# Cointegrated TFP Processes and International Business Cycles

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Cointegrated TFP Processes and

International Business Cycles

*Pau Rabanal, Juan F. Rubio-Ram탭rez,*

*and Vicente Tuesta*

© 2009 International Monetary Fund

WP/09/212

**IMF Working Paper**

Research Department

**Cointegrated TFP Processes and International Business Cycles1**

**Prepared by Pau Rabanal, Juan F. Rubio-Ramírez, and Vicente Tuesta**

Authorized for distribution by Gian Maria Milesi-Ferretti

September 2009

**Abstract**

**This Working Paper should not be reported as representing the views of the IMF.**

The views expressed in this Working Paper are those of the author(s) and do not necessarily represent

those of the IMF or IMF policy. Working Papers describe research in progress by the author(s) and are

published to elicit comments and to further debate.

A puzzle in international macroeconomics is that observed real exchange rates are highly

volatile. Standard international real business cycle (IRBC) models cannot reproduce this fact.

We show that TFP processes for the U.S. and the “rest of the world,” is characterized by a

vector error correction (VECM) and that adding cointegrated technology shocks to the

standard IRBC model helps explaining the observed high real exchange rate volatility. Also

we show that the observed increase of the real exchange rate volatility with respect to output

in the last 20 year can be explained by changes in the parameter of the VECM.

JEL Classification Numbers: E32, F32, F33, F41

Keywords: International Business Cycles, Real Exchange Rates, Cointegration

Author’s E-Mail Address: [email protected]; [email protected];

[email protected]

1 We thank Larry Christiano, Martin Eichenbaum, Jesús Gonzalo, Jim Nason, Fabrizio Perri, Gabriel Rodríguez,

and Barbara Rossi for very useful comments. NSF support is acknowledged. Pau Rabanal is an economist in the

IMF Research Department. Juan Rubio-Ramirez is an associate professor in the Economics Department at Duke

University. Vicente Tuesta is an economist in Deutsche Bank and a visiting professor at CENTRUM Católica..

2

Contents Page

I. Introduction.......................................................................................................................4

II. The Great Moderation and Real Exchange Rate Volatility ..............................................8

III. The

Model.......................................................................................................................11

A.

Households.............................................................................................................13

B.

Firms ......................................................................................................................14

B.1

Final

goods

producers................................................................................14

B.2

Intermediate

goods

producers ....................................................................15

B.3

The processes for TFP ...............................................................................15

C.

Market

Clearing .....................................................................................................16

D.

Equilibrium ............................................................................................................16

D.1

Equilibrium

definition................................................................................16

D.2

Equilibrium

conditions...............................................................................17

E.

Balanced Growth and the Restriction on the Cointegrating Vector ......................20

IV. Estimation of the VECM ................................................................................................21

A.

Data ........................................................................................................................22

B. Integration and Cointegration Properties ...............................................................23

C.

The

VECM

Model .................................................................................................26

V. Results.............................................................................................................................28

A.

Parameterization ....................................................................................................28

B. Matching Real Exchange Rate Volatility ..............................................................29

C.

Intuition..................................................................................................................34

D. Matching the Increase in Real Exchange Rate Volatility ......................................40

E.

The “Backus-Smith Puzzle” ..................................................................................42

VI. Concluding

Remarks.......................................................................................................44

References................................................................................................................................46

Appendix

A. Normalize Equilibrium Conditions.................................................................................51

B. TFP

Series.......................................................................................................................53

Figures

1.

Standard Deviation of HP-Filtered Data. USA and UK ...................................................9

2.

Standard Deviation of HP-Filtered Data. Canada and Australia ....................................10

3

3.

TFP Processes for the US and the “Rest of the World”..................................................23

4.

Impulse Response to a Home Country TFP shock. Model with Stationary

TFP

Shocks .........................................................................................................35

5.

Impuse Response to a Home-Country TFP shock. Model with Stationary

TFP

Shocks .........................................................................................................36

6.

