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SOURCES OF BUSINESS CYCLES IN AN ECONOMY
WITH MONEY, REAL SHOCKS, AND NOMINAL RIGIDITY
—A STUDY OF THE UNITED STATES: 1954 — 1991

By

Keshin T swei

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Economics

1998

ABSTRACT

SOURCES OF BUSINESS CYCLES IN AN ECONOMY
WITH MONEY, REAL SHOCKS, AND NOMINAL RIGIDITY
——A STUDY OF THE UNITED STATES: 1954 — 1991

By

Keshin Tswei

This dissertation examines the sources of postwar US economic fluctuations
in a VAR framework with cointegration constraints. A theoretical macroeconomic
model consisting of equations that describe the labor and goods market behavior
and featuring an ex ante nominal wage contract is used to guide the empirical setup
in this study. The theory prescribes five variables for the economic system, real
output, real balances, real wages, nominal interest rates and inflation and postulates
two cointegrating relationships, the velocity of money and the ex post real interest
rate. Overidentifying restriction tests for the structural restrictions derived from the
theory indicate that the steady-state structure of the system is consistent with the
postwar US data but the dynamic structure is not. As a result only the permanent
shocks specified by the theory, the nominal shock, the technology shock and the
labor-market shock, are identified whereas transitory shocks are unidentified. The
long-run effects of the permanent shocks on the five variables are constrained by their
long-run multipliers derived from the theory. Their contributions to the shorter-run
economic fluctuations are documented in this study and the results are consistent

with several prominent studies in the literature.

COpyright by
Keshin Tswei
1998

Dedicated to my parents, Tsui Chih—An and Li Chiu—Chi.

iv

ACKNOWLEDGMENTS

This dissertation is an important milestone of my education and my life. I
am indebted to many people who have contributed to its successful completion. My
deepest appreciation goes to Professor Robert Rasche, my dissertation advisor, for
guiding me through the entire work and being very generous to me. I admire not
only his economics expertise but also his supporting attitude toward students. I
am grateful to the other members of my committee, Professor Christine Amsler and

Professor Jeffrey Wooldridge, for their invaluable comments.

Teachers, fellow students and friends at MSU and elsewhere have enriched
my life in various ways. They are too many to list here but I thank them all.
Professor Chung Ching—Fan had helped me on my second-year paper and has been
encouraging. J ingYi has been caring and enduring over the course of my dissertation
work. My appreciation also extends to the economics department and the university
for bestowing me much more than just a Ph.D. My stay in America has allowed me

to appreciate the richness of its culture and values.

Most importantly, I thank my wonderful parents, Tsui Chih-An and Li Chiu-
Chi, for their immeasurable love and manifold supports. All the things I have
accomplished and those I will in the future I would attribute to my mom and dad.
My sisters, sister-in-laws and brothers also have been supportive of my study abroad

and they deserve a lot of credits from me.

TABLE OF CONTENTS

LIST OF TABLES ......................................................... viii
LIST OF FIGURES ......................................................... ix
CHAPTER 1
INTRODUCTION ............................................................ 1
CHAPTER 2
VECTOR AUTOREGRESSION AND THE COMMON-TRENDS MODEL 7
2.0 Introduction ......................................................... 7
2.1 Vector Autoregression Model ......................................... 7
2.1.1 Structural VAR and Reduced-Form VAR ...................... 7
2.1.2 Moving-Average Representation ............................... 9
2.1.3 Structural VAR Identification ................................ 10
2.2 Cointegration and the Error-Correction Model ...................... 12
2.3 The Common-Trends Model ....................................... 16
2.4 Permanent and Transitory Shocks Identification .................... 21
2.5 Example of a Common-Trends Model .............................. 26
CHAPTER 3
AN ECONOMIC MODEL CHARACTERIZED BY COINTEGRATION ...... 31
3.0 Introduction ........................................................ 31
3.1 An Economy with Wage Contract .................................. 31
3.2 Solution of the Rational Expectation Model ........................ 33
3.3 The Complete Model Specification ................................. 34
3.4 The Vector Error—Correction Representation ........................ 37
3.5 The Vector Moving-Average Representation ........................ 42
CHAPTER 4
SPECIFICATION, ESTIMATION AND IDENTIFICATION OF
THE ECONOMIC MODEL ................................................. 45
4.0 Introduction ........................................................ 45
4.1 Data Analysis and Unit-Root Tests ................................ 45
4.2 Specification for a VAR With Cointegration ........................ 57

vi

4.3 Testing for Two Cointegration Relations ............................ 63

4.3.1 Johansen’s Cointegration Rank Tests ......................... 64

4.3.2 Horvath and Watson’s Cointegrating Vector Test ............. 65

4.4 Estimation of the VECM .......................................... 67

4.5 Estimation of the Overidentifying Theoretical Model ................ 72

4.5.1 The Complete Model Structure .............................. 72

4.5.2 The Long-Run Model Structure .............................. 75
CHAPTER 5

MACROECONOMIC IMPULSE ANALYSIS ................................. 82

5.0 Introduction ........................................................ 82

5.1 Effects of the Permanent Technology Shock ......................... 84

5.2 Effects of the Permanent Labor-Market Shock ...................... 86

5.3 Effects of the Permanent Nominal Shock ........................... 88
CHAPTER 6

SUMMARY AND CONCLUSIONS ......................................... 100

APPENDIX A. RATIONAL EXPECTATION SOLUTIONS FOR

A MODEL ECONOMY .................................... 103

APPENDIX B. THE EX POST REAL INTEREST RATE EQUATION ..... 105

APPENDIX C. DATA SOURCES AND DEFINITIONS ..................... 107

LIST OF REFERENCES ................................................... 109

vii

LIST OF TABLES

Table 1 - Augmented Dickey-Fuller Unit-Root Tests on Levels

and First-Differences ............................................. 54
Table 2 — Augmented Dickey-Fuller Unit-Root Tests on Velocity

and Ex Post Real Interest Rate ................................... 56
Table 3 - Maximum Likelihood Functions of VAR

Specifications ..................................................... 60
Table 4 - Likelihood Ratio Tests for VAR Dummy

Specifications ..................................................... 61
Table 5 - Likelihood Ratio Tests for VAR Lag—Length

Specifications ..................................................... 61
Table 6 - Residual Analysis for VECMs with Four and Six Lags .............. 62
Table 7 - Cointegrating Rank Test on Unrestricted VAR with

Four and Six Lags ................................................ 63
Table 8 - Horvath/Watson Tests of Pre-specified Cointegrating Vectors ....... 66
Table 9 - Parameter Estimates for VECM .................................... 70
Table 10 - Percentage of Forecast-Error Variance Attributed to

the Technology Shock ............................................ 85
Table 11 - Percentage of Forecast-Error Variance Attributed to

the Labor Market Shock .......................................... 87
Table 12 - Percentage of Forecast-Error Variance Attributed to

the Nominal Shock ............................................... 89

viii

LIST OF FIGURES

Figure 1 - Private-Sector Output, 51:1-94:4 ................................... 47
Figure 2 - First-Differenced Private-Sector Output, 51:1-94:4 ................. 47
Figure 3 - Real Balances, 51:1-94:4 ........................................... 48
Figure 4 - First-Differenced Real Balances, 51:1-94:4 .......................... 48
Figure 5 - Real Wages, 51:1-94:4 ............................................. 49
Figure 6 - First-Differenced Real Wages, 51:1-94:4 ............................ 49
Figure 7 - 3-Month Treasury Bill Rate, 51:1-94z4 ............................. 50
Figure 8 - First-Differenced 3-Month Treasury Bill Rate, 51:1-94:4 ............ 50
Figure 9 - Inflation Rate, 51:1-94:4 ........................................... 51
Figure 10 - First-Differenced Inflation Rate, 51:1-94:4 ......................... 51
Figure 11 - Velocity of Money, 51:1-94:4 ...................................... 52
Figure 12 - Ex Post Real Interest Rate, 51:1-94:4 ............................. 52
Figure 13.1 - The Response of Output to Technology Shocks .................. 91
Figure 13.2 - The Response of Real Balances to Technology Shocks ........... 91
Figure 13.3 - The Response of Velocities to Technology Shocks ................ 91
Figure 13.4 - The Response of Nominal Balances to Technology Shocks ....... 92
Figure 13.5 - The Response of Real Wages to Technoloy Shocks ............. 92
Figure 13.6 - The Response of Nominal Wages to Technology Shocks .......... 92
Figure 13.7 - The Response of Nominal Interest Rates to

Technology Shocks ............................................ 93
Figure 13.8 - The Response of Inflation to Technology Shocks ................. 93
Figure 13.9 - The Response of Ex Post Real Rates to Technology Shocks ...... 93
Figure 14.1 - The Response of Output to Labor-Market Shocks ............... 94
Figure 14.2 - The Response of Real Balances to Labor-Market Shocks ......... 94
Figure 14.3 - The Response of Velocities to Labor-Market Shocks ............. 94
Figure 14.4 - The Response of Nominal Balances to Labor-Market Shocks ..... 95
Figure 14.5 - The Response of Real Wages to Labor-Market Shocks ........... 95

ix

Figure 14.6 - The Response of Nominal Wages to Labor-Market Shocks ....... 95
Figure 14.7 - The Response of Nominal Interest Rates to

Labor-Market Shocks ......................................... 96
Figure 14.8 - The Response of Inflation to Labor-Market Shocks .............. 96
Figure 14.9 - The Response of Ex Post Real Ratas to Labor-Market Shocks 96
Figure 15.1 - The Response of Output to Nominal Shocks .................... 97
Figure 15.2 - The Response of Real Balances to Nominal Shocks .............. 97
Figure 15.3 - The Response of Velocities to Nominal Shocks .................. 97
Figure 15.4 - The Response of Nominal Balances to Nominal Shocks .......... 98
Figure 15.5 - The Response of Real Wages to Nominal Shocks ................ 98
Figure 15.6 - The Response of Nominal Wages to Nominal Shocks ............ 98
Figure 15.7 - The Response of Nominal Interest Rates to

Nominal Shocks .............................................. 99
Figure 15.8 - The Response of Inflation to Nominal Shocks ................... 99
Figure 15.9 - The Response of Ex Post Real Rates to Nominal Shock ......... 99

Chapter 1
INTRODUCTION

In this dissertation the sources of the post-war macroeconomic fluctuations in
the United States are investigated in a vector autoregression framework with cointe-
gration constraints. The common-trends model is employed to identify permanent
innovations in the economic system before their individual contributions to business
cycles can be chronicled. Five nonstationary I(1) macroeconomic aggregates with
significant business-cycle characteristics are examined: the private-sector real out-
put (yt), real money balances (mm), real wages (wrt), short-term nominal interest
rates (Rt) and inflation rate (7rt). They are included based on an expanded theo-
retical model adapted from Blanchard and Fischer’s (1989) business-cycle model.
The model provides equations descriptive of labor and goods market behavior and a
nominal wage contract to feature nominal rigidity in an economy. From the solutions
of the theoretical model we find two cointegrating relations, the velocity of money
(yt - mm) and the ex post real interest rate (Rt — m). It also provides a well defined
set of simultaneous relationships among the five variables which are later rejected
by the post-war US. data. The model nevertheless has long-run information that
is useful to identify the permanent shocks that constitute the common stochastic

trends.

For almost two decades vector autoregression (VAR) has been a very popular
tool for macroeconomic studies. Prior to its introduction, researchers typically had
been criticized for estimating large systems of equations with strong over-identifying

restrictions. VAR in contrast is a more explorative or descriptive approach to em-

1

pirical analysis. To illustrate, for a vector of 1(0) variables wt, write its VAR form
compactly as

A (L) :17; = {it (1.1)

where at is the error vector. Estimable parameters of the VAR contained in the
lag-operator polynomial, A (L), are not restricted except for A (0) = I, an identity
matrix. Rather, an understanding about the system is gained primarily by analyzing
the impulse-response functions and the forecast-error variance decomposition. The
analyses are done on the structural vector-moving—average (VMA) representation of
the VAR,

Amt = R (L) Vt. (1.2)

Vector Vt consists of mutually-independent structural innovations that are propa-
gated through the system to cause fluctuations. R (L) is an infinite-order polynomial
matrix that contains the structural impulse-response functions. The reduced-form

VMA for the first-difference of 2:; can be obtained from (1.1) as
Am 2 C (L) E; (1.3)

where C (L) is also an infinite-order polynomial matrix. The task of identifying

structural impulses from reduced-form residuals amounts to finding an unique matrix
F such that
V; = F at. (1.4)

Then the impulse-response functions are available as
R(L) = C (L) F‘1 (1.5)

which comes from equating (1.2) and (1.3) and then substituting into (1.4).

Traditionally there existed a dichotomy in studying economic growth and busi-

ness cycles (King, Plosser and Rebelo 1988a). Stochastic innovations were thought

to be responsible for business cycles whereas growth was considered driven by deter-
ministic trends. Later, the notion that many economic time-series contain stochastic
growth trends, advocated by Beveridge and Nelson (1981) and Nelson and Plosser
(1982), gained prominence. This suggests that stochastic shocks that form the trends
may in fact also cause short-run fluctuations (King, Plosser and Rebelo 1988b). The
subsequent progress in cointegration research further recognized that different non-
stationary variables may share common stochastic trends. Stock and Watson (1988)

formally outline a common-trends model of the form
x, = A7} + B (L) at. (1.6)
where :rt is n x 1 and rt is the k x 1 common stochastic trends
Tt = Tt—l + Vf- (1.7)

A is a n x k common-trends loading matrix which brings the impact of rt onto
zt. Thus Art represents the permanent or nonstationary component of 113,. The
n x n lag-polynomial B (L) is stable so that B (L) 5, is a stationary component
of 23;. Equation (1.7) shows the common trends are driven by the k—dimensional
structural innovations, VtP, that exert permanent effects. In this dissertation, the
permanent innovations are specified as a technology, a permanent nominal shock and
a labor-market shock. The other 1' (E n — k) innovations in this equation system
produce only transitory effects and are denoted V? such that u; = (11,“ 11,“). It is
interacting to note that in (1.7) the innovations to the permanent trends, i.e., VtP,

are not independent of the disturbances, st, in light of u, = F5, in (1.4). Clearly,

the dichotomy between growth and cycles mentioned above no longer stands.

One major contribution of the common-trends model to VAR analyses is that
it provides special identification schemes that are lacking in VAR models containing

only transitory shocks. The fact that only permanent shocks deliver impacts in the

long run means long-run restrictions are available to identify permanent shocks. To
show this, we first note that over the long run the transitory component in (1.6)

effectively dr0ps out so that the first difference of (1.6) is
Amt = Autp. (1.8)

Thus the loading matrix A also represents the long-run impacts of permanent shocks
on the first-difference of the variables. Comparing (1.8) with the structural long-run

VMA , Ax, = R (1) Vt, reveals that the long-run multiplier matrix is

12(1) = [A20] (1.9)
Then notice the long-run version of (1.5) is

R (1) F = C (1) (1.10)

where C (1) is estimable from a reduced-from VMA. Define F 5 [FL F,’,_,,]’ and

substitute (1.9) into (1.10) to get
AF), = C (1). (1.11)

Here we see in (1.11) that knowledge about the common-trends loading matrix
A provides very useful linear restrictions to identify the permanent shocks since
Vt? = ert.

In this dissertation the conjecture about the form of A is provided by the

mentioned theoretical model. It proposes a form of the long-run equation Amt =

Autp as
— Ag; 1 _ a c 0)
Amn a c 0 VtteCh
Aw'rt = b e 0 143”" , (1.12)
AR, 0 0 1 14mm”
_ Am J _ 0 0 1‘

 

 

 

 

where a, b, c and e are functions of behavioral parameters in the economic model.
Equation (1.12) says that the technology shock and the labor-market shock have
zero long-run impact on the nominal variables, R and 7r. On the other hand, the
nominal shock has zero long-run effect on the real variables, y, mr and wr. Thus,

as is in King et a1. (1991), a long-run nominal neutrality is featured in the model.

The theoretical model adapted from Blanchard and Fischer (1989) is used to
guide most empirical setup in this study. The model has the advantage of includ-
ing both the real-business-cycle and the Keynesian assumptions about the economy.
Two permanent real shocks are treated as important sources of economic fluctua—
tions. A forward-looking wage-setting rule serves to account for the prevalent wage
rigidity phenomenon in the economy. In Chapter 3, the wage-contract model is
presented and solved by the rational expectation techniques. The solutions are con-
sistent with both the Keynesian and the RBC predictions of price and real-wage
movements over business cycles. The vector error-correction model (VECM) and
the VMA representation of the wage-contract model are also derived in Chapter 3.
In the process we obtain the simultaneous structure of F as well as the long-run

multiplier of permanent shocks, or A.

Chapter 2 provides a review of VARs with cointegration constraints and of the
VECM representation. The common-trends representation and how it provides extra
information for identification are covered in detail. Methods to identify permanent
shocks and transitory shocks are provided. An application of the common-trends
methodology is demonstrated with an example from Rasche (1992). In Chapter 4 an
analysis is presented of the stationarity of the time series using unit-root tests and
graphs. Then various specifications of dummy variables and lag lengths are tested
to estimate an optimal VAR model. Cointegration-rank tests are also done to ensure

that two conintegrating vectors can be imposed in estimation. As a result, a VECM

with three lag terms and five dummy variables is estimated. Lastly in Chapter
4 the identification of the structural-form VMA with restrictions on A as stated
above is discussed . Chapter 5 covers the analysis of impulse-response functions and
forecast-error variance decompositions. Chapter 6 presents a summary of findings,
remarks on potential contributions and shortcomings of this study and concludes

the dissertation.

Chapter 2

VECTOR AUTOREGRESSION AND
THE COMMON-TRENDS MODEL

2.0 Introduction

This chapter provides a review of the econometric methodology required for
the empirical analysis in this dissertation. A brief discussion on VAR modeling
techniques is provided in Section 2.1 with attention focused on various identification
approaches. The theory of a multivariate system characterized by cointegration and
the vector error-correction representation is introduced in Section 2.2. The common-
trends model approach which is useful in identifying permanent economic impulses
is covered in Section 2.3. A detailed review of the methods of permanent and
transitory shock identification is covered in Section 2.4. The presentation of Section
2.4 closely follows the approach in Chapters 3 and 4 of Hoffman and Rasche (1996).
An implementation of the identification methods is illustrated in Section 2.5 by an

example from Rasche (1992).
2.1 Vector Autoregression Model
2.1.1 Structural VAR and Reduced-Form VAR

Vector autoregression (VAR), first advocated by Sims (1980), has become one
of the most widely applied time-series techniques by macroeconomists. The VAR
approach is in spirit compatible to Frisch’s (1933) view that macroeconomic time
series are the result of the interaction of stochastic economic impulses and an implicit

propagation mechanism in the economy. With VARs, economists can identify the

7

role of individual disturbance in generating the business cycles and discern their

dynamic effects on the economy.

To illustrate the strategy of VARs, let yt be a n x 1 vector of I(1) variables

that has a finite p order autoregressive representation
A(L) yt = #‘l'Et- (2-1)

where A (L) E I — AIL — AgL2 — — ApLP is a matrix-polynomial in the lag
operator. The usual assumption about A (L) is that all roots of the polynomial
equation |A (L)| = 0 lie outside the unit circle in the complex domain. The mean
of 3;; is denoted ,u and the error term at is assumed independently and identically

normally distributed.

5t N lid N (0, E)
E(5t) = 0, E(€te;) = E

E(5t5;) = 0, t aé 3.

Equation (2.1) is actually estimated with data and is a reduced-form model. Our
ultimate interest is to uncover the structural relationships in the economy that

determine the dynamics of the variables.

The structural-form VAR is as

B (L) y; = 0 -+- Vt (2.2)

or
Boyt = 0 + B. (L) yt—l + Vt (2.3)
where 0 is the vector of means, B (L) 5 Bo — BIL — B2112 — - -- - Bpr’ is a n x n

polynomial in the lag operator, and B‘ (L) is defined by the equation

3‘ (L) 5 Bl +BgL+ ---+B,,LP“.

Contained in B (L) are the structural economic relations that represent the propaga-
tion mechanism mentioned above. The disturbance term Vt represents the exogenous

impulses that shock the economy and has the distribution assumption,

Vt N lid N (0, D)
E(ut) = 0, E0414) = D, where D is diagonal

E(utz/;) = 0, t aé 3.

The zero—covariance assumption for Vt, implied by a diagonal D, is essential to

isolating the individual influence of innovations on the variables.
2.1.2 Moving-Average Representation

If the stability condition of the polynomial matrix A (L) is satisfied, i.e., the
polynomial equation IA (L)| = 0 has all roots outside unit circle, then A (L) has a
inverse as

A (L)“ = C (L) a [flog-Li

where C (L) is an infinite-order matrix-polynomial. Then yt has a vector moving-

average (VMA) representation or Wold representation,
yt = 5 + C (L) 5t, (2.4)

where 6 = A‘1 (1)/1. Here C,- for j from 0 to 00 are the impulse-response matrix
because each of the matrix elements measures the impact on yt over different time-

horizons of a unit change in the error term at,

 

ay't
C-- = i“ s=0,1,---,oo
”3 65,-,

where CU, is the (2', j)th element of 0,.

