Measuring the Effect of Money: Test, Estimation and Identification

Measuring the Effect of Money: Test, Estimation and Identification

Mau-Ting Lin?

Department of Economics, National University of Singapore

June 17, 2003


Abstract





This paper provides a new approach of measuring the effects of money in both the long-
run and the short-run horizons. The key identifying assumptions used to identify and
measure the effect of money are long-run neutrality and long-run homogeneity.


The first chapter shows that both long-run propositions imply certain linear restrictions to
be imposed on the cointegrating space. By testing the validness of such linear restrictions,
both long-run propositions are tested. Compared with the previous long-run tests, the
cointegration test in this paper does not depend heavily on the auxiliary assumptions,
including identification restrictions and the correct selection of macroeconomic variables
to be included in the empirical work.


The second chapter shows that the linear restrictions imposed by the long-run
propositions can be used to identify the monetary shocks when the long-run proposition
is overidentifying. In such a case, it is proved that the monetary shocks can be identified
with the following three identifying assumptions: (1) a monetary shock is long-run
neutral and homogeneous; (2) monetary shocks are not correlated with other structural
shocks; and (3) the long-run effect of money is linearly independent from the long-run
effects of other structural shocks.

? Email: mtlin@bu.edu. This paper is the first two chapters of my dissertation in Boston University. I am
indebted to Robert G. King, Pierre Perron and Simon Gilchrist for their advice throughout the writing of
this paper, to John Chao for his assistance in PIC programming, and to Christopher Otrok for his data
provision. I also wish to thank Zhongjun Qu and Ferhan Salman for helpful comments and discussion. The
usual disclaimer applies.

---

---

Contents
List of Tables
v
List of Figures
vi
Chapter 1.
New Tests of Long-Run Monetary Neutrality and Homogeneity
1
1.
Introduction
1
2.
An Overview of Long-Run Tests
4
3.
Why cointegration is a valuable basis for New LR Tests
13
4.
Estimation with Long-Run Neutrality Restrictions
28
5.
Test Statistic
36
6.
Empirical Application
39
7.
Conclusion
50
Chapter 2.
Long-Run Identi?cation When the Long-Run Proposition is Over-Identifying
52
1.
Introduction
52
2.
An Overview of Long-run Identi?cation
54
3.
Long-Run Identi?cation When the Long-Run Proposition is Over-Identifying
61
4.
Identifying Permanent Structural Shocks
66
5.
VAR Model of International Monetary Transmission: An Application
70
6.
Conclusion
78
Appendices to Chapter 1
95
1.
Solution to a Simple Macro Model
95
2.
Solution to the Simple Macro Model in Section 3.3.4
96
iii

---

3.
Algorithm for the Maximum Likelihood Value of Lemma 2
96
4.
Unrestricted Maximum Likelihood Estimation
97
5.
R-Fold Replications
98
6.
Data Source: Friedman and Schwartz Data
98
7.
Data Source: Post-WWII Quarterly Data
99
Appendices to Chapter 2
101
1.
Computation of the ? Conditional on Monetary Shocks
101
2.
Computation of the Persistence of UIP Deviations
101
3.
Data Source
102
Bibliography
104
iv

---

List of Tables
1
Unit root test for money stock
80
2
Friedman and Schwartz data: Geweke test
80
3
Friedmand and Schwartz data: Unrestricted Estimation of the VECM
81
4
Friedmand and Schwartz data: Cointegration test
81
5
Post-WWII Quarterly data: Geweke test
82
6
Post-WWII Quarterly data: Cointegration test
82
7
VECM Selections
83
8
Long-run neutrality test
83
9
The estimated long-run responses of foreign exchange rates
83
10
The UIP Regression
84
11
Sample Estimated Serial Correlation of UIP Deviations
84
12
UIP regression
84
13
Serial Correlation of UIP Deviations Conditional on the U.S. Monetary
Shocks
85
v

