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Weak instruments and estimated monetary policy rules
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Weak Instruments and Estimated Monetary Policy Rules Omer Bayar Ph.D. PII: DOI: Reference:
S0164-0704(18)30107-1 https://doi.org/10.1016/j.jmacro.2018.10.004 JMACRO 3068
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Journal of Macroeconomics
Received date: Revised date: Accepted date:
14 March 2018 9 October 2018 20 October 2018
Please cite this article as: Omer Bayar Ph.D. , Weak Instruments and Estimated Monetary Policy Rules, Journal of Macroeconomics (2018), doi: https://doi.org/10.1016/j.jmacro.2018.10.004
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Weak Instruments and Estimated Monetary Policy Rules
Title Weak Instruments and Estimated Monetary Policy Rules
Abstract
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Contact 1800 Lincoln Avenue Evansville, IN 47722 [email protected] 812-488-2867
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Author Omer Bayar, Ph.D. Associate Professor of Economics, Guthrie May Endowed Chair in Business Schroeder School of Economics University of Evansville
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Empirical monetary policy rules provide benchmarks for policy evaluation by determining how
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interest rates respond to a small set of macro variables. These rules are based on forward-looking models that require instruments for consistent estimation. The use of standard, albeit weak,
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instruments leads to identification problems in forward-looking rules, preventing reliable policy inference. There are methods to improve instrument strength. When policy rules are estimated
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with stronger instruments, results are more precise and in closer alignment with theory than
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standard estimates.
Keywords: forward-looking policy rule; weak instruments; robust estimation; instrument selection JEL Classification: E52; E58; C26
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1. Introduction
A monetary policy rule is an empirical relationship between a short-term interest rate, such as the federal funds rate, and a small number of macro indicators, including inflation and the output gap. Typically, the lagged interest rate is also added to capture the persistence observed in policy
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rate changes. These rules provide benchmarks for policy evaluation by determining the response
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coefficients on inflation and the output gap as well as the partial adjustment coefficient on the lagged rate, which may be interpreted as capturing deliberate interest rate smoothing or omitted
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serially correlated shocks.
Estimated policy rules are usually built on forward-looking models, since central banks
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set interest rates based on macro forecasts due to informational and operational lags in policy transmission. In this setting, exogenous instruments allow the consistent estimation of forward-
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looking rules. However, in the presence of weak instruments, that is, instruments that are not sufficiently correlated with endogenous variables, point estimates are biased and traditional hypothesis tests perform poorly, as shown by Stock et al. (2002), Dufour (2003), and Stock and Yogo (2005).
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Robust estimation methods have been used to highlight weak identification of empirical policy rules. Mavroeidis (2010) focuses on determinacy, or the Taylor-rule principle, which requires that the nominal interest rate be raised more than inflation to prevent fluctuations due to self-fulfilling expectations. Based on the instruments standard in the literature (lags of inflation,
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output gap, and policy rate), the author reports identification weakness for inflation and output gap coefficients in the post-1979 U.S. data. Similarly, Inoue and Rossi (2011) and Mirza and Storjohann (2014) document weak identification of the Federal Reserve (Fed) policy rules for the same period.1
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The present study employs the Anderson-Rubin (AR) statistic to obtain estimates for inflation and the output gap that are robust to weak instruments as well as missing instruments. Although both issues are of concern in practice, existing robust applications to empirical policy
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rules typically rely on methods that do not address the problem of missing instruments. The AR test also provides robust estimates for the partial adjustment coefficient on the lagged rate, which
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informs on interest rate persistence, another subject largely omitted in previous studies.2 Present results highlight the difficulty of obtaining useful inference when the policy rule
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is estimated with standard, albeit weak, instruments. The robust region for inflation and the
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output gap is large, and violation of the Taylor-rule principle cannot be ruled out, as the region includes below-unity inflation responses. Further, the robust range for the partial adjustment
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coefficient is wide. Because this coefficient determines the speed with which the central bank responds to macro developments, it is difficult to ascertain the pace of policy adjustment. 1
For related work on identification problems in empirical macroeconomics, see Lubik and Schorfheide (2004) and Canova and Sala (2009) who examine dynamic stochastic general equilibrium (DSGE) models, and Mavroeidis (2004), Dufour et al. (2006), Nason and Smith (2008), and Kleibergen and Mavroeidis (2009) who focus on the New Keynesian Phillips Curve (NKPC). 2 An exception is Dufour et al. (2013) who apply a multivariate version of AR test to the New Keynesian model that includes a policy rule with partial adjustment. However, the authors rely primarily on standard instruments of lagged values without providing diagnostics on instrument strength, whereas the present study improves the identification of robust estimates through selection of stronger instruments. 3
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Sensitivity analysis shows that these results are driven by weak instruments, not data or model selection.3 The precision of response and partial adjustment coefficients can be enhanced by raising the strength of instruments. The study considers three alternative methods: a variant of hard
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thresholding, a Wald-based approach, and principal component analysis. These methods yield stronger instruments, instruments with greater explanatory power for endogenous variables than the standard set.
When the policy rule is re-estimated with stronger instruments, the robust region for
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inflation and output gap coefficients is smaller, adhering to the Taylor-rule principle. Similarly, the robust range for the partial adjustment coefficient is narrower, helping enhance inference on interest rate persistence. These findings indicate that instrumentation can be improved without
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imposing undue computational difficulty on the analyst to yield results that are more precise and in closer alignment with economic theory than standard estimates.
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Enhancing the precision of policy rule estimates is of considerable practical importance. Improved estimates will guide market participants, as the U.S. economy continues to normalize
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and the Fed withdraws its policy accommodation over the next few years. They will deepen the
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understanding of persistence, since smoothing is a choice likely reflecting a desire for interest rate stability, while serial correlation may be indicative of omitted factors, leading to better
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predictions of policy actions and lower economic volatility. They will enhance the estimation of underlying parameters when embedded in structural macro models, improving model forecasts. They will also shed light on the pace of the Fed’s response to non-monetary disturbances such as
3
The present study focuses on identification in single-equation models: specifically, the monetary policy rule. See Dufour et al. (2013) for an examination of identification and inference in single- versus multi-equation systems. 4
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technology and oil price shocks, which feed into interest rate setting via their impact on inflation and the output gap.4 The study is organized as follows. The second section outlines the persistence debate before presenting the forward-looking rule and discussing weak instruments. Estimation results
instrument strength and the fifth section concludes.
2. Model
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are reported and interpreted in the third section. The fourth section discusses methods to improve
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The section starts with a brief review of the literature on interest rate persistence. Then, the forward-looking monetary policy rule is introduced and the robust estimation procedure is described.
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2.1 Persistence
Changes in short-term policy interest rates display persistence. Empirical policy rules shed light
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on the observed persistence by determining how the policy rate responds to inflation, the output gap, and the lagged policy rate. A distinctive feature of these rules is a large estimated coefficient
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on the lagged rate.5 The interpretation of this coefficient is subject to debate between two views:
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partial adjustment (interest rate smoothing) and omitted serially correlated shocks. The partial adjustment approach states that a large estimate on the lagged rate implies a
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slow response to macro developments.6 Gradual adjustment is considered desirable for a number of reasons: it may moderate financial market volatility by preventing abrupt policy reversals, which builds credibility (Goodfriend, 1991); it may be optimal under model and data uncertainty
4
See Coibion and Gorodnichenko (2012) for further discussion on the benefits of enhanced policy rule estimates. Clarida et al. (2000) and Sack and Wieland (2000) report quarterly estimates of around 0.8 in the U.S. data. 6 The typical estimate of 0.8 suggests that only 20 percent of a desired interest rate adjustment is achieved after one quarter and around 60 percent after a full year. 5
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(Sack, 2000); and it may help take advantage of the expectations channel in the presence of forward-looking agents, thereby reducing the need for large rate changes (Woodford, 2003). By contrast, Rudebusch (2002) states that the lagged rate is not a fundamental component of the policy rule. Rather, its significance can be attributed to omitted serially correlated factors
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to which the central bank responds: financial market uncertainty, liquidity and credit conditions, asset price volatility, etc.7 There is also indirect evidence against gradual adjustment. A large partial adjustment coefficient should make future interest rate changes highly predictable, an insight that contradicts term structure evidence on forward and futures rates, as documented by
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Rudebusch (2002, 2006) and Rudebusch and Wu (2008). 2.2 Policy Rule
The policy rule is a nested specification that accounts for interest rate smoothing and serially
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correlated shocks, as both mechanisms can explain the observed persistence. The nested rule features first-order interest rate smoothing (
) and first-order residual serial correlation (
),
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in line with English at el. (2003), Gerlach-Kristen (2004), Consolo and Favero (2009), and Bayar
)̂
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(
(1)
PT
(2014).8
In this expression,
is the short-term policy rate, ̂ is the desired Taylor-rule rate, is the serial correlation coefficient,
is a serially correlated
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the partial adjustment coefficient,
is
7
Households and firms in the U.S. became reluctant to borrow and spend, and financial institutions to lend, in the early 1990s. Similar disruptions occurred after the Russian debt default and Asian financial crisis in the late 1990s. The burst of stock market bubble and events of September 11 led to further financial difficulty in 2001. For studies including these shocks in estimated policy rules, see Gerlach-Kristen (2004) and Bayar (2015). 8 The nested model is more general than previous robust applications to estimated policy rules. Mavroeidis (2010), Inoue and Rossi (2011), and Mirza and Storjohann (2014) focus on pure partial adjustment models unsuitable for discussing persistence due to the exclusion of residual serial correlation. 6
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error term, and
is a policy shock orthogonal to the information set available in period
:
. The central bank sets the policy rate based on forecasts. Accordingly, a forward-looking specification is used for the Taylor-rule rate:
where
)
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( ̂
is the rate of inflation expected to occur from period to
output gap expected in
,
is the inflation target, and
,
(2)
is the
is the level of policy rate when
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inflation and output are at their target levels.
