Monetary Policy and the Uncovered Interest Rate Parity
Puzzle: Theory and Empirical Results for Oceania
By
Alfred V Guender
Department of Economics
University of Canterbury
Christchurch, NZ
Revised November 2013
JEL Code: E4, E5, F3
Key Words: Uncovered Interest Rate Parity (UIP) Puzzle, Target Rule, Optimal
Monetary Policy, Openness, Financial Integration
Abstract:
This paper offers a theory-based explanation for why high interest rate countries
see their currencies appreciate, the so-called UIP puzzle. The central bank bases its
target rule on the lag of the policy instrument and the CPI inflation rate. When combined
with a stylized model of an open economy, the endogenous target rule can account for
the systematic negative relation between the change in the exchange rate and the lagged
interest rate differential. Foreign inflation and the foreign interest rate also affect
exchange rate changes. The model-based behavior of the exchange rate is tested on New
Zealand and Australian data with mixed results.
_____________________________________________________________________________________________________
Mailing Address: Private Bag 4800, University of Canterbury, Christchurch 8140, New Zealand. E-mail:
[email protected] Phone: (64)-3-364-2519 Fax: (64)-3-364-2635.
The author thanks Richard Froyen, Carlos Lenz, Tommaso Manchini, David Mayes, and Philip Meguire for
providing valuable feedback on the subject. Constructive comments were received from two anonymous
referees and the editor of this journal as well as from seminar participants at the Bundesbank, Swiss
National Bank, Free University of Berlin, the University of Giessen, the University of Otago, and the
University of Wuerzburg. The hospitality and research support of the Swiss National Bank and the
University of Antwerp is gratefully acknowledged. All errors are the sole responsibility of the author.
Absent any risk premium, the uncovered interest rate parity (UIP) hypothesis suggests
that a positive interest rate differential (domestic interest rate minus foreign interest rate)
should be compensated in full by a depreciation of the domestic currency over the term to
maturity of the underlying short-term financial asset. In practice, the presumed one-for-one
relationship between the interest rate differential and the expected change in the nominal
exchange rate is not apparent in the data. In fact, many studies report a negative association
between movements in the interest rate differential and observed changes in the nominal
exchange rate. The observed violation of the UIP condition has been labeled the “UIP
Puzzle”. 1 The “Carry Trade” phenomenon whereby high interest rate countries see their
currencies appreciate is symptomatic of the failure of the UIP condition.
There has been a long-standing interest in the role of monetary policy as a key driver of
the linkage between a change in the nominal exchange rate and the interest rate differential.
McCallum (1994) recognizes that interest rate smoothing by the central bank can successfully
explain the widely reported empirical failure of the UIP hypothesis. 2 Employing an
optimizing two-country framework, Backus et al (2010) show the inverse link between the
interest rate differential and the change in the nominal exchange rate to be the result of the
domestic central bank ignoring exchange rate movements in setting the policy instrument.
Paradoxically, the foreign central bank must tighten monetary policy in response to its
currency appreciating for the negative association of the interest rate differential with the
change in the nominal exchange rate to hold.
In both McCallum (1994) and Backus et al (2010) the central bank is viewed as
following a simple ad hoc instrument rule. But as shown by Svensson (2003) and Froyen and
Guender (2012), simple stand-alone instrument rules - such as Taylor rules - are inferior to
optimal target rules as specifications of monetary policy rules. Endogenous target rules
1
See, for instance, Fama (1984), Froot and Thaler (1990), Lewis (1995), Engel (1996), Chinn and Meredith
(2004), Burnside et al (2011). Guender and Cook (2011) focus exclusively on whether uncovered interest rate
parity holds between New Zealand and Australia over the 1986- 2008 (July) sample period. Their empirical
study excludes the immediate period after the float of the Kiwi Dollar as well as the peak and aftermath of the
Global Financial Crisis. Following the standard atheoretical testing procedure, which relies on regressing the
realized change in the nominal exchange rate on the lagged interest rate differential, they investigate whether
UIP holds over the short term (90-day horizon) and over the long term (5-year and 10-year horizon,
respectively). They find no evidence in favor of the UIP hypothesis over a short horizon. Over longer horizons
the evidence is mixed with a 10-year horizon providing the strongest support for the theory. Burnside (2013)
explores the role of New Zealand’s risk premium as a factor in the country’s imbalances. In this context he also
examines whether UIP holds between New Zealand and various other countries, Australia being one of them.
He detects a positive yet statistically insignificant link between NZ-AUS Dollar exchange rate movements and
interest rate differentials over the 1990-2010 period and interprets this as evidence in favor of the UIP
hypothesis.
2
See also Anker (1999).
2
dominate simple ad hoc instrument rules because the former rely on superior information
than the latter. Target rules employ an implicit reaction function which permits the central
bank to respond directly to all shocks of the model while a simple instrument rule responds
only to deviations of target variables from their target levels. Evidently, there is an
informational advantage to being able to optimally react to the shocks of the model rather
than to the target variables proper. 3
Optimal policy considerations lie at the heart of the current paper. The paper derives and
embeds an endogenous target rule in a basic open-economy macro model where uncovered
interest rate parity is presumed to hold. The analytical results presented in the current paper
establish that actual changes in the nominal exchange rate respond to three principal
observable factors: the lagged interest rate differential, the foreign inflation rate, and the level
of the foreign interest rate. The key finding of the paper is related to the interaction between
nominal exchange rate movements and the lagged interest rate differential. Optimizing
behavior on the part of the domestic central bank can explain the UIP puzzle. The change in
the nominal exchange rate is negatively related to the lagged interest rate differential. Two
parameters affect the sensitivity of nominal exchange rate changes to movements in the
lagged interest rate differential. They are: the relative weight the central bank places on the
variability of inflation in its objective function and the degree of openness of the economy.
Greater emphasis on stable inflation and greater openness weaken the link between
movements in the nominal exchange rate and lagged interest rate differentials.
The testable implications of the analytical results are examined on New Zealand and
Australian data in the empirical section of the paper. The two largest economies of Oceania
are highly integrated. There are virtually no barriers to the flow of capital, goods, and labor.
The central banks of both countries share similar monetary policy objectives and do not
systematically intervene in the foreign exchange market to manipulate the bilateral exchange
rate. The empirical test examines whether observed changes of the nominal Kiwi Dollar Australian Dollar exchange rate follow the systematic pattern implied by the model. The
hypothesized relationship between bilateral nominal exchange rate changes, nominal interest
rates in New Zealand and Australia, and the Australian rate of inflation is examined over the
whole sample period and a number of sub-sample periods. Structural changes to the operating
framework of monetary policy in New Zealand and the onset of the Global Financial Crisis
warrant a close investigation of how the various rates interacted over shorter time intervals.
3
Froyen and Guender (2007) provide a textbook analysis of instrument versus target rules.
