Discussion Paper No. 32
UNSTABLE VELOCITY AND THE MONETARY APPROACH TO
EXCHANGE RATE DETERMINATION:
THE CANADIAN－U.S. DOLLAR EXCHANGE RATE
By
Stephen S. Poloz
December, 1984
Economic Research Institute
Economic Planning Agency
Tokyo, Japan
UNSTABLE VELOCITY AND THE MONETARY APPROACH TO
EXCHANGE RATE DETERMINATION:
THE CANADIAN－U.S. DOLLAR EXCHANGE RATE
By
Stephen S. Poloz*
* Research Officer, Department of Monetary and Financial Analysis, Bank of Canada, Ottawa, Canada,
KIA OG9.
This paper was written while the author was a visiting researcher at the Economic Research Institute,
Economic Planning Agency, Tokyo, Japan, November－December 1984. I would like to thank the members
of the World Economic Model Group at the EPA for their hospitality, and to thank the Bank of Canada for
granting leave to undertake the project. I am also grateful to David Longworth, Robert Lafrance and John
Murray for their comments on the project proposal. The views expressed here are those of the author and
no responsibility for these views should be attributed to the Bank of Canada, the Economic Planning
Agency, or to the other individuals named above.
Contents
Page
1.
Introduction ··············································································
1
2.
The Models and Data ··································································
3
3.
Adjusting Reported Money Stock Data for Shifts ···························· 11
4.
Exchange Rate Model Estimation and Simulation Results ··············· 19
5.
Further Testing of the DGA Model ·············································· 35
6.
Conclusion ················································································ 41
Appendix ·························································································· 43
1. INTRODUCTION
The purpose of this paper is to examine the empirical relevance of several alternative
monetary models of exchange rate determination for the Canada－U.S. exchange rate. The
principal motivation behind this study is the observation that most previous research has
failed to take account of well-established results of the demand for money literature in implementing monetary exchange rate models, of which the demand for money function constitutes
an integral part. In particular, it is argued below that instability in the demand for money has
reduced the potential for success of monetary exchange rate models, and that preventing this
distortion from biasing tests of these models requires that the problem of velocity shifts be
dealt with explicitly prior to estimation.
A monetary model of the exchange rate is one where the latter is defined as the relative
price of two monies; early examples include Frenkel (1976) and Bilson (1978a). This study
emphasizes this class of models to the exclusion of the more general asset-price approach of
portfolio balance models and the more traditional trade-flaw approach. The latter theory,
which emphasizes the numerous import and export demand functions which underlie the
demand and supply curves for foreign exchange, is inconsistent with many of the observed
empirical regularities of exchange rates (Mussa, 1979). The portfolio balance approach, which
emphasizes imperfect substitutability of domestic and foreign assets, the supplies of which
interact with a portfolio model of the demands to yield the market-clearing exchange rate
(see Branson, Halttunen and Masson, 1977; Bisignano and Hoover, 1982), is judged to be
inapplicable to the case at hand, since Canadian and U.S. dollar assets are perceived to be
very close substitutes (Boothe et al., 1984). It is fair to say that the monetary approach,
which may be viewed as a special case of the portfolio balance theory, has dominated the
recent exchange rate literature, yet problems with its empirical implementation have prompted
a steady succession of modifications. The result of this process is that more recent monetary
models, while continuing to emphasize the role of money and the demand for money equation
in the process of exchange rate determination, contain elements of both the portfolio balance
and trade flow models; see Driskill (1981) and Hooper and Morton (1982), for example.
The plan of this paper is as follows. In Section 2 we briefly summarize the chronology of
monetary exchange rate models, and comment on the shortcomings which this study addresses.
The data which are used are also described in Section 2, in general terms Section 3 focuses on
the Canadian and U.S. demand for money equations, developing simple models which account
for the level adjustments caused by financial innovation, and then constructing shift-adjusted
money series therefrom. In Section 4 five exchange rate models are estimated and tested, using
both published money stock data and the adjusted series constructed in Section 3 Section 5
－1－
considers two special tests of the preferred model which emerges from the results of Section 4,
while Section 6 offers some concluding remarks and suggestions for further research. Appendix
A provides the detailed data definitions and sources, while Appendix B provides a bibliography.
－2－
2. THE MODELS AND DATA
2.1 A Chronology of Monetary Exchange Rate Models
The purpose of this section is to illustrate the sequence of modifications which have been
made to the simple monetary model during the post decade. This purpose may be met by
restricting attention to the various reduced forms and providing an intuitive discussion of the
elements essential to each. For details of the derivations the reader is reterred to the original
article.
The analysis begins with the domestic and foreign demand for money equations which,
for purposes of illustration only, are left in the following very simple form:
(2.1)
m = p + k0 + k1y － k2R
(2.2)
m* = p* + k0* + k1*y* － k2*R*
where asterisks denote foreign variables, lower-case letters denote natural logarithms, the ki
are positive parameters, m denotes money, y income and R the nominal rate of interest. The
vast majority of work in this area assumes that ki = ki* for all i, an assumption which is
convenient and will be retained for the purposes of this section. Then we may write the above
two equations in relative form:
(2.3)
m－m* = p－p* + k1 (y－y*) － k2 (R－R*)
To close the model we combine (2.3) with the assumption of continuous purchasing power
parity (ppp), which may be represented as s = p－p*, where s is the natural logarithm of the
spot domestic price of one unit of foreign currency. This gives the reduced form equation
for the exchange rate:
(2.4)
s = m－m*－ k1 (y－y*) + k2 (R－R*)
This equation says that depreciations of the domestic currency (increases in s) are caused
by relative expansionary monetary policy, high relative interest rates (representing, through
the Fisher effect, a high relative inflation rate) or low relative income growth.
It is recognized that the most critical assumption underlying (2.4) is ppp which, when
tested with aggregate price indices, has been shown not to hold in the short run (Frenkel,
1981a; Officer, 1980). Despite some clever arguments that this failure is unimportant (see
Bilson, 1978a), most subsequent models have attempted to allow explicitly for deviations from
ppp. The path-breaking article in this spirit is Dornbusch (1976), who assumes that asset
markets adjust instantaneously while prices are sticky. A slightly generalized version which
－3－
allows secular inflation is derived in Frankel (1979). The latter begins with the assumption
of covered interest parity, which stutes that the forward premium (x) is equal to the interest
differential (x = R－R*), and further assumes that x is a function of the gap between the
current spot and the equilibrium exchange rate (s̅ ) and of the expected long-run inflation
differential, (p－p*):
(2.5)
x = －k3 (s－s̅ ) + (p－p*)
Combining (2.5) with covered interest parity and eliminating x gives an expression for the
short-run spot rate:
(2.6)
s = s̅ －1/k3 (R－R*－p + p*)
Notice that the term in parentheses is the real interest differential. The exchange rate is
determined by ppp in the long run, which in turn is assumed to be determined by a long-run
version of the relative demand for money equation (2.3) with (R－R*) replaced by (p－p*).
However, it is assumed that the long run values of m, y and p may be taken as their current
actual levels, which amounts to assuming that these variables follow a random walk, so the
model essentially takes (2.4) with (R－R*) replaced by (p－p*) as a definition of s̅ , substitutes
this into (2.6) and collects terms:
(2.7)
s = m－m*－ k1 (y－y*) － 1/k3 (R－R*) + (1/k3 + k2)(p－p*)
For our purposes (2.7) will be referred to as the Frankel model. The overshooting result of
Dornbusch (1976) may be derived from (2.7) by noting that with p and y given in the shortrun and m exogenous, the endogenous variable in the relative demand for money equation
(2.3) is (R－R*). Using (2.3) to eliminate (R－R*) from (2.7) yields a true reduced form which
is free of the simultaneity between (R－R*) and s.
(2.8)
s = (1 + 1/k2 k3)(m－m*) － (k1 + k1/k2 k3)(y－y*) + (k2 + 1/k3)(p－p*)
－1/k2 k3 (p－p*)
Notice that the theoretical coefficient on (m－m*) now exceeds unity. For our purposes we
will refer to equation (2.8) as the Dornbusch model.
An alternative means of allowing for short-run deviations from ppp has been suggested
in Bilson (1978a). Bilson defines (2.4) as the equilibrium exchange rate (s̅ ), which assumes
that ppp holds and that the price level is determined by the interaction of money demand
and exogenous money supply, and assumes that the exchange rate adjusts gradually to s̅
according to a partial adjustment hypothesis.
－4－
(2.9)
s－s-1 = k4 (s̅ －s-1) , 0 < k4 < 1
Assuming that (2.4) defines s̅ and combining with (2.9) yields:
(2.10)
s = k4 (m－m*)－ k4 k1 (y－y*) + k4 k2 (R－R*) + (1－k4) s-1
Function (2.10) we will refer to as the Bilson model. Notice that in this model, in contrast
with the Dornbusch and Frankel models, the coefficient on the relative money supply is
expected to be less than unity. At this point it is convenient to discuss a popular non-monetary
model of the exchange rate, where (2.9) is assumed to hold but the demand for money
equation is thought to be unreliable. Thus, rather than using (2.4) to define s̅ , the nonmonetary model uses ppp (s̅ = p－p*) so that the reduced form equation is given by:
(2.11)
s = k4 (p－p*) + (1－k4) s-1
In practice, any number of other exogenous variables which are perceived to influence the
exchange rate in the short run may also be added to equation (2.11), such as real interest rate
differentials or news about the current account, for example. Included in this class of model
is the exchange rate equation which is in place currently in the Canadian Model of the EPA,
as described in Cockerline (1984).1)
To this point the models have assumed perfect capital mobility, so that any gradual
adjustment takes place only in the goods market. This assumption is relaxed in the next class
of model, known as stock-flow models, which contain elements of portfolio balance exchange
rate theories. The relevant articles are Niehans (1977) and Driskill (1981). The approach
drops the covered interest parity assumption of the Dornbusch (1976) model, replacing it
with a net foreign asset demand function (F), a net trade flows equation (T) and a balance
of payments identity, given by (2.12－2.14), respectively:
(2.12)
F = k5 (x－R + R*)
(2.13)
T = k6 (s－p + p*) － k7 (y－y*)
(2.14)
F = F-1 + T + A
1) The Cockerline equation begins with a form like (2.11) but constrains the adjustment to ppp somewhat by
defining ppp as an eight-quarter moving average of (p－p*) One problem with this equation is that one of
the additional explanatory variables is net official monetary movements, or intervention, which itself is
largely determined by the exchange rate Thus, it is not surprising that intervention is correlated with the
exchange rate; however, the problem of simultaneity is likely to be severe Indeed, the sign which is
obtained indicates that the exchange rate causes intervention rather than the reverse, at least for quarterly
data
－5－
where A includes all other autonomous flows, which are presumed constant. In addition, the
model requires two equations from the real economy, which define the process of price level
adjustment.
(2.15)
(p－p*)+1 － (p－p*) = k8 [(d－d*) － (y－y*)] + (p－p*)
(2.16)
(d－d*) = k9 (s－p + p*) + k10 (y－y*) － k11 (R－R* －p+p*)
where d represents the logarithm of aggregate demand and we have followed Lafrance and
Racette (1984) in adding a secular inflation term to (2.15). The latter two equations originate
with Dornbusch (1976), and have been written in relative form, implying that identical specifications exist for both the domestic and the foreign economy. Equations (2.15) and (2.16) may
be solved for (p－p*):
(2.17)
(p－p*) = k12 (p－p*)-1 + k13 (y－y*)-1 + k14 (m－m*)-1
+ k15 (p－p*)-1 + k16 s-1
The detailed expressions underlying the coefficients are not needed for our purposes and
therefore have been ignored. The stock-flow model solves (2.12－2.14) for x, combines the
result with (2.5) and solves for s̅ . The latter is combined with the long-run form of (2.4)
(where R－R* is replaced with (p－p*) to eliminate s̅ and the result is combined with (2.17).
Finally, the remaining (R－R*) term is solved out using (2.3) and terms are collected to yield
an expression of the following form:
(2.18)
s = k17 + k18 s-1 + k19 (m－m*) + k20 (m－m*)-1 + k21 (p－p*)-1
+ k22 (y－y*) + k23 (y－y*)-1 + k24 (p－p*) + k25 (p－p*)-1
Driskill (1981) points out that the true reduced form exchange rate equation of Dornbusch
(1976), which may be found by combining (2.17) with (2.8) to eliminate (p－p*) from the
latter, is identical in form with (2.18) but has different inplications for the coefficients. In
particular, although k18 + k19 + k20 + k21 = 1 for both interpretations, the Dornbusch view
implies that k18 < 0, k19 > 1, k20 < 0 and k21 < 0, whereas the Driskill view implies that
k18 < 1, k19 > 0, k20
0 and k21 > 0. In other words, the Driskill stock-flow model may be
regarded instead as a Dornbusch sticky-price model with perfect capital mobility but with a
freer dynamic structure.
As noted above the principal motivation behind the sticky-price and stock-flow models
is the dropping of the continuous ppp assumption. However, no explicit allowance is, made
in the various models for permanent or even long-lived deviations of the real exchange rate
from control. Frankel (1982) and Hooper and Morton (1982) have attempted to modify this
－6－
aspect of the monetary model by reintroducing the current account into the exchange rate
equation. Frankel does so by assuming that wealth is an argument of the demand for money
equation, and that wealth will be influenced by accumulation of net foreign assets which, by
definition, is the cumulated current account. Hooper and Morton introduce the cumulated
current account in two ways, arguing that it can effect expectations of the long-run real
exchange rate, as well as the risk premium, a wedge which they introduce into the covered
interest parity relationship. Adding the cumulated current account to any of the models
described above will enable a test of these hypotheses, but discriminating between them is
very difficult.
To conclude this subsection, then, in the work to follow we propose to examine the
empirical properties for the Canada U.S. exchange rate of five models: the non-monetary
model based on ppp (2.11), the Frankel model (2.7), the Dornbusch model (2.8), the Bilson
model (2.10), and the Driskill model (2.18) whih itself may be viewed as a generalization of
the Dornbusch model (2.8). In each case the cumulated current account variable will also be
tested The innovations which this study brings to the exchange rate literature are discussed
in detail in the next subsection.
2.2 Outstanding Issues
A detailed review of the empirical evidence relating to these models would serve little
purpose here, a selection of articles which implement these models may be found in the bibliography, and for an excellent survey the reader is referred to Khan and Willett (1984). It is
fair to conclude from this evidence that these models have empirical content, and that each
of the modifications which have been described above have resulted in improved specifications.
However, the overall success of these models can at best be described as mixed, with the
strongest evidence against the approach provided by the work of Meese and Rogoff (1983a, b).
These authors have compared the out-of-sample forecasting accuracy of (2.4), (2.7) and (2.7)
plus the cumulated current account with that of the simple random walk model. The random
walk model consistently provides a superior forecasting performance.
Most of this literature may be criticised on the grounds that the reduced form exchange
rate equations which have been employed imply assumptions regarding the underlying demand
for money equations which are not supported by the data, as demonstrated in the extensive
demand for money literature. The point has been made by, for example, Hakkio (1982) and
by Smith and Wickens (1984). The problems are essentially four in number.
First, a convenient assumption is that the domestic and foreign variables enter the
exchange rate equation in differential or relative form. This assumes that the demand for
money equation parameters are the same for both countries; for the sticky-price and stock-
－7－
flow models this assumption is extended to other structural equations as well. The importance
of this assumption was first discussed in Haynes and Stone (1981), and they found that the
results of Frankel (1979) were altered in important ways once the restrictions were dropped.
In later work the assumption has been dropped by, for example, Frankel (1982), and Meese
and Rogoff (1983b). Lafrance and Racette (1984) were unable to reject this hypothesis for
the Canada－U.S. exchange rate over the 1970’s.
The implications of rejecting the equal-coefficients hypothesis are far-reaching. Consider
in particular the implications of rational expectations for simple monetary models such as
(2.4), as discussed in Mussa (1976), Barro (1978), Driskill and Sheffrin (1981) and Hoffman
and Schlagenhautf (1983). Assuming uncovered interest perity allows one to replace the
interest differential with the expected change in the exchange rate, and implies that the current exchange rate depends upon the current expected future value of itself. Assuming modelconsistent or rational expectations, therefore, leads to repeated forward substitutions, until
the result is obtained that the current spot exchange rate depends on the expected future
path to infinity of the exogenous variables of the equation. Clearly, this argument rests on
acceptance of the equal coefficients hypothesis; it also rests on the assumption of uncovered
interest parity, which has been placed in doubt by the recent exchange market efficiency
literature (see Longworth, 1981; Boothe, 1983, and Longworth, Boothe and Clinton, 1983).
In any case, operationalizing rational expectations requires assumptions about the future paths
of the exogenous variables. A common assumption, used by Frankel (1979), Driskill and
Sheffrin (1981) and (after first testing the assumption) Lafrance and Racette (1984) is that
(y－y*) follows a random walk while (m－m*) follows a random walk around (p－p*) which
also follows a random walk. Then, for purposes of estimation the actual current values are
used. Testing the equal-coefficients hypothesis, then, requires that one have available expected
series for the individual variables, rather than for the differentials. The most obvious means of
doing so would be to construct time series models of the series in question, as in Hoffman and
Schlagenhauf (1983), but a model-consistent series may be generated from the reduced form
of a structural macro model, if one is available.
A second problem with the standard treatment of the relative money demand equation is
related to the first, and that is that the two money demand equations are assumed to contain
the same explanatory variables. Even in the simplest of money demand models, as in Goldfeld
(1973) and Clinton (1973) for the U.S. and Canada, respectively, there have been differences
in specification documented in the literature. If a simple common form is desired for ease of
implementation of an exchange rate model, then some assessment of the empirical cost of
doing so should be offered.
A third problem with equation (2.3) is that the two demand for money equations are
－8－
assumed to be static, which runs counter to volumes of evidence. The demand for money
literature has allowed explicitly for dynamics in estimation, usually by including a lagged
dependent variable. This implies that lagged real balances should appear in the reduced-form
equation for the exchange rate as well. The empirical importance of this point is easily tested.
A fourth problem with the assumptions underlying equations (2.1) and (2.2) is that the
demand for money conventionally defined has not been stable over time, particularly in the
cases of the U.S. and Canada. For detailed discussion the reader is referred to Goldfeld (1976),
Judd and Scadding (1982) and Simpson (1984) for the U.S.; Landy (1980), Bank of Canada
(1982, 25－29) and Freedman (1983) for Canada. The main driving force behind these instabilities seems to be financial innovation, the main implication of which is that traditional
measures of liquidity must be redefined. Some effort to do so has been made in both
countries, leaving definitions of money whose demand equations are still subject to level
adjustments, both in the past and currently. This problem may have very important implications for the monetary approach to exchange rate modelling. Suppose, for example, that (2.1)
represents the Canadian demand for money, and (2.2) that for the U.S., and suppose that
the latter shifts downward while the former does not. Assuming that m* is allowed to fall, it
will be lower with p*, y* and R* unchanged, and the various monetary models would predict
an increase in s which never materializes because money supply has changed one-for-one with
demand. Notice that under a regime of money stock targeting m* would not decline but in
the long run the foreign price level and hence s would change by the amount of the downward shift. However, the relationship between s and the measured money supply still will
have been displaced by the amount of the shift. The implication is that money stock measures
which enter exchange rate equations should be corrected for shifts to enable undistorted tests
of their performance.
The problem of money demand instability in this context has been alluded to by Frankel
(1981, 1982), Messe and Rogoff (1983b) and Bilson (1978a, 1979b). However, explicit treatment of the problem is restricted to the latter two papers. In the first Bilson has added a time
trend, and in the second a squared time trend, to the relative demand for money equation,
and therefore to his final exchange rate equation as well, to account for the perceived downward drift in relative money demand. Although these variables turn out to be significant
determinants of the Dollar/DM exchange rate it seems likely that more precise estimates could
be obtained if one were to analyze the demand for money equations separately so as to determine the exact timing and nature of the shifts and then altering the model appropriately.
To sum up this discussion, then, there seems to be some scope for improving upon the
performance of monetary models of the exchange rate through more conscientious prior
modelling of the demand for money. In the following sections this approach will be imple-
－9－
mented to the maximum extent possible in developing a monetary model of the Canada－U.S
excharge rate.
2.3 The Data
Details of data sources are provided in Appendix A. All mnemorics to be used in the
presentation below may also be found there. For real income and the price levels GNP and the
GNP deflators are used, and inflation rates are four-quarter differences of the logs of the GNP
deflators. The cumulated current account for Canada cumulates the data from 1926 to avoid
any level problems, the Canada/U.S. exchange rate is presumed to be independent of the
overall U.S. trade balance. GNP, the GNP deflator and the current account are quarterly and
seasonally adjusted at source. The exchange rate is the closing spot rate. The money supply
variables are discussed in detail in Section 3. Interest rates for both countries are 90-day commercial paper rates. Money supply, interest rate and exchange rate data are monthly at source,
and only the money supply is seasonally adjusted; the monthly series are converted to quarterly by selecting the third monthly observation from each quarter and collapsing.
－10－
3. ADJUSTING REPORTED MONEY STOCK DATA FOR SHIFTS
In this section we document the methods which were used to shift-adjust the Canadian and
U.S. money stock series. In both cases the estimated adjustments are based on a particular
model of the demand for money. Hence, readers who have reservations about the underlying
demand for money equations naturally will object to the particular shift adjustments which
have been employed. To some extent this is inevitable, but we have tried to minimize this
problem by using simple models of the demand for money which, although imperfect, are
widely regarded as adequate.
(a) CANADA
From late-1975 until the autumn of 1982 the Bank of Canada was following announced
targets for M1, which comprises currency outside banks and non-government demand deposits
net of private sector float. Some outright economization of transactions balances by large
firms during 1976－1977 led to a small downward shift in the M1 demand equation, but otherwise M1 appeared to be a stable function of nominal spending and interest rates. However,
after 1979 the demand for M1 began to shift downward quite rapidly. This was due to institutional developments on both the personal and the non-personal side (see Bank of Canada,
1982, 25－29; Freedman, 1983):
(i) On the non-personal side, the cash management techniques which had been developed
for large firms in the mid-1970s began to spread to much smaller firms. These developments included consolidation of geographically dispersed corporate accounts into one
central account, automatic transfer of excess balances into interest-bearing notice
accounts near the end of each business day, and automatic paydowns of credit lines
with excess balances. The adoption of these arrangements became widespread in mid1981, when market rates of interest reached historical peaks.
(ii) On the personal side, the introduction of daily interest non-chequable savings accounts in
late-1979 led to some economization of transactions balances. Before this time personal
notice deposits had paid interest only on the minimum monthly balance. Subsequently,
banks began to offer daily interest chequable savings accounts, which offer free chequing
and a market rate of interest when a minimum balance requirement is met. This has led to
a further shift out it M1.
These developments led to the abandonment of the M1 target late in 1982. Although no new
targets have been announced to date, the Bank of Canada has begun to publish a new aggregate,
M1A, which adds to M1 the non-personal notice deposits often used for cash management
－11－
Figure 3.1 M1 and M1A for Canada
purposes, and daily interest chequable savings deposits. The two series M1 and M1A are compared in Figure 3.1. It is clear that the two series are very similar until mid-1981, Based on
the work of Cockerline (1984), the EPA World Economic Model currently uses M1A rather
than M1. Unfortunately, M1A has recently become very difficult to model, because daily
interest chequable savings accounts have been growing very rapidly, largely at the expense of
conventional savings accounts. Hence, the concept of M1A is beginning to over-compensate for
the downward shifts in M1 discussed above. In addition, it is unclear where or when this
process will conclude. Without some survey data it will be very difficult to estimate the
proportion of daily interest chequable savings accounts which relate to transactions motives,
and therefore to model M1A properly. For these reasons it has been decided to use shiftajusted M1, rather than M1A, for the purposes of the present study. Either approach should
provide essentially the same amount of information, but shift-adjusting M1 will be accomplished more readily.
The basic Model which is used for M1 is essentially the same as that used in Clinton (1973)
for M1, and in Cockerline (1984) from M1A. The simple real partial-adjustment model of
money demand, popularized by, for example, Goldfeld (1973), is assumed; this implies that
the natural logarithm of real money balances is a linear function of a constant, logarithms of
－12－
real income and a short-term money market interest rate, and a lagged dependent variable. In
preliminary estimation the data revealed a slight preference for this double-logarithmic form,
as opposed to the semi-logarithmic form used by Cockerline (1984) for M1A. In addition to
these variables we need to model three downward shifts
1976:01 (economization by large
firms), 1979:04 (introduction of daily interest non-chequable savings accounts) and 1981:02
(widespread adoption of cash management techniques by firms and the strong growth in daily
interest chequable savings accounts). The first shift is modelled as a 0－1 dummy variable in
the Cockerline (1984) M1A equation, we have retained this method and have followed the
same approach for the other two shifts, adding 0－1 dummy variables which become unity in
1979:04 and 1981:02, respectively. This approach may be criticised on the grounds that it
is too simple and can only approximately capture the various shifts. However, using the forecast errors of the unadjusted equation to formulate better-fitting shift variables would open
us to the criticism of data mining. Ideally, one would allow all parameters to shift by including
slope-dummy interaction variables. However, in practice it is rarely possible to distinguish
empirically between intercept and slope shifts before accumulating many observations of
post-shift data.
Before considering the results it should be noted that the equation does not include
dummy variables to account for the effects of postal strikes. This is because the use of end-ofquarter data virtually eliminates this problem from the data. Formal definitions of all data
used below are given in Appendix A.
The M1 equation with shift dummy variables included was estimated using all available
data following the 1967 Bank Act revision, 1968:02 － 1984:02. The following result was
obtained:
(3.1)
log(M1/PGNP) = －1.44220 + 0.25377* log (GNP)－0.05182*log (RCP)
(3.46)
(4.25)
(5.11)
+ 0.70640 * log (M1.－1/PGNP.－1)－0.02940 * DSHIF76
(9.85)
(2.94)
－0.01710 * DSHIF79－0.03789 * DSHIF81
(1.49)
(3.07)
R2C = 0.96185
SE = 0.01691
DW= 2.48
Dynamic RMSE = 1.89%
Long-run elasticities: GNP = 0.8643, RCP = －0.1765
All variables BLOCK = CA
68:02－84:02
All parameters have the expected signs and all except that on the 1979 shift dummy are
significant at the 0.95 level. However, the t-statistic on DSHIF79, at 1.49, is sufficiently high
－13－
to be indicative. The long-run elasticities are within the ranges predicted by theory. The intrasample dynamic RMS simulation error of 1.89% compares favourably with a value for M1A
of 2.29% as reported in Cockerline (1984) over the 78:01－82:04 period. The Durbin－Watson
statistic of 2.48 is worrisome, especially since its value is biased towards two in the presence
of a lagged dependent variable. However, it is believed that this indicates not a systematic
correlation in the estimated residuals but rather the inevitable element of misspecification
which arises by using the simple 0－1 dummy variables to model a phenomenon which obviously is more complex. To verify that this interpretation is valid, equation (3.1) was reestimated
over the 1968:02－1979:03 period, with DSHIF79 and DSHIF81 excluded, and the DurbinWatson statistic was a more respectable 2.15. The full result is given in (3.2):
(3.2)
log(M1/PGNP) = －1.31010 + 0.22328 * log (GNP)－0.05584 * log(RCP)
(3.21)
(3.65)
(6.32)
+ 0.75503 * log (M1.－1/PGNP.－1)－0.02582 * DSHIF76
(10.22)
(2.78)
R2C = 0.97047
SE = 0.01481
All variables BLOCK = CA
DW= 2.15.
68:02－84:02
More importantly, all of the parameters in equation (3.2) are within one standard error of
the corresponding parameters in equation (3.1). This indicates that for practical purposes
the shift dummies adequately capture the intended effects.
Equation (3.1) was simulated dynamically over the entire sample period, 1968:02－
1984:02, twice － once as shown and once with the three shift parameters constrained to
zero. The difference between the two simulated M1 series is our estimate of the total shift
in the demand for M1; this total shift adjustment was then added to actual M1 to generate
shift-adjusted M1. Actual M1 and adjusted M1 are shown in Figure 3.2. Notice that the two
series are the same prior to 1976:01. After this time the two series diverge by an increasing
amount, with shift-adjusted M1 given by the dotted line. The two series exhibit very similar
movements, since the smooth geometric estimates of the shifts affect the level gradually and
preserve the short-term variations of the original series. It seems likely from the magnitude
of the gap between actual and shift-adjusted M1 that the two series will generate non-trivially
different results when used as explanatory variables in exchange rate equations below.
(b) UNITED STATES
The U.S. monetary aggregate M1 now comprises currency, demand deposits and other
chequable deposits, the latter category consisting mainly of negotiable order of withdrawal
－14－
Figure 3.2 Actual and Shift-Adjusted M1 for Canada
(NOW) accounts which became available nationwide in January 1981. Thus, U.S. M1 is now
similar in concept to Canadian M1A. The strong growth in NOW accounts since 1981 has
been at the expense of both demand deposits and savings deposits. Hence, just as was the case
for Canadian M1A, adding NOW accounts to old M1 over-compensates for the reduction in
demand deposits due to switching to NOWs. Unlike Canadian M1A, however, which has
never been adopted as a policy target, in the United States “new” M1 has become official.
The details of the various innovations in the U.S. monetary sector during the late 1970’s and
early 1980’s are well-summarised in Brayton (1983), so need not occupy us further here.
In addition to the effect of NOW accounts, of course, one must consider the welldocumented downward shift in the demand for M1 which occurred in 1974－1976 (see
Goldfeld, 1976, see also Judd and Scadding (1982) for a review of the literature which
Goldfeld’s ‘missing money’ prompted). This shift may be handled conveniently using a 0－1
binary variable which becomes unity in 1974:01, as was done for the 1976 shift in Canadian
M1 above. This gives the estimated shift a smooth and gradual interpretation. The net upward
shift in M1 demand due to the nationwide introduction of NOW accounts, on the other hand,
is more difficult to model. For this purpose we have chosen to draw on the work of Brayton
(1983), who has constructed individual models for the three major components of M1
－15－
cur-
rency, demand deposits, and other chequable deposits. The simultaneous downward and
upward shifts in demand deposits and other chequable deposits, respectively, have been
captured using a variable which represents the fraction of the population holding NOW
accounts Brayton constructs this variable, which he denotes ‘ALPHA’, as the product of three
elements, the income-weighted fraction of the U.S. for which NOW accounts were available;
the demand deposit weighted fraction of the population eligible to hold NOW accounts; and
the proportion of those eligible to hold NOW accounts who actually open NOW accounts. The
variable therefore depends essentially upon the number of states which offered NOW accounts,
beginning with New Hampshire and Massachusetts in 1974, and ending with the entire nation
in 1981. The other two elements of ALPHA are estimated from survey evidence. A plot of
the series ALPHA is given in Figure 3.3; for additional details the reader is referred to Appendix C of Brayton (1983).
Figure 3.3 Now Availability Variable (ALPHA) for U.S.
The M1 equation which is estimated for the purposes of this study is essentially that of
Goldfeld (1973), modified to account for the two shifts. Thus, the logorithm of real M1
balances is regressed on a constant, the logarithms of real GNP, the 90-day commercial paper
rate and the rate on savings deposits; the 1974, 0－1 dymmy variable, ALPHA, and the lagged
－16－
dependent variable. In preliminary estimation the usefulness of a time trend and the rate on
NOW accounts was tested without success. This means that the equation as estimated is
without an own-rate of interest, which, since the introduction of NOW accounts, has been
non-zero. However, it is unlikely that the own-interest elasticity could be distinguished
empirically from the effect of ALPHA itself. The somewhat arbitrary starting point of 1968:
02 was chosen, simply because this is the same as that used for the Canadian equation. Ultimately, the exchange rate equations will be estimated from a later start date in any case. The
final equation, estimated to 1984:02, is as follows.
(3.3)
log(M1/PGNP) = 0.32068 + 0.06874* log (GNP)－0.01654* log (RCP)
(1.37)
(2.76)
(4.28)
－0.03302* log (RSD) + 0.86846 * log (M1.－1/PGNP.－1)
(1.87)
(16.94)
－0.01192* DSHIF74 + 0.09828 * ALPHA
(2.78)
(3.21)
R2C = 0.93084
SE = 0.00801
DW= 2.37
Dynamic RMSE = 0.83%
Long-run elasticities. GNP = 0.5227, RCP = －0.1257, RSD = －0.2511
All variables BLOCK = US
68:02－84:02
All coefficients have the expected signs and all except that on RSD are significant at the 0.95
level. The coefficient on RSD, however, with a t-statistic of 1.87, is indicative. The long-run
elasticities of 0.52 for income and －0.13 and －0.25 for the two interest rates are within the
ranges given by theory. The intrasample dynamic RMS simulation error of 0.83% compares
favourably with that presented in Brayton (1983), which was 0.94% for the 1970:01－1981:04
period, and was based on the sum of the three component equation simulations. As was the
case for the Canadian M1 equation, the Durbin－Watson statistic is a little high considering that
its value is biased towards 2 in the presence of a lagged dependent variable. However, as was
the case for Canada, we hesitate to correct for this by assuming an AR1 error structure because
we suspect that the problem is concentrated in the period of shifting, which can only be
approximately modelled. To verify that this was indeed the case, the equation was reestimated
over the 1968:02－1979:04 period over which ALPHA is virtually inactive. The results were
as follows:
(3.4)
log(M1/PGNP) = 0.14712 + 0.06695 * log (GNP)－0.01679 * log (RCP)
(0.53)
(2.43)
(3.82)
－0.03614 * log (RSD) + 0.90346 * log (M1.－1/PGNP.－1)
(1.57)
(13.09)
－17－
－0.01078 * DSHIF74 + 0.12986 * ALPHA
(2.34)
(0.79)
R2C = 0.93241
All variables
SE = 0.00616
DW= 2.19
BLOCK = US
Here the Durbin-Watson statistic is a more respectable 2.19. More importantly, the estimated
parameters of (3.4) are all within one standard error of those presented in (3.3). Hence, it
would seem that ALPHA does an adequate if imperfect job of explaining the NOW shift.
Equation (3.3) was simulated dynamically over the entire sample period twice － once as
written and again with the coefficients on the 1974 shift and on ALPHA constrained to zero.
The difference between the simulated values of M1 in the two cases is our estimate of the total
shift in M1 demand. This estimate of the shift is then added to actual M1 to form shiftadjusted M1. These two series are shown in Figure 3.4. Notice that in this case the upward
shift due to NOWs has been sufficient to more than offset both the 1974 downward shift in
demand deposits and that due to switching into NOWs. Hence, the adjusted series, given by the
dotted line in Figure 3.4, crosses the actual series (given by the solid line) at the beginning of
1982. The difference between the two series is sufficient to generate quite different predictions of the exchange rate within the context of a monetary model. The empirical relevance of
this distinction will be examined in the next section.
:
Figure 3.4 Actual and Shift-Adjusted M1 for U.S.
－18－
4. EXCHANGE RATE MODEL ESTIMATION AND
SIMULATION RESULTS
In the following five subsections the five models discussed at the end of Section 2.1 above
are estimated, respectively. In each case we consider both the model as written in theory and a
generalized form which imposes only those restrictions which are acceptable to the data. The
intra- and post-sample simulation results of each model are summarized in Table 4.1 at the end
of the Section. In Subsection 4.6 the collection of results is discussed with the objective of
choosing a model which is preferred overall. All data definitions are given in Appendix A.
4.1 Non-Monetary Model
In this model the exchange rate follows a partial-adjustment process towards relative ppp
and reacts in the short run to the relative nominal rate of interest and the cumulated current
account. The latter is a negative number over the sample period, so rather than taking its
logarithm we have normalized it on nominal GNP. Estimation of this and all other models
uses the sample period 1970:03－1982:04. The estimation results for this simple model are as
follows:2)
(4.1) log (s) = －1.65720 + 0.33943 * log (CA－PGNP/US－PGNP) + 0.66057* log(s.－1)
(4.94)
(4.92)
(9.58)
－0.06039* log(CA－RCP/US－RCP)－0.64761* (CA－CCA/CA－NGNP)
(3.81)
(3.30)
R2C = 0.30976 (transformed)
SE = 0.01743
DW= 1.84
70:03－82.04
According to equation (4.1) the speed of adjustment to ppp is relatively fast, on the order of
34% per quarter. The coefficient on the cumulated current account is statistically significant
and of the expected sign
a current account surplus, which increases the variable CA－CCA,
induces an appreciation of the Canadian dollar relative to the U.S. dollar. The coefficient on
the relative nominal interest rate is statistically significant but bears a sign opposite to that
suggested by the simple monetary theory. In the latter a higher nominal interest rate relative
to the foreign rate is associated with a higher relative inflation rate and therefore a rising ex2) In order to impose the theoretical constraint that the coefficients on log (CA－PGNP/US－PGNP) and
log (s.－1) be a1 and (I－a1), respectively, equation (4.1) was estimated in the form (log (s) －log (s.－1))
= a0 + a1 * log ((CA－PGNP/US－PGNP)/s.－1) + . . .
Because of this transformation the R2C statistic is not comparable with those of other equations to be
presented below. However, because s is still treated as the endogenous variable for simulation, the RMS
simulation errors are comparable.
－19－
change rate. Perhaps the opposite effect is found because the Canadian and U.S. inflation
rates are sufficiently highly correlated so that the relative interest rate term is dominated by
movements in relative real interest rates. When added to equation (4.1) the Canada－U.S.
relative inflation rate was not statistically significant, thus providing some support for this
interpretation. We should also note that the parameter in question may be subject to simultaneous equations bias, with endogeneity coming by way of the monetary policy reaction
function. The extent of this problem will be examined in Section 5 below.
Simulation results are provided in the first row of Table (4.1). The intrasample dynamic
simulation, with a RMS of 2.04%, shows a degree of precision which is similar to that of the
exchange rate equation of Cockerline (1984). The equation predicts the four observations of
1983 quite well, but does not capture the sudden depreciation which occurred during the first
half of 1984. These results will serve as a point of reference for comparison of the monetary
models below.
4.2 Frankel Model
In this model the nominal exchange rate is regressed on the relative money supply, relative
real income, relative nominal interest rates and the relative inflation rate. Since the equation
includes the money supply as an exogenous variable, there are two versions of the model;
Version 1 uses actual published money supply data, while Version 2 uses the shift-adjusted
money supplies constructed in Section 3 above. In order to capture possible movements in
the underlying real exchange rate we have also included the cumulated current account as an
explanatory variable. The estimation results are as follows; both versions have been estimated
using Cochrane－Orcutt AR－1 estimation.
(4.2) Version 1 － Unadjusted Money
log (s) = 2.05430 + 0.08014 * log (CA－MI/US－MI) － 0.44182* log(CA－GNP/
(1.45)
(0.59)
(1.48)
US－GNP)
－ 0.04038 * log (CA－RCP/US－RCP) － 0.01751 * log (CA－PDOT/
(2.08)
(1.01)
US－PDOT)
－ 0.93125 * (CA－CCA/CA－NGNP)
(1.13)
R2C = 0.13142
SE = 0.01955
70:03－82:04
－20－
DW = 2.38
RHO = 0.98862
(4.3) Version 2 - Adjusted Money
log (s) = 0.33092 + 0.52229 * log (CA－MIADJ/US－MIADJ) － 0.55940* log
(0.33)
(8.75)
(2.30)
(CA－GNP/US－GNP)
－0.04788 * log(CA－RCP/US－RCP) － 0.02578 * log (CA－PDOT/US－PDOT)
(2.23)
(1.48)
－1.25700 * (CA－CCA/CA－NGNP)
(2.29)
R2C = 0.79295
SE = 0.01905
DW = 2.03
RHO = 0.56475
70:03－82:04
All of the parameters have the expected signs except that on the relative inflation rate, which
should be positive according to Frankel’s model. One possible explanation for this result is
that during a major part of our estimation period increases in inflation often gave rise to expectations of tighter monetary policy, and hence higher real interest rates. It is immediately
apparent that Version 2 is statistically superior to Version 1. The degree of statistical
significance is much higher for Version 2, particularly with respect to the relative money
supply variable itself. This result lends support to the arguments in favour of shift-adjustment
of money stock data presented in Section 2 above. A similar comparison is true of the simulation results of equations (4.2) and (4.3), which are presented in Table 4.1. Although the postsample simulation results for the two equations are very similar, the adjusted version clearly
fits the estimation period much better.
Implicit in equations (4.2) and (4.3) is the assumption that domestic and foreign variables
bear coefficients which are of opposite sign but equal in absolute value. A joint test of these
restrictions yields F-statistics of 11.18 and 9.78 for versions 1 and 2, respectively, which, when
compared with the critical F value at the 0.95 level of 2.61, implies rejection of these restrictions by the data. The unrestricted versions of (4.2) and (4.3) were then taken as our new
maintained hypotheses; we then tested the joint significance of three additional variables not
found in these equations but implied by the estimated demand for money equations of Section
3, namely, the lagged real domestic and foreign money supplies, and the U.S. savings deposit
interest rate. For both versions these three variables were jointly insignificant at the 0.95
level. It should also be noted that in versions of (4.2) and (4.3) in which the domestic-foreign
equality constraints have been relaxed no correction for autocorrelation was necessary. Having
discarded the lagged real money stocks and the U.S. savings rate as explanatory variables, then,
we proceeded to impose statistically acceptable restrictions on the generalized equations so
as to improve efficiency. The final estimated equations are as follows.
－21－
(4.4) Version 1 － Unadjusted Money
log (s) = 4.77460 + 0.22170*log (CA－MI) + 0.29661 * log (US－MI)
(4.27)
(2.77)
(4.70)
－1.19870* log (CA－GNP)
(7.05)
+ 0.74577* log (US－GNP) + 0.02891* log (US－RCP) － 0.05949* log
(6.38)
(3.45)
(8.12)
(CA－PDOT/US－PDOT)
R2C = 0.97402
SE = 0.01395
DW = 2.15
70:03－82:04
(4.5) Version 2 － Adjusted Money
log (s) = 5.24240 + 0.21816* log (CA－MIADJ) + 0.20969* log (US－MIADJ)
(7.79)
(3.02)
(1.64)
－1.17190 * log (CA－GNP)
(13.78)
+ 0.70520* log (US－GNP) + 0.03869* log (US－RCP) － 0.05425* log
(6.49)
(5.35)
(6.61)
(CA－PDOT)/US－PDOT)
R2C = 0.97472
SE= 0.01377
DW=2.05
70:03－82:04
Both equations retain the same set of restrictions and therefore have the same form. The two
versions are now very similar in terms of statistical significance and explanatory power, but
the summary statistics remain slightly better for Version 2. The most disturbing aspect of the
results is that the coefficient on the U.S. money stock bears a sign opposite to that predicted
by the theory. The Canadian short-term interest rate is not individually significant; possibly
this is because most of the information contained in that variable may be found either in
US－RCP or in CA－PDOT. Attempts to retain CA－RCP by putting it in the form of a relative
interest rate (CA－RCP/US－RCP), were rejected by the data. The cumulated current account
variable also dropped out of the specification. Tests of the final sets of restrictions implicit in
(4.4) and (4.5) relative to the generalized forms of (4.2) and (4.3) yield F-statistics of 0.73 and
0.13, respectively, which should be compared with the critical value at the 0.95 level of 2.84
(F(3, 40)).
The simulation results of these two versions of the Frankel model are given in Table 4.1.
Comparing these with the previous results we note that the intrasample simulation errors
have been reduced by generalizing the equations and then imposing only statistically accept-
－22－
able restrictions. The two versions perform similarly in this respect. In post-sample simulation,
both models tend to overpredict the exchange rate; that is, to predict more depreciation
relative to what actually occurred. The errors mode by the version which uses adjusted money,
however, are much smaller than those of Version 1. On the other hand, both equations (4.4)
and (4.5) perform much worse in post-sample simulation than do their more restrictive
counterparts, (4.2) and (4.3). This may be because the post-sample simulation period is too
short for the bias implicit in (4.2) and (4.3) to make itself evident; instead, the efficiency
which is gained in exchange for this bias dominates the summary statistics. The same reasoning
is consistent with the opposite ranking which is found for the intrasample period, which is
comparatively long. Thus, the ranking of the various models explored so far and to be investigated below may depend upon how long the typical forecast period is likely to be once the
model is implemented.
4.3 Dornbusch Model
The Dornbusch model addresses the problem of endogeneity in the relative interest rate
term of the Frankel model by solving out the variable using the relative demand for money
function. Although the interpretation of the various parameters changes as a result, the outcome in terms of explanatory variables is to replace the relative interest rate variable with
the ratio of the domestic to the foreign price level. The estimation results for the two versions
(using unadjusted and adjusted money) of this model are given below. As was the case for
the Frankel model, it was necessary to use Cochrane－Orcutt AR－1 estimation for both versions.
(4.6) Version 1 － Unadjusted Money
log (s) = 3.33590 + 0.12347*log (CA－MI/US－MI) － 0.63693* log (CA－GNP/
(1.16)
(0.88)
(2.19)
US－GNP)
－0.07259*log (CA－PGNP/US－PGNP) － 0.02126*1og (CA－PDOT/
(0.15)
(0.96)
US－PDOT)
R2C = 0.05508
SE = 0.02039
DW = 2.37
RHO = 0.99202
70:03－82:04
(4.7) Version 2 － Adjusted Money
log(s) = 0.55810 + 0.35895*log (CA－MIADJ/US－MIADJ) － 0.99589*log
(0.51)
(2.81)
(4.46)
(CA－GNP/US－GNP)
+ 0.53006*log (CA－PGNP/US－PGNP) － 0.05854*log (CA－PDOT/
(2.04)
(3.32)
US－PDOT)
－23－
R2C = 0.73315
SE = 0.01952
DW = 2.15
RHO = 0.62140
70:03－82:04
The cumulated current account variable was tested but was not statistically significant in either
equation. The coefficients on the money stock and real income variables bear the expected
signs; however, as with the Frankel model, the inflation rate term obtains a negative sign,
which is contrary to the theory. In addition, in the Dornbusch model the relative price level
is expected to bear a negative sign, a situation which is true of Version 1 but not of Version 2.
More importantly, only the relative income term attains statistical significance in Version 1,
while the overall degree of significance in Version 2 is much higher. There is no evidence in
either equation of the Dornbusch overshooting hypothesis.
The simulation results from equations (4.6) and (4.7) are given in Table 4.1. As noted
previously for the Frankel model, the intrasample performance of Version 2 is much stronger
than that for Version 1. However, in this case we see that Version 1 is much more precise in
post-sample simulation than is Version 2. Indeed, the post-sample RMS simulation error of
Version 1, at 1.36%, is the lowest observed from all of the models considered thus far.
Both equation (4.6) and equation (4.7) impose linear restrictions on the parameters which
are statistically unacceptable to the data. In particular, testing the implied set of restrictions
that the domestic and foreign variables enter the equations with coefficients equal in absolute
value but opposite in sign results in F-statistics for Versions 1 and 2 of 10.93 and 8.01, respectively, which should be compared with the critical value at the 0.95 level of 2.61. Accordingly,
these generalized forms of equations (4.6) and (4.7) were adopted as our new maintained
hypotheses. Subsequently, the joint significance of three additional variables suggested by our
money demand equations － domestic and foreign lagged real money balances, and the U.S.
interest rate on savings deposits － was tested and rejected. We then imposed sequentially a
series of statistically acceptable restrictions on the two generalized forms so as to increase
efficiency. The final equations are as follows:
(4.8) Version 1 － Unadjusted Money
log(s) = 4.12720 + 0.11634*log (CA－MI) + 0.38137*log (US－MI)
(3.64)
(1.59)
(6.56)
－1.19760*log (CA－GNP) + 0.91049*log (US－GNP)
(6.41)
(6.76)
－0.04988*log (CA－PDOT) + 0.07723*log (US－PDOT)
(5.72)
(7.62)
R2C= 0.97124
SE = 0.01468
DW = 2.23
－24－
70:03－82:04
(4.9) Version 2 － Adjusted Money
log(s) = 6.12730 + 0.33976*log (CA－MIADJ)
(7.71)
(12.82)
－1.54210*log(CA－GNP) + 1.18230* log(US－GNP)
(13.63)
(10.75)
－0.03879*log (CA－PDOT) + 0.09215* log (US－PDOT)
(4.50)
(8.80)
R2C = 0.97000
SE = 0.01500
DW = 2.25
70:03－82:04
Notice that relaxing the restrictions implicit in (4.6) and (4.7) enables the use of ordinary least
squares in estimation for both versions. In neither version has the relative price level term been
retained, and the U.S. money supply variable has been dropped from Version 2. As was the
case with the Frankel model, the U.S. money supply bears a coefficient of sign opposite to
that predicted by the theory in Version 1. The restrictions implicit in (4.8) and (4.9) relative
to the generalized forms of (4.6) and (4.7) were tested; the F-ratios were 0.56 and 0.72, which
should be compared with F (3, 40) = 2.84 and F (4, 40) = 2.61, respectively However, notice
that when compared with the final estimates of the Frankel model, equations (4.4) and (4.5),
it is evident that equations (4.8) and (4.9) omit a statistically significant explanatory variable,
US－RCP. Thus, according to the Frankel model, equations (4.8) and (4.9) are misspeeified.
The simulation results for the two final Dornbusch equations are given in Table 4.1. The
intrasample performance of these equations is much better than that of their more restrictive
counterparts. The post-sample performance of Version 1, however, is worse than for its
restricted form, whereas the post-sample performance of equation (4.9) is much better than
that for (4.7), and must be regarded as very good, in general. This latter result is quite surprising given the point made above with respect to misspecification relative to the Frankel model.
4.4 Bilson Model
The Bilson model is similar in spirit to the non-monetary model considered above, where
the exchange rate follows a partial-adjustment process around ppp; however, the Bilson
model solves the relative price level out of the reduced form equation using the relative
demand for money equation. As a result, the exchange rate is regressed on the relative money
supply, relative real income and nominal interest rates, and the lagged exchange rate. To allow
for changes in the equilibrium real exchange rate we have also included the cumulated current
account. Because of the presence of the lagged dependent variable, correction for autocorrelation was unnecessary. The estimation results for two versions, the first using published money
supply data and the second using shift-adjusted money, are as follows:
－25－
(4.10) Version 1 － Unadjusted Money
log(s) = －1.13110 + 0.08739* log(CA－MI/US－MI) + 0.16536*log(CA－GNP/
(1.30)
(1.45)
(0.72)
US－GNP)
－0.04339*log (CA－RCP/US－RCP) － 0.51424* (CA－CCA/CA－NGNP)
(2.38)
(1.67)
+ 0.87421*log (s.－1)
(16.56)
R2C = 0.94885
SE = 0.01958
DW = 1.92
70:03－82:04
(4.11) Version 2 － Adjusted Money
log(s) = －0.97586 + 0.22963* log (CA－MIADJ/US－MIADJ) － 0.00228* log
(1.68)
(4.75)
(0.02)
(CA－GNP/US－GNP)
－0.06872*log (CA－RCP/US－RCP) － 0.86943* (CA－CCA/CA－NGNP)
(4.13)
(3.58)
+ 0.55619* log (s.－1)
R2C = 0.96458
SE = 0.01629
DW = 2.00
70:03－82:04
As was the case for previous models, we note that the equation which uses shift-adjusted
money fits the data better than does Version 1. Once again the relative interest rate takes on
a negative sign, which is contary to the theory. Relative real output is not statistically significant in either equation, while the relative money stock is only significant in Version 2. We
also note that the coefficient on the relative money stock lends support to the interpretation
of the Bilson model, in which it is the speed of adjustment to ppp. as opposed to the Dornbusch model, where by the overshooting hypothesis this parameter is expected to exceed
unity.
The simulation results for this pair of equations are given in lines (4.10) and (4.11) of
Table 4.1. As we have noted with previous models, a stronger intrasample performance is
rendered by Version 2. Indeed, equation (4.11) has the lowest intrasample RMS percentage
simulation error than any of the other basic models examined so far, 1.65%. Moreover, unlike
previous models, this dominance by Version 2 over Version 1 extends to the post-sample
simulation as well. The post-sample simulation of equation (4.11) is very good except for
1984:02, where it fails to predict the magnitude of the sudden depreciation which occurred
at that time. Indeed, the equation predicts a 1% appreciation in 1984:02, rather than the
3% depreciation which occurred.
Tests of the restrictions implicit in (4.10) and (4.11) that domestic and foreign variables
－26－
enter with coefficients equal in absolute value but opposite in sign yielded F-statistics of
9.09 and 2.48 for Versions 1 and 2, respectively, which should be compared with a critical
value at the 0.95 level of 2.84. Thus, in the case of Version 2 we cannot reject these restrictions. However, we subsequently tested the relative inflation rate as an additional explanatory
variable in (4.40) and (4.11), thus allowing the equations to distinguish between movements
in real and nominal rates of interest, and found empirical support for such inclusion, especially
in equation (4.11). Then, retesting the equal-but-opposite-signs hypothesis for these two expanded versions of equations (4.10) and (4.11) resulted in F-statistics of 11.19 and 4.34,
respectively, which, when compared with the critical value at the 0.95 level of 2.61, imply
rejection of the hypothesis for both versions. Thus, for both Versions 1 and 2 our maintained
hypotheses became (4.10) and (4.11) with the domestic and foreign variables entered separately, and with the domestic and foreign inflation rates added as explanatory variables. At this
point we tested the joint significance of lagged real domestic and foreign money and the U.S.
savings deposit rate, as suggested by the work of Section 3, and found these variables to be
jointly insignificant for both versions. We then imposed statistically acceptable vestrictions
on our maintained hypotheses in sequence, beginning with the most acceptable and stopping
when no additional linear restriction was acceptable to the data. The final equations which
result are as follows:
(4.12) Version 1 － Unadjusted Money
log(s) = 2.44650 + 0.12793* log (CA－MI) + 0.27475* log (US－MI)
(2.34)
(1.71)
(3.29)
－0.74073* log (CA－GNP) + 0.47748* log (US－GNP) － 0.02990* log
(3.83)
(2.88)
(2.02)
(CA－RCP/US－RCP)
－0.04321* log (CA－PDOT/US－PDOT) + 0.22950* log (s.－1)
(4.06)
(1.94)
R2C = 0.97094
SE = 0.01476
DW = 2.17
70:03－82:04
(4.13) Version 2 － Adjusted Money
log(s) = 2.77860 + 0.13655* log (CA－MIADJ) + 0.19684* log (US－MIADJ)
(3.17)
(1.81)
(1.42)
－0.66282*log (CA－GNP) + 0.35214* log (US－GNP) － 0.04558* log
(4.12)
(2.21)
(3.12)
(CA－RCP/US－RCP)
－0.03025* log (CA－PDOT/US－PDOT) + 0.31540* log (s.－1)
(3.23)
(3.22)
R2C = 0.97168
SE = 0.01457
DW = 2.21
－27－
70:03－82:04
The only variable which eventually was dropped from the equations was the cumulated current
account, and it is clear that the equality restrictions were being rejected because of the money
stock and real income coefficients. As was the case in previous models, the U.S. money stock
enters with the wrong sign. The other parameters either take on the expected sign or may be
interpreted as discussed for the Frankel model above. Notice that relaxing the unsupported
restrictions has reduced the reliance of these equations on the lagged dependent variable for
their explanatory power. The final set of restrictions implicit in (4.12) and (4.13), relative to
generalized versions of (4.10) and (4.11) which include both domestic and foreign inflation
as explanatory variables, were tested and found to be acceptable to the data; the relevant
F-statistics were 2.38 and 2.22 for Versions 1 and 2, respectively, which should be compared
with critical values at the 0.95 level of F (3, 39) = 2.85.
The simulation results for equations (4.12) and (4.13) may be found in Table 4.1. There
we note that the intrasample performance is about the same for both versions, and is comparable in precision with that of the Frankel model. Both versions have improved their intrasample performance relative to their restricted counterparts, (4.10) and (4.11). Post-sample
performance is improved for Version 1 but worsened for Version 2, and both (4.12) and
(4.13) simulate post-sample worse than their Frankel and Dornbusch counterparts, and also
worse than the non-monetary model. Both equations (4.12) and (4.13) overpredict the
exchange rate throughout the post-sample simulation period.
4.5 Driskill Model
As demonstrated in Section 2 above the Driskill model is essentially a generalization of the
Dornbusch model. The logarithm of the nominal exchange rate is regressed on logarithms of
the relative money supply, the lagged relative money supply, relative and lagged relative real
income, lagged relative price level, current and lagged relative inflation, and the lagged
exchange rate Estimates of the two versions of this model over the 1970:03－1982:04 period
are as follows:
(4.14) Version 1 － Unadjusted Money
log(s) = －0.36544 + 0.18188* log (CA－MI/US－MI) － 0.15480* log
(0.56)
(1.27)
(1.07)
(CA－MI.－1/US－MI.－1)
－0.51423* log (CA－GNP/US－GNP) + 0.14054* log (CA－GNP.－1/
(1.76)
(0.43)
US－GNP.－1)
－0.01090* log (CA－PDOT/US－PDOT) － 0.01246* log (CA－PDOT.－1/
(0.62)
(0.59)
US－PDOT.－1)
－28－
+ 0.41085* log (CA－PGNP.－1/US. PGNP.－1) + 0.70551* log (s.－1)
(2.72)
(6.98)
R2C = 0.95129
SE = 0.01911
DW = 2.02
70:03－82:04
(4.15) Version 2 － Adjusted Money
log(s) = 0.48463 + 0.23064* log (CA－MIADJ/US－MIADJ) － 0.01464* log
(0.83)
(1.53)
(0.09)
(CA－MIADJ.－1/US－MIADJ.－1)
+ 0.45864* log (CA－GNP/US－GNP) － 0.12372* log (CA－GNP.－1/
(1.70)
(0.38)
US－GNP.－1)
－0.02924* log (CA－PDOT/US－PDOT) － 0.01162* log (CA－PDOT.－1/
(1.66)
(0.59)
US－PDOT.－1)
+ 0.27119* log (CA－PGNP.－1/US－PGNP.－1) + 0.50299* log (s.－1)
(1.80)
(4.09)
R2C = 0.95706
SE = 0.01794
DW = 1.98
70:03－82:04
The cumulated current account variable was tested in preliminary estimation and was not
statistically significant in either version. With the exception of the lagged relative real income
variable the two sign patterns are the same, and these results, although in many cases not
statistically significant, lend support to the Driskill interpretation of this reduced form, as
opposed to that of Dornbusch. Interestingly, the theoretical implication that the coefficients
on relative money, lagged relative money, lagged relative prices and the lagged exchange rate
sum to unity, although not imposed in estimation, would be easily accepted by the data, for
the point estimates sum to 1.14 and 0.99 for versions 1 and 2, respectively. There is no
evidence of autocorrelation.
The simulation results for equations (4.14) and (4.15) are given in Table 4.1. As we have
noted for the Frankel and Dornbusch models, Version 2 simulates better than Version 1 over
the estimation period, but the reverse is true in the post-sample period. However, we would
not expect (4.14) and (4.15) to simulate well post-sample, given the inclusion of several
statistically insignificant variables.
The restrictions implicit in (4.14) and (4.15) that the domestic and foreign variables enter
the equations with coefficients equal in absolute value but opposite in sign were tested and
rejected by the data; the calculated F-ratios were 4.81 and 3.57 for Versions 1 and 2, respectively, which should be compared with a critical value at the 0.95 level of 2.30. Since the
equations already include the lagged money stocks and lagged price levels we tested only
－29－
the U.S. savings deposit rate in addition, and it was found to add no significant explanatory
power to either equation. Thus, our maintained hypotheses became (4.14) and (4.15) with
the domestic and foreign variables entered separately. On these equations we imposed in
sequence restrictions which individually and jointly were acceptable to the data at the 0.95
level. The final equations which emerged from this process are as follows:
(4.16) Version 1 － Unadjusted Money
log(s) = 4.52690 + 0.15257* log (CA－MI/US－MI) + 0.51504* log (US－MI.－1)
(4.08)
(2.16)
(11.97)
－1.26300* log (CA－GNP) + 0.92880* log (US－GNP)
(6.94)
(7.25)
－0.03218* log (CA－PDOT) + 0.05681* log (US－PDOT)
(2.41)
(4.18)
－0.02184* log (CA－PDOT.－1/US－PDOT.－1)
(1.67)
R2C = 0.97405
SE = 0.01395
DW = 2.15
70:03－82:04
(4.17) Version 2 － Adjusted Money
log(s) = 6.09790 + 0.21631* log (CA－MIADJ/US－MIADJ) + 0.42716* log
(8.10)
(2.58)
(6.11)
(US－MIADJ.－1)
－1.45990* log (CA－GNP) + 1.05450* log (US－GNP)
(13.22)
(8.61)
+ 0.04801 * log (US－PDOT) － 0.03676 * log (CA－PDOT.－1/
(4.19)
(4.32)
US－PDOT.－1)
R2C = 0.97352
SE = 0.01409
DW = 2.09
70:03－82:04
The only difference between the two final forms is that CA－PDOT has been dropped from
Version 2 but not from Version 1. Lagged real income, the price levels and the lagged exchange
rate have all been dropped from the equations. Other parameters generally bear the expected
signs, except those on the inflation rate variables. One possible explanation for this result, as
noted above for another equation, is that over a substantial portion of our sample period
an increase in U.S. inflation prompted expectations of tighter U.S. monetary policy, higher
real interest rates and an appreciating U.S. dollar. The final sets of restrictions implicit in
(4.16) and (4.17) relative to generalized versions of (4.14) and (4.15) were tested, and resulted
in F-ratios of 0.36 and 0.46 for Versions 1 and 2, respectively, which should be compared
－30－
with critical values at the 0.95 level of F (8, 34) = 2.23 and F (9, 34) = 2.19, respectively.
There is no evidence of first-order autocorrelation in either equation.
The simulation results of equations (4.16) and (4.17) are given in Table 4.1. As with
previous models the intrasample results have been improved somewhat by relaxing the restrictions which are rejected by the data and eliminating variables which are not statistically
significant. In post-sample simulation equation (4.16) performs worse than its more restricted
counterpart, whereas equation (4.17) performs much better. Indeed, equation (4.17) records
the best post-sample simulation results of any of the models tested.
4.6 Discussion
Several new insights have emerged from the analysis of this section.
First, it is clear that monetary models which use shift-adjusted money supply data fit
the sample period better and have better statistical properties overall than do models which
use unadjusted money supply data. This was true of all four models which use the money
supply as an explanatory variable.
Second, we have seen that the residual autocorrelation found in most exchange rate
models may be eliminated by relaxing the constraint that domestic and foreign variables
enter the reduced form with coefficients which are equal in absolute value and opposite in
sign, a constraint which was rejected by the data for every model considered. The interpretation of these rejections depends on which theoretical model is the maintained hypothesis;
however, which element of the underlying structure is the cause of this rejection is immaterial
from an econometric standpoint. In each case linear restrictions were rejected in favour of a
more general form whose data matrix contained the same information and whose structure
was equally consistent with the underlying theory.
Third, we have noted that for many of the equations examined relaxing unacceptable
restrictions and imposing acceptable ones improves the intrasample fit but worsens the postsample predictive performance. This may be an indication of one or both of the following
factors: (a) the post-sample period may be too short for the bias caused by inappropriate
restrictions to outweigh the reduction in variance which results, or (b) the equations for which
this problem arises are misspecified in any case, in the sense that they exclude statistically
significant variables, so that although the restrictions implicit in these equations are acceptable
to the data, relative to a misspecified alternative, the end result is that the final equations are
simply more precisely biased than their original counterparts, so their predictive ability suffers.
The estimation and simulation results of the Driskill model, which does not seen to be affected
by this problem and which includes regressors omitted by the other models, lend support to
hypothesis (b).
－31－
The superior performance of the Driskill model should not be surprising, since this model
begins with the most general maintained hypothesis of those considered. The final equation
says that the exchange rate is determined by the relative shift-adjusted money supply, the
lagged U.S. shift-adjusted money supply, domestic and foreign real income, foreign inflation,
and lagged relative inflation. Thus, equation (4.17) is essentially a Dornbusch exchange rate
model with extra terms which allow a freer determination of the equations dynamics. The
signs of the coefficients on the inflation rate variables are bothersome but may be rationalized.
Equation (4.17) is based on a well-considered theoretical structure, has strong statistical
properties, achieves the (apparent) minimum intrasample RMS error of 1.3%, and records the
strongest post-sample simulation performance. For these reasons, the Driskill-generalizedadjusted (hence forth, DGA) model (4.17) is our preferred working model.
One of the questions which has been posed in the literature is whether any exchange rate
model can outperform the simple random walk model. The latter is given as follows:
(4.18) log(s) = log (s.－1)
Table 4.1 gives the results of both dynamic and static simulations of model (4.18). The most
relevant comparison is between (4.17) and the static simulation of (4.18), since the DGA
Figure 4.1 Intrasample and Post-sample Simulation of DGA Model
－32－
model does not include a lagged dependent variable. There we see that the DGA model outperforms the static random walk model both within and post-sample. However, it is interesting
to note that in post-sample simulation the static random walk model outperforms 15 out of
17 models in terms of percentage RMS error.
A plot of the intrasample and post-sample simulation of the DGA model (4.17) is
provided in Figure 4.1. Actual data are represented by the solid line, the intrasample simulation by the dotted line, and the post-sample simulation by the dashed line. There is a gap
between the intrasample and post-sample simulations at 1982:04.
－33－
Table 4.1
Dynamic Simulation Errors*
(Percent)
Intra-sample
Equation
MEAN
RMS
Post-sample
83:01
83:02
83:03
83:04
84:01
84:02 MEAN
RMS
(4.1)
Non-monetary
0.07
2.04 －0.30
0.59
0.21 －1.15 －4.08 －7.53 －2.04
3.54
(4.2)
Frankel-basic-unadjusted
2.56
3.65 －0.08
0.59
0.34
0.73 －1.51 －4.38 －0.72
1.93
(4.3)
Frankel-basic-adjusted
0.03
2.17
0.05
0.78
1.15
1.74 －0.63 －4.34 －0.21
2.01
(4.4)
Frankel-general-unadjusted
0.01
1.29
3.16
4.68
4.91
8.15
5.78
3.74
5.07
5.32
(4.5)
Frankel-general-adjusted
0.01
1.27
1.78
2.87
2.75
5.40
2.93
0.82
2.76
3.09
(4.6)
Dornbusch-basic-unadjusted
3.82
5.45 －0.31
(4.7)
Dornbusch-basic-adjusted
(4.8)
(4.9)
0.93
1.02
1.79
0.01 －2.45
0.17
1.36
－0.02
2.36
0.61
2.45
3.96
6.57
4.35
1.99
3.32
3.83
Dornbusch-general-unadjusted
0.01
1.36
3.03
4.33
4.63
7.52
5.61
3.88
4.83
5.04
Dornbusch-general-adjusted
0.01
1.41
1.01
1.52
1.47
3.23
1.72 －0.32
1.44
1.78
(4.10) Bilson-basic-unadjusted
0.73
2.76 －1.94 －2.54 －4.24 －6.47 －10.34 －15.00 －6.75
8.18
(4.11) Bilson-basic-adjusted
0.02
1.65
0.21
1.37
1.25
0.35 －2.30 －6.12 －0.87
2.78
(4.12) Bilson-general-unadjusted
(4.13) Bilson-general-adjusted
0.00
1.31
3.58
6.11
6.71
9.08
7.19
4.77
6.24
6.48
－0.01
1.29
2.61
4.80
4.93
6.01
3.90
1.11
3.90
4.22
0.01
2.21
0.34
2.32
2.89
3.53
3.19
1.14
2.24
2.51
－0.02
1.81
1.57
3.84
4.99
7.14
6.68
4.55
4.79
5.14
(4.16) Driskill-general-unadjusted
0.01
1.27
1.63
4.03
4.62
6.60
6.47
3.98
4.56
4.86
(4.17) Driskill-general-adjusted
0.01
1.30 －1.08
1.03
0.58
0.26
1.82 －1 22
0.23
1.11
(4.18) Random Walk-dynamic
－3.51
8.84 －0.67
0.12 －0.28 －1.25 －3.71 －6.70 －2.08
3.18
(4.18) Random Walk-static
－0.32
2.09 －0.67
0.80 －0.40 －0.98 －2.48 －3.11 －1.14
1.73
(4.14) Driskill-basic-unadjusted
(4.15) Driskill-basic-adjusted
* Intrasample refers to 1970:03－1982:04 sample period; all errors are (estimated-actual) as a percentage of actual.
Notice that only for the equations which include a lagged dependent variable will a dynamic simulation differ from
a static one; this is the case for (4.1),(4.10)－(4.13) and (4.14)－(4.15).
－34－
5. FURTHER TESTING OF THE DGA MODEL
5.1 Functional Form
As noted in Section 3, when estimating the demand for money equations there was a
slight preference for a double-logarithmic (DL) functional form over a semi-logarithmic (SL)
one on the basis of fit. As a result, interest rates were also used in logarithmic form when
estimating the exchange rate equations; by extension, and since one would like to decompose
the effect of interest rates into real and nominal parts for some models, inflation rates have
been treated similarly. Most theoretical work, on the other hand, has assumed an SL form,
with both interest rates and inflation rates entered in level form. Moreover, although this has
not been a problem during the period under investigation, it is possible that in future the
inflation rates will become negative, in which case taking a logarithm will require adding
a positive constant to the original series. The purpose of this subsection is to examine the
importance of this issue for the special case of the DGA model (4.17).
The DGA model was reestimated after changing the inflation rates from logarithms to
levels, with the following result:
(5.1)
log(s) = 6.51640 + 0.20172* log (CA－MIADJ/US－MIADJ) + 0.45423* log
(8.80)
(2.26)
(6.22)
(US－MIADJ.－1)
－1.47870* log (CA－GNP) + 1.01820* log (US－GNP)
(12.98)
(7.63)
+ 0.00812* US－PDOT － 0.00458* (CA－PDOT.－1/US－PDOT.－1)
(4.43)
(3.90)
R2C = 0.97092
SE = 0.01476
DW = 1.97
70:03－82:04
The standard error of equation (5.1) is only slightly (about 5%) higher than that for equation
(4.17), and there is no substantive difference between the two equations. The parameter
estimates other than those on the inflation rates are all within one standard error of their
counterparts in equation (4.17). The similarity between the two sets of results may be underlined by considering the simulation results of equation (5.1), which are as follows:
Simulation Results for Equation (5.1), percent
(a) Intrasample
(b) Post-sample
mean
0.01
RMS
1.36
83:01
－0.47
83:02
1.83
－35－
83:03
1.62
83:04
1.03
84:01
1.31
84:02
－1.20
mean
0.69
RMS
1.32
Not surprisingly, given that its standard error is slightly higher, the SL equation has a slightly
higher intrasample RMS simulation error. The same comparison is valid for post-sample simulation, but the precision of these predictions remains high. Thus, although this is a very limited
test of the DL/SL issue, it seems unlikely that the choice of functional form has seriously
affected the inferences made in previous sections.
5.2 Endogenity and Expectations
We noted in Section 2 that in some of the models considered there exists the potential
for simultaneous equations bias. Indeed, given that the exchange rate is determined within
a two-country structural macro model, strictly speaking all of the right-hand variables should
be treated as endogenous; if the small-country assumption applies to one of the economies,
then the foreign variables may be treated as exogenous. The degree of endogeneity depends
also on the varying lag lengths in the various relationships. For example, if the exchange rate
is given a non-zero weight in the reaction function of the monetary authorities, we would
expect the simultaneity between the exchange rate and the short-term rate of interest to be
higher than, say, that between the exchange rate and real income or the rate of inflation. In
addition, to the extent that the nominal rate of interest is manipulated in order to affect
the exchange rate, the money supply also becomes increasingly endogenous.
From an econometric viewpoint the problem of endogeneity means that shocks which
impact on the economy and affect the exchange rate also affect to varying degrees the
variables on the right-hand side of the exchange rate equation. This means that these explanatory variables may be correlated with the error term in the equation, which gives rise to
simultaneous equations bias. The correct means of addressing this problem in estimation is
to estimate a full structural macromodel using a full information method. Such an approach
is beyond the scope of the present study. A more practical means in a single-equation context
is to use a two-step procedure, whereby the explanatory variables are regressed on a set of
instruments and the fitted values used in the estimation of the exchange rate equation. This is
the approach which is used below.
There is a second reason for considering such an approach. Most of the models considered
above, including the DGA model, have used some assumptions about expectations in their
－36－
derivation. In particular, most of the models used expected versions of the relative demand
for money equation to derive the expected price level and therefore the expected exchange
rate as given by expected ppp, toward which the actual exchange rate is assumed to adjust.
This means that the reduced form exchange rate equations in fact should contain expected
values of the money supply, output and the inflation rate; actual values have been used by
assuming that there are equal coefficients across countries and that the relative variables
follow random walks. However, in the DGA model we were forced by the data to relax the
equal coefficients restriction, and it is much less likely that the individual variables follow
random walks.
The correct means of estimation in this situation, then, is to estimate the entire structure
with the expectations derived by solving the model; that is, rational or model-consistent
expectations. Since this requires estimating a multi-equation model, this, too, is beyond the
scope of this study. However, once in a single-equation context, we note that from an econometric viewpoint the endogeneity problem discussed above and the expectations problem
are essentially the same. To account for the expectations problem we would like to replace
the actual values with unbiased expected values in estimation; that is, the expected values
would differ from the actual values by a random error. The econometric problem with using
actual values is that the included random error may, in fact, be correlated with the residuals of
the exchange rate equation, which is a classical errors-in-variables situation. This problem is
therefore not distinguishable from the endogeneity problem in a single-equation context.
To account for both problems, therefore, we have conducted the two-step procedure
described above. A convenient means of performing the first step is to construct a time series
model for each variable. Indeed, one can approach the efficiency of model-consistent expectations by estimating the exchange rate equation and the time series equations jointly, subject to
the cross-equation constraints implied by rational expectations, such as in Hoffman and
Schlagenhauf (1983). However, this procedure requires computing capabilities beyond those
which were available when this study was completed. Using the two-step procedure means
that the variance-covariance matrix which is estimated at the second stage are incorrect, but
in any case our present concern is with the problem of bias.
In order to reestimate the DGA model, we require estimated series for shift-adjusted
Canadian and U.S. money, Canadian and U.S. GNP, and the U.S. inflation rate. The Canadian
inflation rate enters the equation in lagged form and therefore is treated as exogenous.
Univariate time series models for each variable were fitted over the 1970:03－1984:02 sample
period for orders up to eight, and the Akaike final prediction error criterion was used to
choose the final order for each series. The results are presented in Table 5.1. It is evident that
the time series equations fit the data very well. Only the U.S. shift-adjusted money stock may
－37－
Table 5.1
Time Series Models of The Explanatory Variables*
(1970:05－1982:04)
CA－MIADJ
CA－GNP
US－MIADJ
US－GNP
US－PDOT
Constant
0.15490
(2.84)
0.31578
(2.43)
0.07657
(3.09)
0.07500
(0.78)
0.70365
(2.27)
LAG 1
0.73131
(5.64)
1.30080
(10.22)
0.98750
(232.30)
1.33320
(10.26)
1.42380
(10.29)
LAG 2
0.25636
(1.99)
－0.32734
(2.65)
－
－
－0.34297
(2.65)
－0.36943
(1.68)
LAG 3
－
－
－
－
－
－
－
－
0.19584
(1.04)
LAG 4
－
－
－
－
－
－
－
－
－0.85260
(4.61)
LAG 5
－
－
－
－
－
－
－
－
0.75068
(3.47)
LAG 6
－
－
－
－
－
－
－
－
－0.25838
(1.86)
LAG 7
－
－
－
－
－
－
－
－
－
－
LAG 8
－
－
－
－
－
－
－
－
－
－
R2C
0.99836
0.99333
0.99898
0.98998
0.94141
SE
0.01759
0.01032
0.00841
0.01123
0.49592
DW
2.09
1.98
1.88
2.08
2.02
RMS(%)
1.71
1.00
0.82
1.10
0.46
* All except US－PDOT were estimated in logarithms.
－38－
be approximated by a random walk.
The fitted values from these equations were used to replace their counterparts in the DGA
equation, which was reestimated and resimulated with the following results;
(5.2)
log(s) = 6.02370 + 0.17462* log (CA－MIADJE/US－MIADJE) + 0.50327 * log
(6.25)
(1.68)
(6.21)
(US－MIADJ.－1)
－1.32540*log (CA－GNPE) + 0.81442*log (US－GNPE)
(10.29)
(6.29)
+ 0.02702 + log (US－PDOTE) － 0.03386*log (CA－PDOT.－1/
(2.11)
(3.62)
US－PDOT.－1)
R2C = 0.96095
SE = 0.01711
DW = 1.86
70:03－82:04
Simulation Results, Percent
(a) Intrasample － mean
0.01
RMS
1.59
83:01
1.98
83:02
2.21
83:03
2.91
83:04
1.84
84:01
1.68
84:02
0.04
mean
1.78
RMS
1.98
(b) Post-sample
As anticipated, there has been some loss of precision in the parameter estimates, but only
that on the relative money supply has become statistically insignificant at the 0.95 level (it
remains significant at the 0.90 level). The standard error of the equation has been increased
by about 20%. Nevertheless, five of seven coefficient estimates lie within one standard error
of their counterparts in equation (4.17) and the remaining two are within two standard errors.
Thus, the bias contained in equation (4.17) due to errors-in-variables or endogeneity must be
regarded as statistically small. The precision of the simulation has been reduced is well, but is
still very good in an absolute sense.
It is likely that a more precise and more appealing test of these issues could be performed
once such an equation has been placed within the context of a complete structural model.
Nevertheless, the above results are suggestive that the DGA model will not be severely affected
－39－
by these considerations. It bears repeating, however, that this conclusion would come less
easily for other exchange rate models which have short-term interest rates as explanatory
variables. For such models to succeed it will be necessary to estimate jointly at least the
exchange rate equation and the monetary policy reaction function.
－40－
6. Conclusion
The purpose of this study was to examine the applicability of the class of monetary models
of exchange rate determination to the Canada－U.S. exchange rate. As there had been previous
studies purporting to do the same, the principal motivation of this study was the observation
that previous researchers had failed to take account of well-established results of the demand
for money literature when estimating monetary exchange rate models. The most important of
these was the failure to recognize the problem of instability in the demand for money equation.
The first step in the analysis, then, was to shift-adjust the money supply data; this proved to be
straightforward, as conventional partial-adjustment models of the demand for money were
found to explain the 1968－1984 data adequately once known structural shifts had been
accounted for. Both measures of money were then used in a full comparison of four different
monetary models.
The conclusions regarding exchange rate models which follow from this study may be
stated very simply. First, the hypothesis that monetary models of the exchange rate would
have more attractive statistical properties if shift-adjusted money measures were used in place
of published data was confirmed in every case. Second, the hypothesis of equal coefficients
across countries was rejected in every case; relaxing this restriction was found to eliminate the
problem of first-order serial correlation when it arose in some of the restricted models. Third,
the cumulated current account made statistically significant contributions to the equations
only when the unacceptable equal-coefficients restrictions were imposed; the variable does not
appear in any monetary equation which is acceptable to the data. Fourth, using these results
it was possible to develop an exchange rate equation based on the Driskill (1981) stock-flow
hypothesis with secular inflation which explains the sample period very well and which
predicts the six post-sample observations very accurately. Furthermore, this performance was
found to dominate that of the random walk model, to be independent of the double-log/
semilog functional form issue, and to be only slightly perturbed by accounting for endogeneity
in the explanatory variables. Taken together, these results indicate that the monetary approach
to exchange rate determination has empirical content and may be usefully applied to the
Cnada－U.S. exchange rate.
There are, of course, a number of potentially fruitful avenues for further research. The
concept of shift-adjusted money might usefully be applied elsewhere, in monetary policy
reaction functions, for example. In addition, a number of interesting implications of the various
theoretical models were not tested here and, more importantly perhaps, the implications of
rational expectations have been tested in only the most rudimentary fashion. Both of these
limitations were imposed by software constraints. In any case, the most satisfactory means of
－41－
invoking the rational expectations hypothesis is within the context of a complete structural
macro model. Indeed, it may be counter-productive to invoke rational expectations on a singleequation basis for one equation which will then form part of a structural model, for the
remainder of which expectations continue to be based on perfect foresight or adaptive behavioural assumptions. It is hoped that this study has provided a theoretically attractive
working model of the Canada－U.S. exchange rate which will be useful both before and after
the introduction of rational expectations into the EPA World Economic Model.
－42－
Appendix A － Data Definitions and Sources
In this appendix CANSIM refers to Statistics Canada main database, and BCR refers to the
Bank of Canada Review.
S
: Closing spot price of U.S. dollar in Canadian dollars, end of month data
converted to quarterly by choosing every third end of quarter observation.
(CANSIM B3414, BCR Table 65)
CA－M1
: Canadian MI, average of Wednesday monthly seasonally adjusted data
converted to quarterly by choosing every third end of quarter observation.
(CANSIM B1627, BCR Table 9)
CA－MIADJ
: Shift-adjusted quarterly Canadian MI
CA－GNP
: Canadian quarterly real GNE, seasonally adjusted at annual rates
(CANSIM D40593, BCR Table 53)
CA－PGNP
: Canadian quarterly GNE deflator, seasonally adjusted
(CANSIM D40625, BCR Table 54)
CA－NGNP
: CA－GNP * CA－PGNP/100
CA－RCP
: Canadian 90-day prime corporate paper rate, last Wednesday of each month,
converted to quarterly data by choosing every third end of quarter observation.
(CANSIM B14017, BCR Table 20)
CA－PDOT
: 100*[log (CA－PGNP)－log (CA－PGNP.－4)]
CA－CCA
: Cumulated Canadian current account balance; cumulated from 1926 using
annual data, then quarterly seasonally adjusted data from 1950.
(quarterly current account series CANSIM D60555, BCR Table 69)
CA－DSHIF76 : = 0 until 1975 : 04, = 1 afterward
CA－DSHIF79 : = 0 until 1979 : 03, = 1 afterward
CA－DSHIF81 : = 0 until 1981 : 01, = 1 afterward
US－MI
: U.S. MI, seasonally adjusted monthly average of weekly data, converted to
quarterly by choosing every third end of quarter observation.
(taken from EPA database)
US－MIADJ
: Shift-adjusted quarterly U.S. MI.
US－GNP
: U.S. quarterly seasonally adjusted GNP
(taken from EPA database)
US－PGNP
: U.S. quarterly seasonally adjusted GNP deflator
(taken from EPA database)
－43－
US－RCP
: U.S. 90-day commercial paper rate, adjusted to annual yield, last Wednesday
of the month, converted to quarterly by taking every third end of quarter
observation.
(CANSIM B54412, BCR Table 20)
US－PDOT
: 100* [log (US－PGNP)－log (US－PGNP.－4)]
US－RSD
: Quarterly U.S. Maximum rate paid on savings deposits, taken from DRI
database (mnemonic RMSD)
US－DSHIF74 : = 0 until 1973 : 04, = 1 afterward
US－ALPHA
: Availability variable for Now accounts, based on Brayton (1983)
(taken from EPA database).
－44－
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－48－