One-quarter-ahead forecasts of inflation and output gap (
) are used, in keeping
with the baseline cases from Clarida et al. (2000) and Consolo and Favero (2009), to obtain: (
)[
(
(
)(
(
)]
).
(
)
(3)
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where
)
Equation (3) connects the current policy rate to lagged policy rates, current inflation and
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output gap, and the expected future values of inflation and output gap. Expected values are then replaced by the ex-post observed levels ( )[
)[
(
CE
(
(
)
PT
(
The error term
and
(
)
(
) to receive the estimated rule.
)]
(
) )]
is a linear combination of forecast errors for rule arguments and
orthogonal to information available in period . Endogeneity arises in this setting, since
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(4)
is
correlated with the ex-post observed values. Ideally, consistency is ensured by using exogenous instruments, instruments that are orthogonal to
. The standard practice is to add the lags of
inflation, the output gap, and the policy rate as instruments. 2.3 Weak Identification
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The absence of variation in the Taylor-rule rate ̂ may cause identification difficulty. Because and
enter the second and third terms of equation (4) symmetrically, there needs to be sufficient
variation in the first term to distinguish between partial adjustment and serially correlated shocks. This is typically not a concern in the monetary policy context, as inflation and the output
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gap display large variability due to interest rate and other exogenous shocks.9
There is a second source of weak identification related not to the variability of inflation and output gap but to how well that variability is captured in the estimated model. The problem arises when instruments included to account for movements in endogenous variables are weakly
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correlated with such movements. As a result, point estimates are biased and standard hypothesis tests are unreliable, as test statistics have incorrect size and confidence intervals are inaccurate. In the presence of potentially weak instruments, one approach is to test instrument
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strength and proceed with the forward-looking rule only if instruments are reasonably strong.10 The alternative is robust estimation. The present study uses test statistics whose null distribution
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does not depend on instrument strength, based on methods originally proposed by Anderson and Rubin (1949) and applied by Dufour et al. (2006), Dufour and Taamouti (2007), and Nason and
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education.
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Smith (2008) to the NKPC, the relationship between trade and economic growth, and returns to
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The AR test builds on the linear instrumental variable regression framework: (5) (6)
9
The point, made by Blinder (1986) for inventory models and English et al. (2003) for nested policy rules, is not uncontroversial. Mavroeidis (2010) argues that adherence to Taylor-rule principle in the post-1979 era may have removed self-fulfilling dynamics and reduced the effect of various shocks on inflation and output gap. The resulting decline in the variability of these two variables may then cause identification weakness in the estimated rule. 10 For applications to empirical policy rules, see Consolo and Favero (2009) and Bayar (2014). 8
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where ( ,
is a ,
dependent variable ( ),
,
),
is a
is a
matrix of included exogenous variables
matrix of endogenous variables (
matrix of excluded exogenous variables, or instruments, with
,
and
), and
is a
denoting the sample size
the structural equation (5). The test is based on subtracting
from both sides of equation (5), where
of coefficients chosen by the analyst, substituting in (
where , , and
)]
[ (
)]
)
, testing
(
is independent of (
) and
) equals the standard -statistic for the null hypothesis ). The robust confidence interval for
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(
(7) (8)
in equation (8) is equivalent to
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in equation (5). Formally, if
statistic
(
in the transformed equation (8) stand for the corresponding terms from above.
The AR test states that, provided testing
is a vector
from equation (6), and rearranging to get:
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[
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and the number of instruments respectively.11 The goal here is to obtain consistent estimates for
[
], the AR
, which follows
is then a set of values for which the
analyst fails to reject the null at a given significance level. The process amounts to collecting the that are not rejected at that significance level.
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values of
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The AR test is applied as follows to construct a two-dimensional confidence region for inflation and the output gap. Response coefficients are assigned values
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inside the ranges [
] and [
that vary by 0.1
] respectively, in keeping with Mavroeidis (2010), to create a
grid of 2000 points. Then, the policy rule is estimated at each point inside the grid. Those regressions in which the test does not reject the null
at 10 percent level
establish a 90 percent robust confidence region. 11
End-of-quarter data are used for the policy rate in estimation. As a result, current inflation and output gap, and , are known at the time is set, making them exogenous. 9
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The AR test may face power loss in the presence of many instruments. Newer methods are available to raise power, including those from Wang and Pivots (1998), Kleibergen (2002), and Moreira (2003).12 These methods feature in existing robust applications to policy rules despite their limitations. For one, the AR test is robust to instrument quality in finite samples,
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unlike newer methods that are based on asymptotic properties. More importantly, newer methods are not robust to missing instruments, or specification problems in modeling the endogenous variables in the first stage, properties enjoyed by the AR test. To see why, suppose that equation
̃̃
Equations (7) and (8) then become: [
(
)]
[ (
)]
where
and ̃
)]
(
)
. As a result, the null distribution of
(9)
(10) (11)
(
) remains at
), indicating that the test is robust to the exclusion of relevant instruments.13
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(
under
̃[ ̃ (
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̃̃
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(6) is incorrectly specified and a second matrix of relevant instruments ̃ should also appear.
The problem of missing instruments is critical in the context of policy rules. Although it
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is standard to use lagged rule arguments, a number of studies consider more than the standard
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set: interest spreads, commodity prices, money growth rates, economic forecasts, factors, etc.
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Rarely does the estimated rule include all relevant instruments that could be included, nor should
12
Kleibergen and Mavroeidis (2009) discuss newer methods in the generalized method of moments (GMM) context: S statistic from Stock and Wright (2000), a generalization of AR test to GMM; KLM statistic from Kleibergen (2005); JKLM statistic, the difference between S and KLM statistics; and MQLR statistic, an extension of likelihood ratio test from Moreira (2003) to GMM. 13 Dufour and Taamouti (2007) show that newer methods are subject to large size distortions such that the issue of missing instruments may be as important empirically as the issue of weak instruments. Dufour (2003, 2009) present further discussion on the advantages of AR test relative to newer methods. 10
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it include too many instruments, which would lead to well-known empirical issues: overfitting of endogenous variables, biased parameter estimates, low power for specification tests, etc.14 3. Estimation The section starts with a description of the sample. Then, estimation results are reported and
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sensitivity tests are discussed. 3.1 Data
The sample includes data from the fourth quarter of 1987 to the fourth quarter of 2005, the tenure of former chairman Alan Greenspan. The beginning date is motivated by the fact that policy was
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designed to manage monetary aggregates, not interest rates, over much of the preceding Volcker era. The end date keeps with Mavroeidis (2010) and Mirza and Storjohann (2014) who restrict the analysis to the period before the financial crisis. Monetary policy is thought to have remained
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largely consistent throughout the sample period.
The policy rate is the target funds rate observed on the last business day of each quarter
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to avoid reverse causality issues. Inflation is the percentage change in core personal consumption expenditures (PCE) price index, the Fed’s preferred measure, over the past year. The output gap
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is the percentage deviation of real gross domestic product (GDP) from Congressional Budget
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Office (CBO) estimates of the potential GDP. These choices follow Sack and Wieland (2000), Rudebusch (2002, 2006), English et al. (2003), etc. All datasets are available from the Federal
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Reserve Bank of St. Louis. The estimated model (4) features the first two lags of federal funds rate,
and
,
making them exogenous variables included in the structural model. Instead, four lags of inflation and the output gap in addition to the third and fourth lags of the funds rate are used, which yields 14
While including more information improves efficiency in general, bias due to overfitting may be large enough to offset efficiency gains, with weak instruments exacerbating the efficiency-bias tradeoff. See Roodman (2009) for a review of the relevant literature and Wooldridge (2002) for a textbook treatment. 11
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a set of 10 instruments. The use of lagged values of rule arguments as instruments is standard in the literature and applied by Rudebusch (2002), English et al. (2003), and Mavroeidis (2010) among many others. 3.2 Results
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Table 1 reports GMM point estimates for the policy rule from equation (4) with robust errors. The inflation response exceeds unity at
, indicating that the Fed increases the funds
rate by 1.23 percent in response to a 1 percent rise in inflation, which satisfies the Taylor-rule
by
.