However, we are interested in the impulse responses of the structural innova-

tions. Unlike an unit change in 8: that does not have intuitive meanings, the effect

10

of one unit or one standard-deviation economic shock is interpretable. Similar to
the analysis above, if the stability condition of B (L) is satisfied, its inverse exists
as

B“ (L) = R (L) 2 23.1012ij (2.5)
where R (L) is an infinite-order polynomial matrix. The set of matrices Rj in (2.5)
is the impulsaresponse function and can be defined by

= ayi t+s
5113- t

 

Rijs s=0,1,---,oo.
The vector moving-average representation of the structural—form VAR is then
yt = 6 + R (L) Vt. (2.6)
2.1.3 Structural VAR Identification

To find out the relation between the reduced-form VAR and the structural-

form VAR, premultiply (2.3) by B0‘1 to obtain its reduced form as
y, = B519 + Bg‘B“ (L) yH + 80—112,. (2.7)

By comparing (2.1) and (2.7), we see their parameters can be related by

B (L) = BOA (L) (2.8)
Vt = Boat (2.9)
D = 130233. (2.10)

Thus, the issue of identifying the structural-form from the reduced-form VAR amounts
to locating the unique matrix B0 (to be referred to as the identification matrix) so

that the left-hand-side in equations (2.8) to (2.10) are obtained.

Here we experience the same identification problem as is encountered in the

simultaneous-equation modell. To exactly identify the structural form as that in

 

1To show this, we pre—multiply (2.1) through by any n x n nonsingular matrix H to get

HA(L)yt =Hfl+HEt.

11

equation (2.3), we have to limit the choice of the identification matrix to Bo. This
initially requires 122 restrictions to solve for the n2 elements in Bo. Normalizing
the diagonal elements of Bo to unity reduces the required restrictions to n (n -— 1).
Equation (2.10) also provides luff—Q zero restrictions because D is diagonal. Thus
this standard VAR model will require 51211 additional restrictions on B0 based on

theory or sound economic intuition in order to achieve exact identification.

The first generation of VAR practitioners often applied Cholesky decompo-
sition of the reduced-form covariance matrix 2 to achieve identification. That is,
the upper off-diagonal elements of Bo, in (2.10), are assumed zero. This method
provides the additional EVE—’11 restrictions required for exact identification. This
practice, however, implies a particular recursive ordering in the contemporaneous
relations of the variables. This set of identification restrictions was originally pro-
posed by Wold (1954). The choice in most cases cannot be justified on theoretical
grounds and therefore is often arbitrary. Improved identification schemes were de-
vised later based on specific structural assumptions consistent with economic anal—
ysis. Bernanke (1986) and Sims (1986) provide two examples in which the contem-

poraneous relations of variables are set up this way.

To illustrate this later approach, in equations (2.5) and (2.6) we set L to zero
to get the contemporaneous relations between the reduced-form and the structural-
form VAR as

C (0) at = R (0) Vt (2.11)

01‘

5t = R (0) Vt (2.12)

since C (0) = A(O)"l = In by the form of A(L) in (2.1)2. As long as economic

 

Then it is easy to verify the above equation has exactly the same reduced form as that in (2.7).
2By comparing (2.12) to (2.9), we note that BO 2 12(0)”.

12

. . —1 . . . .
reasonmgs provrde at least in?) such contemporaneous restrictions in equation

(2.12), we can achieve just- or over-identification of the structural VAR model.

With increasing emphasis on the nonstationary nature of many economic time-
series, it is natural to add long-run restrictions for identification that are unavailable
to VARs with purely stationary variables. One long-run restriction implied by nom-
inal neutrality is frequently used for identification (King et al. 1991). It requires
that a permanent inflation shock has zero long-run effect on real variables. Another
example is Blanchard and Quah (1989) in which the demand shock has no long-run
impact on output. In each case, the assumption provides one zero-restriction on the

cumulative multiplier matrix

R“) = ZfioRj

which measures the long-run impacts of various shocks occurring in the distant
past. This technique, first credited to Blanchard and Quah (1989), is later used by
King et al. (1991) and Gali (1992). Compared to the practice of using Cholesky
decomposition or even the contemporaneous restrictions as in (2.11), this technique

is often more justified by economic theories about the long-run effects.
2.2 Cointegration and the Vector Error-Correction Model

In Section 2.1, we discussed the standard VAR methodology where 3;: is as-
sumed to be multivariate covariance-stationary. In this section we will deal with
the change in the formulation of VARs when y, represents a vector of nonstationary
variables. More specifically, elements of 3;; here are integrated of order one, or I(1).
We first provide a formal definition on the order of integration adapted from Engle

and Granger (1986).

Definition 1 A series with no deterministic component which has a stationary,

invertible, ARMA representation after differencing d times, is said to be integrated

13

of order d, denoted :12, ~ I (d).

Among economic time-series that are of the same I (1) order, there can be
certain linear combinations of the series that are I (0). In this case, the variables
are said to be cointegrated. The following definition of cointegration is also from

Engle and Granger (1986).

Definition 2 The components of the vector x; are said to be cointegrated of order
d, b, denoted 2:, ~ CI (d, b), if all components of wt are I (d) and there exists a non-
zero vector B so that fi'rt ~ I (d — b), b > 0. The vector fl is called the cointegrating

vector.

For the case where d = b = 1, cointegration means if the components of y, are
all I (1), then fl’y, ~ I (0) is stationary. The relation )B’y, is often interpreted as an
economic equilibrium relationship that is true only in the long run. When fl’yt 7E 0,
it is interpreted as a deviation from the long-run equilibrium or an equilibrium error.
The equilibrium error is stationary and reverts to its mean of zero over time. In
that case the equilibrium relationship among the economic variables is restored.
When the dimension of y, is greater than two, there may be multiple independent
cointegrating vectors since it is reasonable for the joint behavior of the variables
to be governed by several equilibrium relations. Gather all the r (> 1) existing
cointegrating vectors to form a n x r matrix B. The rank of B, i.e., r, is called the

cointegrating rank of y,.

We now note a standard VAR(p) model in the level of y,,
A(L)yt =H+5ti

as in equation (2.1) can be written as a VAR(p — 1) model in the first-difference of

y, plus a lag level term. It is called the vector error-correction mode] (VECM)

14

F (L) All/t = -Hyt—1 + M + 5t
or
Ayt = PlAyt—l + P2Ayt-2 + ' ° ° + Fp—lAyt-p-H - Hyt—l + II + 5t
where

HEA(1)=I—A1—A2—°'°—Ap
P(L)=I—F1L—F2L2—"'—Fp_1Lp—1

Fi=_2§:i+1Aji (i=1,°--,p—l).

(2.13)

(2.14)

(2.15)

It is clear that in a VECM the response to the long-run equilibrium error (—Hyt_1)

is expressed separately from terms that represent the short-run movements. This

distinction is an important part of what has come to be known as the Engle—Granger

two-step procedure (Engle and Granger 1986) for VECM estimation. This represen-

tation is also used by Johansen (1988) to develop his maximum-likelihood procedure

for estimating the cointegrating rank and the cointegrating space.

We note that the rank of II should be equal to the number of unit roots in the

polynomial equation |A (A) | = 0 since H = A (1) 3. The following discussion about

the rank of II is broken down to three cases concerning the times-series nature of y,:

A. If y, is a vector of stationary series, because all roots of |A (A) | = 0 are outside

the unit circle, II is of full rank n.

B. If all elements of y, are I (1) and no cointegration exists, then y, does not have

a VECM representation but a pure VAR(p — 1) in the first-difference of y,.

 

3This is by Corollary 4.3 in Johansen (1995).

15

This amounts to that II = 0 and the rank of II is equal to zero”. Indeed for
the original VAR in levels in (2.1), the polynomial equation IA (A) I = 0 has n

unit roots and therefore A (1) = II is a zero—rank matrix.

C. When all the n elements of y, are I (1) and r cointegrating relations exist among
the individual variables, the rank of the n—dimensional II is reduced by r to
equal n—r E k. There are k unit roots for the polynomial equation IA (A) I = 0

and n—k roots outside the unit circle if the equilibrium relationships are stable.

For case C, let ,6 be the n x r matrix consisting of the r cointegrating vectors
then there exists an n x r matrix a such that II = afl’ is the coefficient matrix of
yt_1 in the VECM representation in (2.14). The error-correction matrix a is often
regarded as a speed of adjustment coefficient. It determines how much change there
will be in the y, in (2.14) in pr0portion to the size of the equilibrium error, ,B'yt, in
each period. The size of the total adjustment is equal to afl’yt every periods. These
results are formally established in the influential Granger Representation Theorem

(Engle and Granger 1986) or GRT,

Theorem (GRT) Suppose the n x 1 I (1) vector y, can be expressed as (2.1).
Then the model can be written in VECM form as (2.14). Assume the n x n matrix
H has reduced rank r < n and therefore can be expressed as the product of two
full-column-rank n x r matrices a and ,6, i.e., II = afi’. Furthermore, let the n x k
matrices a _1_ and Bi be the orthogonal complements of a and B so that a’ia = 0

and [316 = 0. Then:

 

4This is because no linear combination of y, (including Hy¢_1) is stationary. Therefore, IIyt-1
should not appear on the right-hand—side of (2.14) because Ayt and all its lag terms on the RHS

of (2.14) are I (0) processes.
5But since VECM is a reduced-form represenatation it is more appropriate to treat a as loading

matrix than the speed of adjustment matrix from a structural point of view.

16

(1) Ag, and ,B’yt are stationary

(2) Ayt has a moving average representation Ayt = C (L) (p + 5,)

(3) 0(1) = 51 (a'irsir‘ a; has rank k6

(4) 3;, has the representation yt 2 yo + C (1) (p + 2::1 5,) + C“ (L) 81 where C* (L)
is defined by C (L) = C (1) + C‘ (L) (1 — L) 7.

There are two versions of proof for GRT. The original proof by Engle and
Granger (1986) deals mainly with a reduced-rank VMA representation of vector
yt. In contrast, Johansen (1991) works on a VAR representation and expresses the
theorem in terms of conditions on parameters for cointegration. The theorem pre-
sented above is adapted from Johansen’s version because in this dissertation a VAR
(VECM) is fitted to the data. Regardless of the approach, the theorem establishes
that a cointegrated system of variables can be represented in three equivalent forms:
a vector autoregression with cointegration constraints, a vector error-correction and

a reduced-rank vector moving-average representation.
2.3 The Common-Trends Model

A univariate time series that contains a unit root in its autoregressive rep-
resentation is said to be driven by one stochastic trend. Let rt be such a process

expressed as

(1— L)A (L) (Ht 1’ [1. + E} (2.16)

where u is the mean of wt and e, is a white noise disturbance term. For ease of

discussion, define z, E A (L) :12, and then ( 2.16) becomes

Zt = Zt_1+ [I + (it. (2.17)

 

8I‘ is defined by I‘ E F (1) and I‘ (L) is given in equation (2.15). It is also immediately evident

that 3’0 (1) = C (1)a = 0.
7R£sult (4) will be discussed in greater details in Section 2.3

17

Successive substitution of the above results in
t
z, = 20 +pt+Ze,. (2.1s)
3:1

Aside from 20, z, is characterized by cumulative trends, at being the deterministic
trend and z§:15, the stochastic trend. In fact, the disturbance term 5, that adds
up to the stochastic trend could itself be a linear combination of several random
shocks. In that case the unit-root process 2, is actually driven not by one but by

several underlying stochastic trends.

Based on this concept, we now discuss the concept of common stochastic-
trends in the context of a VAR for the n—dimensional y,. Given all elements of y,
are I(1), the 11 variables together contain a maximum of n distinct stochastic trends
that can derive from the n-dimensional structural innovations, 11,. The disturbance
terms, 5,, are linear combinations of the structural innovations. It is possible that
the n stochastic trends that affect each element of y, are not independent. If y,
is driven by a reduced number of independent trends then certain elements of y,
must share some common trends. Stock and Watson (1988) formalize the idea by
asserting that a common-trends model (CTM) exists for a cointegrated system of
nonstationary variables. Specifically, for a n—dimensional vector I (1) time series
with r distinct cointegrating relations, the first difference of the variables can be
characterized as driven by n — r common stochastic trendss. Because GRT also
guarantees the equivalence between cointegration and VECM, it follows that a CTM
can be derived for a VECM system and vice versa. Thus the VECM estimation
results computed from the maximum likelihood procedure of Johansen (1988, 1991)

are useful in solving for the common—trends representation.

In presenting the common-trends framework, Stock and Watson (1988) direct

 

8Conversely, if there exists k (= n - r) common trends for the n-dimensional vector, then r

independent linear combinations of the I (1) variables can be found such that they are stationary.

18

their analysis on the vector moving-average representation which can be obtained
by inverting the corresponding VAR model with cointegration constraints. The
inversion method is different from the usual method for VARs without cointegration

considerations and is discussed in Warne (1990). Write the reduced-from VMA as
where 6 = C (1) 11. Recursive substitution of (2.19) results in

y. = y0+6t+C(L)(1+L+L2+---+L‘)5, (2.20)

= y. + C(1)(ut + i a.) + 0* (L) 51 (221)

8:1

Noting in the last step use is made of the relation

0 (L) = 0(1) + 0* (L) (1 — L). (2.22)
where
OWL) = CgL+CfL+CgL2+...
C; = - f: Ck forj=0,1,2,...
k=j+1

Now (2.21) can be written as
y: = 110 + C (1) cm + C” (L) 5t (2.23)

where (p, is a random walk with drift process

Wt -:— cPt—l + II + 51 (2.24)
t
= 900 + 11t+ Z 6.. (2.25)
8:1

Recall from Granger’s Representation Theorem in 2.2 that with r cointegrat-
ing vectors the rank of C (1) is n — r and fl'C (1) = 0. Since C(l) has rank n — r,

there exists a n x r full-column—rank matrix B, such that C (1) B, = 0. Furthermore,

19

a full-column-rank n x k matrix can also be found, denoted Bk, with its columns
orthogonal to the columns of B,. Define A E C (1) B), which has rank k. Create

the nonsingular n x n matrix B = (3,38,). Then
C (1)13 = (A30) -.—. As,c

where S), 5 (11,50) is a kx n selection matrix. Now (2.23) can be rewritten as

vi 110 +C(1)BB“<pt +6" (L) 8:

= yo + A 5,, B‘lcp, + 0* (L) 5,

= yo + Ar, + C“ (L) 5,. (2.26)

The k-dimensional common stochastic-trends r, follows the random walk with drift
process9

7, = 'y + r,-1 + 11f. (2.27)

We see that n x 1 y, is driven by a reduced number (k) of stochastic trends
which, in turn, arise from the innovations uf . Equation (2.26) is the common-trends
representation proposed by Stock and Watson (1988). They use this representation
to develop test methods regarding the number of common trends or, equivalently,
the number of cointegrating vectors in a system of nonstationary variables. Note
the common-trends representation can be regarded as a multivariate extension of
the Beveridge and Nelson (1981) decomposition of a univariate time series into a

permanent component and a transitory component. The second term on the RHS

 

312
BC

r

9Define B‘1 E ( ) where B; is k x n and B: is r x n. Then (2.27) is derived as

Tt = SkB-1<P1=Bi%

3120‘ + ‘Pt—l + 5t)

’Y + Tt—l + "1-

20

of (2.26), Ar,, is the nonstationary or permanent component of y, expressed by a
linear combination of n — r stochastic trends. The third term, C“ (L) 5,, represents

the non-integrated transitory component.

The CTM specified in (2.26) can be called the structural-form CTM. The
reason is that it involves the part of the permanent-shocks 11,13 from the vector
structural-innovations 11,. The other part of 11,, denoted 11;", is called the transitory
shocks because its impacts die out with time. Identifying the structural-form CTM
from the reduced-from CTM in (2.23) is similar in spirit to the problem of the
structural VAR from the reduced-form VAR. Comparing (2.23) to (2.26) provides
the relation

AT, = C (1) (p, (2.23)

or furthermore, by (2.24) and (2.27), the three equations

A7 = 0(1),. (2.29)
Au,” = C(1)5, (2.30)

and
AA’ = C (1) >30 (1)’. (2.31)

Note (2.31) holds because we assume E (11,14) 2 In“).

To identify A which has n - k elements, as many restrictions are required.
First of all, there is one set of restrictions that can be derived from knowledge about
cointegration. Specifically, cointegration requires 8% = 0 which supplies k -r zero

restrictions. The result ,6’ A = 0 obtains because it is required for the cointegration

 

10The covariance of structural innovations equal to In means the diagonal elements of B0 in the
structural-form VAR in 2.1,

Built = 9 + B‘ (L) 311—1 + Vt,

are not normalized to 1. Therefore to identify Bo we now need 122 restrictions instead of n (n - 1).

21

of y, in (2.26). That is, for the equilibrium error,
fi’yt = fi’yo + fl’An + fl’C‘( 1)e,, (2.32)

to be stationary, the trend term, ,6’Ar,, cannot appear in the RHS of (2.26). Sec-
ondly, equation (2.30) provides another 50%;) restrictions. # restrictions are
not supplied because the n x n symmetric matrix AA’ has rank k, the same as that
of C (1). With the two sets of restrictions, there are still 5% restrictions lacking
to exactly identify A. These extra restrictions could be specified for A based on
knowledge about the long-run effects of the structural innovations 11,. One example
mentioned in Section 2.1.3 is the nominal neutrality condition which requires the
long—run impact of nominal shocks on real variables to be zero. How this knowledge
about the total impacts of 11, (specifically 11f) helps identify will be clarified in the

following discussions.
2.4 Permanent and Transitory Shock Identification

In Section 2.3, we consider issues of identifying the common-trends loading
matrix A. This sections covers the issue of identifying the entire cointegrated VAR
system. We learn from Section 2.3 that the n x 1 structural innovation vector 11,
is composed of k permanent shocks, 115’, and n - k transitory shocks 113" so that
11, = [11,” 1137],. The task is to uncover the vector 11, from the reduced-form VAR
residuals, 5,. This is operationally achieved by finding a n—dimensional square matrix
F such that 11, = F5,. Partition F so that F = [F,; Fr’I' and F, is k x n and Fr is

r x n. Then If and 11;? are individually identified as

th = FkEt

We will use a two-step procedure by which an initial candidate of F is first found

before obtaining the final F. The concept of the orthogonal compliment of the

22

cointegration space is essential for the identification.

The Granger Representation Theorem in Section 2.2 provides the reduced-

form total impact matrix as
C (1) = 11, (alFfiQ—l a; (2.33)

where a, and 6, are orthogonal compliments of a and B respectively so that fl’ifl =
c1101 2 0 and I‘ = I — ngllf‘, is from (2.15). The values of C (1), F,- a and B can

be obtained by estimating a VECM. It is then possible to specify the permanent

shocks 11,” and the loading matrix A as

u,” = c115, (2.34)

A

51 (alrfllrl - (2-35)

The specification of (2.34) and (2.35) is based on Autp = C (1)5, in (2.30). Equation
(2.34) and (2.35) implicitly defines

F, s a; = (A’A)‘1 A’C (1). (2.36)

The uniquely derived a, = F, is what defines the stochastic trends. Now, by the

structural-form long-run VMA of the first-differenced variables

Ay, = R (1) 11, = Autp, (2.37)
VP

and by recalling 11, 5 fr , we arrive at a special condition for the long-run
Vt

multiplier of cointegrated VARs as
12(1) = IAEOI . (2.38)

The interpretation of (2.37 ) and (2.38) is that in the long-run only permanent shocks
impact on the economy while effects of transitory impulses die out. A is called the
common-trends loading matrix because it transmits the long-run effect of 11,” onto

variables in the system.

23

From now on, for notational simplicity and without loss of generality, we will
write the loading matrix in (2.35), [3, (a’if‘fljfl , as B, so now A = fi,. Then

(2.33) is written as C (1) = )6 1011 and permanent shocks are still

Vi) = (AME—1 5'10 (1) 81

= 0115, (2.39)

(given C (1) known, knowing either a, or ,6, automatically yields the other). The
identification problem arises here because even with C (1) known and both 01 and
6 given, there are still infinitely many choices of 01 j and B 1”- Restrictions have to
be imposed on a, and/ or B, to accomplish identification. Usually researchers have

better ideas as to the structure of cointegration relations contained in 5.