Impuse Response to a Home-Country TFP shock. Model with Cointegrated

TFP

Shocks .........................................................................................................38

7.

Impuse Response to a Home-Country TFP shock. Model with Cointegrated

TFP

Shocks .........................................................................................................38

Tables

1.

Unit Root tests for TFP...................................................................................................24

2.

Cointegration Statistics I.................................................................................................25

3. Cointegration

Statistics II: Johansen’s test .....................................................................26

4.

Likelihood ratio tests.......................................................................................................27

5. VECM

model ..................................................................................................................27

6a. Results

.............................................................................................................................30

6b. Results

.............................................................................................................................32

6c. Results

.............................................................................................................................33

7. Changing

ρa and κ ...........................................................................................................35

8.

Investment-Specific Technology shocks ........................................................................43

4

**I.**

**Introduction**

A central puzzle in international macroeconomics is that observed real exchange rates are

highly volatile. Standard international real business cycle (IRBC) models cannot reproduce

this fact when calibrated using conventional parameterizations. For instance, Heathcote and

Perri (2002) simulate a two-country, two-good economy with total factor productivity (TFP)

shocks and ﬁnd that the model can only explain less than a fourth of the observed volatility in

real exchange rates for U.S. data. An important feature of their model, following the seminal

work of Backus, Kehoe, and Kydland (1992), is that it considers stationary TFP shocks that

follow a VAR process in levels.1

In this paper we provide evidence that TFP processes for the U.S. and a sample of main

industrialized trade partners have a unit root and are cointegrated. Motivated by this empirical

ﬁnding, we introduce technology shocks that follow a vector error correction model (VECM)

process into an otherwise standard two-country, two-good model. Engle and Granger (1987),

Engle and Yoo (1987), and LeSage (1990) indicate that if the system under study includes

integrated variables and cointegrating relationships, then this system will be more

appropriately speciﬁed as a VECM rather than a VAR in levels. As Engle and Granger (1987)

note, estimating a VAR in levels for cointegrated systems leads to ignoring important

constraints on the coefﬁcient matrices. Although these constraints are satisﬁed asymptotically,

small sample improvements are likely to result from imposing them in the cointegrating

relationships. Falling to impose them affects the small sample estimates and the implied

dynamics.

The presence of cointegrated TFP shocks requires restrictions on preferences, production

functions, and the law of motion of the shocks in order to have balanced growth. The

restrictions on preferences and technology of King, Plosser, and Rebelo (1988) are sufﬁcient

for the existence of balanced growth in a closed economy. However, in a two-country model,

an additional restriction on the cointegrating vector relating the TFP processes is needed. In

1Other studies that consider a VAR in levels are: Kehoe and Perri (2002), Dotsey and Duarte (2007), Corsetti,

Dedola and Leduc (2008a, 2008b), and Heathcote and Perri (2008).

5

particular, we need the cointegrating vector to be (1;

1); which means the ratio of TFP levels

(or, equivalently, the difference of the log-levels of TFP) across countries is stationary. After

presenting evidence for this additional restriction, we show that the VECM speciﬁcation for

TFP processes solves a large part of the real exchange rate volatility puzzle without affecting

the good match for other moments of domestic and international variables. In particular, we

show that our model can generate a real exchange rate volatility more than four times larger

than an equivalent model with stationary shocks calibrated as in Heathcote and Perri (2002).

Why does a model with cointegrated TFP shocks generate higher relative volatility of the real

exchange rate than a model with stationary shocks? The reason is that the VECM parameter

estimates imply higher persistence and lower spillovers than the traditional stationary

calibrations. As we brieﬂy explain below, and later in more detail, higher persistence and

lower spillovers imply higher volatility of the real exchange rate and lower volatility of output.