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principle. Active inflation targeting is complemented with a strong output gap response captured
Both persistence parameters enter significantly at
and
, showing that
interest rate smoothing and serially correlated shocks play separate roles in policy setting.15 An suggests that the Fed takes about 3 quarters
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estimated partial adjustment coefficient of
in eliminating one half of the gap between the actual federal funds rate and the Taylor-rule rate: )
( )
(
)
(
)
ED
(
. Although point estimates are reasonable, the model
does not yet address issues pertaining to instrument strength.
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Before proceeding to robust estimation, the section presents formal evidence of weak
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instruments. The -statistic for the joint significance of instruments in the first stage for each endogenous variable is a common measure. In models with multiple endogenous variables, the
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Wald statistic from Cragg and Donald (1993), CD, offers a multivariate version of the -statistic under
errors. For non-
errors, an alternative is the Kleibergen and Paap (2006) Wald
statistic, KP. In both cases, weak instrument bias is a decreasing function of the test statistic; larger test scores are preferred. 15
These estimates are in line with English at el. (2003), Gerlach-Kristen (2004), Consolo and Favero (2009), and Bayar (2014), unlike Coibion and Gorodnichenko (2012) who consider several tests that provide support for interest rate smoothing only. 12
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Table 2 reports that -statistics are 3.56 for
and 2.02 for
, much less than the
rule-of-thumb value of 10 from Staiger and Stock (1997) for models with a single endogenous variable. Further, the joint CD statistic is 1.84 and KP statistic is 2.32, far below 18.76, 10.58, and 6.23, critical values for 2 endogenous variables and 10 instruments from Stock and Yogo
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(2005) for maximal biases of 5, 10, and 20 percent. The concern that standard instruments of lagged values may not be sufficiently correlated with endogenous variables appears to have empirical basis.16
The study turns to the AR test for estimates that are robust to weak instruments. The test , therefore the corresponding hypothesis
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fails to reject the null
, at 10 percent level
at every shaded grid point in Figure 1. These non-rejection points collectively form a 90 percent robust confidence region for
and
. The region is large, containing many coefficient
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combinations compatible with the underlying econometric structure, which confirms that the policy rule is weakly identified over the Greenspan era. Further, the violation of the Taylor-rule
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principle cannot be ruled out, as the region includes grid points at which
is below unity.
The AR test can be extended to accommodate additional restrictions on vector
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equation (5), including the partial adjustment coefficient . The task is warranted since
from and
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, when weakly identified, contaminate all other model coefficients, as shown by Kleibergen and Mavroeidis (2009). Similarly, Dufour (2003) and Dufour et al. (2006) discuss extensions of
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robust tests to exogenous variables included in the structural model, since weak instrument bias carries over to the rest of the model. The joint hypothesis [(
features in the updated system: )
(
)]
[ (
)]
(
)
(12)
16
Tables 1 and 2 report the point estimates and tests of instrument strength for all specifications from the sensitivity section. These results are in line with the findings discussed here. 13
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(13)
where testing
in equation (13) provides a test of
in equation (5).
The extended AR test is used to build a robust confidence interval for
based on values
. The serial correlation coefficient ( ) is selected to match the average
the output gap (
and
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of point estimates for specifications reported in Table 1. Response coefficients on inflation and ) are chosen to fall roughly in the center of the robust region from
Figure 1; these values are slightly higher than point estimates but not inconsistent with the
individual components of vector
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literature. The exercise is based on the projection approach, which yields confidence intervals for (in this case, ) by finding all values of that component for
which the AR test does not reject the joint null
.
In applying the extended AR test, the partial adjustment coefficient is allowed to vary by ], which is much wider than typical estimates from the literature.
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0.01 inside the range [
Then, equation (13) is run for all values of
is not rejected at 10 percent level produce a 90 percent robust range for
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the null
that fall in this range. Those regressions in which
,
, and .
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jointly with the values selected for
The robust range includes as large a value as
, which suggests that 13.5 quarters
value
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are needed to eliminate one half of the interest rate gap. At the low end, the coefficient takes the , which corresponds to 1.2 quarters, more than 11 times as rapid an adjustment.
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Evidently, the robust confidence interval for
is too wide to sufficiently inform on interest rate
persistence.
3.3 Sensitivity Figure 1 is based on core PCE inflation and the GDP output gap. For sensitivity analysis, alternative measures of inflation and economic slack are used: core consumer price index (CPI)
14
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inflation is combined with the GDP output gap while core PCE inflation is combined with the unemployment gap, that is, the difference between the actual unemployment rate and CBO estimates of the natural rate of unemployment. For comparison, instrumentation is based on the standard set.
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The central bank looks one quarter ahead in evaluating future conditions in Figure 1. It is worthwhile to consider longer forecast horizons: 4 quarters for inflation and up to 2 quarters for output gap (
and
), in line with Clarida et al. (2000) and Consolo and
Favero (2009). The second test continues with standard instruments, using core CPI inflation and
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the GDP output gap for which the robust region is smallest among alternative macro measures, indicating improved identification.
A third test focuses on the time-series structure. In keeping with Clarida et al. (2000) and )
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Coibion and Gorodnichenko (2012), the first-order smoothing and residual correlation ( setup from Figure 1 is extended to include second-order dynamics:
which describes
(14)
ED
)̂
while
PT
(
results if the error term is replaced by
.
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In this test, standard instruments are maintained. The test uses CPI inflation and the GDP output gap and assumes that the central bank looks one quarter ahead (
), the two cases for
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which robust regions are smallest among alternative specifications. As before, the goal is to stack the deck against weak identification. One final test follows Choi (1999), Dueker (1999), and Bayar (2015) in estimating a
monthly policy rule. In this case, the funds rate series is based on levels observed on the last business day of each month. Inflation is the percentage change in monthly realizations of PCE price index averaged over the past year. The output gap is the percentage deviation of monthly 15
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Industrial Production index (available at the St. Louis Fed) from its long-run trend, as determined by the Hodrick-Prescott filter. Instruments include once again the standard set. In all sensitivity tests, the AR-based robust confidence regions are relatively large and include grid points at which the Taylor-rule principle is violated.17 Further, robust ranges for the
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partial adjustment coefficient, which are presented in the last column of Table 1, remain too wide to shed light on persistence. Together, these findings indicate that preceding results are driven by instrument weakness rather than the choice of data or model. 4. Instrument Strength
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The section considers methods to raise instrument strength in the estimated rule. The task is to find stronger instruments by explanatory power, 10 of them to be consistent with preceding analysis, from a larger pool of instruments typically used in the literature and then re-apply the
inflation and the GDP output gap,
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AR test. For comparison, same data and model specification from Figure 1 are used: core PCE ,
. Therefore, any difference in results can
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be attributed to the set of instruments used in estimation. The pool includes standard instruments: third and fourth lags of policy rate, four lags of
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inflation, and four lags of the output gap. Following Clarida et al. (2000) and Consolo and
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Favero (2009), four lags of term structure spread (
), the gap between long-term bond rate
and three-month Treasury bill rate, four lags of commodity price index growth rate (
) are included. Orphanides (2004) and Mirza
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four lags of money supply growth rate for M2 (
), and
and Storjohann (2014) exploit the information available in macro forecasts by using Greenbook data. In the same spirit, two-quarter-ahead forecasts of core CPI inflation (
17
) and GDP
These figures are not reported for brevity. They are available from the author upon request. 16
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growth (
) are added to the pool, which are available from the Federal Reserve Bank
of Philadelphia. This makes a total of 24 instruments in the pool. The analysis begins with hard thresholding from Bai and Ng (2009) to rank instruments
which runs an endogenous variable variables
( ,
,
,
(
or
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by explanatory power for endogenous variables in the regression: (15)
) on a constant, the full set of exogenous
), and one instrument
, where
is assumed to be
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Equation (15) is estimated for both endogenous variables and all instruments Then, separate rankings of instruments are created for
and
.
based on -statistics for
In determining the 10 strongest instruments, the highest-ranked instrument for
.
is selected
first, followed by the highest-ranked instrument for
not yet taken. The resulting set, starting
with the strongest instrument, includes:
,
,
, and
,
,
,
,
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,
,
.
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The forward-looking rule is re-estimated using these instruments. The AR-based robust
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region for inflation and the output gap is smaller, as illustrated by Figure 2, indicating that there are far fewer coefficient combinations for which the model is statistically valid. Importantly, the
CE
region displays adherence to the Taylor-rule principle, a result that proved elusive under standard instruments. Further, the projection-based robust range for the partial adjustment coefficient is
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narrower,
, which allows more precise interpretation of persistence.
Does the finding of improved precision remain intact for coefficient values outside of
Mavroeidis (2010) ranges? The answer is affirmative, as shown by AR tests for broader ranges from Mirza and Storjohann (2014): [ the robust region covers [
] for
] for and [
and [ ] for
] for
. For standard instruments,
; relative to Figure 1, the region is
17
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extended on the left and unchanged on the right for
while extended in both directions for
For stronger instruments from hard thresholding, the robust region covers [ [
] for
] for
; relative to Figure 2, the region is unchanged in both directions for
and while
.
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slightly extended in both directions for
.