---

List of Figures
1
Cointegrating Vector Space and the Long-Run E?ect of Money: A
Bivariate Example
86
2
Cointegrating Vector Space and the Long-Run E?ect of Money: A
Trivariate Example
87
3
Long-Run Proposition Restriction on Cointegrating Vector Space: A
Bivariate Example
88
4
Long-Run Proposition Restriction on Cointegrating Vector Space: A
Trivariate Example
88
5
Long-Run Proposition Test on the Annual Data: Friedman and Schwartz
Method
89
6
The Geometric Relations Between Restricted and Unrestricted Estimates
of the Cointegrating Vectors: Trivariate Case
90
7
The Geometric Relations Between Restricted and Unrestricted Estimates
of the Cointegrating Vectors: Four-variable Case
91
8
Long-Run Proposition Test on Quarterly Data: Friedman and Schwartz
Method
92
9
The Impulse Response Functions
93
10
The Impulse Response Functions of UIP Deviations
94
vi

---

CHAPTER 1
New Tests of Long-Run Monetary Neutrality and
Homogeneity
1. Introduction
The e?ect of money on real activity is one of the central research topics in macroeco-
nomics. Standard macroeconomic models suggest short-run e?ects of money on both real
output and the price level, with the longer-run e?ects on nominal variables but not on real
variables. The research reported in this chapter devises new tests of long-run propositions
about the e?ect of money on economic activity, in particular the long-run neutrality (LRN)
and the long-run homogeneity (LRH). In this chapter, as in much prior work, long-run neu-
trality is de?ned as the implication that a once-for-all change in the level of money should
not have a long-run impact on real variables. Similarly, long-run homogeneity is de?ned
as the implication that such a permanent change in the level of money should a?ect all
nominal variables proportionately in the long run.
There is a lengthy history of e?orts to test LRN and LRH. One notable early strand of
research on these two issues was at the Federal Reserve Bank of St. Louis during the 1970s.
The St. Louis researchers ran regressions of the ?rst di?erence of log output on the current
and lagged values of money growth and then computed a long-run multiplier of money —
the sum of the regression coe?cients — as the basis for LRN and LRH tests. These tests
were part of a larger e?ort by the St. Louis researchers to characterize the empirical e?ects
of monetary and ?scal policy on macroeconomic activity, which was heavily criticized for
its reduced form nature and its lack of concern about the direction of causality (see, e.g.
Ando and Mogiliani (1990)). In the context of LRN and LRH, the force of this criticism is

---

2
that a long-run multiplier which di?ers from its theoretical value — of zero for LRN and of
one for LRH — may be a re?ection of central bank policy response to economic conditions
rather than a rejection of the long-run proposition. Later, in the early 1970s, Sargent
and Lucas pointed out another important di?culty with LRN tests based on an estimated
long-run multiplier: when the economy does not embody any long-run variation in money,
the estimated long-run multiplier does not accurately capture the long-run e?ect of money.
Their argument was a forerunner of the critique that Lucas subsequently made: if long-run
variation in money is not a part of the environment that shapes the behavioral responses of
economic agents, then a reduced form analysis — such as regressions or vector autoregressions
— can never provide an answer about the e?ect of a long-run change in money.
Concern about causality and the Lucas critique cast a shadow over applied research
on long-run (LR) tests for nearly two decades. However, Fisher and Seater (1993) pointed
out that the pessimism was not necessarily justi?ed if economists are concerned about
whether a LR hypothesis held in a particular history and the historical data contained
long-run variation in money. In the situation where money was nonstationary (integrated
of order one), they showed how LRN can be tested via a long-run regression with proper
identifying assumptions made to disentangle the causality between output and money. For
convenience in the discussion below, we call tests based on these two ideas — integration
and identi?cation— second generation tests of LR propositions.1 These second generation
studies principally concerned bivariate relations between variables: they look at relations
between money and output to test LRN and between money and the price level (or nominal
income) to test LRH.
Two drawbacks of the long-run neutrality tests along second generation lines have been
pointed out in the literature. One is that the results of the test are heavily dependent on
identifying assumptions. More formally, identi?cation involves making the correct mapping
between the forecast errors and structural shocks, particularly the monetary shock. The
1Other related research is contained in Geweke (1986), who uses frequency domain methods, and King
and Watson (1997), who use vector autoregression (VAR) methods.