Therefore, as in the case of identifying A in Section 2.3, we make use of the
orthogonal condition fi’fl, = 0. We have to come up first with an initial candidate
for 31, denoted fl 1, such that fl’flfl = 0. This imposes r - k zero restrictions on Bi

toward identifying 31- The tentatively identified permanent innovations are thus
01 _ 01 0 ‘1 01
01,5, — ( .LIBL) IBJ.0(1) 5t. (2.40)

Next, it is necessary to use the conventional assumption of independence
among structural innovations. Inspect the covariance matrix of the tentative per-
manent innovations,

2,. = 2012029,, (2.41)

which is restricted to be a diagonal matrix. But this will rarely be the case when it

is based on the initial 013 and 113. To ensure a diagonal covariance matrix we find

 

t

11This can be easily shown by obtaining 01’ = H01:L and [31 = B ,H '1 and verify 011’ .___, 5, is

also a common trends that satisfies

0 (1) = [31011 = Alai-

24

the lower triangular Cholesky decomposition of E p, denoted 7r, such that
213 = 7er’IT'. (2.42)

Then we officially identify a, and fly by

01, = 01EI(7r')_1

0
Bin.

A1

Note that flflafl’f = fl 1011 = C (1) is satisfied and thus a, and B, are related by
a'1: (MAD—1510(1)-

Now it is straightforward to show that the covariance matrix of the new per-

manent innovations, 11!” = i5,, is diagonal as

P P1 _ 1
E(11,11,) — 01,201,
__ —1 01 0 1-1
— 7r ayEaiUr)

: W—12P(7f’)_l

:01»

where the last step is based on (2.42). Thus we have uniquely determined F, = 01’i

and identify the permanent innovations according to
P _ _ —1 01
11, — er, -— (7r 01) 5,. (2.43)

Whether conditions for identifying 01 1 and B, are met can be checked by
counting the number of available restrictions. Since a, and B, are nx k matrices, n-k
restrictions are needed to exactly identify either. By using Cholesky decomposition
for the symmetric a‘I’Eaf’L, we get flk—J—Q zero restrictions useful in identifying I}, and

al. The initial candidate, 133, exhibits r - k restrictions. Thus, additionalflgile

 

25

nk - Egg—11 — rk) identifying restrictions are required on 63 (or B 1) . This is usually
provided by the long-run neutrality condition that restricts certain stochastic trends
not to affect y, in the long run. This sets the corresponding elements of the common-

trends loading matrix, fly, equal to zero.

The discussion on how to identify F, follows the work of Warne (1990) which
is very similar to the strategy used to identify F,. The assumption of indepen-
dence among structural innovations is still critical for the task. First, independence

between the permanent and transitory shocks requires
Cov (11:11?) = a’iEF,’ = 0. (2.44)

Warne suggests specifying an initial candidate of F, as
F,0 = ao'z-l (2.45)

where a0 is any space spanned by columns of a so that 011010 = 0 and therefore the
independence assumption holds as stated in (2.44) 12. Second, independence among

the elements of transitory shocks also requires the covariance matrix of 11:",
ST E (2053—1223—1010 = 01053—1010, (2.46)

to be diagonal. Similar to discussions on identifying Fk, this requires getting the

lower-triangular Cholesky decomposition, denoted Q, such that
ET = QDTQ' (2.47)
Then we can improve on the initial candidate by getting
F. = Q"‘F.°
= Q-l (01°)le

= (WE-1 (2.48)

 

12About the choice of a0, Englund et al. (1994) recommends an r x n selection matrix U, chosen

so that a0 = (1(Ucit)”1 with U oz a nonsingular matrix.

26
where 01* E a°(Q’)’1. By (2.48) we know the newly identified transitory shocks,
11? = (WE—15,, (2.49)

satisfy the zero-covariance requirement of (2.44). It is also easy to verify that the
covariance of 11;" is the diagonal Dr in (2.47). Now stacking the results derived from
above for F), and F,, we have the unique matrix F to identify the structural shocks

as

Vt = F53

(2121)” 510(1) I .-
Q“ (02012-1 "

 

Still, we need to count the number of available restrictions to determine
whether exact identification is indeed achieved. There are n . 1' elements in F,. Zero
covariance between permanent and transitory shocks entails k - 1' zero restrictions on
F,. The assumption of zero covariance for transitory shocks or, more precisely, get-
ting the Cholesky decomposition of the symmetric ET in (2.47 ), provides additional
dig—12 restrictions on F,. To complete the identification, 4% (z rn — dr—J—ll — rk)
more restrictions need to be furnished. These extra restrictions are typically cast in
a way to produce zero impact effects for certain transitory shocks on variables in the
system. Englund et al. (1994) discuss how zero contemporaneous effect restrictions
can be imposed by choosing the selection matrix U considered above. However,
I choose a different approach (Rasche 1992) that will be presented in Section 2.5
with a numeric example. This approach, without using U matrix, also imposes zero

contemporaneous effect restrictions to identify transitory innovations.
2.5 Example of a Common-Trends Model

The implementation of the identification procedure can be illustrated by a
simple example of term structure of interest rate and money demand study by

Rasche (1992). Four variables were studied with two cointegrating relations present

27

among the variables, real balances (m — p), real output (y), long-term interest rate
(RL) and short-term interest rate (R3). Denote :12:, = [m — p RL y R3] and the 2x4

cointegrating matrix is estimated as

s = (2.50)

 

10—1A
010—1

where A is estimated to be 0.124. The first row of 6 defines a typical money demand
equation with unit coefficient for real output. The second equation describes a

stationary term structure of interest rate relation.

To identify F such that F5, = 11,, an initial candidate for F, denoted F0, is first
selected before eventually achieving identification according to F = H‘1F0 where
H ‘1 is the Cholesky decomposition of FOE (5,52) Fo’. There are 16 (= 42) elements
in F so as many restrictions have to be specified. The Cholesky decomposition

provides ten and six remain to be devised.

The 4x 4 reduced-from long-run multiplier C (1) is obtained by applying result
(3) of the GRT and has the form of

q

' —/\ C1 + 62
0(1) = c‘ (2.50)

C2

Cl

 

 

where c, and 62 are estimated to be

c2 _ .827 .026 .668 —.019

 

clI I—17.734 .080 5.558 —.103I

It is obvious that C (1) has a reduced rank equal to 2. Notice the requirement

,6’ C (1) = 0 is satisfied. Next the initial candidate of F, is set as

F)? = 51’: I c‘ I (2.51)

62

28

and 52 is obtained, by the conditions flflafi' = C (1), as

I- H

—A

.32
n

1
0
2.52
0 1 ( )
0

 

 

I- d

With the form of 63’ chosen as above, the first permanent shock can be interpreted
as a nominal shock that permanently impacts on both the long and the short rate
(second and fourth variables)”. The second permanent shock is interpreted as a real
shock that impacts permanently upon real output (third variable) but not nominal
interest rates. Notice the condition fi’flf’L = 0 is satisfied and this provides four
identifying restrictions. There are two restrictions yet to be specified, one shall be
a zero long-run impact restriction to identify F, and the other a zero first-period-

impact restriction to identify F,.

The initial candidate for F, is chosen to be

F0:

1'

2.53
0100 ( )

 

0001I

This specification defines the first transitory shock to have unit first-period impact
on the fourth variable, the short rate, while the second transitory shock to have
unit first-period impact on the second variable, the long rate. The particular form
of Ff.) chosen above, as will be shown shortly, is essential to imposing the zero
contemporaneous-efl'ect restriction for transitory shock identification. Now stack F:

and F,0 to form

‘ —17.734 .080 5.558 —.103 '

. 2 . 2 . —. 1

F0: 87 06 668 09 . (2.54)
0.0 0.0 0.0 1.0

0.0 1.0 0.0 0.0

 

 

 

13Recall that, in the long run, A2, = Bluf.

29

Then obtain the lower-triangular Cholesky decomposition of FOE (5,52) F 0’ as

Then F is identified by

I 1.0 0.0 0.0 0.0"
0.0085 1.0 0.0 0.0
—1.9296 —19.429 1.0 0.0

_ 0.0782 9.4637 0.6039 1.0,

 

 

F = H‘IFO

p

 

I.

—17.154 0.0950 5.92 —.1138 '
1.0426 0.0270 0.6613 —0.0193
—12.844 0.7080 24.272 .4050

 

—0.7691 0.3095 —21.379 '"0'05282

(2.55)

Now we show how the zero impact-period—effect restriction is embodied in the

form of F,0 in (2.53) in order to exactly identify the two transitory shocks. First

calculate the inverse of F0 as

FO—l =

' —0.0406 0.3380 0.0022 —0.0055‘
0.0 0.0 0.0 1.0
0.0503 1.0785 0.0257 —0.0321
0.0 0.0 1.0 0.0

.. .I

 

 

The arithmetic of inverting a matrix ensures that as long as each row of F,0 has only

one element equal to 1 and all other elements equal to 0 as in (2.54), then it will

turn out that one row in (F0)_1 will have its fourth element as 1 while the other

elements all equal to 0. It will also turn out that a second row in (F0)'1 will have

its third element equal to 1 while the rest are zero. To inspect the impact-period

effects we need to calculate the impact matrix R (0) , using (2.12) in Section 2.1,

12(0) = 1‘-1 : (120)—1 H

 

’—0.0425 0.2421 —0.00111 —0.00554"

0.0782 9.4637 0.6039 1.0

0.00743 0.1763 0.00631 -0.0321
L—1'9296 -—19.4294 1.0 0.0

 

30

With the particular form of (F0)_1 along with the fact that H is lower triangular,
the product of (F0)—1 and H is guaranteed to have a zero element in its fourth
column. Thus our particular form of F,0 carries the assumption that the second
transitory shock has zero contemporaneous effect on the fourth variable, the short

rate.

As for the long-run neutrality restriction, it is assumed that the first perma-
nent shock, the nominal shock, should not have any long-run impact on the third
variable, real output. This requirement imposes a zero restriction on R (1). Specif-
ically, the restriction is imposed on the common-trends loading matrix 5, since by
(2.38), R(1) = I B, 0 I. Recall in Section 2.4 that B, is related to the initial

choice of 63 by

 

51 = 337?

-_A 1-

_ 1 0 1 0

_ 0 1 In, 1]
I1 0-
P-A+7r211.

_ 1 0

— n2, 1
I 1 0.

 

 

Then for the neutrality condition to hold, an, has to be zero 14. This supplies one

overidentifying restriction toward recovery of the structural model.

 

14It will be shown in Chapter 4 that 1r is the same as Bu which is a partition of the 4 x 4 lower-
B 0
triangular matrix B = Bu B where every partition is a 2 x 2 matrix. More discussion
21 22

on this is in Section 4.5 where my identification issues are addressed. Also, a zero restriction
on Bu essentially requires a restricted Cholesky decomposition to be estimated. A RATS SVAR

procedure is useful for such purposes.

Chapter 3

AN ECONOMIC MODEL CHARACTERIZED
BY COINTEGRATION

3.0 Introduction

A simple economic model with a wage contract is presented in Section 3.1.
The rational expectation solution of the model is obtained in Section 3.2 and cyclical
properties of the model are provided. A real interest rate identity is added to the
economic model in Section 3.3 and a resulting econometric system of five equations
is obtained. In Section 3.4, a structural-form vector error-correction representation
of the system is derived, with two cointegration relations present, M2 velocity and
ex post real interest rate. The structural-form moving-average representation is
derived in Section 3.5 and the long-run multiplier of structural innovations is used
to guide identification in Chapter 4. The presentation of this chapter closely follows
that of Chapter 5 of Hoffman and Rasche (1996).

3.1 An Economy With Wage Contract

Horn Chapter 1 we know it is desirable to use identification restrictions that
can be derived from economic theory to identify a common-trend model. For this
purpose we will utilize a model of macroeconomic fluctuations adapted from Blan-
chard and Fischer (1989, p.518). This model contains a wage contract that provides

the model economy with a nominal rigidity.

31

32

y? = mt — pt + vi, (3.1)
y: = 30% " wt + UN), 3 > 0, (3.2)
n? = 7(p, — w, + 01211,) + 113,, y > 0, 0 S a _<_ 1, (3.3)
n: = 6(w, — p,), 15 2 0, (3.4)
w, I E,_1'n.;i = E,_1n§, n, = 71?. (3.5)

The variables y,, n,, 111,, and p, are, respectively, the logarithms of aggregate output,
employment, the nominal wage, and the price level and v1, and v, are supply and
demand shocks. More specifically, since the aggregate supply equation (3.2) is a
variant of the Cobb-Douglas production function, 111, is considered the technology
or productivity shock. The aggregate demand equation (3.1) is in the form of a
velocity equation so the demand shock, v,, alternatively has the interpretation of

velocity shock.

Labor demand is affected by the labor demand shock 113, in addition to shocks
to labor productivity, i.e., 111,. Included in 113, could be exogenous job creations and
eliminations such as that due to input price shocks, increasing market integration
and specialization, demographic changes, and shocks that affect inventory and ca-
pacity utilization. It is critical to recognize that only the portions of these forces that
do not directly affect aggregate supply in (3.2) are included in 113,. They influence
the aggregate supply but only through their effects on the labor market. In (3.4)
labor supply is assumed positively related to the real wage. The cause of nominal
rigidity is revealed in (3.5) where wage contracts are set one-period in advance and
are intended to equate the next period expected labor demand and supply. The
actual employment is assumed to be demand-determined, so 11, = 11?. Thus when

unforeseen shocks take place in the upcoming period only firms can adjust the level

33

of employment and the preset wage cannot be changed. Alternatively, 113, can be
specified as a labor supply shock so 113, is put in (3.4) instead of (3.3). This is
similar to Shapiro and Watson’s (1988) specification of a random walk with drift
labor—supply process. Nothing other than a few mathematical signs will be affected
by this change in the model. For this reason, we can label 113, the labor-market

shock rather than a labor demand shock.
3.2 Solution of the Rational Expectation Model

We can solve the above equation system with Rational Expectation techniques
for the solution of p, w, and y. To save space, we leave the details of the solution
process in Appendix A. Introduce the notation 31?, E E,_1:r, for any variable 2:, so
that 512‘, is the rational expectation of 2:, made at time t—l subject to all information
available then. The solutions for 111,, p, and y, are all expressed as combinations of

the expectation terms, 513,, and expectation error terms, as, — 5,, as follow:

A

A A A g
= m+v+ a—lu + 11
Pt 1 t A“ )11 6+7 31

 

fl

 

 

 

+m (m, — fit) + (v, - 171)] - 5 + 1(111, — 171,). (3-6)
111, = fi,+17,+[fl(a— 1)+a] 1’21,+?:f:173,. (3.7)
91 = 13(1 — (1)1711 — 6 f 71731

7% (m. — m.) + (v. - a) + (us — 22)]- (3-8)

This model has properties resembling those of Benassy’s (1995) model that al-
lows for both the traditional Keynesian and the Real Business Cycle interpretations
of price and real wage movements over business cycles. To see this, first calculate

the real wage according to wr, E 111, — p, to get

 

 

wr, = afi1,+6+ fi3t
1 A A H A
‘87: [<m.—m.)+(a—e)i+ ,Htun—un). (ti-9)

34

As can be seen from (3.6), (3.8) and (3.9), unexpected technology disturbances
cause price p and output y levels to be negatively related and real wages w — p and
output y to be positively related. This results are typical of the RBC prescription
of price and real wage behaviors over the business cycle (Els 1995). On the other
hand, monetary surprises produce positive correlations between p and y and negative
relations between 11) — p and y. This is in line with traditional Keynesian views of
procyclical prices and counter-cyclical real wages due to nominal rigidity (Hairault
and Portier 1993). The observed cyclical patterns of p and w — p in a given time
period are likely to depend on the relative strength of monetary and technology
shocks in that period.

3.3 Complete Model Specification

Now we specify both the technology shock 11,, and the labor demand shock

113, are random walk with drift processes,
“it = 71+ uit—l + 611 i: 1,3,

and consequently

1711 =Ti+uit—1} 3.: 1,3.

“it "' “it = 511

As for the money supply process, we use the specification of Hoffman and Rasche

(1996) and have
m, = 7711-1 + "21

(3.10)
“21 = T2 + “21—1 + 621

It then follows that
77711 = Tnt—l + “21 — 51
mt — 7711 = 621-
The aggregate demand shock in (3.1) is also considered a stationary process with a

moving-average representation, v, = 0(L)e5,, where 6 (L) = 00 + 01L1 + 02L2 + - - - +

OPLP. Innovations 51,, 62,, 63, and 65, are all assumed white noise processes.

35

With all the stochastic terms fully specified, (3.8) can be expressed as

 

U51 E 92 = 5(1— (1) (Ti + ult—l) — (73 + 1131—1)

6+7

 

+ [61, + 62, + 6065,] , (3.11)

3
6 + 1
Here we define 115, E y, to be used below where 115, is seen as a new composite
variable dominated by two random walk with drift processes. The solved wage-

contract model can now be summarized by four equations, (3.11), (3.1), (3.2) and

(3.10), involving four variables, y,, m,, p, and 11),.

To make this theoretical model complete with more nominal variables repre

sented, we use the real interest rate identity
7‘7‘, 5 Rt "‘ fit+1 5 ¢ (L) €6t (3.12)

where R, is the nominal interest rate and 7r,+1 is the inflation rate defined by 7r,+1 =
p,+1 — p,. The real interest rate, rr,, has been found to be a stationary long-run
relationship by Mishkin (1992) and Crowder and Hoffman (1996). Here I assume it
is governed by a stable moving-average process 43 (L) = (be +d>1L1+1233L2 +- - 1+¢qL9

and 56, is white noise.

The real interest rate identity provides one additional equation but two extra
variables to the model. We can get around this problem by eliminating p, in (3.1)
and (3.2) and replace m, and 111, respectively by the real balance, mr, E m, —p,, and
the real wage, wr, E 111, — p,. Now the first equation in (3.10) can be manipulated
to be

mr, = mr,_1 ‘- 7ft '1' ”2t. (3.13)

by using mr, E m, — p, and 7r, 5 p, — p,.,. The last problem lies in the next-period
inflation rate in (3.12) whose ex post form, 1r,+1, is not available for estimation along

with y,, mr,, wr, and R, at time t . The solution is to use (3.12) to acquire a new

36

equation for the stationary process, R, —7r, —mr, +mr,_,, which includes the current
inflation, 7r,. We show this equation in (3.14) below and leave the derivation details
in Appendix B. We note the equation can be given the economic meaning that the
ex post real interest rate, defined as R, — 7r,, is about equal to, in the long run, the

growth in real balances mr, — mr,_1.

Collecting (3.1), (3.2), (3.11), (3.13) and (3.14), we have a five equation system

 

 

y‘ 5 “5t (3.11)
y‘ + fl w” = flu” (3.2)
m7", + 7r, = mr,_.1 + U2, (3.13)
y, _ mr, = 0 (L) 55¢ (3_1)
Rt _ 7ft _ mr, + mr,_1 = 72 - ,3 (1 — a) Ault + 5 + 7Aust
+fi + 1 (6” + €2t) + 112 (L) £5: + ¢ (L) 56,. (3.14)

Defining d E fl(1 — a), g E 3% and h E 3% for (3.14), then the five—equation

economic system above can be presented in matrix form as

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

'10000"y,' '00000"y,_,‘
1 0 flO 0 mr, 0 0 000 mr,_1
0 1 0 0 1 001', = 0 1 000 107,4
1 —1 0 0 0 R, 0 0 0 00 1r?H
L0—101—1_7r,‘ _0—1000‘_7r,_1‘
"0001‘ - '0 000‘-
"1: Aultl
0000 0 000
”2: Au2t
+ 0100 + 0 000
U33 A113
0000 0 000 M
”U,
_0000.-5‘- L—d0h0_‘5‘
F000 0 0 61,1 01
000 0 0 62, 0
+ 000 0 0 63, + 0 (3.15)
0000(L) 0 e5, 0
_990¢(L)¢(L)__Est, _Tz,

 

 

 

 

 

 

37

The coefficient matrix on the LHS of (3.15) contains the contemporaneous structure
of the five I ( 1) variables in the system. This information may be applied to identify
a 5 x 5 F matrix such that FZF’ = D where D, a diagonal matrix, and E are
respectively the covariance matrix of structural innovations and reduced-form errors.
Exact-identification for this model requires 10 parameters in F and 5 in D to be
estimated. For ease of analysis, premultiply the coefficient matrix, denoted F, by a

transformation matrix W so the resulting matrix has unit diagonals as in

F q - I

 

 

 

1000010000
000—10 10000
W-F=O%000 01001
00001 1—1000
-00100..0—101—1.
”10000“
—11000
= {1,0100
0-101—1
.01001_

 

 

This matrix appears to have extra zero and unit restrictions in a Wold causal chain
structure, with the exception that one nonzero element exists in the upper triangular.
There is only one free parameter to estimate and thus there are 9 over-identifying
restrictions. Sets of reduced number of overidentifying restrictions are available such
as that provided by the common-trends loading matrix [3 1- They are discussed in

the following sections.
3.4 The Vector Error-Correction Representation

We first formulate a separate equation system below to solve for the VECM
later. This system is constructed for 11,, for i = 1,2,3,5 based on specification

assumptions in Section 3.2 as

 

 

 

 

 

 

 

38

 

01'

 

 

 

 

 

U11

 

 

 

 

 

(1—L 0 0
0 1 —' L 0 O 71%
0 0 1 — L 0 113,
—dL 0 hL l .1 _ U5; ..
’10 0 0 0“ :‘ ' fl
_ 01 0 0 0 m + a (3m)
00 1 0 0 Q‘ a '
e
_ g g 0 900 0 5t d7, —- hT3
. . €6t . . -
Now the inverse of the polynomial matrix on the LHS of (3.16) is
(1 0 0 0
0 1 0 0
1—L‘1 317
( ) 0 0 1 0 ( )
L dL 0 —hL (1 - L) .
By premultiplying through (3.16) by the square matrix in (3.17), we get
TAM,“ 1 0 0 0 0'
13113, 0 0 1 0 O
_Au5,_ de+g(1—L) g(l—L) —hL 909(1—L) O)
E” '1 0 0‘
€2t 7’ 1
0 1 0
x + 3.18
Est 0 0 1 T2 ( )
e
51 - d 0 —h - T3
L 66‘ .