An alternative ranking of instruments can be obtained in models that feature both
endogenous variables based on Wald statistics from Cragg and Donald (1993) and Kleibergen and Paap (2006). In this exercise, equation (4) is estimated with GMM using two instruments,
from hard thresholding,
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the minimum required in the presence of two endogenous variables. The strongest instrument , is included in all regressions in addition to one of the
remaining 23 instruments from the pool. These remaining instruments are then ranked by their mean CD and KP Wald statistics: , and
.
,
,
,
,
,
,
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,
,
When the policy rule is re-estimated using this set of instruments, the AR-based robust
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region is even smaller, as illustrated by Figure 3, indicating once again that the Fed satisfied the
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Taylor-rule principle during the Greenspan period. Moreover, the robust confidence interval for the partial adjustment coefficient is tighter still,
, further enhancing inference on
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interest rate persistence.
A third alternative is principal component analysis, in line with Jolliffe (2002) and
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Bontempi and Mammi (2015). The analysis reduces instrument count by building uncorrelated linear combinations, or principal components, of those instruments that are ordered such that leading components preserve most of the original information. In keeping with hard thresholding and the Wald-based approach, the analysis is applied to the same pool of 24 instruments. In this case, 24 principal components are constructed first, which capture all of the variation in the pool.
18
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Then, 10 leading components, which explain 93 percent of total variation, are used to replace 10 standard instruments. The forward-looking rule is re-estimated with principal components. The AR-based robust region remains small, as illustrated by Figure 4, suggesting as before that the Fed adhered
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to the Taylor-rule principle during the period of interest. Further, the robust range for the partial adjustment coefficient remains narrow,
, which continues to permit relatively
precise inference on persistence.18
The study uses revised data to keep with the literature, which may lead to misleading
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interpretations of past policy (Orphanides, 2001). In a final test to alleviate concerns regarding measurement error, a real-time policy rule plugs into equation (3) one-quarter-ahead Greenbook forecasts of inflation and the output gap produced by the Fed staff for the Federal Open Market
to yield 90 percent intervals of [
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Committee (FOMC) meetings. The rule is estimated on real-time data from the Philadelphia Fed ] for
,[
] for
, and [
] for .
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Generally, these results are more precise than those for standard instruments but less precise than those for stronger instruments, indicating that the lack of precision is only partly explained by
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measurement error while stronger instruments further improve the identification of policy rule.
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The section closes on a cautious note. The point is not that strongest possible instruments are found; in the realm of macro data, there are likely other variables, observed or constructed,
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with more explanatory power for endogenous variables in the policy rule. Nor is it to claim that hard thresholding, Wald-based approach, or principal components are the most effective methods of choosing instruments. Instrument selection in a data-rich environment has been studied; Bai
18
Power loss does not appear to be a concern, as AR-based regions for stronger instruments, Figures 2-4, are much smaller than KLM-based regions from Mavroeidis (2010) and comparable to, if not smaller than, those from Mirza and Storjohann (2014). 19
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and Ng (2009) discuss boosting, hard thresholding, and information criteria while Bai and Ng (2010) construct factor-based instruments.19 Rather, the goal is to show that the standard practice of using lagged rule arguments as instruments prevents reliable policy inference. The problem is resolved by selection methods that
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raise instrument strength without subjecting the analyst to substantial computational demands. The resulting set of stronger instruments yields policy rule estimates that are more plausible and in closer agreement with theory than standard results. 5. Conclusion
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Robust estimation methods help deal with weak instruments in empirical macro models. One drawback is that robust confidence intervals may be too large to concur with economic theory, which complicates inference on policy behavior. The present study applies alternative methods
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to enhance instrument strength in forward-looking policy rules, which alleviate their weak identification.
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The analysis of principal components from the preceding section is more restrictive than traditional applications to large datasets at the disposal of policymakers, containing series on
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production, prices, employment, interest rates, etc. In this context, the robustness of the AR test
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to missing instruments becomes interesting, because principal components built on large datasets would capture, to a certain degree, the information missing in small instrument sets. Examining
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the extent to which a broader application of principal components may improve identification beyond what is provided here holds promise for future work.
19
Bernanke and Boivion (2003), Favero et al. (2005), Mirza and Storjohann (2014) consider applications to policy rules. Although the latter paper is similar in focus, the present study provides improvement by using the AR test that is robust to weak and missing instruments, estimating nested rules with partial adjustment and residual correlation, and finding evidence of adherence to the Taylor-rule principle for a much smaller instrument set. 20
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The study uses end-of-quarter interest rates, which do not capture the true model, since the target rate is set by the FOMC that meets every 6 weeks. However, policy rules based on the FOMC schedule face a number of limitations: nearly half (19 out of 42) of funds rate changes between 1990 and 2001 occurred outside of the official timetable (13 via conference calls and 6
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without); there were gradual rate changes between 1987 and 1990 that took up to two quarters to complete; there were periods in the late 1980s during which the funds rate was defined in a range rather than as a specific target. These factors complicate the matching of funds rate changes to
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meeting dates, an issue that may be resolved in future work.
Tables
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Forecast horizon
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Funds rate Funds rate
PCE inflation, Industrial production gap
( -value) 0.93 (0.04) 0.74 (0.00) 1.52 (0.00) 0.54 (0.12) 0.77 (0.05) 0.77 (0.00) 0.69 (0.03) 0.93 (0.00)
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PCE inflation, CBO output gap CPI inflation, CBO output gap PCE inflation, Unemployment gap Forecast horizon
( -value) 1.23 (0.09) 1.31 (0.00) 1.61 (0.00) 1.45 (0.04) 1.64 (0.01) 1.33 (0.00) 1.36 (0.00) 1.80 (0.00)
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Table 1: Policy Rule Estimates
( -value) 0.66 (0.00) 0.77 (0.00) 0.76 (0.00) 0.72 (0.00) 0.66 (0.00) 0.77 (0.00) 0.83 (0.00) 0.44 (0.00)
( -value) 0.80 (0.00) 0.56 (0.00) 0.61 (0.01) 0.68 (0.00) 0.72 (0.00) 0.57 (0.00) 0.42 (0.25) 0.95 (0.00)
Robust Range for [0.57, 0.95] [0.60, 0.95] [0.47, 0.95] [0.55, 0.95] [0.59, 0.95] [0.61, 0.95]
GMM point estimates are in first four columns. -values are in parentheses. Instruments are lags of inflation, output gap, and funds rate. AR-based robust confidence intervals for are in fifth column. Robust intervals are not reported in sixth and seventh rows due to appearance of multiple terms under second-order interest rate smoothing.
21
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Table 2: Weak Instrument Tests in GMM
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Forecast horizon
Cragg-Donald Wald Statistic for and
Kleibergen-Paap Wald Statistic for and
2.02
1.84
2.32
4.53
2.56
2.14
2.22
4.01 26.33 ( 25.11 ( 25.20 ( 28.70 (
2.56
2.69
2.42
2.63 12.38 ( 25.03 (
1.39
2.48
1.27
1.28
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3.56
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PCE inflation, CBO output gap CPI inflation, CBO output gap PCE inflation, Unemployment gap Forecast horizon
-Statistic for
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-Statistic for
) ) ) )
) )
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Funds rate
4.97
2.72
2.31
2.40
4.54
3.68
2.47
2.59
5.43
1.51
1.28
1.38
Funds rate
PCE inflation, Industrial production gap
Wald statistics are compared to critical values in Stock and Yogo (2005) tables. Forecast horizon is unless otherwise stated.
22
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Figures
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Figure 1: Baseline
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Shaded area represents the joint AR-based confidence region for response coefficients in the forward-looking policy rule at 90 percent level.
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Figure 2: Hard Thresholding
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PT
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Shaded area represents the joint AR-based confidence region for response coefficients in the forward-looking policy rule at 90 percent level.
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Figure 3: Wald-Based Approach
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PT
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Shaded area represents the joint AR-based confidence region for response coefficients in the forward-looking policy rule at 90 percent level.
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Figure 4: Principal Component Analysis
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PT
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Shaded area represents the joint AR-based confidence region for response coefficients in the forward-looking policy rule at 90 percent level.
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Favero, C., M. Marcellino, and F. Neglia. 2005. Principal Components at Work: The Empirical Analysis of Monetary Policy with Large Data Sets. Journal of Applied Econometrics, 20, 603620.
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Kleibergen, F. and S. Mavroeidis. 2009. Weak Instrument Robust Tests in GMM and the New Keynesian Phillips Curve. Journal of Business and Economic Statistics, 27, 293-311. Lubik, T. and F. Schorfheide. 2004. Testing for Indeterminacy: An Application to U.S. Monetary Policy. American Economic Review, 94, 190-217.