3
This paper also carries out a test of the UIP hypothesis. A restricted form of the modelbased equation is estimated, one that is consistent with the conventional test of the UIP
hypothesis. The conventional test is based on the notion that it is innocuous to replace the
forward-looking expectation of the nominal exchange rate with the realized exchange rate
(plus an error term) and maintain the assumption that the interest rate differential is the only
factor behind observed exchange rate changes. However, as emphasized above, estimating
such a narrow specification of the test equation ignores the effect of other variables on
observed exchange rate changes. As such, one could argue that the conventional test equation
of the UIP hypothesis suffers from an omitted variable bias if the model captures reality.
The empirical results suggest that over the whole sample period (1985q2 – 2012q1) the
exchange rate conforms to two key predictions of the model. First, the Kiwi Dollar
appreciated vis-à-vis the Australian Dollar if interest rates in New Zealand were higher than
in Australia; and second, a rise in the Australian rate of inflation caused the Kiwi Dollar to
rise in value against the Australian Dollar. Because of the change to the operating procedure
of monetary policy in New Zealand in March 1999, the model-based prediction of exchange
rate behavior is also examined in two sub-sample periods. During the Cash Settlement
Balances period (1985q2 – 1999q1) there is substantial evidence for the inverse relationship
between the change in the bilateral exchange rate on the one hand and the lagged interest rate
differential and the Australian rate of inflation on the other. Following the switch to the
Official Cash Rate System in March 1999, the model-based prediction of exchange rate
changes fares much worse. The hypothesized inverse relationship between the exchange rate
change and the lagged interest rate differential ceases to exist. Instead, the empirical results
suggest that in the sub-sample period prior to the outbreak of the Global Financial Crisis the
Kiwi Dollar sharply depreciated against the Australian Dollar if interest rates in New Zealand
were higher than in Australia. The positive relationship between the lagged interest rate
differential and the observed change in the nominal exchange rate does not carry over to the
sub-sample period whose beginning marks the outbreak of the Global Financial Crisis in
2007.
Further scrutiny reveals, however, that the results reported for the whole sample period
are not robust to the length of the sampling interval. The inverse relationship between
nominal exchange rate movements and the interest rate differential disappears if the
beginning of the sample period is pushed back by one year, i.e. if the four quarters
4
immediately following the float of the Kiwi Dollar are dropped from the analysis. Testing the
restrictions imposed by UIP uncovers almost no support for the theory.
The remainder of the paper is structured as follows. Section II introduces the model.
Section III determines the optimal policy setting under a target rule and derives the key
equation of the model which identifies the factors that drive changes in the nominal exchange
rate. Section IV presents the empirical findings of the test of exchange rate behavior. Section
V concludes.
II. The Model
This section lays out the model that serves as the frame of reference for examining the
UIP puzzle. The linear model consists of four equations and a policy rule which is discussed
in the next section. Variables marked by an asterisk denote the foreign counterpart of the
domestic variable. Foreign variables are treated as exogenous random variables. The first
equation is the definition of the policy instrument (xt).The policy instrument is defined as the
difference between the domestic interest rate (it ) and the foreign interest rate (𝑖𝑡∗ ) . Adopting
this convention simplifies the analysis and allows us to compare our results directly with
those reported by McCallum (1994). The second equation is the UIP condition in nominal
terms. It allows for the existence of a risk premium (𝜌𝑡 ). The two remaining equations
describe the behavior of inflation. Equation (3) is a condensed equation of the rate of
domestic inflation. It is obtained by combining three different elements: an open economy IS
equation, UIP in real terms, and a Phillips curve. 4 The rate of domestic inflation is inversely
related to the policy instrument but reacts positively to the difference between expected
domestic inflation and expected inflation abroad in period t+1. The composite stochastic
disturbance captures the effect of demand-side and cost-push shocks on domestic inflation.
Equation (4) is the definition of CPI inflation.
𝑥𝑡 = 𝑖𝑡 − 𝑖𝑡∗
(1)
𝑥𝑡 = 𝐸𝑠𝑡+1 − 𝑠𝑡 + 𝜌𝑡
(2)
∗ )
𝜋𝑡 = −𝛼𝑥𝑡 + 𝛼(𝐸𝑡 𝜋𝑡+1 −𝐸𝑡 𝜋𝑡+1
+ 𝑢𝑡
(3)
∗
𝑢𝑡 = 𝜅(𝑣𝑡 −𝑎1 (𝑖𝑡∗ − 𝐸𝑡 𝜋𝑡+1
) + (𝑎2 − 𝑎1 𝛾)𝜌𝑡 ) + 𝑤𝑡
4
𝛼 = 𝜅((1 − 𝛾)𝑎1 + 𝑎2 )
For further details on the derivation of equation (3), the reader is referred to part C of the appendix.
5
𝜋𝑡𝐶𝑃𝐼 = (1 − 𝛾)𝜋𝑡 + 𝛾(Δ𝑠𝑡 + 𝜋𝑡∗ )
(4)
0≤𝛾≤1
III. Optimal Policy Based on a Target Rule
III.1 The Central Bank’s Objective and the Specification of Monetary Policy
The aim of the central bank is to minimize the variability of the CPI inflation rate and the
policy instrument:
𝑉(𝑥𝑡 ) + 𝜇𝑉(𝜋𝑡𝐶𝑃𝐼 )
(5)
𝜇 = aversion to inflation variability relative to variability of policy instrument.
Associated with the quadratic objective function is a linear target rule. This rule embodies
a systematic relationship between the variables that appear in the objective function. The
specific form that the target rule takes depends in part on the way monetary policy is
implemented. For the case at hand, the linear target rule is a simplified version of the one that
underlies optimal policy from a timeless perspective. 5 Employing this particular target rule
has two important advantages. First, it permits the derivation of closed form solutions of the
endogenous variables of the model and, second, it identifies the key parameters in the UIP
puzzle.
The specification of the target rule takes the following form:
𝜃1 𝑥𝑡 + 𝜃2 𝑥𝑡−1 + 𝜋𝑡𝐶𝑃𝐼 = 0
(6)
𝜃1 and 𝜃2 are relative weights that the central bank treats as policy parameters. They
represent the importance that the central bank attaches in the target rule to the policy
instrument in the current and previous period, respectively, compared to the current rate of
CPI inflation. Being endogenous and of “higher order” in the sense that it is only one step
below the objective function, the target rule locks 𝑥𝑡 , 𝑥𝑡−1 and 𝜋𝑡𝐶𝑃𝐼 into a systematic
relationship which guarantees that the central bank implements monetary policy optimally. 6
The target rule is enforced by the implicit reaction function which illustrates the optimal
5
Woodford (2003) and Froyen and Guender (2007) describe policy from a timeless perspective.
Part A of the appendix shows how the target rule is determined from an intertemporal perspective.
6
6
response of the policy instrument to the shocks of the model and current as well as past
feedback variables:
1
∗
𝑥𝑡 = (1−𝛾)𝛼 ( 𝜃2 𝑥𝑡−1 + (1 − 𝛾)(𝛼(𝐸𝑡 𝜋𝑡+1 − 𝐸𝑡 𝜋𝑡+1
) + 𝑢𝑡 ) + 𝛾(∆𝑠𝑡 + 𝜋𝑡∗ ))
(7)
where ut = κ vt − a1(it* − Et π t*+1 + (a2 − a1γ )ρt ) + wt
A noteworthy feature of the endogenous target rule represented by equation (6) is that it
looks deceptively similar to a simple ad hoc instrument rule of the sort employed by
McCallum (1994). 7 Were the central bank to employ a simple instrument rule, then the
policy instrument would respond to its own lagged value and to deviations of the rate of CPI
inflation from target (presumed to be zero):
𝑥𝑡 = 𝜆2 𝑥𝑡−1 + 𝜆1 𝜋𝑡𝐶𝑃𝐼
(8)
The task of the central bank is to choose the optimal speed with which to respond to a
deviation of the CPI inflation rate from target and the optimal degree of interest rate
smoothing. It is important to note here that the simple instrument rule does not explain the
interest rate smoothing behavior on the part of the central bank. Interest rate smoothing is
arbitrarily assumed.
Despite the apparent similarity, there are two fundamental differences between the
endogenous target rule (eqn. (6)) and the instrument rule (eqn. (8)). The two policy rules
differ concerning the interpretation of the policy parameters and the information set upon
which the two rules are conditioned. First, notice that it is conceivable that the current setting
of the policy instrument does not appear in the target rule (eqn. (6)). This happens when
𝜃1 = 0. In sharp contrast, 𝑥𝑡 always appears in the simple instrument rule (eqn. (8)) as it
embodies the specification of monetary policy: it conveys the central bank’s reaction to
deviations of CPI inflation from target and its intention to smooth the interest rate. Second, it
is evident that the monetary policy response under the target rule approach is conditioned on
superior information. According to equation (7), the setting of the policy instrument adjusts
to provide the optimal response to the composite shock 𝑢𝑡 . The ad hoc simple instrument rule
in equation (8) cannot do so as it responds only to observed deviations of CPI inflation from
target. The target rule approach is thus built on the assumption that the central bank is in a
7
In McCallum’s specification of the simple instrument rule, the policy instrument responds to the change in
the nominal exchange rate.
7
position to observe and to respond to the shocks of the model, a clear informational
advantage over the simple instrument rule.
III.2 Model Solution and Optimal Policy
To solve the model for the endogenous variables, substitute first the definition of the CPI
inflation rate, equation (4), into the target rule. Next, eliminate the current-period rate of
domestic inflation by substituting equation (3) into the target rule. The policy instrument can
be eliminated by the amended UIP condition where the expected change in the exchange rate
next period has been disposed of with the help of the putative solution for the exchange rate.
Finally, solve for the change in the nominal exchange rate. Using the solution for the change
in the nominal exchange rate, we can solve for the remaining endogenous variables of the
model. The solutions appear in equations (9) – (12).
𝛾
𝑥𝑡 = 𝛾+𝜃 𝜌𝑡
(9)
2
𝜋𝑡 = − �
𝜅��(1−𝛾)𝑎1 +𝑎2 �𝛾−(𝑎2 −𝑎1 𝛾)(𝛾+𝜃2 )�
𝛾+𝜃2
𝜃
� 𝜌𝑡 + 𝜅(𝑣𝑡 − 𝑎1 𝑖𝑡∗ ) + 𝑤𝑡
1
𝜋𝑡𝐶𝑃𝐼 = − 𝜃2 𝑥𝑡−1 − 𝛾+𝜃
𝜌𝑡
Δ𝑠𝑡 = −
𝜃2
𝑥
𝛾 𝑡−1
(11)
2
𝜃1 +(1−𝛾)(−𝛼)
𝛾
�
𝛾
𝛾+𝜃2
− [�
(10)
+
(1−𝛾)
𝛾
𝜅(𝑎2 − 𝑎1 𝛾)]𝜌𝑡 −
(1−𝛾)
𝛾
(𝜅(𝑣𝑡 − 𝑎1 𝑖𝑡∗ ) + 𝑤𝑡 ) − 𝜋𝑡∗
(12)
The variances of the CPI inflation rate and the policy instrument are then substituted into
the objective function. The policymaker’s objective is to minimize the expected loss function
by choosing the optimal values of 𝜃1 and 𝜃2 :
min 𝐸[𝐿𝑡 ] = 𝑉(𝑥𝑡 ) + 𝜇𝑉(𝜋𝑡𝐶𝑃𝐼 )
𝜃1 ,𝜃2
(13)
Minimizing expected losses with respect to the two policy parameters yields their
respective optimal value:
𝜃1∗ = 0
1
𝜃2∗ =𝜇𝛾
(14)
8
Substituting the optimal policy coefficients into equation (12) yields the solution for the
change in the nominal exchange rate:
Δ𝑠𝑡 = −
1
𝜇𝛾2
𝑥𝑡−1 − 𝜋∗𝑡 +
(1−𝛾)
𝛾
(𝜅𝑣𝑡 + 𝑤𝑡 )
(1−𝛾)𝜅 𝑎1 ∗
𝛾
(1−𝛾)(−𝛼)
𝑖𝑡 − ��
𝛾
𝜇𝛾2
� 𝜇𝛾2 +1 +
(1−𝛾)
𝛾
𝜅(𝑎2 − 𝑎1 𝛾)� 𝜌𝑡 +
(12’)
The target rule approach offers a solution to the UIP puzzle: the coefficient on 𝑥𝑡−1in
1
equation (12’) equals − 𝜇𝛾2 . The above result is thus consistent with the widely reported
empirical observation that high-interest rate countries see their currencies rise in value. Two
deep parameters determine the strength of the relation between an observed movement in the
nominal exchange rate and the lag of the policy instrument: the central bank’s relative
aversion to CPI inflation variability and the degree of openness of the domestic economy.
The greater the aversion to inflation variability, the weaker the negative association between
the lagged interest rate differential and the change in the nominal exchange rate. Indeed, in
countries where strict CPI inflation targeting is the norm (𝜇 → ∞), there should be no
evidence for a robust link between the two variables. 8 Greater openness also weakens the
relation between exchange rate movements and lagged interest rate differentials. Greater
openness means that a smaller change in the current setting of the policy instrument is needed
to control expected CPI inflation through an expected change in the nominal exchange rate.
Apart from the lagged policy instrument, two other observable variables affect the change
in the nominal exchange rate. According to equation (12’), the foreign rate of inflation and
the current level of the foreign interest rate affect the nominal exchange rate change. In fact, a
one percent increase of the foreign inflation rate should lead to a decrease of the nominal
exchange rate by the same magnitude. An increase in the foreign interest rate should cause
the domestic currency to depreciate.