 

 

To derive the VECM, add one-period-lagged (3.11) (3.2) and (3.13) respectively to

the RHS of (3.15) to obtain

 

1- q

1 0 0 0 0
1 0 5 0 0
0 1 0 0 1
1 —10 0 0
_0 —10 1-4‘

 

 

y,
1711‘,

UN}

7ft

 

OCOHH

 

OMOO

l
p—n

OOOQO

OOOOO

 

 

lit—1
mTt-r
1071-1

Rhl

7Tt— 1

 

d

 

 

 

 

 

 

 

 

 

 

 

-l J
U%&&
uuuu
AAAA
P b
1 l-
10000
0000h
00100
06,00d
+
mu. 1
24222
snap...
ymefl
P b
q ld
00000
00000
00000
004.00
00000
+

 

 

 

 

 

(3.20)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

)
9
J
3
(
la
a 11..
8 Emma... .111...
1235
m meMM uuuu
mfl AA AAAA
1 p L- I
. :1 0000110000
ttttt 0
. 1% 0000_
. is. 00100
x) h 00000 d
t
0000@ 0 0011006r00_
¢ b
m 1 _ -
)D/ 0 000_0 +
DOOM/k an r t. .
0¢. 1 ._ 44444
11 tt
1 11 r t
00000._.n_.n_._._.. t... .wmwflw
00009ymeMWMWWDmtm. .
00009 AAAAT
. .q .F .00001
1
0010
+ _. 1.00001_.
1 1000010010.
0000B. 00000
p .0 000001
6,00 00010
1103,000
1
001__ 11000.0
001
111010 __.
_ 1010 +

 

 

 

Subtract

' 1

 

 

 

 

 

 

 

 

 

 

 

4...........4
r
.43....
.. AAAA
M. .
.C- u
ummamE 00001
Geeeev
ld 00001_.
.m
t
D/C 00000
0000(m
¢fl00110
e
\leLlum 000.1..0
nF .
0¢.l
m __
00000b
01 .
000090 t... t
0000 x)
. g.n0/u AAAA
.r .
+ MW
- .m— 1.
0000h.m 0010_
- )
+ lw00001
806,000
m 11
“001__
t
on
1% 11010
u. .
S
D
e
h
T

40

 

 

 

q

 

 

0 0 P lit—2
.33 (1100022:
0 0 0 —1 1
1 0 Rt—2
.0 1. . 70—21
'dL+g(1—L) g(l—L) —hL 009(1-L)
0 0 0 0
+ 0 1 0 0
0 0 0 0(L)
_ g-d 9 h ML)

 

OOQR
HOP-loo

—d

H

0
0
0
0

¢(L

 

)1

D‘OOO

 

 

l

61¢
€2t
53:
651
€6t

-

 

7'1
7'2
7’3

(3.21)

To obtain the reduced-form VECM, first take inverse of the coefficient matrix

on the LHS of (3.20) as

CO

1
1

0
1’1

EP-
Ohl"
H H C O O

 

O

O

Or—tooo

 

H0O

Now premultiply it through (3.20) and rearrange the error-correction term to get

the reduced-form VECM:

 

 

 

 

 

 

F Ay, _ 0 O
Amr, 1 —1
Awr, = 0 0
AR, 0 1
_ A7r, —1 2
( 0 0 ‘
1 0
1 —1
+ 0 0 [0 0
0 —1
h—1 0 .1
' dL+g(1—L) g(l—L)
dL+g(1—L) g(l—L)
+ _ —fl+dL;g(1—L) ”$9 (1 _ L)
g—d g+l
.-dL-9(1-L) 1-9(1-—L)

 

 

 

 

 

 

0 0 0 ' ' Ay,_1
0 O 0 Amr,_1
0 0 O Awr,_1
0 -1 1 AR,_1
0 0 0 . L AWt_1
P yt—z .
0 0 0 W”
'LUTt_2
0 1 —1
R14
1. 70-2 .
—hL Boy (1 — L)
—hL 009 (1 — L) — 0 (L)
éhL —%Oog (1 — L)
h 10 (L)

l

0
0
0
¢ (L)
0

 

 

 

 

611 d 0 —h
62, d 0 —h 7',
x 63, + -— 1‘“ 0 %h 7’2 . (3.22)
65, —d 2 h 73
L€6t . L —d 1 h, .

 

In the above derivations, a structural vector error-correction representation is
obtained from a fully specified economic model characterized by two cointegration
relations. With the time-series nature of the stochastic shocks specified as in Section
2.3, the existing long-run equilibrium relations between the variables are already
revealed in (3.1) and (3.12). The first cointegration relation is the velocity of money,
y, — mr,, and the second one is the ex post real interest rate, R, — 7r,. The knowledge
about their exact forms will be useful in terms of improving statistical efficiency when
we estimate a reduced-from VECM model in Chapter 4. In other words, the theory
superimposes ‘known’ cointegrating vectors so no parameters need to be estimated
in a VECM. We notice that even though no speed of adjustment parameters are
specified in the theoretical model, a VECM representation is derived that specifies

‘known’ adjustment parameters in Oz.

Now we show that the t0p 3 x 5 partition of the structural matrix F on the LHS
of (3.21) is a common-trends matrix a’i. The structural error-correction coefficient
in (3.21) can be written as Fa where a is the 5 x 2 reduced-from adjustment matrix.

Define F = if where F), and F, are 3 x 5 and 2 x 5 respectively. Then

1'

Fka

Ea
' 1 0
1 0
= 0 1
1 -1
LO -—1 0

Fa:

 

co‘cbo
Hoooo
II
OHCOO
Hoooo

 

 

 

 

 

 

-1 L-l 0 L J

implies Fka = 0. Thus F), is an orthogonal compliment of a and may be treated as a

42

common-trends matrix a’i. If we only impose identification restrictions included in
F], and allow F, to be freely estimated, the number of overidentifying restrictions is
reduced for the model. Such an approach also amounts to ignoring all the coefficient
restrictions in (3.1) and (3.14) which compose the dynamic structure of the economic
model. This is a meaningful approach only if the entire structure of F is rejected
by an overidentifying restriction test. Only in this case would we ask the question
whether the long-run structure of the model alone can be successfully estimated by

the data.
3.5 The Vector Moving-Average Representation

To construct a VMA for the 5-dimensional vector variables, we first write the

structural VAR model in (3.19) in lag operator as

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

H-L 0 0 0 0 ‘ ' y, ‘ '01
1—L 0 fl(1—L) 0 0 mr, 0
0 (1—L)2 0 0 (1—L) 1111', = 0
1 —1 0 0 0 R, 0
_ 0 —(1—L) 0 1 —1 L70 J _ng
(0001‘-A.'0000 0"51,‘
"It
0 0 0 0 AW 0 0 0 0 0 62,
+ 0 100 + 000 0 0 63,. (3.23)
A113,
0 0 0 0 (Aust 0 0 0 0(L) 0 65,
_~d0h0, ‘ _990¢(L) ¢(L) Lem
Substitute (3.18) into (3.23) to get
'1—L 0 0 0 0 y, ' ' d 0 -h'
l—L 0 0(1—L) 0 0 mr, s 0 0 7'1
0 (1—L)2 0 0 (l—L) 101', = 0 1 0 T2
1 -1 0 0 0 R, 0 0 0 T3
0 —(1—L) 0 1 —1 “_7r,‘ _—d 1 h‘
“dL+g(1—L) g(1—L) -hL 009(1—L) 0 l '51,‘
fl 0 0 0 0 62,
+ 0 1 0 0 0 £3, . (3.24)
0 0 0 0(L) 0 65,
_ 9- d g h ML) ¢(L) , 56t ,

 

 

 

 

43

Now construct the inverse of the polynomial matrix on the LHS of (3.24),

 

' 1 0

1 0

—1 1 1
(“Ll ’3 a
0 0

.—<1-—L) o

0
0
0
1
1

0 0
—(1—L) 0

0 0

0 (l—L)
(1--L)2 0 l

 

(3.25)

Premultiply through (3.24) by the matrix in (3.25), without (1 — L)—l, to obtain

the VMA model Ax, = 6 + R(L) 11,. After collecting terms we derive the VMA

representation of the economic model as

 

 

 

 

 

 

P MU P dL+g(1-L) 9(1-L) l
Amr, dL+g(1—L) g(1—L)
Aw'rt = 1 — % (dL +9 (1 — L)) 61; + —%g(1 — L) 62,
Are (g-d)(1-L) 1+g(1-L)

(A2,. _—dL(1—L)—d(1—L)2, _1-g(1—L)2_,

’ —hL ‘ ' 009(1—L) _ 0
-hL (909-9(L))(1-L) 0

+ %L 63; + 33999 (1 — L) 65, +
h(1-L) ¢(L)(1-L) ¢(L)(1-L)
LhL(1-L) _ _—[Oog-0(L)](1-L)2. .

( d 0 —h‘

d 0 —h T1

+ 1—21;d 0 % 72

0 1 0 T3
0 1 0 .

 

 

 

 

 

 

 

 

Est

(3.26)

To find out the long-run multiplier R (1) = [[3, 0] of the structural innovations

12,, set all L equal to 1 in (3.26) to get

' d 0

d 0

R(1)Vt= 1—% 0
0 1

L 0 1

 

611
62:
63:
Est

OOOOO
OOOOO

 

 

 

Est _

(3.27)

44

and the constant-term vector

d 0 —h
d 0 —h 71
6: 1—g 0 g 72 . (3.28)
0 1 0 7'3
L 0 1 0 _

 

 

The 5-dimensional vector 6 forms the deterministic trend component in the level
of x; = [3), mr, 001', R, 7r,] and the 3—dimensional T can be treated as the common
deterministic trends. The particular form of R (1) in (3.27) reveals that innovations
65, and 56,, with no long-term effects on the first-difference of 33,, are transitory
shocks. In fact, they are, respectively, the underlying force that forms the stationary
demand shock in (3.1) and the stationary real interest rate shock in (3.12). On the
other hand, the technology shock 15,, and the labor-market shock 63, have nonzero
long-term effects on real variables Ay,, Amr, and Awr,. The monetary shock 62,
has long—run effects on nominal variables AR, and A7r, but not on real variables.
Thus a long—run neutrality of money is a property of the economic model. The three

permanent shocks constitute three common stochastic trends in the level of 3,.

As considered in Chapter 2, the particular form of )6, contained in R (1) is
valuable to identify common stochastic trends or permanent shocks. In the preceding
sections structural information included in F and its subset oz’i also are shown useful
for identification. Between a, and 51 however, only one is required for identification
by the relation R (1) F = )6 10'; = C (1). As an reduced-form moving average model
or C (1) can be derived from the same theory, the knowledge of either a _L or 0,
yields the other immediately. In principle identification based on a, or 0, should

also have equal statistical power for overidentifying restriction tests.

Chapter 4
SPECIFICATION, ESTIMATION AND

IDENTIFICATION OF
THE ECONOMIC MODEL

4.0 Introduction

This chapter presents an econometric analysis of the five-equation cointegrated
system and the identification of the structural model. The time-series pr0perties of
the variables and two assumed cointegration relations are examined in Section 4.1
using unit-root tests and graphs. Specification tests are conducted in Section 4.2 to
determine an optimal vector error-correction model to estimate. Two cointegrating
vectors, an M2 velocity and an ex post real interest rate, are imposed on all VECMs
specifications considered. A three-lag VECM including a linear time trend on the
levels and five dummy variables is selected. The validity of imposing two cointe-
grating vectors on a VECM is confirmed in Section 4.3 by Johansen’s (1988, 1991)
cointegrating rank tests and Horvath and Watson’s (1995) tests for pre—specified
cointegrating vectors. The VECM estimation results are presented in Section 4.4.
The identification of structural VMAs using dynamic and steady-state restrictions

derived in Chapter 3 is shown in Section 4.5.
4.1 Data Analysis and Unit-Root Tests

Seasonally adjusted quarterly data of the United States from 1951:1 to 1994:4
are studied in this dissertation. Particularly, real output (y,), real balances (mr,),

real wages (1111“,), short-term nominal interest rate (R,) and inflation rate (7r,) are of

45

46

major interest. Variables are in natural logarithms, except for the nominal interest
rate. The measure for real output is the private-sector GDP used by King et al.
(1991) defined as GDP minus the government purchases”. Real balances are equal
to the M2 measure divided by the price deflator. Real wages are defined as the
nominal wage divided by the price deflator. The nominal-wage measure used is the
average hourly earnings of workers in the manufacturing sector. The interest rate on
Treasury Bills of 3-month maturity is used as the short-term nominal interest rate.
Inflation is the log first difference of the price deflator. Both the nominal interast
rate and the inflation rate are on per annum basis. Detailed variable definitions and

data sources are presented in Appendix C.

Before formally undertaking statistical tests to determine the time-series prop-
erties of the variables, it is useful to first graph the variables and make a preliminary
statement. The levels of the five variables in this study are presented in Figures 4.1,
4.3, 4.5, 4.7 and 4.9 while their first-differences are presented in Figures 4.2, 4.4,
4.6, 4.8 and 4.10. The levels of output, real money balances and real wages appear
to be trending upward so they may be generated either by random walk with drift
process or be trend stationary processes. The levels of the nominal interest rate and
inflation, despite not showing any clear trend movement, appear nonstationary be-
cause of changing mean levels and variabilities. They may be generated by random
walk processes without drift or simply be stationary series. If the first differences of
variables are judged to be I (0), that can lend support to the claim that their levels
are I (1) The first—differences of the variables appear to have constant means in the
figures. However, except for that of inflation, the first differences appear to have

changing variability. The inspection seems to suggest that the first differences are

 

15Profassor Rasche points out that this definition of real output is not private-sector GDP but
private-sector gross purchases becuase the former should only exclude government purchases of

labor services from GDP rather than total government purchases.

ELSO

£125

£100

'175

‘150

'125

'100

(L036

0.024 —1 N

1(4le

41000
41012 —‘

41024 ’*

47

de

 

 

 

 

IITIITTlllTrlllllllllITlllllllllllTTTllllTlT

1951 1957 1963 1969 1975 1981 1987 1993

Figure 1 Real Output (Private Sector), 51:1-94:4

First difference of (36p

 

 

 

 

 

L—__
—

 

 

 

 

 

ll (1 ' )1 ”v '1

%»

 

 

 

 

 

 

 

 

41036 11 1111'111 ITI 111 0111 1111 1111 lril 111 111 1111 11

1952 1958 1964 1970 1976 1982 1988 1994

Figure 2 First—differenced Real Output, 51:1-94:4

84)

48

 

7.8 H

715 r

7J1 —

7.2 J

74) —

613'“

 

615

 

(1054

(1045

(1036

(1027

(1018

(1009

(1000

41009

41018

lTlllllllllilllllrillllleilllllllillllllIII

1951 1957 1963 1969 1975 1981 1987 1993

Figure 3 Real Balances, 51:1-94:4

First difference of Mr

 

 

 

 

 

 

 

a 1‘1 “11 111 11111“

 

 

 

11111111111111111111111111111r1111111111111

1952 1958 1964 1970 1976 1982 1988 1994

Figure 4 First-differenced Real Balances, 51:1-94:4

2.34

49

 

2.28 —1

2.22 ~

2.16 —*

2.04 —*

1.98 -*

1.92 —1

 

1.86

 

0.0240

0.0200

0.0160

0.0120

0.0080

0.0040

0.0000

-0.0040

-0.0080

-0.0120

Tilli llllllllilTlITTTTlllllllllllllTTTTTiT

1963

1951

1957

1 969

1975 1 981 1987 1 993

Figure 5 Real Wages, 51:1-94z4

First difference of Wr

 

11

 

.1111

P
Lg

 

 

 

 

ll

'1111‘1“

 

E

11 , 1
J 11 V

 

 

 

 

IlllTllllllllllilililil llTITTllTTlTYllTFTT

1 952

1958

1964

1 970

1 976 1 982 1 988

Figure 6 First-differenced Real Wages, 51:1-94z4

 

 

50

16

 

14 fi

124

10 fl

/

fl

“/V

 

 

 

o TTllllllTlllllTTlITliTjTllTllTlIITTIllllilll

1951 1957 1963 1969 1975 1981 1987 1993

Figure 7 Nominal Interest Rate, 51:1-94z4

First difference of R

 

_A
l

 

o w/WuM.mMAMN\1l/111 M11111 . Jv
V111 11111111 11111111

 

 

 

 

 

 

 

'4 TlilllllllililllIlITTTTTIIlerlllillTlllill

1952 1958 1964 1970 1976 1982 1988 1994

Figure 8 First-differenced Nominal Interest Rate, 51:1-94z4

51

 

 

 

 

 

 

Inf
15
1o—
5—
O 11 v1 ‘1 W
'5 111111111111111111111111—r1—r11111111111111111
1951 1957 1963 1969 1975 1981 1987 1 993

Figure 9 Inflation Rate, 51:1-94z4

First difference of Inf
7.5

 

5.0 -*

2.5 #

i.111111i1,.111.1i.11l “11
11111 1111 111

0.0

 

@—
E
—":

 

-2.5 -‘

 

 

 

-7.5 —‘

-10.0 —1

 

-1 2.5 —‘

 

 

 

45.0 1111111111111111111111111111111111111111111

1952 1958 1 964 1 970 1 976 1982 1988 1994

Figure 10 First—difl'erenced Inflation Rate, 51:1-9424

52

 

 

 

 

 

de - Mr
Q50
045 —
0A0 _
035 ~
030 — \A\W/Nq
025 * //MkJ/V
020 111111 11111 f1111 1FTT11 11111 TITTfiFTTIT1iTITTj41F
1951 1957 1963 1969 1975 1981 1987 1993
Figure 11 Velocity of Money (M2), 51:1-9424
R - Inf
10
5 _—

11-1

 

-10 ~

 

 

 

 

'15 IIIIIIIIIIIIiIIIIIIlIIIIIIIIIrIIIIITTITTIIII

1951 1957 1963 1969 1975 1981 1987 1993

Figure 12 Ex Post Real Interest Rate, 51:1-94z4

53

not stationary I (0) series.

Table 1 shows test results of the Augmented Dickey-Fuller unit-root t and 2

tests (Dickey and Fuller 1979). The tests involve running regressions of the form

1:
xt = a + 6t + pxt_1 + Z bjA$t_j + 6,. (4.1)
j=1

A linear-trend term (St is included in (4.1) if graph-inspection suggests there is a
trend component in the series. Conversely, a regression is run without a trend term
as is the case for interest rates, inflation and all the first differences of the variables.
The tests are conducted using the RATS uradfsrc procedure written by Norman

Morin. The t and 2 test statistics are calculated according to

i—1 .
ti = p“ Z=fl,7’
0i

 

21' = T(fii—1) i=fl,7’

where p,- and 6.- are respectively the estimates of p,- and its standard error. The
subscript p and T indicates whether a test statistic is computed from regressions
run with (T) or without ([1) a trend term included. Both type of statistics (with
subscript T or p) have non-standard distributions so tabulated critical values have to
be consulted. The joint F tests, in cases where no trend term is included, have the
null hypothesis p“ = 1 and 01,, = 0. That is, a random-walk process is hypothesized.
In cases where a trend term is included, the null hypothesis is p, = 1 and 6 = 0, i.e.,
a random walk with drift process. The F statistic is calculated in the usual Wald
form but its asymptotic distribution is again nonstandard and appropriate tables
are to be consulted (Dickey et al 1994). The number of lag differenced variables,
indexed by j in equation (4.1), is determined by the Ljung-Box autocorrelation tests
done on the regression residuals ’6}. Lags are sequentially added until the Ljung-Box
test fails to reject the null of no serial correlation. This procedure is followed because

at has to be white noise for the unit-root test to be valid.