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Weak Instruments and Estimated Monetary Policy Rules Omer Bayar Ph.D. PII: DOI: Reference:
S0164-0704(18)30107-1 https://doi.org/10.1016/j.jmacro.2018.10.004 JMACRO 3068
To appear in:
Journal of Macroeconomics
Received date: Revised date: Accepted date:
14 March 2018 9 October 2018 20 October 2018
Please cite this article as: Omer Bayar Ph.D. , Weak Instruments and Estimated Monetary Policy Rules, Journal of Macroeconomics (2018), doi: https://doi.org/10.1016/j.jmacro.2018.10.004
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Weak Instruments and Estimated Monetary Policy Rules
Title Weak Instruments and Estimated Monetary Policy Rules
Abstract
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Contact 1800 Lincoln Avenue Evansville, IN 47722 [email protected] 812-488-2867
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Author Omer Bayar, Ph.D. Associate Professor of Economics, Guthrie May Endowed Chair in Business Schroeder School of Economics University of Evansville
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Empirical monetary policy rules provide benchmarks for policy evaluation by determining how
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interest rates respond to a small set of macro variables. These rules are based on forward-looking models that require instruments for consistent estimation. The use of standard, albeit weak,
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instruments leads to identification problems in forward-looking rules, preventing reliable policy inference. There are methods to improve instrument strength. When policy rules are estimated
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with stronger instruments, results are more precise and in closer alignment with theory than
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standard estimates.
Keywords: forward-looking policy rule; weak instruments; robust estimation; instrument selection JEL Classification: E52; E58; C26
1
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1. Introduction
A monetary policy rule is an empirical relationship between a short-term interest rate, such as the federal funds rate, and a small number of macro indicators, including inflation and the output gap. Typically, the lagged interest rate is also added to capture the persistence observed in policy
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rate changes. These rules provide benchmarks for policy evaluation by determining the response
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coefficients on inflation and the output gap as well as the partial adjustment coefficient on the lagged rate, which may be interpreted as capturing deliberate interest rate smoothing or omitted
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serially correlated shocks.
Estimated policy rules are usually built on forward-looking models, since central banks
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set interest rates based on macro forecasts due to informational and operational lags in policy transmission. In this setting, exogenous instruments allow the consistent estimation of forward-
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looking rules. However, in the presence of weak instruments, that is, instruments that are not sufficiently correlated with endogenous variables, point estimates are biased and traditional hypothesis tests perform poorly, as shown by Stock et al. (2002), Dufour (2003), and Stock and Yogo (2005).
2
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Robust estimation methods have been used to highlight weak identification of empirical policy rules. Mavroeidis (2010) focuses on determinacy, or the Taylor-rule principle, which requires that the nominal interest rate be raised more than inflation to prevent fluctuations due to self-fulfilling expectations. Based on the instruments standard in the literature (lags of inflation,
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output gap, and policy rate), the author reports identification weakness for inflation and output gap coefficients in the post-1979 U.S. data. Similarly, Inoue and Rossi (2011) and Mirza and Storjohann (2014) document weak identification of the Federal Reserve (Fed) policy rules for the same period.1
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The present study employs the Anderson-Rubin (AR) statistic to obtain estimates for inflation and the output gap that are robust to weak instruments as well as missing instruments. Although both issues are of concern in practice, existing robust applications to empirical policy
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rules typically rely on methods that do not address the problem of missing instruments. The AR test also provides robust estimates for the partial adjustment coefficient on the lagged rate, which
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informs on interest rate persistence, another subject largely omitted in previous studies.2 Present results highlight the difficulty of obtaining useful inference when the policy rule
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is estimated with standard, albeit weak, instruments. The robust region for inflation and the
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output gap is large, and violation of the Taylor-rule principle cannot be ruled out, as the region includes below-unity inflation responses. Further, the robust range for the partial adjustment
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coefficient is wide. Because this coefficient determines the speed with which the central bank responds to macro developments, it is difficult to ascertain the pace of policy adjustment. 1
For related work on identification problems in empirical macroeconomics, see Lubik and Schorfheide (2004) and Canova and Sala (2009) who examine dynamic stochastic general equilibrium (DSGE) models, and Mavroeidis (2004), Dufour et al. (2006), Nason and Smith (2008), and Kleibergen and Mavroeidis (2009) who focus on the New Keynesian Phillips Curve (NKPC). 2 An exception is Dufour et al. (2013) who apply a multivariate version of AR test to the New Keynesian model that includes a policy rule with partial adjustment. However, the authors rely primarily on standard instruments of lagged values without providing diagnostics on instrument strength, whereas the present study improves the identification of robust estimates through selection of stronger instruments. 3
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Sensitivity analysis shows that these results are driven by weak instruments, not data or model selection.3 The precision of response and partial adjustment coefficients can be enhanced by raising the strength of instruments. The study considers three alternative methods: a variant of hard
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thresholding, a Wald-based approach, and principal component analysis. These methods yield stronger instruments, instruments with greater explanatory power for endogenous variables than the standard set.
When the policy rule is re-estimated with stronger instruments, the robust region for
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inflation and output gap coefficients is smaller, adhering to the Taylor-rule principle. Similarly, the robust range for the partial adjustment coefficient is narrower, helping enhance inference on interest rate persistence. These findings indicate that instrumentation can be improved without
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imposing undue computational difficulty on the analyst to yield results that are more precise and in closer alignment with economic theory than standard estimates.
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Enhancing the precision of policy rule estimates is of considerable practical importance. Improved estimates will guide market participants, as the U.S. economy continues to normalize
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and the Fed withdraws its policy accommodation over the next few years. They will deepen the
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understanding of persistence, since smoothing is a choice likely reflecting a desire for interest rate stability, while serial correlation may be indicative of omitted factors, leading to better
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predictions of policy actions and lower economic volatility. They will enhance the estimation of underlying parameters when embedded in structural macro models, improving model forecasts. They will also shed light on the pace of the Fed’s response to non-monetary disturbances such as
3
The present study focuses on identification in single-equation models: specifically, the monetary policy rule. See Dufour et al. (2013) for an examination of identification and inference in single- versus multi-equation systems. 4
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technology and oil price shocks, which feed into interest rate setting via their impact on inflation and the output gap.4 The study is organized as follows. The second section outlines the persistence debate before presenting the forward-looking rule and discussing weak instruments. Estimation results
instrument strength and the fifth section concludes.
2. Model
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are reported and interpreted in the third section. The fourth section discusses methods to improve
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The section starts with a brief review of the literature on interest rate persistence. Then, the forward-looking monetary policy rule is introduced and the robust estimation procedure is described.
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2.1 Persistence
Changes in short-term policy interest rates display persistence. Empirical policy rules shed light
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on the observed persistence by determining how the policy rate responds to inflation, the output gap, and the lagged policy rate. A distinctive feature of these rules is a large estimated coefficient
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on the lagged rate.5 The interpretation of this coefficient is subject to debate between two views:
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partial adjustment (interest rate smoothing) and omitted serially correlated shocks. The partial adjustment approach states that a large estimate on the lagged rate implies a
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slow response to macro developments.6 Gradual adjustment is considered desirable for a number of reasons: it may moderate financial market volatility by preventing abrupt policy reversals, which builds credibility (Goodfriend, 1991); it may be optimal under model and data uncertainty
4
See Coibion and Gorodnichenko (2012) for further discussion on the benefits of enhanced policy rule estimates. Clarida et al. (2000) and Sack and Wieland (2000) report quarterly estimates of around 0.8 in the U.S. data. 6 The typical estimate of 0.8 suggests that only 20 percent of a desired interest rate adjustment is achieved after one quarter and around 60 percent after a full year. 5
5
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(Sack, 2000); and it may help take advantage of the expectations channel in the presence of forward-looking agents, thereby reducing the need for large rate changes (Woodford, 2003). By contrast, Rudebusch (2002) states that the lagged rate is not a fundamental component of the policy rule. Rather, its significance can be attributed to omitted serially correlated factors
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to which the central bank responds: financial market uncertainty, liquidity and credit conditions, asset price volatility, etc.7 There is also indirect evidence against gradual adjustment. A large partial adjustment coefficient should make future interest rate changes highly predictable, an insight that contradicts term structure evidence on forward and futures rates, as documented by
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Rudebusch (2002, 2006) and Rudebusch and Wu (2008). 2.2 Policy Rule
The policy rule is a nested specification that accounts for interest rate smoothing and serially
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correlated shocks, as both mechanisms can explain the observed persistence. The nested rule features first-order interest rate smoothing (
) and first-order residual serial correlation (
),
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in line with English at el. (2003), Gerlach-Kristen (2004), Consolo and Favero (2009), and Bayar
)̂
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(
(1)
PT
(2014).8
In this expression,
is the short-term policy rate, ̂ is the desired Taylor-rule rate, is the serial correlation coefficient,
is a serially correlated
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the partial adjustment coefficient,
is
7
Households and firms in the U.S. became reluctant to borrow and spend, and financial institutions to lend, in the early 1990s. Similar disruptions occurred after the Russian debt default and Asian financial crisis in the late 1990s. The burst of stock market bubble and events of September 11 led to further financial difficulty in 2001. For studies including these shocks in estimated policy rules, see Gerlach-Kristen (2004) and Bayar (2015). 8 The nested model is more general than previous robust applications to estimated policy rules. Mavroeidis (2010), Inoue and Rossi (2011), and Mirza and Storjohann (2014) focus on pure partial adjustment models unsuitable for discussing persistence due to the exclusion of residual serial correlation. 6
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error term, and
is a policy shock orthogonal to the information set available in period
:
. The central bank sets the policy rate based on forecasts. Accordingly, a forward-looking specification is used for the Taylor-rule rate:
where
)
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( ̂
is the rate of inflation expected to occur from period to
output gap expected in
,
is the inflation target, and
,
(2)
is the
is the level of policy rate when
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inflation and output are at their target levels.
