Advanced Seminar in Macroeconomic Research I - wiwi.rub.de fileAdvanced Seminar in Macroeconomic Research I Ruhr-University Bochum Faculty of Management and Economics Chair of International

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

mailto:[email protected]

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

4

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

5Advanced Seminar in Macroeconomic

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


Research I

General Information

Course material on Moodle

Slides

References

Data and Code

Additional Material


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


Research I

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


Research I

Course Outline

Chapter 3: Forecasting

3.1 Introduction and Definitions

3.2 Forecasting Evaluation

3.4 Case Study: Forecasting Exchange Rates


Research I

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


Research I

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


Research I


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


Research I


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.


Research I


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


Research I

Chapter 1: Cointegration


1.2 The Engle-Granger methodology


1.3.1 Basics


1.4 Extensions: Nonlinear cointegration


Research I

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


Research I

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


Research I

𝑌𝑡 = 𝑌𝑡−1 + 𝜀𝑡

𝑌𝑡 = 𝑌0 +

𝑖=1

𝑡

𝜀𝑡



Research I

Figure 4: A stationary time series



Research I

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.


Research I

𝑌𝑡 = 𝑌𝑡−1 + 𝜇0 + 𝜀𝑡

∆𝑌𝑡 = 𝜇0 + 𝜀𝑡


• 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


Research I

𝑌𝑡 = 𝑌0 + 𝜇0𝑡 +

𝑖=1

𝑡

𝜀𝑡


• 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.


Research I




Research I

Figure 6: A trend-stationary process



Research I

Figure 7: A random walk with drift



Research I

Figure 8: Industrial Production, United States



Research I

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


Research I

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)


Research I

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


Research I

• 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


Research I



• 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?


Research I

𝑦𝑡 − 𝛽´𝑧𝑡~ 𝐼(0)



• 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:


Research I

𝑠𝑡 = (𝑝𝑡 − 𝑝𝑡𝑓)



• However, PPP might be only useful as a long-run

benchmark


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.$)



• 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)


Research I



• 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?


Research I



• 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


Research I



Disposition


1.2 The Engle-Granger Methodology


1.3.1 Basics


1.4 Extensions: Nonlinear cointegration


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


Research I

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)


• 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).


Research I


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.


Research I


• 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


Research I


• 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


Research I


• 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:


Research I

∆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


• 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


Research I


• 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


Research I


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)).


Research I

𝑌𝑡 = 𝛽0 + 𝛽1𝑍𝑡 + 𝜀t


• 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).


Research I

∆ 𝜀𝑡 = 𝑑1 𝜀t−1 + 𝑒𝑡


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


Research I


• 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)


Research I


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


Research I

∆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


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


Research I


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)


Research I


• 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


Research I


• 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


Research I


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


Research I

𝑌𝑡 = 𝛽10 + 𝛽11𝑍𝑡 + 𝜀1𝑡

𝑍𝑡 = 𝛽20 + 𝛽21𝑌𝑡 + 𝜀1𝑡


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


Research I

Documents

Advanced Seminar in Macroeconomic Research I - wiwi.rub.de fileAdvanced Seminar in Macroeconomic Research I Ruhr-University Bochum Faculty of Management and Economics Chair of International