54

Table 1

Augmented Dickey-Fuller Tests
for a Unit Root

 

 

Lag t, (t,,) z, (zu) Joint F test
y 1 -2.6023 -16.2625 3.5046
m 2 -2.3811 -5.8579 3.2850
p 2 -1.4618 —2.6930 3.1096
71 1 (—4.24236) (—36.35896) (9.00056)
10 7 -1.9781 -5.8950 2.2940
7717‘ 1 -1.1983 -4.8765 1.2658
1117‘ 0 -0.2084 -0.2708 22.43056
R 11 (—1.5204) (—4.4261) (1.4797)
Ag 0 (—8.87256) (—110.24506) (39.36616)
Am 1 (—5.17606) (—52.76646) (13.39916)
A71 4 (—8.04226) (821.0265) (32.33836)
Aw 5 (—3.62606) (—30.9860°) (6.76826)
Amr 0 (-8.09146) (—93.27226) (32.73956)
A107 2 (—5.16776) (—66.72166) (13.36926)
AR 6 (—6.19166) (365.5331) (19.18216)

 

Notes: 1. Statistics in parenthesis are t“ and z” and are t,

and 23,- otherwise. 2. a, b and c indicate statistical significance at
the 10%, 5% and 1% level respectively. 3. Critical values are shown
in Table 2

As shown in Table 1, the ADF test fails to reject the null of a unit-root
at 10 percent level for yt, mrt, wrt, and R, and therefore supports our claim that
they are I(1) variables. On the other hand, the unit-root hypothesis is strongly
rejected at 1 percent level for Ayt, 71;, Am, Amrt, Awrt and A114 suggesting they
are I(O) stationary. These are almost exactly what we postulated in the theoretical
model presented in Chapter 3, except that the inflation rate, 71¢, is assumed an I( 1)

variable there. Recall inflation is cointegrated with the I( 1) nominal interest rate,

55

Rt, to form the stationary real interest rate relation, R, — 7”er In the literature
there is a debate about whether the order of integration for inflation is I(1) or I(O).
Discussions on this can be found in Baillie, Chung and Tieslau (1996). Likewise,
there are also conflicting findings regarding whether the order of pt is 1(2) or I(1)
in the literature. Despite the unit-root test results, we will still treat 71¢ as an I(1)
variable in the following analysis for two reasons. First of all, there could be a
power of test problem for the unit-root test procedure. Diebold and Rudebusch
(1991) found the Dickey-Fuller test has low power when the true value of p is near
but not equal to one. Furthermore, we have just learned that the nominal interest
rate is 1(1) and we will also learn shortly that the ex post real interest rate, R, —7rt, is
indeed an I(O) cointegration relation. Then 71; being the difference of R, and R, — 71¢

cannot be I(O) since I(1) +I(0) cannot be I(O).

Figures 4.11 and 4.12 graph the two cointegration relations, the M2 velocity
(yt — mm) and the ex post real interest rate (R, — 11,). They appear to be station-
ary despite slightly irregular mean levels relative to the univariate time-series plots
shown in Figures 4.2, 4.4, 4.6 and 4.8. Under this circumstances, we have to rely
on formal tests to make inferences about their true properties. Table 2 shows the
Augmented Dickey-Fuller test results for these two relations. In the top panel, re
gressions are run without a trend term included while one is included in the bottom
panel. We will rely only on results in the top panel because neither R, — 71; nor
yt — mm show any discernible trend movement in Figures 4.11 and 4.12. The bot-
tom panel is included to provide extra reference”. Notice that in both panels the

tests are conducted over two sample periods of difl'erent length. The shorter sample

 

16We briefly mention that the test results for regressions with a trend included. The tests still
strongly reject the unit root hypothesis for the real interest rate. But all t, z and F tests fail to
reject the null of a unit root for the velocity relation at 10 percent level no matter whether the

long or short sample is used.

56

Table 2

Augmented Unit Root Tests for
31¢ — mm and Rt — 7ft

 

Without linear trend in regression

 

 

 

 

Lag t,‘ 2:” F test
The short sample:1953:1-1991z4
y¢ — mm 1 -3.0404" 47.9109" 4.6290"
R¢ — 71¢ 1 -3.89766 -30.78746 7 .60656
The long sample:1951zl-1994z4
y¢ — m7‘¢ 1 -1.5416 -8.6028 1.2536
R¢ — 71¢ l -5.1136c -48.8346c 13.1504c
Sig. level
10 % -2.57 -11.2 3.81
Critical value 5 % -2.88 -14.0 4.63
1 % -3.46 -20.3 6.52

 

With linear trend in regression

 

 

 

 

Lag t, 2, F test
The short sample:1953:1-1991:4
y¢ — mr¢ 1 -.3015 -17.7332 4.7997
R¢ — 71¢ 1 —4.37126 —39.26190 9.55496
The long sample:1951zl—1994z4
y¢ — mr¢ 1 —1.7825 —9.4958 3.7957
R¢ — 71¢ 1 —5.5728C —59.7335"' 15.56266
Sig. level
10 % -3.13 -18.0 5.39
Critical values 5 % -3.43 -21.3 6.34
1 % -3.99 -28.4 8.43

 

Notes: 1. a, b and c indicate statistical significance at the 10%, 5% and 1%
level respectively. 2. Lag indicates the lag lenth of the ADF regression.
3. Critical values are from Hamilton (1994).

57

covers the period from 1953:1 to 1991:4 and the longer sample covers from 1951:1
to 1994:4. Both sample periods are tested because they yield different results, sug-
gesting there could be fundamental shifts in the long-run relationship between the

two samples.

In the t0p panel of Table 2, and over the shorter sample, all t, z and F tests
strongly reject the null of a unit root at 5 percent level for the velocity and at 1
percent level for the real interest rate. Therefore, both cointegration relations are
valid for the shorter sample. The story is somewhat different when the tests are
done over the longer period. While the real interest rate continues to be shown
stationary at a 1 percent significance level, tests on the velocity fail to reject the
unit-root hypothesis at 10 percent level for the longer sample. Thus there appears
to be a regime change in the data generating process for the velocity of M2. This
change causes the cointegrating velocity relationship to break down in the ADF
tests. To confirm, we observe in Figure 4.11 that the velocity, before 1953:1 and
after 1991:4, moves in a more drastic fashion and its mean is also higher than the
mean of observations in the shorter sample. For this reason we will estimate an
empirical wage-contract model using only the short-sample data that upholds both

cointegrating relations.
4.2 Specification for a VAR. with Cointegration

The first question to be answered in specifying a vector-autoregression model
is how many lags to include. It is also customary to add seasonal or regime dummy
variables to capture systematic shiftings in time-series processes. The most widely
applied specification test for such decisions is the likelihood-ratio tests. Write a
standard n—dimensional VAR with p lag-terms and a vector of dummy—variables D¢
as

P
23¢ = Z Aixt—i + [J + ‘I’Dt + 5t (4.2)

i=1

58

where ,u, A,- and \II are parameters to be estimated and the error vector s¢ is as-
sumed distributed as i.i.d. N (0, 2). The likelihood-ratio test statistic Sims (1980)

suggested is

A

(T — c) (In E,

 

 

— M1234) (4.3)
where f), and 2,, are respectively the maximum-likelihood estimates of the error
covariance of the restricted VAR (with a shorter lag and fewer dummy variables)
and the unrestricted VAR. T is the number of observations used and c = 1 + 5 p is
the number of parameters in each equation in the unrestricted VAR. The statistic in
(4.3) has an asymptotic x2 distribution with degree of freedom equal to the number

of restrictions imposed on the system.

If cointegration is an important characteristic of the equation system, then a
cointegrating-rank restriction or even explict cointegrating-vectors need to be im—
posed in estimating a VAR. In such cases, the VAR is usually modeled by its vector

error-correction representation,

p-l
A513,: = [J + Z PiAIE t-i + 0,6, IE t-l '1’ WD; “‘1' 513, (4.4)

1:1

where u and D¢ are constant and dummy variables respectively and the Rs, (1 and
6 can solve for the A,-s in (4.2). Under the assumption of r cointegrating rela-
tions, a and 16 are both 77. x 7' matrix and the rank of II E afl’ is 7. Typically,
Johansen’s (1988, 1991) maximum-likelihood procedure is used to model (4.4). In
cases where specific cointegration vectors are imposed, fl’:c¢_1 become known vari-
ables and the maximum-likelihood estimates of the parameters can be obtained by
running ordinary-least-squares for each of the n equations in (4.4). The OLS is a
valid method because typically no cross-equation restriction is imposed on VARs or

VECMs. The models in this dissertation are estimated using the CATS package.

There are three sets of dummy variables under consideration for D¢ in (4.2).

The first set is labelled DummyO which in fact contains no dummy variables at all

59

so that D¢ = O. The second specification is labelled Dummyl which includes three
time dummies, Den, Dyg¢ and Dan, so that D¢ is a 3 x 1 vector. This specification
has been used by Hoffman and Rasche (1996) in their studies of money-demand
functions. The third set of dummy variables is Dummy2 which includes two time
dummies, D73¢ and D75¢, in addition to the three defined in Dummyl. The three

dummy-variable sets and the five dummy variables are defined below:

DummyO = {No dummy variables}
Dummyl = {D67t7D79t7D82t}

Dummy2 = {D6771 D73 t1 D75t: D7911 D821}

where

0 t<t-
n,,: 3 '=67, 73, 75, 79, 82
1 1121,

and

fig 2 19671 4, 1573 = 1973 2 4, t75 = 1975 2 3,

1979 : 4, 1,, = 1982 : 1.

7579

The dummy variable D73¢ is zero through 1973:3 and one thereafter while D75¢ is
zero through 1975:2 and one thereafter. They are included to account for the 1973-
74 oil price shocks and the sharp increase in the price level that it caused. 0671:
is defined as zero from the initial time period through 1967:3 and one thereafter.
It is included to capture the acceleration of inflation with the Vietnam conflict.
D79¢ is zero through 1979:3 and one thereafter and D32¢ is zero through 1981:4 and
one thereafter. They are included to reflect the New Operating Procedures of the
Federal Reserve in place between 1979z4 and 1981:4 since means of the cointegrating

vectors could shift with this policy change.

Table 3 shows values of the likelihood functions for VARs of different lag-length

and dummy-variable specifications. Since the velocity and the ex post real rate coin-

60

Table 3
Maximum Likelihood Function of VARs

 

 

VAR lags DummyO Dummyl Dummy2
1 -28.7692 -29.1157 -29.4820
2 -30.0543 -30.2567 -30.5817
3 -30.5506 -30.7167 -30.9095
4 -30.8103 -30.9872 -31.2050
5 -31.0312 -31.1817 -31.4047
6 -31.3632 -31.5443 -31.8064
7 -31.5596 -31.7440 -31.9914
8 -31.8002 -31.9795 -32.2156

 

tegration relations are imposed in estimation, we in fact estimate the VECMs. The
likelihood functions are then used to conduct likelihood-ratio tests to first determine
the Optimal dummy-variable setting as shown in Table 4. Then they are used to
test for the optimal lag-length specification as shown in Table 5. Judging by the
test results in Table 4, regardless of the lag-length of VARs, Dummy2 consistently
We therefore accept the Dummy2

fares better than DummyO and Dummyl”.

specification and include all five dummy variables in the VAR model.

The likelihood-ratio test results shown in Table 5 consistently suggest VARs
with either four or six lags is the optimal specification to use. However when a

likelihood-ratio test for a constrained five-lag VECM (six-lag VAR), against a non-

 

17In the case of testing for DummyO versus Dummyl, the degree of freedom is 15 = 3 x 5,
the number of the reduced dummy variables (3) times the number of equations (5). Similary, the
degree of freedom in the test of Dummyl versus Dummy2 is 10 = 2 x 5 and in DummyO versus
Dummy2 equal to 25 = 5 x 5. In the case of testing for models with sequentially shorter lags, e.g.,
p - 1 versus p, the degree of freedom is equal to the number of elements in the parameter matrix

A,,or25=5x5.

Table 4
Likelihood Ratio Tests for Dummy Specification

 

 

 

 

VAR Lags D0 vs. D1 D1 vs. D2 D0 vs. D2
1 49.5595* 51.6384* 100.5048*
2 27.9271* 44.2014* 71.7237*
3 22.1020 25.2489* 47.0185*
4 22.6458* 27.4428* 49.7347*
5 18.5164 26.9842* 45.1996*
6 21.3781 30.4048* 51.4205*
7 20.8429 27.4559* 47.9298*
8 19.3687 25.0255* 44.0356*
d.f. 15(=3x5) 10(=2x5) 25(=5x5)
X2 (a = 10%) 22.31 15.99 34.38
" indicates significant at 10%
Table 5

Likelihood Ratio Tests of Lag Length

 

 

 

VAR lags DummyO Dummyl Dummy2
1 vs. 2 181.2076* 157.4525* 149.5646*
2 vs. 3 67.4886* 61.1867* 42.9392*
3 vs. 4 34.0299 34.6253* 37.2418*
4 vs. 5 27.8284 23.9210 24.1625
5 vs. 6 40.1684* 42.7868* 46.5972*
6 vs. 7 22.7871 22.5684 20.5306
7 vs. 8 26.7044 25.4308 23.7663

 

Critical valuezx2 = 34.38 (a = 10%, d.f. = 5 x 5)
for all cells. " indicates significant at 10%

62

Table 6
Residual Analysis for VECMs with 4 and 6 Lags

 

 

Lags 611 621 631 641 651

Standard dev. 4 0.009 0.007 0.004 0.623 1.451

6 0.008 0.007 0.004 0.569 1.401

R2 4 0.445 0.517 0.411 0.465 0.521

6 0.510 0.561 0.483 0.553 0.554

Normality 4 9.1336 28.6696 0.567 70.8436 4.870

6 13.2996 18.6916 0.223 53.1826 7.7906

ARCH(4) 4 5.496 7.518 15.2606 31.2926 2.843

ARCH(6) 6 7.279 12.210 4.817 26.6046 3.912
Multivariate— 4 LM(4)=28.617 p-value = 0.28
autocorr. 6 LM(4)=23.021 p-value = 0.58

 

Note: The limiting distribution of the normality, ARCH(4), ARCH(6),
and LM(4) test are x2 with d. f. equal to 2, 4, 6 and 25 respectively.
a and 0 indicates rejection of the null at 5% and 1% significance level

respectively.

restricted alternative, is performed, the null is rejected at a significance level less
than 1 percent. There appears to be an internal inconsistency in that VAR(6) with
cointegration is selected in Table 5 to be possibly the Optimal lag-length but at the
same time it is rejected in favor of a VAR(6) without cointegration. On the other
hand, when the VAR lag is reduced to four (three-lag VECM), the VAR with two
cointegration restrictions cannot be rejected at a 10 percent significance level. It
is possible the five—lag VECM is over-parameterized so as to reduce the power of
the likelihood test for cointegration. We therefore will include only four lag terms,

instead of six, in the cointegrated VAR model.

We also note that replacing VAR(6) with VAR(4) does not effectively change

63

Table 7

Cointegrating Rank Test
on Unrestricted VAR with 4 and 6 Lags

 

 

Max-)1 statistic Criticl value

 

 

 

 

 

H0 k=p—7' [=4 [=6 a=10% a=5%
r = 0 k = 5 36.54“ 37.58“ 30.77 33.18
7 = k = 4 25.02“ 33.89“ 24.71 17.17
7 = 2 k = 3 18.11 18.66 18.70 20.78
7‘ = k = 2 13.09“ 13.44“ 12.10 14.04
7 = 4 k = 1 2.36 1.28 2.82 3.96

Trace statistic Critical value
H0 k=p—7‘ [=4 [=6 a=10% a=5%
r = 0 k = 5 95.11“ 104.84“ 65.06 68.91
7‘ = 1 k = 4 58.27“ 67.26“ 43.96 47.18
7' = 2 k = 3 33.56“ 33.38“ 26.79 29.51
7 = 3 k = 2 15.45“ 14.72“ 13.34 15.20
7” = 4 k = 1 2.36 1.28 2.82 3.96

 

Notes: a. and b indicates statistical significance at 10 %

and 5 % respectively.

the properties of the residuals as shown in Table 6. Listed in the table are the
univariate standard deviations, unadjusted R2, normality tests, ARCH tests and
the system-wide autocorrelation LM tests. In terms of R2 and standard deviations,
VAR(6) outperforms VAR(4) simply because it includes more lagged explanatory
variables. This, however, does not result in more favorable pmperties for VAR(6)

in terms of the ARCH, the normality and the serial-correlation test statistics.
4.3 Testing for Two Cointegration Relations

An useful way of confirming the validity of the two derived cointegrating vec-

tors is to conduct formal cointegration tests. Johansen’s cointegrating rank tests

64

(Johansen 1988; Johansen and Juselius 1990) will first be performed. If the conjec-
ture of the rank 7- = 2 is confirmed, even though it does not provide inference on the
coefficients of cointegrating vectors, we can be more confident about the empirical
results of VECM(3) with two cointegration relations imposed. A direct support
for imposing the two specific cointegration relations can be gained by undertaking
Horvath and Watson’s (1995) tests for pre-specified cointegrating vectors. As shown
below, both types of test support the use of cointegrating velocity and real interest

rate relations in estimating the economic model.
4.3.1 Johansen’s Cointegration Rank Tests

There are two types of cointegration-rank tests. The first one is the trace test

which tests the pair of hypotheses about the rank of II E 076’ in (4.4),

H0 : rank(II)_<_7'

H1 : rank(II)=n.

The test procedure is first set 7‘ = 0 and sequentially increase the value for r if
the null is rejected. When the data ceases to reject a null, the particular 7 is then
treated as the cointegrating rank. The second rank test is the maximum eigenvalue

test, or Max-A test, which has the pair of hypotheses,

H0 : rank(11)=7'

H1 : rank(H) = 7+ 1.

The procedure begins with zero for r and sequentially increases its value if the null
is rejected. Again when a null is not rejected, the particular 7' value is inferred as
the number of existing cointegrating vectors. The distribution of both test statistics

are non-standard and simulated distributions are available (Hamilton 1994)”.

 

18We note that the simulated distribution depends on the specification of deterministic terms

65

The tests are conducted on the VECM counterparts of both VAR(4) and
VAR(6). As the test results in Table 7 indicate, the two types of rank test have
conclusions in conflict with each other. For the Max-A test, the 7‘ = 2 hypothesis is
confirmed. The maximum-eigenvalue test rejects both the null of r = 0 and 'r = 1
but fails to reject the null of 7‘ = 2 against the alternative of 7' = 3 for both lag
Specifications. In contrast, the trace test rejects the hypothesis of 7' = 2. Specifically,
the nulls of a cointegrating rank 7' g 0, 1, 2, and 3 are strongly rejected. What is
then suggested by the trace test is a rank of 4 or 5. Since the subsequent null of

r g 4 against 7' = 5 is not rejected, the trace test infers the cointegrating rank is 4.

We will rely on the r = 2 conclusion of the maximum-eigenvalue tests. The
r = 4 conclusion of the trace test lacks both theoretical and intuitive justifications
because it implies only one stochastic trend exists in a fairly complete economic
system including five nonstationary series. It is quite common to find at least two
stochastic trends in a model of more than four I(1) variables such as in Shapiro and
Watson (1988) and Karras (1993). There is indeed no statistical evidence to rule out
the trace test 7" = 4 result in favor of the max-A test 7' = 2 result that is consistent
with our theoretical model in Chapter 3. Nevertheless we are more interested in
the question whether an empirical business-cycle model can be estimated from that

theory and be consistent with the postwar U.S. macroeconomic data.
4.3.2 Horvath and Watson’s Cointegrating Vector Test

Economic theories often imply parameters of 0’s, 1’s and -1’s for cointegrating
relations. In response to this phenomenon, Horvath and Watson (1995) developed
the testing procedures for such explicitly specified cointegrating vectors in the con—

text of a finite-order Gaussian VECM as in (4.4). Interests are focused on the size

 

in the VECM. We use the critical values as reference instead of correct values for our model that

includes five dummy variables.

66

Table 8. Horvath/Watson Tests
of Pre—specified Cointegrating Vectors

 

 

 

70,, = rm, = 0 Critical values Computed
7'01, Talc flak flak 10% 1% Wald statistic
0 1 0 MV 12.49 19.00 20.53126
0 1 0 RR 12.49 19.00 31.5164“
0 2 0 MV, RR 23.51 31.26 56.8986“
1 1 MV RR 12.49 19.00 36.2707“
1 1 RR MV 12.49 19.00 25.00096

 

Note: 0 indicates significance at a 1% level. MV is the cointegrating
M2 velocity and RR stands for the ex post real rate.

of the cointegrating rank 7 = rank(II) and the pair of hypotheses

H0 : rank (II) = 7'0

Ha : rank (11) = 7'0 + 7",, with 70 > 0.

Under the null, 70 is further defined by To = 70,, + 70,, where 70,, is the number
of known cointegrating vectors while 760,, represents the unknown or unrestricted
cointegrating vectors under the null. The number of additional cointegrating vectors
present under the alternative is Ta. Similarly, the extra rank is further divided
according to r, = rm, + n,,, where the subscripts u and k denote unknown and
known, respectively. The Horvath and Watson tests generalize the procedures of
Johansen’s rank tests where no known cointegrating vectors are present. That is,
his rank tests consider only cases with 7‘0,c = n,,, = 0 and the hypotheses of interest

for n,,, = 1 is

Ho : rank (H) = 7'0“

Ha : rank (H) = 70,, + 1

which is shown above as the maximum-eigenvalue test.

67

Table 8 presents the test results on the validity of M2 velocity (MV) and ex
post real rates (RR) cointegration relations. The first test specifies a null of no
cointegration and the alternative of a cointegrating M2 velocity. The computed
Wald statistic, 20.53, is greater than the simulated 1 percent critical value and thus
strongly rejects the no—cointegration hypothesis in favor of a cointegrating velocity
relation. The second test also strongly rejects a null of no cointegration in favor of a
cointegrating ex post real interest rate relation. A comparison of the Wald statistic,
31.52, to that in the first test, 20.53, indicates statistical evidence is stronger for
the cointegrating ex post real rate than that for the M2 velocity. This is consistent
with the nonstationary test results for the two cointegration relations presented in
Table 2. The third test specifies a null of no cointegration and an alternative of two
cointegrating vectors. The test again strongly rejects no—cointegration at less than 1
percent significance level in favor of the two pre—specified cointegration vectors. In
the last two tests shown in Table 8, one of the two cointegrating vectors is specified
under the null while the other one is added under the alternative. Again both
test strongly rejects the nulls of a single cointegration in favor of both cointegration
relations being admitted in a VECM. Thus, the evidence obtained from the Horvath-
Watson tests highly supports the practice of imposing cointegrating velocity and real

interest rate relations as done in the next section.
4.4 Estimation of VECM and VMA

From the analysis in Sections 4.2 and 4.3, we have determined a specific VAR
model to estimate. It has five time dummies and four lag terms. The lag is shortened

to three in the VECM representation of in; = [y¢ m7'¢ wr¢ R¢ 71¢] as

Ax¢ = F1A$¢_1 + F2A$t_2 + F3A$t_3 — afl'$¢_1 + [I + “I’Dt "‘1‘ St. (4.5)

68

The cointegrating vectors derived in Chapter 3,

,_1—100 0
fi—[o 0 01—1l’ (4'6)

is imposed while 07 is not restricted. The CATS procedures, based on Johansen’s
(1988, 1991 and 1995) MLE principle, estimates the other parameters in (4.5). Also
see J ohansen and J uselius (1990, 1992) for empirical applications of the methodology.
The imposed fl and the estimated 67 together provide a summary about the long-run
dynamics of the system whereas the other parameter estimates represent the Short-
run dynamics of the system. The likelihood-ratio test statistic of this specification is
X2 distributed with 6 degree of freedom. The p-value of the test statistic is 0.13 and

therefore the restricted model is not rejected at the 10 percent significance level.

The estimated Speed of adjustment matrix, in tranSpose, is

1 —0.128 0.003 0.031 —4.189 —9.326 ”
, (—3.637) (0.104) (1.903) (—1.695) (—1.618) (47)
a = .
-0.003 0.000 —0.001 —O.266 0.249
1 (—3.632) (0.647) (—1.668) (—4.620) (1.850) ,

 

 

where numbers in parenthesis are t ratios. We notice that values in the second
column of o/ are very small. Their t ratios also suggest that the two elements are
not statistically different from zero. Thus weak exogeneity exists for real balances
which is the second element of the :6¢ vector. The estimated a’ is very different from

the one obtained in the theoretical model,

I

(4.8)

 

0100—1
000—10'

This may indicate that both a and 6 derived in the theory are not concurrently
consistent with the data. By imposing the theoretical ,6 and taking the estimated
a’ as the correct adjustment matrix we are stating that portions of the theoretical
model are misspecified so as to yield the incorrect form of a’ in (4.8). We can

reestimate the VECM model by, in addition to the above 6, further restricting the

69

second row of a to zero. The parameter estimates are presented in Table 8 and
their t ratios in parenthesis. The likelihood-ratio test statistic for this specification
against the alternative of a nonrestricted VAR has a x2 distribution with 8 degree
of freedom. Its p—value = 0.25 is a substantial improvement over the p-value of
0.13 above without weak exogeneity imposed. For this reason we will take this new

restricted version to be the correct specification for subsequent analysis.

Next, we convert the VECM estimation results to obtain its vector moving-

average (VMA) representation as19
A93¢ = 6 + C (L) e¢. (4.9)

More importantly, for the purpose of identifying the common trend space, a 1, and

the factor loading matrix, 511 we want to calculate the long-run multiplier matrix

of the reduced-form errors 6¢ as20

c (1)

201'
i=0
= ,3 (a1.
With C (1) calculated, the long-run covariance matrix of A$¢ can be calculated

as C (1) EC (1)' where )3 is the covariance of the error 5¢. The estimated long-run

multiplier matrix has reduced rank and has the form

 

19This involves first converting the VECM to its VAR representation and then invert A (L), the
lag-polynomial matrix of VAR, to get C (L). The inversion techniques for a reduced-rank A (L)

requires special consideration and is treated in Warne (1990).
20The original formula for C(1) as in (2.33) is 6 _L (a’iI‘B i)’1 611’i before we simplify

54 (alrfifl—l as 31 on page 23.

70

Table 9
Parameter Estimates for VECM

 

 

 

0.0 0.031 —4.189 —9.326
0,: (—3.637) (0.0) (1.903) (—1.695) (—1.618)
—0.003 0.0 —0.001 —0.266 0.249
(—3.632) (0.0) (-1.668) (—4.620) (1.850)
,_ 0.043 0.005 —0.005 1.246 1.908
(4.036) (0.640) (—0.957) (1.660) (1.091)
-0.007 0.006 0.010 0.003 —0.008 ”
(—2.562) (1.659) (1.608) (0.645) (—2.019)
—0.001 —0.002 0.001 -—0.001 —0.003
(—0.602) (—0.651) (0.301) (-0.271) (—1.001)
‘1” —0.001 —0.002 —0.001 —0.001 —0.002
(-0.899) (—0.990) (—0.513) (—0.319) (—-1.147)
—0.539 0.534 1.120 —0.415 —0.231
(—2.818) (1.987) (2.588) (—1.429) (—0.828)
0.420 0.486 —1.204 0.330 0.596
(0.942) (0.774) (—1.193) (0.488) (0.916)_
10.08 0.01 0.01 1.35 1.761
0.01 0.05 0.01 0.47 —4.27
2:10-3x 0.01 0.01 0.02 0.34 —2.02
1.35 -—0.47 0.34 387.40 42.50
11.76 —4.27 —2.02 42.50 210700,

F —0.128

 

q

 

 

 

 

71

Table 9 (cont’d)
Parameter Estimates for VECM

 

0.082 0.230 -0.084 0.003 —0.002 ‘
(0.924) (1.895) (-0.420) (2.701) (-1.755)

0.173 0.123 -0.224 -0.006 —0.001
(2.432) (1.270) (—1.409) (-6.218) (-1.374)

-0.062 0.040 —0.182 -0.001 —0.000

 

 

 

 

I" =
‘ (-1.512) (0.712) (—1.987) (-1.907) (-0.459)
8.439 8.056 —5.187 0.327 -0.111
(1.356) (0.949) (-0.372) (3.851) (-1755)
-1.963 4.334 96.101 0.583 —0.422
_ (—0.135) (0.219) (2.951) (2.945) (-2.865) .
0.067 0.165 -0.280 —0.001 —0.001
(0.742) (1.373) (-1.419) (-1.184) (1.737)
0.076 0.105 -0.012 -0.002 -0.000
(1.049) (1.095) (-0.078) (—2.146) (-0.629)
r = 0.052 0.101 —0.231 0.000 —0.001
2 (1.269) (1.836) (—2.545) (0.014) (—1.409)
5.963 —2.565 -13.681 -0.307 0.011
(0.944) (-0345) (—O.988) (—3.604) (0.191)
—22.491 11.816 48.183 -0.004 —0.100
1(—1.526) (0.602) (1.493) (—0.019) (—0.725) .
—0.116 0.116 0.250 0.002 -0.001 1
(—1.374) (1.083) (1.281) (1.900) (—1.212)
—0.002 0.082 0.135 -0.02 -0000
(-—0.026) (0.962) (0.863) (-2.337) (-0.894)
r _ -0.038 0.139 —0.079 —0.000 0.000
3‘ (—0.969) (2.824) (-0.885) (—0.710) (0.685)
0.177 6.576 14.007 0.245 0.031
(0.030) (0.877) (1.024) (2.813) (0.757)
-11.005 —3.485 45.774 0.465 —0.006
L(-0.797) (—0.199) (1.435) (2.288) (-0.061)-

 

 

 

72

0.536 1.008 —O.819 —0.010 -0.006 '
(1.514) (3.587) (—1.189) (—2.445) (-1.443)

0.536 1.008 —0.819 -0.010 —0.006
(1.514) (3.587) (—1.189) (—2.445) (—1.443)

0.194 0.001 0.844 —0.003 0.001
C 1 = .
( ) (1.815) (0.011) (4.601) (—2.439) (1.155) (4 10)

—21.026 45.150 22.625 0.368 0.199
(—2.343) (6.340) (1.296) (3.736) (1.947)

—21.026 45.150 22.625 0.368 0.199
1 (—2.343) (6.340) (1.296) (3.736) (1.947) .

 

 

where numbers in parenthesis are t ratios. The linear trends in the level of variables

23¢ are calculated as
I

C(1)11=[0.0096 0.0096 0.0035 0.0695 0.0695]

We notice the first row and the second row of C (1) have the same values. This
indicates the total impact of the errors on the first difference of output is the same
as that of real balances, a condition for the stationary velocity relation. Similarly,
a stationary ex post real interest rate relation requires the total effects of the errors
and the same for both the nominal interest rate and the inflation rate. This is
indicated by the identical fourth and fifth rows in (4.10). In contrast, the third
row of C (1) is not linearly dependent on any other row because real wages are not

cointegrated with other variables.
4.5 Estimation of the Overidentifying Theoretical Model
4.5.1 The Complete Model Structure

Identification of a structural-from VMA
AID; = 6 + R (L) 1/¢ (4.11)

from its reduced-form counterpart in (4.9) involves finding a F matrix such that

73

Vt = FEt (4.12)

R(L) = C (L) F“. (4.13)

The n—dimensional structural innovations 11¢ is composed of k-dimensional perma-
nent shocks 11,1” and r-dimensional transitory shocks 14". The covariance of the errors
is E (125;) = E and the innovation covariance E (1411;) = D is a diagonal matrix.
Form the partition F’ = [F,g F1] where F), is k x n and Fr is 7‘ x 71. As discussed in

Chapter 2, the long-run multiplier of (4.8) and (4.11) are

ASL} C (1) 5; = flia'ifi

= R(1) 11¢ = flit/f. (4.14)

Therefore permanent shocks are identified according to th = Fée¢ = a16¢ and
0’1 = (6150-1510(1)-
We begin identification with structural information available in F in Chapter

3. The explicit form of F is

(4.15)
—1
—1 —-1

L d

"6.1
II
o1—tc1—u1—4
ooo‘mo
HOOOO
OHOO

 

 

The top 3 x 5 partition of F is F), which describes the contemporaneous relations
among the variables in the long-run structure of the model. The bottom 2 x 5 par-
tition is F, and represents the contemporaneous relations in the dynamic structure
of the model. Estimation is done by solving an optimization problem for F and D
such that

FEF’=D

74

subject to the pattern of F in (4.15). First normalize F to make its diagonals equal

 

 

 

tolasin

F*=W-F
'10000"10000"
000—10 10600

204000 01001
00001 1—1000
_0010010—101—11
"1000 l
—1100

= 40100. (4-16)
0—101—1
_01001‘

 

 

In this form we notice the three rows that form 07’, are now respectively rows 1, 3
and 5. F is subject to nine overidentifying restrictions and there is only a single free

parameter to estimate in ozi (Fk).

Calculations are done by a RATS SVAR procedure written by Lansarotti and

Seghelini. The result is

 

 

' 1 0 0 0 0 '
—1 1 0 0 0
F“ = -0.1487 0 1 0 0 (4.17)
0 —1 0 1 —1
_ 0 1 0 0 1 .
"0.0089 0 0 0 0 '
0 0.0105 0 0 0
13% = 0 0 0.0039 0 0 (4.18)
0 0 0 1.5497 0
1 0 0 0 0 1.4484_

 

Note that if the estimated model is not rejected by an overidentifying restriction
test, we then need to premultiply the estimated F“ by W"1 so it returns to its theo-

retical form. However, the model is rejected by the overidentifying restriction test at

75

effectively zero significance level. The test statistic is 477.99 for )8 distribution with
9 degrees of freedom. Thus the theoretical structure of F or the complete model in

Chapter 3 as a whole is rejected by the data.

We now want to answer the question whether the data is consistent with parts
of the model. The first three equations in (3.15) represent the steady-state structure
of the model. They apply in equilibrium and hence are considerably less restricting
than the dynamic parts of the model or the last two equations in (3.15). Therefore,
we now inquire whether permanent shocks can be identified out of the long-run

structure of the model, in particular, by using the long-run multiplier.
4.5.2 The Long-Run Model Structure

The common-trends loading matrix 61 derived in Chapter 3, in the form of

the long-run multiplier of permanent shocks Arc¢ = fill/t? , is

 

l Ayt l a C 0
Amr¢ a c 0 14““
A1117} = b e 0 111““ . (4.19)
AR¢ 0 0 1 14‘0“““1
_ A7r¢ . 1 0 0 1 .

 

 

 

where a, b, c, and e are constants”. This equation indicates that the technology
shock and the labor-market shock have zero long-run impact on the nominal vari-
ables, R and 71. In contrast, the nominal shock has zero long-run effect on the real
variables, y, mr and 1177'. Estimation of the long-run model based on ,8, is accom-
plished by specifying an initial 5.01 and then find a k x k lower-triangular matrix T

with unit principal diagonal such that [31 = fiflT is of the form in (4.19). Select the

 

 

0 -1 0
0 -1 0
initial choice as 611 = 1 O . To find out what restrictions are necessary for
0 0 1
1. 0 1 .1

 

“According to (3.28), a = d, b = 1 - %, c = -—h and e = g.

76

 

1 0 0
T: t2, 1 O ,obtain
t31 7332 1
181. = fliT
'0 —1 0'
0 —1 0 1 0 0
= 1 1 0 t21 1 0
0 0 1 1531 [32 1
_0 0 1_
' —12, —1 0'
—t21 —1 0
= 1+121 1 0 (4.20)
1531 7532 1
_ t31 7532 1,

 

 

From (4.20) it is obvious that to derive a 61 of the form as in (4.19), we have to
impose two zero restrictions on T, i.e., t3, = Ln = 0.

Note that C(l) in (4.10) can be expressed as C(1)’ = [ 6’, 0’1 c’2 cf, 6', ]
where c,- for z' = 1, 2, 3 are all 1 x 5 row vectors. The initial common-trends matrix

cz’i is obtained as

a1 = (1214117100)

 

' 0.5 0.5 1 0 0 cl+c2
= —0.5 —0.5 0 0 0 0(1): —c,
_ 0 0 0 0.5 0.5 03
0.7313 1.0090 0.0244 —0.0125 —0.0043
= —0.5370 —1.0080 0.8190 0.0097 0.0057 . (4.21)
1—210261 45.1500 22.6276 0.3734 0.1933

The eventually identified permanent shocks are subject to the same restrictions in
T according to

Vt? E 07112 = T‘la‘i'a (4.22)

in order for C (1) = ,Bfla‘i’ = 51.01 to hold.

77

We will demonstrate the technique of identification in the setting of a trans-
formed VECM. Premultiply the VECM in (4.5), omitting constant and dummy

variable terms to save space, by a full rank n x n permutation matrix

 

 

 

 

 

 

 

 

 

WC 1
W = 1 ( ) (4.23)
W2
where W, is k x n and W2 is 7' x n. Then we have
W C 1 W C 1 3
1 ( ) Ax, = 1 ( ) [Z PiAxt—i ‘1‘ 016,114 '1' 5}]
W2 W2 i=1
W C 1 3 0 *
= 1 ( ) Zn A414 + 1 + 5” (4.24)
2 i=1 W205 Ilia—1 53¢
"' W C 1
where 6; E E” E 1 ( ) 13¢. Notice the k x k covariance E (61,511) 2
52t W2

 

 

2'; = W10 (1) EC (1)’ W,.

B 0
Define a, E Bu¢ where B = 11 B 0 D
21 22 T

Bu and 822 are lower-triangular matrices with unit principal diagonals. Also both

 

]andE[u¢1/;]=D= DP 0 ].

Dp and DT are diagonal matrices. Then we have the relation 81111,? = 61‘, and the

first k equations in (4.24),

3
W10 (1) A23¢ = W10 (1) ZFiAmt—i + 81111:),

i=1

can be expressed as

3
Bfi1W1C(1)A:r¢ = Bf,‘ W10 (1) Z 1“,A:c,-,- + 14’. (4.25)

1—1

Working on only the k transformed long-run equations we can identify permanent
shocks by
14” = Bfi'WlC (1) a, (4.26)

78
and have E (ufuf’) = Bf 11231311" = Dp that is diagonal.

To begin estimation first specify

 

 

l 0
W1 = —1 0 (4.27)
0 0 0 0 1
and so
. 01 .
1 1 0 0 c1
W100) = —1 0 0 0 c2
0 0 0 0 1 c3
. C3 .
c1 + C2
= —cl =a‘1'. (4.28)
C3
With W1C (1) = 03’ established in (4.28), then (4.26) can be written as
VtP = Bflla‘i'et. (4.29)

Comparing (4.29) to the permanent shocks equation 14” = T‘la‘i’st in (4.22), we

find that T = B”. Thus the loading matrix 31 in (4.20) is also identified as

1 0 0
E; = 53811. In estimating B” = bgl 1 0 we also need to impose the
(731 532 1

restrictions b31 = b32 = 0 as is done to T in (4.20) in order for fli to match its

theoretical form.

We now demonstrate how the permanent technolog, the labor-market and
the nominal shocks are identified by the long-run multipliers which are implicitly
defined in the moving-average representation of W1 Ax, (Rasche 1997) or

WIAIL't = W10(L) 5t
= W10 (1) 83 + W1 (1 — L) C’ (L) 6;

= But/f + W1 (1 - L) 0* (L) at. (4.30)

79

In the long run (4.30) becomes W1 Amt = But/f or

r q

 

 

All:
1 0 0 Amrt
—1 0 0 0 0 Awrt
0 0 0 1 AR,
_ Am _
1 Ayt 1 0 View
= —1 0 0 Awrt = 021 1 0 Viabor . (4.31)
0 0 1 Am 0 0 1 1230mm“
—1
l
Premultiply through the second equation in (4.31) by —1 0 0 to get
0
4y. -1... —1 o «496*!
Awrt = 1 +b21 1 0 Vial)"
Am 0 0 1 ugwmina‘

Thus the permanent technology shock is identified as having a long-run multiplier on
real output equal to —b21 (it turns out —b21 = 0.906), the labor-market shock having
a unitary long-run multiplier for real wages and the permanent nominal shock also

a unitary long-run multiplier for inflation.

The covariance matrix of the first k equation in (4.24) is calculated as

2‘; = BnDijl=WlC(l)ZC(1)’W{
2.258 —2.046 —8.825

= 10-4x —2.046 2.019 8.810 . (4.32)
—8.825 8.810 1313

The decomposition of 21' into B” and Dp is computed with SVAR. The results are

1.0 0 0
Bu = -0.906 1.0 0 (4.33)
O 0 1.0

80

and
2.58 0 0
Up = 10-4 x 0 0.1654 0
0 0 1313

The overidentifying restriction test for the estimation is x2 distributed with two
degrees of freedom. The test statistic is 4.523 with a p-value of 0.1042 and thus the
overidentifying restrictions of the common-trends model is not rejected by the data

at a 10 percent significance level.

The permanent shocks are then identified as

1.0 0 0
uf’ = BfllW1C(1)5t=Bfllo/i'5t= 0.906 1.0 0
0 0 1.0

0.7313 1.0090 0.0244 —0.0125 —0.0043

x —0.5370 —1.0080 0.8190 0.0097 0.0057 5t
—21.0261 45.1500 22.6276 0.3734 0.1933
73.13 100.9 2.44 —1.25 —0.43
= 10-2x 12.57 —9.3644 84.111 —0.1628 0.1803 5.. (4.34)
—2102.6 4515 2262.8 37.34 19.33

The common-trends loading or the long-run multiplier of permanent shocks is cal-

culated as

 

51 = fi1311
'0 —1 01
0 —1 0 1.0 0 0
= 1 1 0 —0.906 1.0 0
0 0 1 0 0 1.0
_0 0 14
"0.906 -1 “

 

 

81

The steady state equilibrium of the long-run economic model is expressed as

 

Ayt l

Amrt

Awrt
AR:

Am

 

 

"0.906 —1
0.906 —1
0.094 1
0 0
0 0

0'
0
0
1
1..

 

tech
V t

ugabor . (4.35)

nominal
”1

We find that the numeric values of the elements in £1 are all within the range for

parameters stated in the theoretical model in (3.28) and (3.2)-(3.4).

Chapter 5
MACROECONOMIC IMPULSE ANALYSIS

5.0 Introduction

This chapter presents the impulse response function and forecast-error vari-
ance decomposition analysis with respect to the identified permanent shocks or
common stochastic trends (Ahmed and Park 1994). Innovation accounting is also
extended to nominal balances and nominal wages which are not explicitly modeled in
the previous Vector Error-Correction Models but are nonetheless integral elements

of the economic analysis to be conducted.

The long-run responses of variables to the permanent shocks are restricted by
the common-trends loading matrix stated in (4.35). As a result, real variable move-
ments are strongly influenced by real permanent shocks in the long run in the sense
that a high percentage of the forecast-error variance is explained by the technology
and the labor-market shock. Specifically, the technology shock has long—run posi-
tive effects on both output, real balances and real wages. The labor-market shock
also impacts positively on real wages but negatively on output and real balances.
On the other hand, nominal variables in the long run are exclusively dominated by
the permanent nominal shock also in a variance-decomposition sense. The nominal
interest rate, inflation, nominal balances and nominal wages all respond positively
to the permanent nominal shock. Such distinct dichotomy between the real and the
nominal shock effects is only reasonable while the economy is in its steady state
equilibrium.

The permanent shocks, as also found in other studies, are important sources of

82

83

fluctuations in the short term. Specifically, the technology shock accounts for about
40 to 60 percent of the output variance on one- to two—year horizons. King et al.
(1991) and Mellander et al. (1992) arrived at similar percentages. More significantly,
the technology shock accounts for about 80 percent of the real balance forecast
variance in the very short run, comparable to about 70 percent found by King et al.
(1991). The labor-market shock explains about 20 percent of the output variance
within the first year. This compares to the finding of Shapiro and Watson (1988)
that at least 40 percent of the output variance is accounted for by a labor-supply
shock. Labor-market shocks explain 40 percent of the real wage variance in the
impact quarter and its influence increases thereafter. As for the permanent nominal
shock, it accounts for a high proportion of the nominal interest rate variability. This
is similar to the finding of Englund et al. (1994) but different from the extremely
low percentages found by King et al. (1991). The nominal trend accounts for at
most 30 percent of the inflation variance within the first two years. It is at least 30

percent in King et al. (1991) on the same horizon.

One seemingly unsatisfactory aspect of the impulse analysis in this chapter is
that the parameter estimates are measured very imprecisely. The standard errors of
the impulse response parameters and variance decompositions are calculated with
the asymptotic distribution approach used by Giannini (1992) and Warne (1990).
Their large size renders most of the parameter estimates insignificant. This is, how-
ever, not a phenomenon new to studies employing the VAR methodology and has
been discussed in Runkle (1987). We can still gain valuable knowledge from the gen-
eral patterns revealed regarding the dynamic effects of stochastic impulses on the
economy. More importantly the impulse-response patterns and the variance decom-
positions are amenable to economic interpretations consistent with the theoretical

model in Chapter 3, as is presented in the following sections.

 

84

5.1 Efl'ects of the Permanent Technology Shock

The impacts of the technology shock on five variables in the VECM model,
nominal balances, nominal wages and two cointegration relations, are illustrated
by their impulse response functions (IRFs) plotted in Figures 13.1 to 13.9. As is
discussed in the identification setup of Chapter 4, in equation (4.29), the technology
shock is normalized to produce equal long-run effects, 0.9, on real output and real
balances. We observe that the responses of output, real and nominal balances are all
positive across the horizon. Output and real balances respond to technology shocks
in a very timely fashion, reaching major fractions of their long-run responses only in
six quarters. The short-run impacts on the velocity are negative, indicating output

increases at a rate slower than real balances whose increase is aided by falling prices.

The long-run effect of the technology shock on real wages is normalized to 0.09
as shown in (4.29). In the first three years, the response of real wages overshoots
its long-run steady-state level as shown in Figure 13.5. This is most likely due to
the mechanism of delayed wage adjustment to price changes even though nominal
wages steadily decrease. This in turn indicates prices fall at a much faster rate than
nominal wages decrease in the beginning. This conjecture about the response of
prices is borne out by the IRF of inflation with respect to technology shocks shown
in Figure 13.8 which shows a steep drop of the inflation rate in the first year. It is
reasonable that rising productivity tends to produce a disinflationary effect. It in
turn could lower the inflation premium charged by the nominal rate. The temporary
effect on the nominal interest rate is indeed negative, as shown in Figure 13.7, but
smaller than the negative effect on inflation. As a result, there is a positive response

of the ex post real interest rate as shown in Figure 13.9.

Percentages of the forecast-error variance of variables attributable to the tech-

nology shock are shown in Table 10. In the long run, output and real balance error-

85

Table 10

Percentage of Forecast-Error Variance Attributed to
the Permanent Technology Shock

 

 

 

Forecast Real Nominal Real Nominal Interest

Horizon Output Balances Balances Wages Wages Rate Inflation
0 10.25 63.70 35.50 12.10 1.80 17.98 31.76
1 14.54 77.43 48.69 10.31 7.30 20.20 29.17
2 23.53 82.42 49.92 19.53 6.75 21.36 28.65
3 36.46 85.94 52.98 25.31 6.97 20.00 28.10
4 44.23 87.55 53.89 29.90 7.19 18.20 26.40
8 68.31 89.20 49.35 35.02 7.65 12.79 21.59
12 78.48 89.16 37.23 31.84 7.67 10.44 19.01
16 83.32 89.32 26.71 27.24 7.37 9.02 17.20
24 87.62 89.40 15.51 20.14 6.14 7.06 14.50
36 89.28 89.33 8.24 14.07 4.43 5.35 11.81
48 89.53 89.26 5.03 10.91 3.22 4.32 9.98

 

variances are predominantly accounted for by the technology shock. The variability
of output over the near term is already strongly affected by technology shocks.
About 40 to 60 percent of the error variance is explained during the second year.
King et al. (1991) and Mellander et al. (1992) arrived at similar percentages for their
technology shocks. The technology shock accounts for as high as over 60 percent of
the forecast variance of real balances even in the impact quarter. The percentage
increases on longer horizons. This is a more dramatic result compared to that of
King et a1. (1991) where about 70 percent of real balance variance is explained by a
permanent technology shock. In comparison, its role in affecting nominal balances
is not nearly as huge although still very critical for up to four years. We observe
that even though in the long run the technology shock plays no role in affecting the
nominal interest rate and inflation, it accounts for 20 to 30 percent of the forecast

variance in both during the first year. Real wage variability is explained at most

86

about 30 percent by the technology shock on any horizon. A bulk of the source of
the real wage variance comes from the labor-market shock. On the whole nominal

wages are only slightly affected by the technology shock.
5.2 Effects of the Permanent Labor-Market Shock

Figures 14.1 to 14.9 plot the impulse response patterns of variables to the
labor-market shock. The shock is normalized to yield in the long-run an unit impact
on real wages and a negative unit impact on both output and real balances. The
effects on the growth of nominal balances, in Figure 14.4, are smaller than that of
nominal wages, in Figure 14.6, although both are steadily increasing before reaching
an almost constant rate. Due to a labor-market shock, real balances, in Figure 14.2,
decrease while real wages, in Figure 14.5, increase. This implies that the growth
rate of prices (the inflation level) increases at a pace faster than that of nominal

balances and slower than the growth of rate nominal wages.

According to the theoretical wage-contract model, an upward adjustment of
nominal wages is made slower by a wage contract mechanism. When a positive
labor-market shock strikes, real wages are rising gradually. During the adjustment
process, the wage contract allows firms to hire extra workers at their discretion when
real wages are still sufficiently low relative to a new expected equilibrium real wage
rates. For this reason there is a short-run boost in output growth as is apparent in
Figure 14.1. As the nominal wage level catches up and as the short-run aggregate
supply curve, whose position depends negatively on nominal wages, shifts up the
positive output effect diminishes. Eventually the growth rate of output becomes
negative to reflect a new tighter labor market condition caused by the labor-market
shock. It is interesting to observe that while the long-run negative unitary response
of real balances comes into place in about two years, it takes more than seven years

for real wages and output to reach their respective equilibrium response of 1 and -1.

87

Table 11

Percentage of Forecast-Error Variance Attributed to
the Permanent Labor-Market Shock

 

 

Forecast Real Nominal Real Nominal Interest

Horizon Output Balances Balances Wages Wages Rate Inflation
0 29.95 3.23 0.04 40.75 90.46 0.31 18.58
1 26.69 2.96 0.39 52.27 80.14 0.43 20.04
2 22.32 2.60 1.02 46.44 75.84 2.83 19.73
3 18.80 2.68 1.40 44.19 66.96 5.95 20.89
4 16.45 3.42 1.39 43.79 62.57 7.67 20.63
8 8.44 6.43 1.86 47.11 51.71 9.39 20.73
12 5.77 7.78 3.45 55.38 45.09 8.64 19.65
16 4.87 8.29 4.86 63.28 40.19 7.87 18.43
24 4.89 8.91 4.62 73.70 31.68 6.58 16.11
36 6.05 9.42 3.16 81.56 22.19 5.19 13.37
48 7.07 9.71 2.16 85.34 15.95 4.27 11.39

 

The long-run impacts on the nominal interest rate and inflation are restricted
to zero. In the short term they respond positively to the labor-market shock as
shown in Figures 14.7 and 14.8. Upon impact, the inflation effect is the highest
and it then gradually diminishes as the output level adjusts. The peak response of
the nominal rate occurs one year after an impact and then the response tapers off.
There is a huge negative initial effect in the ex post real rate. It rapidly disappears

as an inflation premium is factored into the nominal rate.

The role of the labor-market shock in accounting for the forecast-error vari-
ances is shown in Table 11. Three major points are worth noting. First, as expected
by its design, the labor market shock is a dominant source of variability in real wages,
accounting from an initial 40 percent to 85 percent in the 48th quarter. Moreover,
the proportions of the nominal wage variance explained within two years are even

higher than that of the real wage, with a high of 90 percent in the impact quarter.

88

Second, the pr0portion of the output forecast variance is 30 percent initially and
goes down as the horizon lengthens. Third, the labor market shock consistently
explains a non-trivial portion of the inflation forecast variance, between 10 and 20
percent. There are no significant explanatory power from the labor-market shock

for the variances of real balances, nominal balances and the nominal interest rate.
5.3 Effects of the Permanent Nominal Shock

The impulse responses to the permanent nominal shock are plotted in Figures
15.1 to 15.9. The nominal shock is designed to produce an unitary effect in the
long run on both the nominal interest rate and the inflation rate. Since nominal
neutrality is dictated by the wage-contract model, the total impacts on output, real
balances and real wages are zero in the long run. Even temporary impacts on the
three real quantities are very short-lived and disappear completely in less than three
years. The nominal shock temporarily increases both production in the economy
and the buying power of money as shown in Figures 15.1 and 15.2. There is an
obvious delay in real output growth relative to the real balance increase because
the peak response of output is about one year later than that of the real balance.
Consequently there is an initial negative impact on the velocity from the permanent

nominal shock as in Figure 15.3.

The IRFs of nominal balances and nominal wages, shown in Figures 15.4 and
15.6 respectively, are monotonically increasing at a constant rate. The reason for
this originates from the identities by which the responses of nominal balances and

nominal wages are obtained,

Am, 5 Amrt+7rt (5.1)

Aw, E A'LUTt'l'fl't. (5.2)

The long-run effects of the nominal shock on both Amrt and Awrt are restricted

 

89

Table 12

Percentage of Forecast-Error Variance Attributed to
the Permanent Nominal Shock

 

 

 

Forecast Real Nominal Real Nominal Interest

Horizon Output Balances Balances Wages Wages Rate Inflation
0 0.02 33.06 52.34 1.73 6.95 19.46 2.95
1 1.67 17.50 36.80 1.23 8.86 25.53 9.36
2 2.26 10.96 29.71 0.87 13.48 27.10 13.45
3 3.25 7.85 28.54 1.52 19.66 32.78 16.47
4 3.54 5.75 28.29 1.22 22.72 39.86 21.33
8 2.23 2.56 34.52 0.70 31.06 52.32 30.67
12 1.48 1.69 46.04 0.52 37.24 59.45 37.08
16 1.12 1.28 55.78 0.38 42.58 64.70 42.35
24 0.75 0.86 69.42 0.23 54.10 71.95 50.69
36 0.49 0.59 81.73 0.14 67.77 78.50 59.47
48 0.37 0.45 88.19 0.09 76.85 82.53 65.59

 

to be zero by the requirement of nominal neutrality. On the other hand, the long-
run shock effect on the first-difference of inflation, Am, is restricted to 1.0, making
the nominal shock effects on the level of inflation, 7n, infinitely cumulating. The
responses of nominal balances and nominal wages are thus dominated by this cumu-
lative impact on the inflation level. In the long run the cumulative trend in nominal
balances and nominal wages cancels out that in inflation. This allows nominal neu-

trality to hold with respect to real balances and real wages in the model.

The short-term effects on real wages, in Figure 15.5, are initially positive and
then are erased in about three years. At first glance, the nominal interest rate and
inflation have very similar response patterns. The effects are fairly moderate in
the first year, overshoot the long-run unitary levels in the second year and reach
the long-run near the beginning of the third year. However the fact is that their

exact short-run response paths are very diflerent. This is evidenced by the volatile

90

response pattern of the ex post real rate during the first three years shown in Figure

15.9.

The forecast-error variance decompositions for the permanent nominal shock
are shown in Table 12. Four observations are particularly worth noting. First, the
nominal shock is a very important source of variability for the nominal interest rate
and inflation on horizons longer than twelve quarters. This is not surprising since
the nominal shock is identified by having long-run impacts only on nominal rates
and inflation. Second, nominal balances and nominal wages are also dominated by
the nominal shock over the longer term. This is consistent with the above discussion
that shows the IRFs of both variables with respect to the nominal shock are domi-
nated by the cumulative inflation level responses to the nominal shock. Third, over
the very short run the nominal shock is less important for explaining the inflation
error variance. Lastly, the nominal shock accounts for essentially none of the error
variance for output and real wages. It also explains very little of the real-balance
variance beyond an one-year horizon. These results seem to suggest the permanent
shock which is identified by a long-run neutrality condition is also quite neutral in

the short run.

91

IRF of 611;) to chh shocks

 

 

 

096
”/’/‘4’
once .K’fihfid’
064-1
048—1 /
l
032 —4 /
/
/
/
01641 YYYTTYYYYTTIIYTTTYYIYITIYYYYYTTYfiTYTIIYYYYTYYTYYTTTT
o 6 12 1B 24 30 36 42 48

Figure 13.1 The Response of Output to Technology Shocks

IRF 01 Mr to Tech shocks

 

()9 d

08 H

()5 M

 

 

 

 

 

04

Figure 13.2 The Response of Real Balances to Technology Shocks

TTTTT

IRF of de-Mr to Tech shocks

 

-000

—005 ‘

~010‘

~020

~025

-080

-035

~040

Figure 13.3 The Response of Velocities to Technology Shocks

 

 

IIIIrTVIlTYTTIIIIIIIYYYYIIIITYTYTYYTTTI

Y
6 12 18 24 30 36 42 48

 

 

 

92

IRF of M2 to Tech shocks
250

 

200 ~ \

175 —1

 

 

 

 

 

100 TTYYIIVI[YYI'ITTYTTYYTYTVYITFYYYTT'YYYTTY‘TWTTYYYYV

O 6 1 2 1 8 24 30 36 42 48

Figure 13.4 The Response of Nominal Balances to Technology Shocks

IRF of Wr to Tech shocks

 

 

 

 

o 1 75
/ \\
o 150 A / \-.
\‘
\
\\
\
0.125 —< \
\\
\‘x
\.
0 100 _‘ \
\\
\\\,
\\
-\\
o 075 —< 0‘“"\\\7
2\‘\\
XM~~
0050 YYTYVVYITY‘YYYYY‘TWfin7II'YIY YYYYY TYYTYTTYY YYYYYYY I'lfi
o 6 12 18 24 30 36 42 4a

Figure 13.5 The Response of Real Wages to Technology Shocks

IRF of W to Tech checks

 

 

 

 

-1BO rrriwrrrrrrrrrvtlrrrrvrrrrrITYfirrrrrrrTfTrTTrfirrrrr

O 6 1 2 1 8 24 30 36 42 48

Figure 13.6 The Response of Nominal Wages to Technology Shocks

93

IRF of Fl to Tech checks

 

 

 

1254
15.o~
-17,5—<
[I
200— I
'225 IYTYWTVTTTVTTTIIIIIIIIIIITYIIYIIYYTIllrrrTIIIIIIIYYT

 

O 6 1 2 1 8 24 30 36 42 48

Figure 13.7 The Response of Interest Rates to Technology Shocks

IRF 01‘ Int to Tech shocks

 

 

/_________

.24 _1 \J/
-32 ~

.40 _4

 

 

 

'56 YYTYTYYYYIIIIYIIIVI IIIII 7111111111IITTTVTTIYTIYIYTTT

0 6 1 2 1 8 24 30 36 42 48

Figure 13.8 The Response of Inflation to Technology Shocks

IRF oi Fl-lnf to Tech shocks

 

35-1

25—4

20—1

h
——‘_—

 

5 4 /\/\/“/\\
V

 

 

 

'5 IYYYTYIITIIIIIVIIIT'TTIYT l FITTTIIIIIYVITTIIITTITM

0 6 1 2 1 8 24 30 36 42 48

Figure 13.9 The Response of Ex Post Real Rates to Technology Shocks

040

-000

~080

-120

94

IRF of de to Labor shocks

 

 

 

 

 

\
¥\\
\
\\
\\
\\
\\\
\ \
\
r1111111111rrrrrwrrrrrvIrrrrrrrrTTTwrtrr11711111rrrr
O 6 12 18 24 30 36 42 48

Figure 14.1 The Response of Output to Labor-Market Shocks

IRF of Mr to Labor shocks

 

 

 

 

Figure 14.2 The Response of Real Balances to Labor-Market Shocks

IRF oi Gap-Mr to Lebor shocks

 

125 *

100—4

075 ‘

050‘4

0254

 

 

x_

 

000

1

IIIWTYYTWTITTTTTTTIIYYIVYIYTIYTVIYYTYIITIIIIYITTTT

6 12 18 24 30 36 42 48

Figure 14.3 The Response of Velocities to Labor-Market Shocks

560

95

IRF of M2 to Labor ahocka

 

4ao~

400-—1

320 —

240 —‘

 

 

 

 

111171111Irrrrrrrrrrrrrr1rrrirrrrrrvrrrrrrTTfi—TrrrtI
O 6 12 18 24 30 36 42 48

Figure 14.4 The Response of Nominal Balances to Labor-Market Shocks

IRF of Wr to Labor shocks

 

1o -«

08—1

074

06—1

05—(

 

 

04

 

vaer 11111 r rrrrrr vrrrrrrrrrrvvvrrrrrrrrwrrvIrrrrtYr

0 6 1 2 1 8 24 30 36 42 48

Figure 14.5 The Response of Real Wages to Labor-Market Shocks

1 600

1400

1200

1 000

800

600 -

400

Figure 14.6

IRF of W to Labor shocks

 

 

 

 

M
_.
I"
d 1/
I
/
/
,
I
//
__( 1/
x“
f.
/
I'I/
ITTTYYIYYTIIIITTYTYIIITYYI[VIIIIYYTIIIYITTYrT—lIYTTY
O 6 12 18 24 30 36 42 48

The Response of Nominal Wages to Labor-Market Shocks

96

IRF of Fl to Labor ahocka

 

40—4

30—1

1o—< K.“

[I
0

 

 

 

-10

 

YT‘TTTTYIYIIIIYTYITIYTTIYTTTTY ITIYIITTTTTTITWIVIYY

6 1 2 1 8 24 30 36 42 48

Figure 14.7 The Response of Interest Rates to Labor-Market Shocks

IRF of In! to Labor ahocka

 

40—

0 17 YYYYYY TYTVIYYYYIVYYTYIVIT 1111111 rllTT‘TTTTYIITYIY‘Y

0 6 1 2 1 8 24 30 36 42 48

 

 

 

Figure 14.8 The Response of Inflation to Labor-Market Shocks

IRF of R-Inf to Labor ahocke

_20 4 /

 

 

—100 -<

-120 —‘

-14o —

 

 

 

-160 YlIlIIIYjIIIYII'llYIIIIIIIIIIIIIIYTYIIIIIIIIYIIIIII

O 6 1 2 1 8 24 30 36 42 48

Figure 14.9 The Response of Ex Post Real Rates to Labor-Market Shocks

97

IRF of de to Nomlnal ahocka
0.0072

 

0.0060 ~
00048 —<
\

\

00036 - 1‘
\‘1

00024 a ‘\

’/”\
00012 ~ \ \-—~

 

0 .0000 l

 

 

 