The intuition behind this result is as follows. In the standard IRBC model with no spillovers,

when productivity increases at home, home households feel richer and output, consumption,

and investment increase, while labor rises because of the upsurge in marginal productivity. As

output at home grows, the demand for intermediate goods produced in the foreign country

also soars. Provided that the elasticity of substitution between home and foreign intermediate

goods is low enough, the terms of trade deteriorate and the real exchange rate depreciates,

reﬂecting greater world scarcity of the foreign intermediate good relative to the home one. As

the persistence of TFP shocks increases, home country households feel richer and supply less

labor and capital. This has two main effects. First, it lowers the initial increase of home output

and, hence, its volatility. Second, it causes a larger demand increase for the foreign

intermediate good and, hence, a larger terms of trade deterioration and real exchange rate

depreciation. As a result, higher persistence in TFP shocks implies higher relative volatility of

the real exchange rate with respect to output.

When spillovers of TFP shocks across countries are introduced in the model, a “news”

channel arises. This channel has the opposite effect than the one described above. Since

foreign country households know that productivity will eventually increase in their country,

6

they feel richer and supply less labor and capital but demand more consumption goods and,

therefore, demand more intermediate good from the home country. Thus, the foreign

intermediate good is, relatively, less scarce and the real exchange rate would tend to

depreciate less than in a model with no spillovers. Faster spillovers amplify these effects and

lead to a lower relative volatility of the real exchange rate with respect to output.

Therefore, the mechanism we just described requires high persistence of each of the TFP

processes, as well as high persistence in their difference (i.e., a slow transmission of shocks

across countries), in order to explain high relative volatility of the real exchange rate with

respect to output. This is what comes out of our parameter estimates. Estimating a VECM

introduces a unit root in the system. What is also crucial for our results is that we estimate a

very slow speed of convergence to the cointegrating relationship, implying that the second

largest root of the system is also very close to, but inside, the unit circle.

Another very well documented empirical fact is the substantial decline in the volatility of

most U.S. macroeconomic variables during the last 20 years. That change in the cyclical

volatility is known as the “Great Moderation.”2 In this paper, we report that, for most

industrialized countries, the Great Moderation has not affected the real exchange rate as

strongly as it has affected output. As a result, the ratio of real exchange rate volatility to

output volatility has increased. We also show that the increase in the relative volatility of the

real effective exchange rate of the U.S. dollar coincides in time with a weakening of the

cointegrating relationship of TFP shocks between the U.S. and the “rest of the world.”3 More

important, we conﬁrm that if we allow for a fading in the cointegrating relationship of the size

estimated in the data, the model can jointly account for the observed increase in the relative

volatility of the real exchange rate and the substantial decline in the volatility of output.

An important problem of IRBC models is that the co-movement between the real exchange

2Some early discussion of the Great Moderation can be found in Kim and Nelson (1999). A discussion of

different interpretations for this phenomenon and some international evidence can be found in Stock and Watson

(2002) and Stock and Watson (2005), respectively.

3In section IV we describe the set of countries that compose our deﬁnition of “rest of the world.”

7

rates and the ratio of consumption levels across countries does not match the one observed in

the data (Backus and Smith, 1993). Even when considering cointegrated TFP shocks the

model still generates a correlation close to one, while in the data the correlation is negative

and close to zero. For this reason, we consider two extensions of the benchmark model that

allow us to better ﬁt this correlation without affecting relative volatility of the real exchange

rate. In particular, we consider a taste shock as in Heathcote and Perri (2008) and an

investment-speciﬁc technology shock as in Raffo (2009). As also shown by these authors in

stationary environments, both type of shocks help us accomplish the objective.

Our paper relates to two important strands of the literature. On the one hand, it connects with

the literature stressing the importance of stochastic trends to explain economic ﬂuctuations.

King, Plosser, Stock, and Watson (1991) ﬁnd that a common stochastic trend explains the

co-movements of main U.S. real macroeconomic variables. Lastrapes (1992) reports that

ﬂuctuations in real and nominal exchange rates are due primarily to permanent real shocks.

Engel and West (2005) show that real exchange rates manifest near–random-walk behavior if

TFP processes are random walks and the discount factor is near one, while Nason and Rogers

(2008) generalize this hypothesis to a larger class of models. Aguiar and Gopinath (2007)

show that trend shocks are the primary source of ﬂuctuations in emerging economies. Alvarez

and Jermann (2005) and Corsetti, Dedola, and Leduc (2008a) highlight the importance of

persistent disturbances to explain asset prices and real exchange rate ﬂuctuations, respectively.

Also, Lubik and Schorfheide (2006) and Rabanal and Tuesta (2006) introduce random walk

TFP shocks to explain international ﬂuctuations and Justiniano and Preston (2008) suggest

that in order to explain the comovement between Canadian and US main macroeconomic

variables it is important to introduce correlations between the innovations of several structural

shocks. However, these papers do not formalize a VECM, test for cointegration, or estimate

the cointegrating vector and the short run dynamics of the system.

On the other hand, our paper also links to the literature analyzing different mechanisms to

understand real exchange rate ﬂuctuations. Some recent papers study the effects of monetary

shocks and nominal rigidities. Chari, Kehoe, and McGrattan (2002) are able to explain real

exchange rate volatility in a monetary model with sticky prices and a high degree of risk

8

aversion. Benigno (2004) focuses on the role of interest rate inertia and asymmetric nominal

rigidities across countries. Other papers use either non-tradable goods, pricing to market, or

some form of distribution costs (see Corsetti, Dedola, and Leduc 2008a, 2008b; Benigno and

Thoenissen 2007, and Dotsey and Duarte 2007). Our model includes only tradable goods with

home bias, which is the only source of real exchange rate ﬂuctuations. Our choice is guided

by evidence that the relative price of tradable goods has large and persistent ﬂuctuations that

explain most of the real exchange rate volatility (see Engel 1993 and 1999). Fluctuations of

the relative price of non-tradable goods accounts for, at most, one-third of the real exchange

rate volatility (see Betts and Kehoe 2006, Burnstein, Eichenbaum, and Rebelo 2006, and

Rabanal and Tuesta 2007).

The rest of the paper is organized as follows. Section II documents the increase of the real

exchange rate volatility with respect to output for most industrialized countries. In Section III

we present the model with cointegrated TFP shocks. In Section IV we report estimates for the

law of motion of the (log) TFP processes of the United States and a “rest of the world”

aggregate. In Section V we present the main ﬁndings from simulating the model, leaving

Section VI for concluding remarks.

**II.**

**The Great Moderation and Real Exchange Rate Volatility**

In this section, we present evidence that in the period known as “the Great Moderation,” the

relative volatility of the real exchange rate (measured as the real effective exchange rate) with

respect to output (measured as real GDP) has increased in the United States, the United

Kingdom, Canada, and Australia. In Figures 1 and 2 we present the standard deviation of the

HP-ﬁltered output, the standard deviation of the HP-ﬁltered real exchange rate, and the ratio

of the two for these four countries. We compute the standard deviation of rolling windows of

40 quarters.

Let us ﬁrst focus on the U.S. economy. Figure 1 shows a substantial decline in the volatility of

output, from 2.3 percent standard deviation in the window 1973:1-1982:4, to 0.8 percent in

9

Figure 1: Standard Deviation of HP-Filtered Data. USA and UK.

USA-SD(GDP)

UK-SD(GDP)

.024

.020

.022

.016

.020

.018

.012

.016

.014

.008

.012

.004

.010

.008

.000

85

90

95

00

05

85

90

95

00

05

USA-SD(REER)

UK-SD(REER)

.08

.068

.064

.07

.060

.056

.06

.052

.048

.05

.044

.040

.04

.036

.032

.03

.028

85

90

95

00

05

85

90

95

00

05

USA-SD(REER)/SD(Y)

UK-SD(REER)/SD(Y)

7

11

10

6

9

5

8

7

4

6

3

5

4

2

3

1

2

85

90

95

00

05

85

90

95

00

05