IV. An Empirical Test of Model-Based Exchange Rate Behavior
For an empirical examination of exchange rate behavior we turn our attention to Australia
and New Zealand, two countries whose economies are closely integrated. The close links
between the two countries extend across all sectors of the economy. Notably,
8
Under strict inflation targeting (𝜇 → ∞), the coefficient on 𝜌𝑡 in eqn. (9) approaches unity. This suggests that
the policy instrument completely offsets the effect of the stochastic risk premium. With 𝑥𝑡 = 𝜌𝑡 this implies
further from eqn. (2) that 𝐸𝑡 ∆𝑠𝑡+1 must be zero for UIP to hold. This is indeed the case. Update eqn. (12’) by
one period, take conditional expectations dated t and let 𝜇 → ∞. 𝐸𝑡 ∆𝑠𝑡+1 = 0 results. The UIP condition is also
satisfied for 0 ≤ 𝜇 < ∞ as in this case 𝑥𝑡 and 𝐸𝑡 ∆𝑠𝑡+1 share the burden of adjusting to the stochastic risk
premium.
9
•
There are no barriers to the flow of capital between the two countries.
•
Since the passage of the Closer Economic Relations Act of 1983 the two countries
have maintained free trade.
•
Labor mobility between the two countries is extremely high.
•
New Zealand’s banking sector is dominated by Australian-owned banks. Therefore
the default risk on bank bills is virtually the same in both countries.
•
The monetary policy objectives of the Reserve Bank of Australia and the Reserve
Bank of New Zealand are the same: to control inflation over the medium term.
•
Movements of the business cycle are fairly synchronous in the two countries.
•
Business practices and legal arrangements are virtually identical.
•
Currency risk between the two countries is negligible. Australia and New Zealand
share a common risk exposure to extreme events abroad. 9
Summary Measures of Nominal Interest Rates, Inflation Rates, and the Exchange Rate
Table 1 reports summary information about inflation and nominal short-term interest rates
in New Zealand and Australia over the whole sample period (1985q2 – 2012q1) and several
sub-periods. 10 The last column of the table also reports summary information about the
bilateral exchange rate. The reason for carving up the whole sample period into sub-periods is
due to the occurrence of two identifiable exogenous events. 11 First, the Reserve Bank of New
Zealand changed its operating procedure in March 1999. To account for the switch from a
quantity-based to a price (interest rate)-based operating procedure, which led to a dramatic
reduction in nominal interest rate volatility, the whole sample period is divided up into two
sub-periods. In the first sub-period the implementation of monetary policy rested on the Cash
Settlement Balances framework. This operating procedure gave way to the Official Cash Rate
(OCR) system on March 17th, 1999. Despite a few alterations to the original set-up, the OCR
system has remained the operating procedure of the Reserve Bank of New Zealand to this
day. Second, the 1999q2-2012q1 period is split further into two sub-periods. By many
accounts the Global Financial Crisis began in the third quarter of 2007. 12 The period 1999q29
See Burnside (2013) on this.
By March 1985 both the New Zealand and Australian Dollar were floating currencies. As a result, we chose
the second quarter of 1985 as the beginning of the sample period.
11
This obviates the need for relying on stability tests such as the Quandt-Andrews breakpoint test. One might
think that the passage of the Reserve Bank Act of 1989 may have had an effect on New Zealand financial
market variables in early 1990. But this did not turn out to be the case. (See Figure 1 and 2.)
12
th
Bullard (2010), Mishkin (2011) and others date the beginning of the crisis to August 7 2007 when BNP
Paribas closed one of its money market funds.
10
10
2007q2 is designated as the relatively calm period prior to the outbreak of the Global
Financial Crisis. With the onset of the Global Financial Crisis in the third quarter of 2007 and
the subsequent Sovereign Debt Crisis, the period 2007q3 – 2012q1 marks a period of
heightened uncertainty and fragility in world financial markets. 13
In each cell, the top number represents the standard deviation and the bottom number
represents the mean (in italics) of the relevant variable. What is remarkable is that the
volatility of bilateral exchange rate changes is relatively stable across the sub-periods while
the volatility of the nominal interest rate and CPI inflation in New Zealand fell sharply after
the switch of the operating procedure in March 1999. The standard deviation of the NZ 90day bank bill rate fell from 5.94 % to 1.93 % and the standard deviation of CPI inflation
decreased from 5.25 % to 1.03 %. Mean inflation decreased by more than half, from 5.54% to
2.63 %. By comparison, volatility of interest and inflation rates also decreased substantially
in Australia, though the magnitude of the respective decrease was less. Interestingly, in the
wake of the Global Financial Crisis the standard deviation of inflation rose sharply in New
Zealand while it fell in Australia. The increase in the volatility of the 90-day bank bill rate
was also slightly more pronounced in New Zealand than in Australia over the 2007q3 –
2012q1 sub-period relative to the pre-crisis period.
Figure 1 attests to the relative stability of nominal exchange rate changes since the early
1990s. According to Figure 2, the interest rate differential exhibited far greater volatility at
the beginning of the sample period and the sub-period prior to the introduction of the Official
Cash Rate system in March 1999 (marked by vertical spike).
Econometric Analysis
The starting point of the econometric analysis of the Kiwi Dollar-Australian Dollar
exchange rate is equation (12’). To examine the fit of the model-based exchange rate
equation, we propose the following regression equation (updated by one period):
𝐴𝑈
𝐴𝑈
𝑠𝑡+1 − 𝑠𝑡 = α + 𝛽𝑖 (𝑖𝑡𝑁𝑍 − 𝑖𝑡𝐴𝑈 ) + 𝛽𝜋𝐴𝑈 𝜋𝑡+1
+ 𝛽𝑖 𝐴𝑈 𝑖𝑡+1
+ 𝜖𝑡+1
13
(12”)
The effect of the Global Financial Crisis on Oceania was limited. Australia avoided entering a recession
altogether while New Zealand experienced a mild and rather short-lived recession. Still, the crisis rattled
financial markets in Oceania to such an extent that the two central banks took emergency measures to
safeguard the banking system.
11
Three identifiable variables appear on the right-hand side of the regression equation. They
are: the lagged interest rate differential (difference between NZ and Australian 90-day bank
bill rates in period t), the Australian rate of inflation, and the Australian 90-day bank bill rate.
Ordinary Least Squares is used to estimate the regression equation over the whole sample
period (1985q2 – 2012q1) and several sub-periods. 14 The coefficient estimates of the above
regression equation along with standard regression diagnostics appear in Table 2. Several
observations are in order. First, over the whole sample period, the empirical results confirm
the model-predicted negative association of exchange rate changes with lagged interest rate
differentials: the point estimate of the coefficient on the lagged interest rate differential is
negative and statistically significant at the 5 percent level. The coefficient estimate on the
Australian interest rate is also negative as predicted by the model and statistically significant
but the point estimate of -3.898 is (in absolute terms) too large to be consistent with a value
of -1. The level of the Australian interest rate has no bearing on the bilateral exchange rate.