One-quarter-ahead forecasts of inflation and output gap (
) are used, in keeping
with the baseline cases from Clarida et al. (2000) and Consolo and Favero (2009), to obtain: (
)[
(
(
)(
(
)]
).
(
)
(3)
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where
)
Equation (3) connects the current policy rate to lagged policy rates, current inflation and
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output gap, and the expected future values of inflation and output gap. Expected values are then replaced by the ex-post observed levels ( )[
)[
(
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(
(
)
PT
(
The error term
and
(
)
(
) to receive the estimated rule.
)]
(
) )]
is a linear combination of forecast errors for rule arguments and
orthogonal to information available in period . Endogeneity arises in this setting, since
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(4)
is
correlated with the ex-post observed values. Ideally, consistency is ensured by using exogenous instruments, instruments that are orthogonal to
. The standard practice is to add the lags of
inflation, the output gap, and the policy rate as instruments. 2.3 Weak Identification
7
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The absence of variation in the Taylor-rule rate ̂ may cause identification difficulty. Because and
enter the second and third terms of equation (4) symmetrically, there needs to be sufficient
variation in the first term to distinguish between partial adjustment and serially correlated shocks. This is typically not a concern in the monetary policy context, as inflation and the output
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gap display large variability due to interest rate and other exogenous shocks.9
There is a second source of weak identification related not to the variability of inflation and output gap but to how well that variability is captured in the estimated model. The problem arises when instruments included to account for movements in endogenous variables are weakly
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correlated with such movements. As a result, point estimates are biased and standard hypothesis tests are unreliable, as test statistics have incorrect size and confidence intervals are inaccurate. In the presence of potentially weak instruments, one approach is to test instrument
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strength and proceed with the forward-looking rule only if instruments are reasonably strong.10 The alternative is robust estimation. The present study uses test statistics whose null distribution
ED
does not depend on instrument strength, based on methods originally proposed by Anderson and Rubin (1949) and applied by Dufour et al. (2006), Dufour and Taamouti (2007), and Nason and
CE
education.
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Smith (2008) to the NKPC, the relationship between trade and economic growth, and returns to
AC
The AR test builds on the linear instrumental variable regression framework: (5) (6)
9
The point, made by Blinder (1986) for inventory models and English et al. (2003) for nested policy rules, is not uncontroversial. Mavroeidis (2010) argues that adherence to Taylor-rule principle in the post-1979 era may have removed self-fulfilling dynamics and reduced the effect of various shocks on inflation and output gap. The resulting decline in the variability of these two variables may then cause identification weakness in the estimated rule. 10 For applications to empirical policy rules, see Consolo and Favero (2009) and Bayar (2014). 8
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where ( ,
is a ,
dependent variable ( ),
,
),
is a
is a
matrix of included exogenous variables
matrix of endogenous variables (
matrix of excluded exogenous variables, or instruments, with
,
and
), and
is a
denoting the sample size
the structural equation (5). The test is based on subtracting
from both sides of equation (5), where
of coefficients chosen by the analyst, substituting in (
where , , and
)]
[ (
)]
)
, testing
(
is independent of (
) and
) equals the standard -statistic for the null hypothesis ). The robust confidence interval for
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(
(7) (8)
in equation (8) is equivalent to
M
in equation (5). Formally, if
statistic
(
in the transformed equation (8) stand for the corresponding terms from above.
The AR test states that, provided testing
is a vector
from equation (6), and rearranging to get:
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[
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and the number of instruments respectively.11 The goal here is to obtain consistent estimates for
[
], the AR
, which follows
is then a set of values for which the
analyst fails to reject the null at a given significance level. The process amounts to collecting the that are not rejected at that significance level.
PT
values of
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The AR test is applied as follows to construct a two-dimensional confidence region for inflation and the output gap. Response coefficients are assigned values
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inside the ranges [
] and [
that vary by 0.1
] respectively, in keeping with Mavroeidis (2010), to create a
grid of 2000 points. Then, the policy rule is estimated at each point inside the grid. Those regressions in which the test does not reject the null
at 10 percent level
establish a 90 percent robust confidence region. 11
End-of-quarter data are used for the policy rate in estimation. As a result, current inflation and output gap, and , are known at the time is set, making them exogenous. 9
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The AR test may face power loss in the presence of many instruments. Newer methods are available to raise power, including those from Wang and Pivots (1998), Kleibergen (2002), and Moreira (2003).12 These methods feature in existing robust applications to policy rules despite their limitations. For one, the AR test is robust to instrument quality in finite samples,
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unlike newer methods that are based on asymptotic properties. More importantly, newer methods are not robust to missing instruments, or specification problems in modeling the endogenous variables in the first stage, properties enjoyed by the AR test. To see why, suppose that equation
̃̃
Equations (7) and (8) then become: [
(
)]
[ (
)]
where
and ̃
)]
(
)
. As a result, the null distribution of
(9)
(10) (11)
(
) remains at
), indicating that the test is robust to the exclusion of relevant instruments.13
ED
(
under
̃[ ̃ (
M
̃̃
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(6) is incorrectly specified and a second matrix of relevant instruments ̃ should also appear.
The problem of missing instruments is critical in the context of policy rules. Although it
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is standard to use lagged rule arguments, a number of studies consider more than the standard
CE
set: interest spreads, commodity prices, money growth rates, economic forecasts, factors, etc.
AC
Rarely does the estimated rule include all relevant instruments that could be included, nor should
12
Kleibergen and Mavroeidis (2009) discuss newer methods in the generalized method of moments (GMM) context: S statistic from Stock and Wright (2000), a generalization of AR test to GMM; KLM statistic from Kleibergen (2005); JKLM statistic, the difference between S and KLM statistics; and MQLR statistic, an extension of likelihood ratio test from Moreira (2003) to GMM. 13 Dufour and Taamouti (2007) show that newer methods are subject to large size distortions such that the issue of missing instruments may be as important empirically as the issue of weak instruments. Dufour (2003, 2009) present further discussion on the advantages of AR test relative to newer methods. 10
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it include too many instruments, which would lead to well-known empirical issues: overfitting of endogenous variables, biased parameter estimates, low power for specification tests, etc.14 3. Estimation The section starts with a description of the sample. Then, estimation results are reported and
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sensitivity tests are discussed. 3.1 Data
The sample includes data from the fourth quarter of 1987 to the fourth quarter of 2005, the tenure of former chairman Alan Greenspan. The beginning date is motivated by the fact that policy was
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designed to manage monetary aggregates, not interest rates, over much of the preceding Volcker era. The end date keeps with Mavroeidis (2010) and Mirza and Storjohann (2014) who restrict the analysis to the period before the financial crisis. Monetary policy is thought to have remained
M
largely consistent throughout the sample period.
The policy rate is the target funds rate observed on the last business day of each quarter
ED
to avoid reverse causality issues. Inflation is the percentage change in core personal consumption expenditures (PCE) price index, the Fed’s preferred measure, over the past year. The output gap
PT
is the percentage deviation of real gross domestic product (GDP) from Congressional Budget
CE
Office (CBO) estimates of the potential GDP. These choices follow Sack and Wieland (2000), Rudebusch (2002, 2006), English et al. (2003), etc. All datasets are available from the Federal
AC
Reserve Bank of St. Louis. The estimated model (4) features the first two lags of federal funds rate,
and
,
making them exogenous variables included in the structural model. Instead, four lags of inflation and the output gap in addition to the third and fourth lags of the funds rate are used, which yields 14
While including more information improves efficiency in general, bias due to overfitting may be large enough to offset efficiency gains, with weak instruments exacerbating the efficiency-bias tradeoff. See Roodman (2009) for a review of the relevant literature and Wooldridge (2002) for a textbook treatment. 11
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a set of 10 instruments. The use of lagged values of rule arguments as instruments is standard in the literature and applied by Rudebusch (2002), English et al. (2003), and Mavroeidis (2010) among many others. 3.2 Results
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Table 1 reports GMM point estimates for the policy rule from equation (4) with robust errors. The inflation response exceeds unity at
, indicating that the Fed increases the funds
rate by 1.23 percent in response to a 1 percent rise in inflation, which satisfies the Taylor-rule
by
.
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principle. Active inflation targeting is complemented with a strong output gap response captured
Both persistence parameters enter significantly at
and
, showing that
interest rate smoothing and serially correlated shocks play separate roles in policy setting.15 An suggests that the Fed takes about 3 quarters
M
estimated partial adjustment coefficient of
in eliminating one half of the gap between the actual federal funds rate and the Taylor-rule rate: )
( )
(
)
(
)
ED
(
. Although point estimates are reasonable, the model
does not yet address issues pertaining to instrument strength.
PT
Before proceeding to robust estimation, the section presents formal evidence of weak
CE
instruments. The -statistic for the joint significance of instruments in the first stage for each endogenous variable is a common measure. In models with multiple endogenous variables, the
AC
Wald statistic from Cragg and Donald (1993), CD, offers a multivariate version of the -statistic under
errors. For non-
errors, an alternative is the Kleibergen and Paap (2006) Wald
statistic, KP. In both cases, weak instrument bias is a decreasing function of the test statistic; larger test scores are preferred. 15
These estimates are in line with English at el. (2003), Gerlach-Kristen (2004), Consolo and Favero (2009), and Bayar (2014), unlike Coibion and Gorodnichenko (2012) who consider several tests that provide support for interest rate smoothing only. 12
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Table 2 reports that -statistics are 3.56 for
and 2.02 for
, much less than the
rule-of-thumb value of 10 from Staiger and Stock (1997) for models with a single endogenous variable. Further, the joint CD statistic is 1.84 and KP statistic is 2.32, far below 18.76, 10.58, and 6.23, critical values for 2 endogenous variables and 10 instruments from Stock and Yogo
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(2005) for maximal biases of 5, 10, and 20 percent. The concern that standard instruments of lagged values may not be sufficiently correlated with endogenous variables appears to have empirical basis.16
The study turns to the AR test for estimates that are robust to weak instruments. The test , therefore the corresponding hypothesis
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fails to reject the null
, at 10 percent level
at every shaded grid point in Figure 1. These non-rejection points collectively form a 90 percent robust confidence region for
and
. The region is large, containing many coefficient
M
combinations compatible with the underlying econometric structure, which confirms that the policy rule is weakly identified over the Greenspan era. Further, the violation of the Taylor-rule
ED
principle cannot be ruled out, as the region includes grid points at which
is below unity.
The AR test can be extended to accommodate additional restrictions on vector
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equation (5), including the partial adjustment coefficient . The task is warranted since
from and
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, when weakly identified, contaminate all other model coefficients, as shown by Kleibergen and Mavroeidis (2009). Similarly, Dufour (2003) and Dufour et al. (2006) discuss extensions of
AC
robust tests to exogenous variables included in the structural model, since weak instrument bias carries over to the rest of the model. The joint hypothesis [(
features in the updated system: )
(
)]
[ (
)]
(
)
(12)
16
Tables 1 and 2 report the point estimates and tests of instrument strength for all specifications from the sensitivity section. These results are in line with the findings discussed here. 13
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(13)
where testing
in equation (13) provides a test of
in equation (5).
The extended AR test is used to build a robust confidence interval for
based on values
. The serial correlation coefficient ( ) is selected to match the average
the output gap (
and
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of point estimates for specifications reported in Table 1. Response coefficients on inflation and ) are chosen to fall roughly in the center of the robust region from
Figure 1; these values are slightly higher than point estimates but not inconsistent with the
individual components of vector
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literature. The exercise is based on the projection approach, which yields confidence intervals for (in this case, ) by finding all values of that component for
which the AR test does not reject the joint null
.
In applying the extended AR test, the partial adjustment coefficient is allowed to vary by ], which is much wider than typical estimates from the literature.
M
0.01 inside the range [
Then, equation (13) is run for all values of
is not rejected at 10 percent level produce a 90 percent robust range for
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the null
that fall in this range. Those regressions in which
,
, and .
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jointly with the values selected for
The robust range includes as large a value as
, which suggests that 13.5 quarters
value
CE
are needed to eliminate one half of the interest rate gap. At the low end, the coefficient takes the , which corresponds to 1.2 quarters, more than 11 times as rapid an adjustment.
AC
Evidently, the robust confidence interval for
is too wide to sufficiently inform on interest rate
persistence.
3.3 Sensitivity Figure 1 is based on core PCE inflation and the GDP output gap. For sensitivity analysis, alternative measures of inflation and economic slack are used: core consumer price index (CPI)
14
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inflation is combined with the GDP output gap while core PCE inflation is combined with the unemployment gap, that is, the difference between the actual unemployment rate and CBO estimates of the natural rate of unemployment. For comparison, instrumentation is based on the standard set.
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The central bank looks one quarter ahead in evaluating future conditions in Figure 1. It is worthwhile to consider longer forecast horizons: 4 quarters for inflation and up to 2 quarters for output gap (
and
), in line with Clarida et al. (2000) and Consolo and
Favero (2009). The second test continues with standard instruments, using core CPI inflation and
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the GDP output gap for which the robust region is smallest among alternative macro measures, indicating improved identification.
A third test focuses on the time-series structure. In keeping with Clarida et al. (2000) and )
M
Coibion and Gorodnichenko (2012), the first-order smoothing and residual correlation ( setup from Figure 1 is extended to include second-order dynamics:
which describes
(14)
ED
)̂
while
PT
(
results if the error term is replaced by
.
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In this test, standard instruments are maintained. The test uses CPI inflation and the GDP output gap and assumes that the central bank looks one quarter ahead (
), the two cases for
AC
which robust regions are smallest among alternative specifications. As before, the goal is to stack the deck against weak identification. One final test follows Choi (1999), Dueker (1999), and Bayar (2015) in estimating a
monthly policy rule. In this case, the funds rate series is based on levels observed on the last business day of each month. Inflation is the percentage change in monthly realizations of PCE price index averaged over the past year. The output gap is the percentage deviation of monthly 15
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Industrial Production index (available at the St. Louis Fed) from its long-run trend, as determined by the Hodrick-Prescott filter. Instruments include once again the standard set. In all sensitivity tests, the AR-based robust confidence regions are relatively large and include grid points at which the Taylor-rule principle is violated.17 Further, robust ranges for the
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partial adjustment coefficient, which are presented in the last column of Table 1, remain too wide to shed light on persistence. Together, these findings indicate that preceding results are driven by instrument weakness rather than the choice of data or model. 4. Instrument Strength
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The section considers methods to raise instrument strength in the estimated rule. The task is to find stronger instruments by explanatory power, 10 of them to be consistent with preceding analysis, from a larger pool of instruments typically used in the literature and then re-apply the
inflation and the GDP output gap,
M
AR test. For comparison, same data and model specification from Figure 1 are used: core PCE ,
. Therefore, any difference in results can
ED
be attributed to the set of instruments used in estimation. The pool includes standard instruments: third and fourth lags of policy rate, four lags of
PT
inflation, and four lags of the output gap. Following Clarida et al. (2000) and Consolo and
CE
Favero (2009), four lags of term structure spread (
), the gap between long-term bond rate
and three-month Treasury bill rate, four lags of commodity price index growth rate (
) are included. Orphanides (2004) and Mirza
AC
four lags of money supply growth rate for M2 (
), and
and Storjohann (2014) exploit the information available in macro forecasts by using Greenbook data. In the same spirit, two-quarter-ahead forecasts of core CPI inflation (
17
) and GDP
These figures are not reported for brevity. They are available from the author upon request. 16
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growth (
) are added to the pool, which are available from the Federal Reserve Bank
of Philadelphia. This makes a total of 24 instruments in the pool. The analysis begins with hard thresholding from Bai and Ng (2009) to rank instruments
which runs an endogenous variable variables
( ,
,
,
(
or
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by explanatory power for endogenous variables in the regression: (15)
) on a constant, the full set of exogenous
), and one instrument
, where
is assumed to be
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Equation (15) is estimated for both endogenous variables and all instruments Then, separate rankings of instruments are created for
and
.
based on -statistics for
In determining the 10 strongest instruments, the highest-ranked instrument for
.
is selected
first, followed by the highest-ranked instrument for
not yet taken. The resulting set, starting
with the strongest instrument, includes:
,
,
, and
,
,
,
,
M
,
,
.
ED
The forward-looking rule is re-estimated using these instruments. The AR-based robust
PT
region for inflation and the output gap is smaller, as illustrated by Figure 2, indicating that there are far fewer coefficient combinations for which the model is statistically valid. Importantly, the
CE
region displays adherence to the Taylor-rule principle, a result that proved elusive under standard instruments. Further, the projection-based robust range for the partial adjustment coefficient is
AC
narrower,
, which allows more precise interpretation of persistence.
Does the finding of improved precision remain intact for coefficient values outside of
Mavroeidis (2010) ranges? The answer is affirmative, as shown by AR tests for broader ranges from Mirza and Storjohann (2014): [ the robust region covers [
] for
] for and [
and [ ] for
] for
. For standard instruments,
; relative to Figure 1, the region is
17
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extended on the left and unchanged on the right for
while extended in both directions for
For stronger instruments from hard thresholding, the robust region covers [ [
] for
] for
; relative to Figure 2, the region is unchanged in both directions for
and while
.
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slightly extended in both directions for
.
An alternative ranking of instruments can be obtained in models that feature both
endogenous variables based on Wald statistics from Cragg and Donald (1993) and Kleibergen and Paap (2006). In this exercise, equation (4) is estimated with GMM using two instruments,
from hard thresholding,
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the minimum required in the presence of two endogenous variables. The strongest instrument , is included in all regressions in addition to one of the
remaining 23 instruments from the pool. These remaining instruments are then ranked by their mean CD and KP Wald statistics: , and
.
,
,
,
,
,
,
M
,
,
When the policy rule is re-estimated using this set of instruments, the AR-based robust
ED
region is even smaller, as illustrated by Figure 3, indicating once again that the Fed satisfied the
PT
Taylor-rule principle during the Greenspan period. Moreover, the robust confidence interval for the partial adjustment coefficient is tighter still,
, further enhancing inference on
CE
interest rate persistence.
A third alternative is principal component analysis, in line with Jolliffe (2002) and
AC
Bontempi and Mammi (2015). The analysis reduces instrument count by building uncorrelated linear combinations, or principal components, of those instruments that are ordered such that leading components preserve most of the original information. In keeping with hard thresholding and the Wald-based approach, the analysis is applied to the same pool of 24 instruments. In this case, 24 principal components are constructed first, which capture all of the variation in the pool.
18
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Then, 10 leading components, which explain 93 percent of total variation, are used to replace 10 standard instruments. The forward-looking rule is re-estimated with principal components. The AR-based robust region remains small, as illustrated by Figure 4, suggesting as before that the Fed adhered
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to the Taylor-rule principle during the period of interest. Further, the robust range for the partial adjustment coefficient remains narrow,
, which continues to permit relatively
precise inference on persistence.18
The study uses revised data to keep with the literature, which may lead to misleading
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interpretations of past policy (Orphanides, 2001). In a final test to alleviate concerns regarding measurement error, a real-time policy rule plugs into equation (3) one-quarter-ahead Greenbook forecasts of inflation and the output gap produced by the Fed staff for the Federal Open Market
to yield 90 percent intervals of [
M
Committee (FOMC) meetings. The rule is estimated on real-time data from the Philadelphia Fed ] for
,[
] for
, and [
] for .
ED
Generally, these results are more precise than those for standard instruments but less precise than those for stronger instruments, indicating that the lack of precision is only partly explained by
PT
measurement error while stronger instruments further improve the identification of policy rule.
CE
The section closes on a cautious note. The point is not that strongest possible instruments are found; in the realm of macro data, there are likely other variables, observed or constructed,
AC
with more explanatory power for endogenous variables in the policy rule. Nor is it to claim that hard thresholding, Wald-based approach, or principal components are the most effective methods of choosing instruments. Instrument selection in a data-rich environment has been studied; Bai
18
Power loss does not appear to be a concern, as AR-based regions for stronger instruments, Figures 2-4, are much smaller than KLM-based regions from Mavroeidis (2010) and comparable to, if not smaller than, those from Mirza and Storjohann (2014). 19
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and Ng (2009) discuss boosting, hard thresholding, and information criteria while Bai and Ng (2010) construct factor-based instruments.19 Rather, the goal is to show that the standard practice of using lagged rule arguments as instruments prevents reliable policy inference. The problem is resolved by selection methods that
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raise instrument strength without subjecting the analyst to substantial computational demands. The resulting set of stronger instruments yields policy rule estimates that are more plausible and in closer agreement with theory than standard results. 5. Conclusion
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Robust estimation methods help deal with weak instruments in empirical macro models. One drawback is that robust confidence intervals may be too large to concur with economic theory, which complicates inference on policy behavior. The present study applies alternative methods
M
to enhance instrument strength in forward-looking policy rules, which alleviate their weak identification.
ED
The analysis of principal components from the preceding section is more restrictive than traditional applications to large datasets at the disposal of policymakers, containing series on
PT
production, prices, employment, interest rates, etc. In this context, the robustness of the AR test
CE
to missing instruments becomes interesting, because principal components built on large datasets would capture, to a certain degree, the information missing in small instrument sets. Examining
AC
the extent to which a broader application of principal components may improve identification beyond what is provided here holds promise for future work.
19
Bernanke and Boivion (2003), Favero et al. (2005), Mirza and Storjohann (2014) consider applications to policy rules. Although the latter paper is similar in focus, the present study provides improvement by using the AR test that is robust to weak and missing instruments, estimating nested rules with partial adjustment and residual correlation, and finding evidence of adherence to the Taylor-rule principle for a much smaller instrument set. 20
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The study uses end-of-quarter interest rates, which do not capture the true model, since the target rate is set by the FOMC that meets every 6 weeks. However, policy rules based on the FOMC schedule face a number of limitations: nearly half (19 out of 42) of funds rate changes between 1990 and 2001 occurred outside of the official timetable (13 via conference calls and 6
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without); there were gradual rate changes between 1987 and 1990 that took up to two quarters to complete; there were periods in the late 1980s during which the funds rate was defined in a range rather than as a specific target. These factors complicate the matching of funds rate changes to
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meeting dates, an issue that may be resolved in future work.
Tables
CE
Forecast horizon
AC
Funds rate Funds rate
PCE inflation, Industrial production gap
( -value) 0.93 (0.04) 0.74 (0.00) 1.52 (0.00) 0.54 (0.12) 0.77 (0.05) 0.77 (0.00) 0.69 (0.03) 0.93 (0.00)
ED
PT
PCE inflation, CBO output gap CPI inflation, CBO output gap PCE inflation, Unemployment gap Forecast horizon
( -value) 1.23 (0.09) 1.31 (0.00) 1.61 (0.00) 1.45 (0.04) 1.64 (0.01) 1.33 (0.00) 1.36 (0.00) 1.80 (0.00)
M
Table 1: Policy Rule Estimates
( -value) 0.66 (0.00) 0.77 (0.00) 0.76 (0.00) 0.72 (0.00) 0.66 (0.00) 0.77 (0.00) 0.83 (0.00) 0.44 (0.00)
( -value) 0.80 (0.00) 0.56 (0.00) 0.61 (0.01) 0.68 (0.00) 0.72 (0.00) 0.57 (0.00) 0.42 (0.25) 0.95 (0.00)
Robust Range for [0.57, 0.95] [0.60, 0.95] [0.47, 0.95] [0.55, 0.95] [0.59, 0.95] [0.61, 0.95]
GMM point estimates are in first four columns. -values are in parentheses. Instruments are lags of inflation, output gap, and funds rate. AR-based robust confidence intervals for are in fifth column. Robust intervals are not reported in sixth and seventh rows due to appearance of multiple terms under second-order interest rate smoothing.
21
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Table 2: Weak Instrument Tests in GMM
CE
Forecast horizon
Cragg-Donald Wald Statistic for and
Kleibergen-Paap Wald Statistic for and
2.02
1.84
2.32
4.53
2.56
2.14
2.22
4.01 26.33 ( 25.11 ( 25.20 ( 28.70 (
2.56
2.69
2.42
2.63 12.38 ( 25.03 (
1.39
2.48
1.27
1.28
ED
3.56
PT
PCE inflation, CBO output gap CPI inflation, CBO output gap PCE inflation, Unemployment gap Forecast horizon
-Statistic for
M
-Statistic for
) ) ) )
) )
AC
Funds rate
4.97
2.72
2.31
2.40
4.54
3.68
2.47
2.59
5.43
1.51
1.28
1.38
Funds rate
PCE inflation, Industrial production gap
Wald statistics are compared to critical values in Stock and Yogo (2005) tables. Forecast horizon is unless otherwise stated.
22
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AC
CE
PT
ED
M
Figures
23
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M
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Figure 1: Baseline
AC
CE
PT
ED
Shaded area represents the joint AR-based confidence region for response coefficients in the forward-looking policy rule at 90 percent level.
24
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Figure 2: Hard Thresholding
AC
CE
PT
ED
Shaded area represents the joint AR-based confidence region for response coefficients in the forward-looking policy rule at 90 percent level.
25
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Figure 3: Wald-Based Approach
AC
CE
PT
ED
Shaded area represents the joint AR-based confidence region for response coefficients in the forward-looking policy rule at 90 percent level.
26
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M
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CR IP T
Figure 4: Principal Component Analysis
AC
CE
PT
ED
Shaded area represents the joint AR-based confidence region for response coefficients in the forward-looking policy rule at 90 percent level.
27
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Bai, J. and S. Ng. 2010. Instrumental Variable Estimation in a Data Rich Environment. Econometric Theory, 26, 1577-1606.
Bayar, O. 2014. Temporal Aggregation and Estimated Monetary Policy Rules. The B.E. Journal of Macroeconomics (Contributions), 14, 553-577.
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Bayar, O. 2015. An Ordered Probit Analysis of Monetary Policy Inertia. The B.E. Journal of Macroeconomics (Contributions), 15, 705-726. Bernanke, B. and J. Boivion. 2003. Monetary policy in a data-rich environment. Journal of Monetary Economics, 50, 525-546. Blinder, A. 1986. More on the Speed of Adjustment in Inventory Models. Journal of Money, Credit, and Banking, 18, 355-365.
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Bontempi, M. and I. Mammi. 2015. Implementing a strategy to reduce the instrument count in panel GMM. