressive differenced terms to include for
autocorrelation:
• Critical values are provided by MacKinnon (1994;1996)
• Further modification by Elliot et al. (1996) to improve the
distinction between a trend stationary and a non-
stationary series
• For further reference on unit root tests see Enders
(2009)
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Δ𝑦𝑡 = 𝜇 + 𝜑𝑦𝑡−1 +
𝑖=2
𝑝
𝛽𝑖 Δ𝑦𝑡−𝑖+1 + 𝜀𝑡
1.1.3 Unit root testing
1.1.4 The spurious regression problem and the
concept of cointegration
• Why is a distinction between stationary and
nonstationary series important?
• Suppose two stochatisc processes, 𝑌𝑡 and 𝑧𝑡 which
contain unit roots
• Granger and Newbold (1974): Regression of a unit root
processes on another independent unit root processes
will produces an apprarently significant relationship
although despite the fact that none exists in reality
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• In such a case, the error term is not stationary and
statistical inference is invalid because of biased standard
errors (Enders, 2009)
• „Spurios regression“
• If the error term which results from a regression with I(1)
regressors is stationary, a cointegrating relationship is
observed
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1.1.4 The spurious regression problem and the
concept of cointegration
• Alternative strategy of differencing ignores a possible
long-run relationship between both series
• Root idea of cointegration: Even if two series are
nonstationary (I(1)), there might exist a linear
combination between them which is stationary (I(0):
• Why is this concept interesting from an economic point of
view?
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𝑦𝑡 − 𝛽´𝑧𝑡~ 𝐼(0)
1.1.4 The spurious regression problem and the
concept of cointegration
• Many economic concepts might be considered as a long-
run steady state equilibrium rather than a condition which
holds continously
• Example: Purchasing power parity postulates a
proportional relationship between the nominal exchange
rate and the price differential:
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𝑠𝑡 = (𝑝𝑡 − 𝑝𝑡𝑓)
1.1.4 The spurious regression problem and the
concept of cointegration
• However, PPP might be only useful as a long-run
benchmark
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Research I
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1975 1980 1985 1990 1995 2000 2005 2010
ln(CPIger/CPIu.s.) ln(DM/U.S.$)
1.1.4 The spurious regression problem and the
concept of cointegration
• In terms of cointegration analysis we might want to ask
the following questions when analyzing PPP
Is there a cointegrating relationship beetween the
nominal exchange rate and the price differential?
If this is the case, are both series moving strictly
proportional in the long-run?
Distinguishing „weak“ and „strong“ PPP (Beckmann,
2011)
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1.1.4 The spurious regression problem and the
concept of cointegration
• Alternative way for testing PPP: Testing for a unit root in
the real exchange rate
• Main general questions we want to adress:
How can we assess whether a long-run relationship
between two (or more) series exists?
How can the cointegrating parameters be estimated?
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1.1.4 The spurious regression problem and the
concept of cointegration
• Two different approaches to test for cointegration and
estimate the corresponding parameters:
The Engle and Granger two step methodology:
Estimation of (possible) cointegrating vector and testing
for stationarity of residuals
The Johansen methodology: Estimation of cointegrating
vector as part of a Vector Error Correction model
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1.1.4 The spurious regression problem and the
concept of cointegration
Disposition
1.1 Introduction and definitions
1.2 The Engle-Granger Methodology
1.3 The Cointegrated VAR approach
1.3.1 Basics
1.3.2 Modelling Cycle
1.4 Extensions: Nonlinear cointegration
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Research I
1.1.1 Long-run relations and error correction
• Definition of cointegration by Engle and Granger (1987):
The components of a vector
are said to cointegrated of order d,b, denoted by
if:
1.) All components of 𝑥𝑡 are integrated of order d
1.) There exists a vector such that the
linear combination is
integrated of order (d - b) with b > 0.
The vector is called cointegrating
vector39
Advanced Seminar in Macroeconomic
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𝑥 (𝑡= 𝑥1𝑡 , 𝑥2𝑡, , 𝑥3𝑡, , … 𝑥𝑚)′
)𝑥𝑡~𝐶𝐼(𝑑, 𝑏
𝛽 = (𝛽1, 𝛽2, … . . 𝛽𝑛)
𝛽𝑥𝑡 = (𝛽1 𝑥1,𝑡 + 𝛽2 𝑥2,𝑡 … . . 𝛽𝑛𝑥𝑛,𝑡
𝛽 = (𝛽1 , 𝛽2 , … . . 𝛽𝑛)
1.1.1 Long-run relations and error correction
• The original definition of Engle and Granger (1987)
implies:
Cointegration refers to a linear combination of
nonstationary variables. See Chapter 5 for some
thoughts on nonlinear cointegration.
Cointegration refers to variables that are integrated of
the same oder. Two quantities might for example be both
integrated of order two (I(2)). Multicointegration
corresponds to the case of equilibrium relationships
between variables of different orders (Enders, 2009; Lee
and Granger,1990).
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1.1.1 Long-run relations and error correction
If 𝑥𝑡 has n nonstationary components, there may be as
many as n-1 linearly independent cointegrating vectors.
We will come back to the issue of multivariate
cointegration in Chapter 3.
• Most applications focuse on the case where each
variable is integrated of order one (I(1)) and the
cointegrating relationship is stationary. The reason is that
traditional time series analysis applies for stationary
variables. In addition, most economic variables are not
integrated of an higher order than unity.
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1.1.1 Long-run relations and error correction
• For this reason, the term cointegration is frequently used
for variables which are CI (1,1). Our simple definition in
Chapter 1 also relies on this idea.
• Stock and Watson (1988): Cointegrated variables share
common stochastic trends
• Common stochastic trends might for example result from
technology shocks
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1.1.1 Long-run relations and error correction
• Cointegrated variables are influenced by the extent of
deviation from long-run equilibrium
• Example: If PPP holds as a long-run equilibrium, the
nominal exchange rate should respond to deviations
from PPP
• Short run dynamics of the variables which are influenced
by the deviations from the equilibrium are captured by
the error correction mechanism
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1.1.1 Long-run relations and error correction
• Granger representation theorem: For any set of I(1)
variables, error correction and cointegration are
equivalent representations
• A simple error correction model for 𝑌𝑡 in case of
cointegrating relationship between 𝑌𝑡 and 𝑍𝑡 is given by
. In order to avoid
misspecification, lags of the differences are usually
included. This leads to the following equation:
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∆𝑌𝑡 = 𝑎0 + 𝑎1 𝑌𝑡−1 − 𝛽0 − 𝛽1𝑧𝑡−1 + 𝜂𝑧𝑡
∆𝑌𝑡 = 𝑎0 + 𝑎1 𝑌𝑡−1 − 𝛽0 − 𝛽1𝑧𝑡−1 +
𝑖=1
𝑛
𝑎11 𝑖 ∆𝑌𝑡−𝑖 +
𝑖=1
𝑛
𝑎12 𝑖 ∆𝑍𝑡−𝑖 +𝜂𝑧𝑡
1.1.1 Long-run relations and error correction
• Summing up: Main idea of Engle and Granger (1987):
Separating out short-run and long-run dynamics
Long-run dynamics are captured by estimating the long-
run relationship as the first step
Short-run dynamics are captured by adjustment
dynamics and the terms in first differences in the error
correction form
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1.1.1 Long-run relations and error correction
• Steps for applying the Engle and Granger (1987)
methodology:
Step 1: Pretest all variables for their order of integration
using unit root test. If both variables are stationary,
standard time series methods apply. If they are
integrated of different order, they are not cointegrated in
the usual sense. However, one might want to consider
the concept of multicointegration
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1.1.1 Long-run relations and error correction
Step 2: If the series and are integrated of the same
order, proceed by estimating a possible long-run
relationship of the following form:
• If the variables are cointegrated the OLS estimator, is
super-consistent (Stock and Watson, 1988).
• If 𝑌𝑡 and 𝑍𝑡 are integrated of the same oder (I(1)) and
form a cointegrating relation via the long-run coefficients,
the resulting error term is stationary (I(0)).
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𝑌𝑡 = 𝛽0 + 𝛽1𝑍𝑡 + 𝜀t
1.1.1 Long-run relations and error correction
• Hence, the next step includes applying a unit-root test to
the fitted residuals:
• Note that an intercept term should not be included since
𝜀t is a residual from a regression equation
• If 𝑒𝑡 in the equation above does not seem to be white
noise, an augmented form of the test should be used
(See Chapter 1).
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∆ 𝜀𝑡 = 𝑑1 𝜀t−1 + 𝑒𝑡
1.1.1 Long-run relations and error correction
If the hypothesis 𝑑1 = 0 cannot be rejected, the residual
series contains a unit root and we conclude that the
series 𝑌𝑡 and 𝑍𝑡 are not cointegrated
spurious regression
• On the opposite, a rejection suggests that the residuals
are stationary and both are cointegrated
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1.1.1 Long-run relations and error correction
• Note that standard critical values are not valid. The
original regression is based on the idea of minimizing the
sum of squared residuals. Hence, the procedure is
biased towards finding a stationary process
• Valid critical values are provided by Mac Kinnon (1991)
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1.1.1 Long-run relations and error correction
Step 3: If the variables are cointegrated, the following
error correction model can be estimated:
Besides the error correction term, those two equations
constitute a usual VAR model in first differences. Hence,
standard techniques like OLS can be applied
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∆𝑌𝑡 = 𝑎0 + 𝑎𝑦 𝜀𝑡−1 +
𝑖=1
𝑛
𝑎11 𝑖 ∆𝑌𝑡−i +
𝑖=1
𝑛
𝑎12 𝑖 ∆𝑍𝑡−i +𝜂𝑦𝑡
∆𝑍𝑡 = 𝑎2 + 𝑎𝑧 𝜀𝑡−1 +
𝑖=1
𝑛
𝑎11 𝑖 ∆𝑌𝑡−i +
𝑖=1
𝑛
𝑎12 𝑖 ∆𝑍𝑡−i +𝜂𝑧𝑡
1.1.1 Long-run relations and error correction
Step 4: Assess model adequacy and perform diagnostics
In terms of cointegration, the speed of the adjustment
coefficients is of particular interest. If the adjustment
coefficient turns out to be zero, the corresponding
variable does not respond to long-run deviations.
In our example, this might be the case for 𝑍𝑡 as the right-
hand side variable
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1.1.1 Long-run relations and error correction
However, for 𝑌𝑡, we expect a significant and negative
adjustment coefficient
This would imply that 𝑌𝑡 reacts to deviations from the
long-run path.
Example: The nominal exchange rate should react to
deviations from PPP while this would not be necessarily
the case for prices (which might be sticky in the short
run)
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1.1.1 Long-run relations and error correction
• Further inference and diagnostics, such as impulse
response, might be carried out in a similiar fashion to a
usual VAR (not covered in this course)
• Note: In oder to achieve white noise residuals for the
error correction system, estimation of a seemingly
unrelated regression might be required
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1.1.1 Long-run relations and error correction
• Caveats of the Engle and Granger methodology (Enders,
2009):
Procedure does not allow for a systematic estimation
when multiple cointegrating vectors are present
The choice of the left-hand side variable is arbitrary. For
the same dataset, it is possible that one equation
indicates cointegration while the other does not
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1.1.1 Long-run relations and error correction
As an example, the following two cointegrating
regressions should result in identical error terms:
However, this result only holds asymptocially if the
sample grows infinitely large
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𝑌𝑡 = 𝛽10 + 𝛽11𝑍𝑡 + 𝜀1𝑡
𝑍𝑡 = 𝛽20 + 𝛽21𝑌𝑡 + 𝜀1𝑡
1.1.1 Long-run relations and error correction
Another disadvantage is the application of a two step
procedure. Errors introduced in the first step (estimation
of long-run relationship) transmit into the second step
(error correction mechanism)
Modified single estimators and multivariate methods are
able to circumwent some of these shortcomings
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