At 0.06 the adjusted R2 is rather low, suggesting that the right-hand side variables can explain
only a small fraction of the movements in the exchange rate.
Second, a simple robustness check tough raises doubts about the apparent link between
exchange rate changes and lagged interest rate differentials. Consider the results reported in
the shaded second row of Table 2. If the sample size is shortened by one year and the
regression is re-estimated over the 1986q2 – 2012q1 period, the coefficient estimate on the
lagged interest rate differential is statistically insignificant and the adjusted R2 drops to 0.01.
It is thus evident that events in 1985 are the driving force behind the statistical significance of
𝛽𝑖 reported for the whole sample period. 15 At the same time, the negative impact of the
Australian inflation rate on the nominal exchange rate change also holds over the shortened
sample interval.
Third, there is no concrete empirical evidence in favor of the UIP proposition over the
whole sample period. Equation (12”) nests the standard test specification of the UIP and
14
The use of OLS can be justified on the grounds that there are no feedback effects from changes in the Kiwi
Dollar-Australian Dollar exchange rate to the rate of inflation and the interest rate in Australia.
15
In 1985 the drive towards restructuring the New Zealand economy was well under way. Against this
backdrop there was substantial uncertainty in financial markets regarding the government’s borrowing needs
and wage bargaining. Over a span of six months (March through September), the yield curve in New Zealand
shifted up dramatically, especially at the short end, due to heightened concern about rising inflationary
expectations. At the same time the Kiwi Dollar appreciated markedly. In late 1985 and early 1986 the Reserve
Bank of New Zealand introduced new liquidity management procedures to combat the extreme volatility of
short-term interest rates, a measure which proved to be successful. Short-term market interest rates became
somewhat less volatile and began to ease as a consequence.
12
rational expectations hypothesis. The joint hypothesis that 𝛽𝑖 = 1, 𝛽𝜋𝐴𝑈 = 0, 𝛽𝑖 𝐴𝑈 = 0 is
rejected emphatically on the basis of an F-test. 16 The value of the F-test-statistic is 15.27 with
a corresponding p-value of 0.01.
Splitting up the whole sample period into two sub-periods provides an interesting
contrast. The empirical results generated for the first sub-period (1985q2 – 1999q1) during
which the Cash Settlement Balances system was in operation mimic those obtained for the
whole sample period. The estimated coefficients on the lagged interest rate differential and
the Australian rate of inflation remain negative and statistically significant. The only notable
difference between the first sub-period and the whole sample period is the substantially better
fit of the estimated regression equation over the first sub-period. The adjusted R2 rises to 0.14
in the first sub-period. Inspection of the fourth (shaded) row reveals, however, that the caveat
for the whole sample period also applies to the first-subsample period: once the first four
observations are eliminated from the sample period, the predictive power of the regression
equation drops markedly; worse, none of the right-hand side variables is significantly
correlated with the change in the nominal exchange rate. Once again the restrictions imposed
by the UIP condition can be safely rejected for the first sub-period.
A vastly poorer fit of the regression is obtained for the second sub-period. Importantly,
the coefficient on the lagged interest rate differential is positive and close enough to unity so
that the UIP hypothesis cannot be rejected in the OCR period. The value of the F-statistic
drops to 0.35 with a corresponding p-value of 0.78. However, on the whole, the regression
results for this sub-period are not very informative. There is no discernible systematic
relationship between the right-hand side variables and the change in the nominal exchange
rate, a result implying that the level of the exchange rate exhibited near random walk
behavior during the 1999q2 – 2012q1 sub-period.
Further examination of the 1999q2 – 2012q1 sub-period reveals a robust positive
connection between interest rate movements and subsequent changes in the nominal
exchange rate in the period prior to the outbreak of the Great Financial Crisis (1999q2 –
2007q2). At a rate of almost 7 to 1, the Kiwi Dollar depreciated excessively relative to the
Australian Dollar in the wake of a positive interest rate differential. This finding is clearly at
variance with the model-based prediction of exchange rate behavior. Notice though that the
New Zealand currency appreciated sharply in response to increasing inflation in Australia, an
16
The constant term is ignored.
13
observation that conforms to the prediction of the model. 17 The empirical results for the subperiod are not consistent with the prediction of the UIP proposition. The joint null hypothesis
for the three coefficients can be safely rejected at the 1 percent level. While low at an
adjusted R2 of 0.05, the predictive content of the estimated regression during this sub-period
is far better than for the subsequent 2007q3 – 2012q1sub-period. Even though the joint
hypothesis about the coefficients underlying the UIP proposition cannot be rejected the
estimated regression equation for the most recent sub-period is utterly uninformative and
indicative of the near-random walk behavior of the nominal exchange rate mentioned
above. 18
V. Conclusion
The standard test of the joint hypothesis of UIP and rational expectations consists of a
regression of the change in the nominal exchange rate on the lagged interest rate differential.
Employing this test regression, scores of empirical papers reject the UIP hypothesis for a
large number of countries over different sample periods because the estimated coefficient on
the interest rate differential turns out to be statistically insignificant or statistically significant
but negative.
Using an open economy setting where the central bank attempts to minimize fluctuations
in the CPI inflation rate and the policy instrument, this paper offers an explanation for why
high interest rate countries see their currencies appreciate. This observation is generally
interpreted as evidence against uncovered interest rate parity. This interpretation is, however,
unwarranted as the UIP condition, which is based on the expected change in the exchange
rate, is satisfied in the modeling framework. When applied to data, the standard test of the
UIP hypothesis neglects to take account of the effect of monetary policy and foreign factors
on the observed change in the nominal exchange rate.
The optimizing approach provides a rationale for why the lag of the policy instrument –
the interest rate differential - appears in the endogenous target rule. When the target rule is
17
If the level of the Australian inflation rate and the level of the Australian 90-day bank bill rate appear in the
regression for the 1999q2 – 2007q2 sub-period instead of first differences, then the fit of the regression is
2
even better as the adjusted R increases to 0.13.By the late 1990s inflation had been far less erratic than in the
mid and late 1980s. As such inflation and by implication nominal interest rates may actually behave more like
mean-reverting stationary processes during the sub-period in question. For details on the time series
properties of the regression variables the reader is referred to the notes attached to Table 2.
18
In a related context, Burnside (2013) reports that tests of the UIP hypothesis for the New Zealand Dollar
against the Australian Dollar, Japanese Yen, and the US Dollar uncover more support for the theory if the
Global Financial Crisis (2008-2010 period) is included in the sample. His study looks at the 1990-2010 period.
14
combined with the building blocks of the open economy model – one of which is the
assumption of uncovered interest rate parity plus the existence of an exogenous risk premium
– the current change in the nominal exchange rate responds
•
negatively to an increase in the lag of the interest rate differential
•
negatively to a rise in the foreign rate of inflation and
•
positively to a rise in the level of the foreign interest rate.
The unobservable risk premium plus the shocks of the model appear as an additional factor
that impact on the exchange rate.
New Zealand and Australia, two closely linked countries, provide the backdrop for an
empirical test of exchange rate behavior.
Over a sample period spanning 25 years the model-based prediction of exchange rate
behavior fits the observed pattern of exchange rate movements fairly well but only if the
sample period includes 1985. Over shorter intervals, there is scant evidence in favor of the
model-based prediction of the exchange rate. Only the Australian rate of inflation bears an
inverse relationship to the change in the nominal exchange rate as predicted by the model in
the sub-period prior to the introduction of the Official Cash Rate system in 1999 and in the
sub-period prior to the onset of the Global Financial Crisis. A nested test of the UIP
hypothesis over the whole period produces no support for the theory. At the same time, there
is one episode – the Global Financial Crisis and Sovereign Debt Crisis – during which
momentum for a one-for one positive relationship between interest rate differentials and
nominal exchange rate movements begins to build. Only time will tell whether this linkage
will evolve into a statistically significant relationship.
One clear result emerges. The connection between interest rate movements and exchange
rate movements remains an elusive one, even in countries such as New Zealand and Australia
where institutional and legal frameworks are similar and capital moves freely.
15
References:
Anker, Peter (1999), “Uncovered Interest Rate Parity, Monetary Policy, and Time-Varying
Risk Premia,” Journal of International Money and Finance, 18, pp. 835 -851.
Backus, David K, Federico Gavazzoni, Chris Telmer, and Standley E. Zin, “Monetary Policy
and the Uncovered Interest Rate Parity Puzzle.” Unpublished paper (2010).
Bullard, James (2010), “Three Lessons for Monetary Policy from the Panic of 2008,” Federal
Reserve of St. Louis Review, 92 (3), 155-164.
Burnside, Craig, Martin Eichenbaum, Isaac Kleshchelski, and Sergio Rebelo (2011), “Do
Peso Problems Explain the Returns to the Carry Trade?” Review of Financial Studies, 24(3),
853-891.
Burnside, Craig (2013), “New Zealand’s Risk Premium,” New Zealand Economic Papers, 47
(1), pp. 27-52.
Chinn, Menzie and Guy Meredith (2004), “Monetary Policy and Long Horizon Uncovered
Interest Rate Parity,” IMF Staff Papers, 51 (3), pp. 409-430.
Engel, Charles (1996), “The Forward Discount Anomaly and the Risk Premium: A Survey of
Recent Evidence,” Journal of Empirical Finance, 3, pp. 123-192.
Fama, Eugene (1984), "Forward and Spot Exchange Rates." Journal of Monetary Economics,
14(3), pp. 319-38.
Froot, Kenneth and Richard Thaler (1990), “Anomalies: Foreign Exchange,” Journal of
Economic Perspectives, 4 (3), pp. 179-192.
Froyen, Richard T. and Alfred V. Guender, Optimal Monetary Policy under Uncertainty,
Edward Elgar, Cheltenham, UK, 2007.
Froyen, Richard T. and Alfred V. Guender (2012), “Instrument versus Target Rules as
Specifications of Optimal Monetary Policy,” International Finance, 15 (1), pp. 99-123.
Guender, Alfred V. and Bevan Cook (2011), “Monetary Policy Implementation and
Uncovered Interest Rate Parity: Empirical Evidence from Oceania,” New Zealand Economic
Papers, 45 (3), pp. 209-229.
Lewis, Karen K (1995), “Puzzles in International Financial Markets,” in G.M. Grossman and
K. Rogoff (eds.), Handbook of International Economics, vol. III, Amsterdam: North-Holland,
pp. 1913-1971.
McCallum, Bennett T. (1994), “A Reconsideration of the Uncovered Interest Parity
Relationship,” Journal of Monetary Economics, 33, pp. 105-132.
Mishkin, Frederic S. (2011), “Over the Cliff: from the Subprime to the Global Financial
Crisis, “ Journal of Economic Perspectives, 25 (1), pp. 49-70.
Svensson, Lars E. O. (2003), “What Is Wrong with Taylor Rules?” Journal of Economic
Literature XLI, (2), pp. 426-477.
Woodford, Michael, Interest and Prices, Princeton University Press, Princeton, 2003.
16
𝜋 𝑁𝑍
𝑖 𝑁𝑍
𝜋 𝐴𝑈
𝑖 𝐴𝑈
Standard Deviation
∆𝑠1
Mean
1985q12012q1
4.10
2.52
5.47
4.29
0.0365
4.17
3.77
8.95
7.88
-0.0018
5.25
3.17
5.94
4.81
0.0389
5.54
4.42
11.93
10.18
-0.0039
1.03
1.17
1.93
1.03
0.0340
2.63
3.00
5.65
5.34
0.0012
0.77
1.23
0.98
0.63
0.0323
2.48
3.09
6.33
5.45
-0.0014
1.30
1.01
2.61
1.51
0.0372
2.97
2.96
4.51
5.17
0.0038
(Whole Per.)
1985q11999q1
(CSB Per.)
1999q22012q1
(OCR Per.)
1999q22007q2
(Pre-GFC Per.)
2007q32012q1
(GFC Per. +)
Table 1: Summary Measures (%)
Note: The respective annualized CPI inflation rate and 90-day bank bill data were obtained
from the database maintained by the Reserve Banks of Australia and New Zealand. The
nominal exchange rate was retrieved from the RBNZ database.
CSB Per. = Period during which Cash Settlement Balances Regime was in Operation2
OCR Per. = Official Cash Rate Period (current operating procedure)
Pre-GFC Per. = Period before Outbreak of Global Financial Crisis
GFC Per. + = Period since Outbreak of Global Financial Crisis
1
First difference of the log of the bilateral exchange rate (units of NZ Dollars per Australian Dollar)
Strictly speaking, the CSB system was introduced only in 1986. 1985 was chosen as the beginning of the
sample period because by then both currencies were floating.
2
17
Figure 1
DS
Nominal Exchange Rate Changes(Log Differences)
.15
.10
.05
.00
-.05
-.10
-.15
86
88
90
92
94
98
96
00
02
04
06
08
10
04
06
08
10
Figure 2
DIFI
Interest Rate Differential
.03
.02
.01
.00
-.01
-.02
Notes:
86
88
90
92
94
96
98
00
02
1. The Reserve Bank Act of 1989, which enshrines in law the independence of the Reserve Bank of
New Zealand, took effect on February 1, 1990. There are no indications that its passage had a
material effect on the behavior of nominal exchange rate changes or the interest rate differential.
18
Table 2: An Empirical Assessment of the NZD -AUD Exchange Rate.
Sample Period
1985q2 – 2012q1
1986q2 – 2012q1
1985q2 - 1999q1
1986q2 – 1999q1
1999q2 – 2012q1
1999q2 – 2007q2
2007q3 – 2012q1
α
𝛽𝑖
𝛽𝜋𝐴𝑈
𝛽𝑖𝐴𝑈
F-Statistic (J,N-K) for UIP Test
𝐻0 : 𝛽𝑖 = 1, 𝛽𝜋𝐴𝑈 = 0, 𝛽𝑖 𝐴𝑈 = 0
0.0006
-0.7549**
-3.898***
2.308
15.27***
(0.0023)
(0.3580)
(1.100)
(1.532)
(3, 104)
0.0004
-0.2442
-2.475*
1.788
4.02***
(0.0023)
(0.3786)
(1.344)
(2.151)
(3, 100)
0.0006
-0.9015*
-5.794**
2.847
7.67***
(0.0042)
(0.5401)
(2.662)
(2.085)
(3,52)
0.0007
-0.4996
-3.437
2.282
3.32**
(0.0049)
(0.6916)
(2.370)
(2.715)
(3, 48)
-0.0004
1.220
-1.229
-0.944
0.35
(0.0039)
(1.222)
(1.653)
(2.796)
(3,48)
-0.0163*
6.926***
-1.915***
0.6359
6.79***
(0.0087)
(1.802)
(0.6393)
(3.394)
(3,29)
0.0049
0.9579
2.732
-2.285
0.10
(0.0035)
(1.141)
(5.794)
(5.187)
(3,15)
Adj 𝑅 2
DW
N
0.06
1.94
108
0.00
1.99
104
0.14
1.69
56
0.01
1.67
52
-0.04
2.33
52
0.05
2.03
33
-0.16
2.44
19
Notes:
1.
2.
3.
Newey-West standard errors in parentheses. * (**) [***] indicates significance at the 10 (5) [1] percent level. In each cell, the asterisk denotes rejection at the
respective significance level of the hypothesis that the estimated coefficient equals zero.
Augmented Dickey-Fuller (ADF) tests for non-stationarity were carried out prior to estimation. Over the whole sample period the change in the bilateral
exchange rate and the nominal interest rate differential were found to be stationary. The null hypothesis that either series contained a unit root could be
rejected at the 1 percent level. In sharp contrast, ADF tests for the Australian inflation rate and the Australian 90-day bank bill rate could not reject the nullhypothesis of a unit root in the data. Both series were first differenced prior to estimating the regression to create a balanced regression.
The equation estimated is:
𝐴𝑈
𝐴𝑈
𝑠𝑡+1 − 𝑠𝑡 = α + 𝛽𝑖 (𝑖𝑡𝑁𝑍 − 𝑖𝑡𝐴𝑈 ) + 𝛽𝜋𝑎𝑢 𝜋𝑡+1
+ 𝛽𝑖 𝐴𝑈 𝑖𝑡+1
+ 𝜖𝑡+1
𝑠𝑡+1 − 𝑠𝑡 = change in the quarterly exchange rate of the New Zealand Dollar per Australian Dollar in period t+1
𝑖𝑡𝑁𝑍 − 𝑖𝑡𝐴𝑈 = difference in the quarterly yield of the 90-bank bill rate in New Zealand relative to Australia in period t
𝐴𝑈
= first difference of quarterly CPI inflation rate in Australia in period t+1
𝜋𝑡+1
𝐴𝑈
= first difference of quarterly 90-day bank bill rate in Australia in period t+1
𝑖𝑡+1
The quarterly yield is computed as 𝑖 𝑗 = log(1 + 𝑥 𝑗 /4) where x represents the annual yield in decimal form (j=NZ, AU).
Ordinary Least Squares is used to estimate the coefficients. The Australian rate of inflation and interest rate are considered to be exogenous regressors in the
following sense. Both variables are viewed as affecting the Kiwi-Australian Dollar spot exchange rate. But the bilateral exchange rate has no bearing on the
Australian rate of inflation and the Australian short-term interest rate because of the small size of New Zealand relative to Australia.
20
Appendix: A. The Intertemporal Optimization Problem
The policymaker seeks to minimize an intertemporal loss function that consists of squared
deviations of the CPI inflation rate and the policy instrument. After manipulating equation (3), we can
express the constraint in terms of the CPI inflation rate and the policy instrument. 19 The policy
problem that the central bank faces can then be stated as:
min
𝐶𝑃𝐼
𝜋𝑡
𝑥𝑡
∞
2
𝐸0 � 𝛽 𝑡 (𝑥𝑡2 + 𝜇𝜋𝑡𝐶𝑃𝐼 )
𝑡=0
(15)
subject to
𝐶𝑃𝐼
∗
𝜋𝑡𝐶𝑃𝐼 = 𝛼𝐸𝑡 𝜋𝑡+1
− 𝛼𝑥𝑡 − 𝛼𝐸𝑡 𝜋𝑡+1
+ 𝛾(∆𝑠𝑡 + 𝜋𝑡∗ ) + 𝛼𝛾𝜌𝑡 + (1 − 𝛾)𝑢𝑡
(16)
The policymaker implements monetary policy from a timeless perspective, a form of optimal policy
under commitment.
The Lagrangean takes the following form:
2
ℒ = 𝐸0 {𝑥02 + 𝜇𝜋0𝐶𝑃𝐼 + 𝛿0 �𝛾(𝜋0∗ + Δ𝑠0 ) + 𝛼𝜋1𝐶𝑃𝐼 − 𝛼(𝑥0 + 𝜋1∗ ) + 𝛼𝛾𝜌0 + (1 − 𝛾)𝑢0 − 𝜋0𝐶𝑃𝐼 �
2
+𝛽[𝑥12 + 𝜇𝜋1𝐶𝑃𝐼 + 𝛿1 �𝛾(𝜋1∗ + 𝑥0 − 𝜌0 ) + 𝛼𝜋2𝐶𝑃𝐼 − 𝛼(𝑥1 + 𝜋2∗ ) + 𝛼𝛾𝜌1 + (1 − 𝛾)𝑢1 − 𝜋1𝐶𝑃𝐼 �]
+⋯}
(17)
Policy from a timeless perspective imposes special conditions on the initial period. The constraint
in period -1 is not binding for policy choice in period 0. In addition, there is no effect of 𝑥−1 on ∆𝑠0 .
Suppose the policymakers sets policy for period t. The optimizing condition for the rate of CPI
inflation in period t is given by
2𝜇𝜋𝑡𝐶𝑃𝐼 − 𝛿𝑡 + 𝛿𝑡−1 𝛼 = 0
for t=0,1,2…
(18)
and 𝛿−1=0 for the initial period.
The optimizing condition for the policy instrument for period t is:
2𝑥𝑡 − 𝛿𝑡 𝛼 + 𝛽𝛿𝑡+1 𝛾 = 0 for t=0,1,2…
(19)
𝛿𝑡 denotes the Lagrange multiplier in period t.
Combining the two optimizing conditions yields the target rule:
∞
𝜇�𝛼
𝑗=0
𝑗
𝐶𝑃𝐼
𝜋𝑡−𝑗
∞
= −�
𝑗=0
𝛼𝑗
𝑥
(𝛽𝛾)𝑗+1 𝑡−𝑗−1
(20)
A parsimonious representation of the above target rule is obtained by restricting the number of
terms of the sequence to one on both sides of eqn. (20) and setting 𝛽 = 1:
19
The derivation of the constraint appears in part B of the appendix.
𝜋𝑡𝐶𝑃𝐼 = −
1
𝑥
𝜇𝛾 𝑡−1
(21)
Eqn. (21) is the same target rule that appears in Section III which discusses the implementation of
monetary policy via the target rule approach from a more heuristic angle. Eqn. (13) results if, first,
one multiplies the intertemporal loss function by (1-𝛽) and, second, takes the limit as 𝛽 → 1.
A numerical optimization procedure (DYNARE) can be used to determine the variances of the
endogenous variables and the policy instrument under the globally optimal rule, i.e. under policy from
a timeless perspective. For 𝑎1 = 0.5, 𝑎2 = 0.25, 𝛾 = 0.4, 𝜅 = 0.1, 𝜇 = 1, and unit variances for all
exogenous disturbances, the variances of the relevant variables are shown in Table A1.
Table A1: Results Based on Policy from a Timeless Perspective
V(x)
0.0191
V(𝜋 𝐶𝑃𝐼 )
0.1187
V(Δ𝑠)
4.0216
E(L)=V(x)+𝜇𝑉(𝜋 𝐶𝑃𝐼 )
0.1378
V(𝛿1 )
0.0763
V(𝛿2 )
0.4770
Table A2 shows the variances of the policy instrument, the CPI inflation rate and the change in the
nominal exchange rate under commitment based on the analytical results reported in Section III of the
paper. It is apparent that these variances are extremely close to those reported in Table A1.
Table A2: Results Based on Simplified Target Rule
V(x)
0.0190
V(𝜋 𝐶𝑃𝐼 )
0.1189
V(Δ𝑠)
4.0213
E(L)=V(x)+𝜇𝑉(𝜋 𝐶𝑃𝐼 )
0.1379
B. Derivation of Budget Constraint
Multiply the domestic inflation equation by (1 − 𝛾):
∗ )
(1 − 𝛾)𝜋𝑡 = (1 − 𝛾)(−𝛼𝑥𝑡 + 𝛼(𝐸𝑡 𝜋𝑡+1 −𝐸𝑡 𝜋𝑡+1
+ 𝑢𝑡 )
(22)
Add and subtract 𝛾(∆𝑠𝑡 + 𝜋𝑡∗ )on the left-hand side of above equation and restate resulting expression
as:
∗ )
∗ )
𝜋𝑡𝐶𝑃𝐼 =𝛾(∆𝑠𝑡 + 𝜋𝑡∗ ) + 𝛼(1 − 𝛾)𝐸𝑡 𝜋𝑡+1 + 𝛼𝛾(𝐸𝑡 ∆𝑠𝑡+1 + 𝐸𝑡 𝜋𝑡+1
− 𝛼(𝐸𝑡 ∆𝑠𝑡+1 + 𝐸𝑡 𝜋𝑡+1
+
(1 − 𝛾)(𝑢𝑡 − 𝛼𝜌𝑡 )
(23)
Making use of the definition of the CPI inflation rate and the UIP condition on the right-hand side
allows us to express eqn. (23) as the policymaker’s constraint:
𝐶𝑃𝐼
∗
𝜋𝑡𝐶𝑃𝐼 = 𝛼𝐸𝑡 𝜋𝑡+1
− 𝛼𝑥𝑡 − 𝛼𝐸𝑡 𝜋𝑡+1
+ 𝛾(∆𝑠𝑡 + 𝜋𝑡∗ ) + 𝛼𝛾𝜌𝑡 + (1 − 𝛾)𝑢𝑡
(24)
C. Derivation of Domestic Inflation Equation
We begin with a simple specification of the Phillips curve and an open-economy IS relation: 20
20
Typically, forward-looking expectations of inflation appear in the Phillips curve. But they are not necessary to
generate an expectations channel through which monetary policy works in the present model. As shown
below, the forward-looking expectation of inflation eventually enters the domestic inflation equation via the IS
22
=
π t κ yt + ut
(
(25)
)
yt =
−a1 it − Et π tCPI
+1 + a2 (qt − E t qt +1 ) + vt
(26)
𝑞𝑡 = real exchange rate
Assume perfect exchange rate pass-through. This allows us to replace the CPI inflation rate with
𝜋𝑡𝐶𝑃𝐼 = 𝜋𝑡 + 𝛾∆𝑞𝑡 .
(27)
Using eqn. (27) and imposing real UIP allows us to eliminate 𝐸𝑡 ∆𝑞𝑡+1 from IS:
(
)
−a1 ( it − Et π t +1 ) − (a2 − a1γ ) it − it* − (Et π t +1 − Et π t*+1 ) − ρt + vt
yt =
(28)
Next, add to and subtract a1 (it* − Etπ t*+1 ) from eqn. (28). After making use of the definition of the
policy instrument and substituting the IS relation into the Phillips curve, we can restate the rate of
domestic inflation as:
πt =
−α xt + α Et π t +1 − Et π t*+1  + κ vt − a1(it* − Et π t*+1 ) + (a2 − a1γ )ρt  + wt
(29)
=
α κ [a1(1 − γ ) + a2 ] > 0
The presence of the forward-looking expectation of inflation in this equation implies that monetary
policy works partly through the expectations channel.
The above equation can be simplified to read:
πt =
−α xt + α Et π t +1 − Et π t*+1  + ut
(30)
where 21
ut = κ vt − a1(it* − Et π t*+1 + (a2 − a1γ )ρt ) + wt .
relation. Adding 𝛽𝐸𝑡 𝜋𝑡+1 to eqn. (25) would merely enhance the potency of the expectations channel. The
omission of 𝐸𝑡 𝑦𝑡+1 from eqn. (26) is of greater consequence. An analytic solution of the UIP puzzle exists only
if this term does not appear in eqn. (26).
21
Here we see that if the Phillips curve included 𝛽𝐸𝑡 𝜋𝑡+1 then the coefficient on 𝐸𝑡 𝜋𝑡+1 in eqn. (30) would
change to (𝛼 + 𝛽). The constraint facing the policymaker would change slightly. However, the target rule (eqn.
(21)) would remain unaffected. This alternative derivation is available upon request from the author.
23