'00012 1111!IIIIIT'IIIIIlllTYTI1111VFTTYTTTTTTIYTTTYTTYTYTTIII

0 6 1 2 1 8 24 30 36 42 48

Figure 15.1 The Response of Output to Nominal Shocks

IRF of Mr to Nomlnal ahocka

 

 

 

 

 

00125
1
00100—4
00075——1
)
0.0050“ \
00025—« \
‘1
.-//f
00000 YYrTTYYYT-“TIYTIUTYTYTYYYYYYTYYYYYYYYTTTTYYTYYTYIYYTIY
0 6 12 18 24 30 36 42 48

Figure 15.2 The Response of Real Balances to Nominal Shocks

IRF oi Gap-Mr to NomInal checks
0 004

o 002 —-1 \\.\
\7»,
\\\\_/—— —d‘

0 000 r

 

 

-0 002 *1

—O 004 -+

0.006 —1

-O 008 —1

-0 010 —

-0.012 -‘

 

 

 

~0.014 111117111ItrrrvrrT—rWTIVTTrrrirIvfifrrrri—Twrrrr7rrrr

0 6 1 2 1 8 24 30 36 42 48

Figure 15.3 The Response of Velocities to Nominal Shocks

98

IRF of M2 to NomIn-I shocks

 

20‘

 

 

 

O YYYITIYIYllIYVIVTlIVTYIYTYTYITVVYYTTlYl’TT’YTYTrrjlIl

0 6 1 2 1 8 24 30 36 42 48

Figure 15.4 The Response of Nominal Balances to Nominal Shocks

IRF of Wr to Nominal shocks

 

 

 

 

 

00020
00015 —‘
00010 --4
00005 _J
"____'__,__,.
00000 ‘ __,_#~
\ ///'

1 ,r/

\ /'
*00005 rrrrrrrr \r’l rrrrrrrrrr rwrrrrrrrrrrrrrt‘lrvtvrrrrvvrvrvYr

O 6 12 18 24 30 36 42 48

Figure 15.5 The Response of Real Wages to Nominal Shocks

IRF of w to Nomlml shocks

 

4O -* /,/’ ’
30 — /

. /
20 "1 ’1/

 

 

 

o IYYYTYYYYIIIYIYVIIIIYIITllIleTYTTI1TTTIIIYYVYTIYIT

0 6 1 2 1 8 24 30 36 42 48

Figure 15.6 The Response of Nominal Wages to Nominal Shocks

99

IRF of Fl to NomIn-l shocks

 

 

 

 

 

 

07 wrrrrrrrrr.rrrrrrrrrrlrITTrrrITITYTIYIFTrrrrT—rrrIr—r—r
() 6 12 18 24 30 36 42 48

Figure 15.7 The Response of Interest Rates to Nominal Shocks

IRF of IrflI to Nomlnal shocks

 

128—

 

1 12 -l ’\ \K
096 #

OBO—J

 

 

 

064 YYYYYYYYYYYYYYYYY T TTTTTT T YYYYYYYYYYY YIYYTYTYYYTYYYYY

Figure 15.8 The Response of Inflation to Nominal Shocks

IRF 01' R-Inf to Nomlnal shocks

005 —4 LA A
/\
_ i h” L.
l ' \_
I

-005 4 V

 

 

‘I-"‘

030 «

 

 

 

‘035 llYYIlTWTTfTTYTllIYTTrrTYlTTTTTTTlTITYIIITTYTYITIYT

O 6 1 2 1 8 24 30 36 42 4B

Figure 15.9 The Response of Ex Post Real Rates to Nominal Shocks

Chapter 6
SUMMARY AND CONCLUSION

This study of postwar US. economic fluctuations using the common-trends
methodology is guided by a simple theoretical model that has properties suitable
for business cycle studies. It includes a wage-contract equation to partly account
for the wage rigidity characteristics in the economy. In addition, the model allows
for cyclical real wage and price behaviors that are consistent with predictions from
both the Keynesian and the real business cycle theories. In this model we are able
to identify the roles of the permanent shocks as sources of economic fluctuations.
Particularly, the permanent labor-market shock is featured to capture a separate
effect on the aggregate supply that is independent of that from the technology shock.
All variables used have stochastic-trend components and are widely considered to
have significant cyclical properties. Among them, real wages, to my knowledge, has

not been modeled previously in the common-trends model framework.

Two cointegration relations implied by the wage contract model are confirmed
by the unit-root tests, cointegration rank tests and Horvath and Watson’s (1995)
cointegrating-vector tests. They are then imposed in estimating a VAR model that
includes five time dummies. The acquired empirical model cannot be rejected by
a likelihood-ratio test at a 10 percent significance level. A first attempt to identify
both permanent and transitory structural shocks based on the contemporaneous
relations from the theoretical model is not successful. In the next attempt we are
able to identify the permanent shocks using the long-run structure available in the

theoretical model though transitory shocks are not identified. Identification of the
100

 

101

permanent shocks depends critically on their long-run multipliers predicted by the
theoretical model. Findings on the significance of the permanent shocks as sources
of fluctuations for output, real balances, inflation and the nominal interest rate
are generally consistent with earlier studies employing the same cointegrated VAR

methodology.

The common drawback suffered by VAR studies is also present in this study.
Possibly due to over-parametrization (as discussed by Runkle 1987), I have found
the standard errors of the structural impulse—response functions and the forecast
error variance decompositions to be relatively large. The dynamic patterns revealed
in such analyses can nevertheless improve our knowledge about the dynamic impacts

of the permanent shocks on the economy.

In the long run, variations in output and real balances, which form the velocity
relation, are dominated by the technology shock. Also for the long run, the labor-
market shock dominates the variability in real wages. Given that long-run nominal
neutrality is prescribed by the theory, long-run variations in the nominal interest

rate and inflation are explained almost exclusively by the permanent nominal shock.

Similar to findings by others, I find permanent shocks to be important sources
of fluctuations even in the short run. Specifically, the nominal trend accounts for
a high proportion of the nominal interest rate variability in the short run. This
is similar to the finding of Englund et al. (1994) but different from the extremely
low percentages found by King et al. (1991). The impact of the nominal shock on
inflation is at least 30 percent within the first two years in King et al. (1991). I find
the nominal trend accounts for at most 30 percent of the inflation variance within
the same horizons. The technology shock accounts for about 40 to 60 percent of the
output variance on one- to two-year horizons. King et al. (1991) and Mellander et

al. (1992) arrived at similar percentages. More significantly, the technology shock

102

accounts for about 80 percent of the real balance forecast variance in the very short
run, comparable to about 70 percent found by King et al. (1991). The labor-
market shock already explains 40 percent of the variance in the impact quarter and
only increases its influence on real wages thereafter. I found the labor-market shocks
explain about 20 percent of the output variance within the first year. This compares

to at least 40 percent of the output variance found by Shapiro and Watson (1988).

APPENDICES

APPENDIX A

RATIONAL EXPECTATION SOLUTIONS FOR
A MODEL ECONOMY

First define it E Et_1:rt for any variable xt, i.e., it is the rational expectation
of :13; made at time t subject to all information available at time t — 1. First apply
rational expectation on the labor demand and labor supply equation and set them

equal to obtain the contract wage.

Et—1{CS (wt — 1%)} = Et—l {7 (Pt — wt + mm) + Ust}

(6 + 7) (wt — fit) = OI’Yfiu + fist

 

A A 1 A
wt = pt+auu+ 7213, (a.1)

6 +
Note in the above a 5 33:1”; and Et_1wt E a, = wt since current wages are set in the

last period. Then set aggregate supply and aggregate demand equal,

mt“Pt+'Ut = fi(pt_wt)+flult

Pt = fi—iT-I (mt + ,Bwt — 571” + ’Ut) (302)

Now substitute equation (a.1) into equation (a2) and apply Et_1 on both side to

get
= —- m an —u — u
Pt ,8 + 1 t Pt 1t 6 + 7 3t 1t t
fit 2' fit + ’11} + ,8 (a — 1) 1’1” + fl fi3t (a3)

 

6 + 7
Substitute (a.3) back into (a.1) to solve for wt in terms of predetermined terms as

A A A +1A
wt = mt +1}; + [fl (a— 1) +0.] U” + ?+WU3t. (3.4)

 

The solution for pt is obtained by substituting equation (a.4) into equation (a2) as

A

 

 

pt 2 fit + '17; + 3(0 —' 1)fi1t+ 6 + 7113;
1 A A A
+_fl +1 [(mt — mg) + (U: — 1%)] — 5 i 1(u1t - U1t)- (3-6)

103

104

The solution for real wages, 101', 5 wt — pt, is then

A

wrt E wt—ptzaiiu+6+7u3t

1
fl+1

 

 

[(mt — m.) + (2),: — 50] +

 

5 A

u — u . 3.9
fl + 1( n It) ( )
Finally substitute (3.6) into the aggregate supply equation (3.2) to solve for the

output level as

 

yt = 50" (1)77” — 6575173:
fl [(mt — fit) + ("Ut — 17;) +(U1t — I210] . (3.8)

+—
3+1

APPENDIX B
THE EX POST REAL INTEREST RATE EQUATION

The real interest rate identity includes the next period ex ante inflation rate
which is to be solved in terms of currently available variables. Before doing that we
first define real balances by mrt E mt — pt and inflation by 7rt E pt — pt_1, then the

real balance process can be written as
mrt = mrt_1 — 7r; + uzt. (3.13)
Move forward one period on (3.13) and take expectation to get
7?,“ = fig,“ — Wt+1+ mrt. (b.1)
Substitute (b.1) into (3.12) to get
Tt = 7ATt+1 + ‘13 (L) 6(St
= an“ — T7171“ + mm + ¢ (L) est. (b.2)

We also need to solve for r’n‘rHl. Before doing that we subtract m, in (3.13), from
both sides of (b.2) because we then can obtain an operational contemporaneous or

ex post real-interest-rate equation as
Tt — 7ft = fi2t+1'— WHI + mTt + ‘15 (L) Est + 7’th - mTt-l — U2t
or
T; — 7r, — mm + mrt_1 = 52ng — 212; + [mm — mm“) + ()5 (L) 66; (b.3)

The last equation will be used as one of the solution equations to form the equation

system of Chapter 3.

Now we want to further solve for mrt — Wt+l in equation (b.3). First obtain

mrt from the aggregate demand equation, (3.1), and (3.11) as

 

m1} = ,6 (1 — a) (7'1 + u1t_1) — (T3 + U3t_1)

6+7
105

106

+

 

,8 :1 [61¢ + €2t + 0063] — 0(L)€5t.

and then move one—period forward to get

 

mrt+1 = ,3 (1 — a) (71 + u”) — (7'3 + u3t)

6+7

+——e +6 +06 —6Le .
fi-l-l [ n+1 2t+l o 5t+1] ( ) 5t+1

Take expectation conditional on information available at time t to get

 

Wt+1= 5(1— a)('r1+ 11”)- ’6 7 (T3 + 1143;) — C (L) 65¢

6+

where C (L) = [6 (L) — 60] L‘l. Subtracting (b.5) from (b.4) we have

 

 

 

m'rt — 771.73.“ = —,B (1 — a) Ault + 6 +7AU3t + [B +1(€1t+ €21)
+¢ (L) €5t
where w (L) E C (L) — 0 (L) + $011. Now (b.3) follows immediately as
7'; '— 7ft — mrt +m'rt_1 = T2 — B (1 — a)Au1¢+ 6 +7AU3;

 

+ (611+ €2t) + 'l/J (L) 651 + 43 (L) 66‘. (3.14)

[3
fi+1

APPENDIX C

DATA SOURCES AND DEFINITIONS

All data are obtained from Citibase database except for M2 nominal balances
data between 1951:1 and 1958z4 and nominal wages data. Certain data may be
monthly from the data source and are averaged to form their quarterly counterparts.

Data spans from 1951:1 to 1994:4. and are seasonally adjusted.

The three real measures examined in this study, real output, real balances and
real wages can be derived from subtracting the price deflator form their respective
nominal measures. The price deflator is in turn derived by subtracting the real

output from the nominal output measure, as described in detail below.

Real output (y) is defined as the real domestic product minus the real gov-

ernment purchase. Using the Citibase symbols, it is
y = ln(GDPQ — GGEQ).
Both GDPQ and GGEQ are measured in 1987 dollar value.

To calculate the price deflator we need the measure of the nominal output
which is defined as the logarithm of nominal GDP minus the nominal government
purchase, or ln(GDP-GGE) in symbols. The price deflator (p) is calculated accord-

ing to

 

_ 1n GDP — GGE
’0 ‘ GDPQ — GGEQ

and note p = O in 1987.
All series have been converted to their natural logarithm except for the nom-

inal interest rate data.

The source of the nominal money supply data is the monthly Citibase M2
series which is available for 1959:1-1994:12. The M2 series for the period 1951:1-

1958zl2 is provided by Professor Robert Rasche. He estimated the series based
107

108

on data reported in Banking and Monetary Statistics: 1941-1970 published by the
Board of Governors of the Federal Reserve System in 1976. The complete monthly
series are then averaged to obtain the quarterly observations. The real money bal-
ances (mr) is defined as

mr = ln (M2) — p.

The source of the nominal wage data is the monthly average hourly earnings
of production workers in the manufacturing sector (Series ID EESOOOOOOfi) available
from the Bureau of Labor Statistics. The reason of this particular series being chosen
is because it has complete observations of the sample period studied. The monthly
data is then averaged to derive the quarterly nominal wages (W). The quarterly
real wages is defined as

wr = In (W) —p.

The source of the nominal interest rate is the monthly observations of the
3—month Treasury Bill rate (FYGM3) in the secondary market measured in annual
percentage. It is not seasonally adjusted. The quarterly nominal interest rate (R)

is just the quarterly average of the monthly series.

Finally, following King et al. (1991), price inflation (7r) is measured in annual

percentage rate according to

7r = 400 x (pt — pt_1).

LIST OF REFERENCES

LIST OF REFERENCES

Ahmed, Shaghil, and Jae Ha Park, 1994, Sources of Macroeconomic Fluctuations
in Small Open Economies, Journal of Macroeconomics, 16, 1-36.

Baillie, Richard T., Chung, Ching-Fan Chung, and Margie A. Tieslau, 1996, Ana-
lyzing Inflation by the Fractionally Integrated ARFIMA-GARCH Model, Jour-
nal of Applied Econometrics, 11, 23-40

Benassy, Jean-Pascal, 1995, Money and Wage Contracts in an Optimizing Model
of the Business Cycle, Journal of Monetary Economics, 35, 303,315.

Bernanke, Ben, 1986, Alternative Explanations of the Money-Income Causality,
Carnegie-Rochester Conference Series on Public Policy, 25, 49—100.

Beveridge, Stephen, and Charles Nelson, 1981, A New Approach to Decomposi-
tion of Economic Time Series into Permanent and Transitory Components
with Particular Attention to Measurement of the ‘Business Cycle’, Journal of
Monetary Economics, 7, 151-74.

Blanchard, Olivier, and Stanley Fischer, 1989, Lectures on Macroeconomics, Cam-
bridge, Mass.: MIT Press.

Blanchard, Olivier, and Danny Quah, 1989, The Dynamic Effects of Aggregate
Demand and Supply Disturbances, American Economic Review, 79, 655—73.

Crowder, William J ., and Dennis L. Hoffman, 1996, The Long-Run Relationship
Between Nominal Interest Rates and Inflation: the Fisher Equation Revisited,
Journal of Money, Credit, and Banking, 28, 102-18.

Dickey, David A., and Wayne A. Fuller, 1979, Distribution of the Estimators for
Autoregressive Time Series with a Unit Root, Journal of the American Statis-
tical Association, 74, 427-31.

Dickey, David A., Dennis W. Jansen, and Daniel L. Thornton, 1994, A Primer on
Cointegration with an Application to Money and Income, in B. Ghaskara Rao
(ed.) Cointegration for the Applied Economist, New York: St. Martin’s Press.

109

110

Diebold, Francis, and Glenn Rudebusch, 1991, On the Power of Dickey-Phller Tests
Against Fractionally Alternatives, Economic Letters, 35, 155—60.

Els, Peter J.A. van, 1995, Real Business Cycle Models and Money: A Survey of
Theories and Stylized Facts, Weltwirtschaftliches Archiv, 131, 223-64.

Engle, Robert F., and C.W.J. Granger, 1987, Cointegration and Error Correction:
Representation, Estimation and Testing, Econometrica, 55, 251-76.

Englund, Peter, Anders Vredin, and Anders Warne, 1994, Macroeconomic Shocks
in an Open Economy—A Common Trends Representation of Swedish Data
1871-1990, in V. Bergstrom and A.Vredin (ed.) Measuring and Interpreting
Business Cycle, New York: Oxford University Press.

Frisch, R., 1933, Propagation Problems and Impulse Problems in Dynamic Eco-
nomics, in Johan Akerman (ed.) Economic Essays in Honour of Gustav Cassel,
London: George Allen, 171-205.

Gali, Jordi, 1992, How Well Does the IS-LM Model Fit Postwar US. Data?, The
Quarterly Journal of Economics, 107, 209-38.

Giannini, Carlo, 1992, Tapics in Structural VAR Econometrics, Berlin: Springer-
Verlag.

Hairault, Jean-Olivier, and Hanck Portier, 1993, Money, New-Keynesian Macroe-
conomics and the Business Cycle, European Economic Review, 37, 1533-68.

Hamilton, James D., 1994, Time Series Analysis, Princeton, New Jersey: Princeton
University Press.

Hoffman, Dennis L., and Robert H. Rasche, 1996, Aggregate Money Demand Func-
tions: Empirical Applications in Cointegrated Systems, Boston: Kluwer Aca—
demic Publishers.

Horvath, Michael T. K., and Mark W. Watson, 1995, Testing for Cointegration
When Some of the Cointegrating Vectors Are Prespecified, Econometric T he-
ory, 11, 984-1014.

Johansen, Soren, 1988, Statistical Analysis of Cointegration Factors, Journal of
Economic Dynamics and Control, 12, 231-54.

—, 1991, Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian
Vector Autoregressive Models, Econometrica, 59, 1551-80.

—, 1995, Likelihood-Based Inference in Cointegrated Vector Autoregressive Mod-
els, New York: Oxford University Press.

111

Johansen, Soren, and Katarina Juselius, 1990, Maximum Likelihood Estimation
and Inference on Cointegration—with Applications to the Demand for Money,
Oxford Bulletin of Economics and Statistics, 52, 169-210.

—, and —, 1992, Testing Structural Hypotheses in a Multivariate Cointegration
Analysis of the PPP and the UIP for the UK, Journal of Econometrics, 53,
211-44.

Karras, Georgios, 1993, Sources of US. Macroeconomic Fluctuations: 1973-1989,
Journal of Macroeconomics, 15, 47-68.

King, Robert, Charles Plosser, and Sergio Rebelo, 1988, Production, Growth and
Business Cycles: I The Basic Neoclassical Model, Journal of Monetary Eco—
nomics, 21, 195-232.

King, Robert, Charles Plosser, and Sergio Rebelo, 1988, Production, Growth and
Business Cycles: II. New Directions, Journal of Monetary Economics, 21, 309-
41.

King, Robert, Charles Plosser, James Stock, and Mark Watson, 1991, Stochastic
Trends and Economic Fluctuations, American Economic Review, 81, 795-818.

Mellander, E., A. Vredin, and A. Warne, 1992, Stochastic Trends and Economic
Fluctuations in a Small Open Economy, Journal of Applied Econometrics, 7,
369-94.

Mishkin, Frederic S., 1992, Is Fisher Effect For Real? A Reexamination of the
Relationship Between Inflation and Interest Rates, Journal of Monetary Eco-
nomics, 30, 195-215.

Nelson, Charles R., and Charles I. Plosser, 1982, Trends and Random Walks in
Macroeconomic Time Series, Journal of Political Economy, 71, 405-22.

Rasche, Robert H., 1992, Money Demand and the Term Structure: Some New
Ideas on an Old Problem, Working Paper No. 9211, Department of Economics,
Michigan State University.

—, 1997, The Identification Problem in VAR and VEC Models, Manuscript, De-
partment of Economics, Michigan State University.

Runkle, David E., 1987, Vector Autoregressions and Reality, With comments by
Christopher A. Sims, Olivier J. Blanchard, and Mark W.Watson, Journal of
Business and Economic Statistics, 5, 437-54.

Shapiro, Matthew, and Mark Watson, 1988, Sources of Business Cycle Fluctua-
tions, NBER Macroeconomics Annual, 111-48.

112

Sims, ChristOpher A., 1980, Macroeconomics and Reality, Econometrica, 48, 1-48.

—, 1986, Are Forecasting Models Usable for Policy Analysis?, Federal Reserve
Bank of Minneapolis Quarterly Review, 10, 2-16.

Stock, James H., and Mark W. Watson, 1988, Testing for Common Trends, Journal
of the American Statistical Association, 83, 1097-107.

Waren, Anders, 1990,Vector Autoregression and Common Trends in Macro and
Financial Economics, Ph.D. thesis, Stockholm School of Economics.