---

3
importance of this set of assumptions (and the fragility of neutrality tests with respect
to them) is most apparent in the VAR analysis of King and Watson (1997), where the
sensitivity of the long-run neutrality tests to various identifying assumptions is graphically
displayed. Yet, while such a VAR approach to neutrality testing has become popular2, its
application also requires that the researcher select the list of variables properly. This is
because the vector of variables used in a second generation study must reveal the shocks in
the economy correctly to the researcher. More formally, it must be possible to map between
the forecast errors and the true structural shocks. If the researcher is studying a subvector
of economic activity, then there are many reasons that this mapping may be infeasible.
So accurate speci?cation of the data vector is an essential part of second generation tests.
Thus, any rejection of LRN or LRH in a VAR context can signal that the theory is wrong
or that the identi?cation and/or variable selection assumptions are incorrect.
In this chapter, I develop a LR test based on cointegration concepts which depends on
a basic identifying assumption shared with the second generation tests: there must be an
independent source of nonstationary variation in the monetary time series. But my test
does not require either of the other maintained assumptions of the second generation tests:
it can evaluate LRN and LRH without a parametric identifying assumption and without
correct speci?cation of the macroeconomic data vector.
Turning to the details, I follow the approach of second generation tests in working with a
vector autoregression that is nonstationary in levels, so that there can be the stochastic trend
in money that these studies and my approach both rely on. But, in contrast to these earlier
studies, I suppose that there may be cointegration among the macroeconomic variables and
study a system with three or more variables so as to test LRN and LRH. To be precise,
I employ a vector error correction (VEC, henceforth) model of a form that is standard in
cointegration analysis. With an application of the Granger representation theorem3, I prove
2e.g. Bernanke and Mihov (1998), Serletis and Koustas (1998), and Bae and Ratti
(2000).
3Please refer to Chapter 4 of Johansen(1995).

---

4
that any LR hypothesis can be interpreted as a set of linear constraints on the orthogonal
cointegrating space, which in turn imposes restrictions on the cointegrating space. This
orthogonality condition is independent of the conventional identifying assumptions employed
in the second generation LR tests.
The LRN and LRH hypothesis constrain the cointegrating vector space, with a sacri-
?ce in degrees of freedom which is dictated by the particular hypothesis. Exploiting this
property, I show how to construct a likelihood ratio test for a particular LR hypothesis.
When the degrees of freedom sacri?ced is larger, the LR hypothesis is stronger, i.e., more
constraining on the estimated model. Hence, the LR hypothesis that I derive in this chap-
ter can be used to test LRN against LRH since, as I formulate these hypotheses, LRH is a
stronger hypothesis involving a greater sacri?ce in degrees of freedom.
I apply my LR tests to two di?erent data sets for real output, nominal interest rate,
the price level and nominal money stock: one is an annual data set based on the monetary
history of Friedman and Schwartz (1982), which covers 1940-1975, and the other is a post
WWII quarterly data set which covers 1959:1-2002:2. For the latter period, I split the data
into two subsamples: one is a pre-Volcker sample and the other is a post-1983 sample. For
all samples, I did not reject long-run neutrality (LRN). Long-run homogeneity was rejected
in the pre-Volcker sample of quarterly data, but not in the other two samples.
2. An Overview of Long-Run Tests
There has long been interest in testing propositions about the long-run link between
money and real or nominal variables, which are at the heart of classical macroeconomics.
In this section, the history of such tests is reviewed and the alternative approach taken in
this chapter is highlighted.
To begin, it is useful to review the two basic long-run propositions considered in this
chapter. In the long run, a permanent change in the level of the money stock is assumed

---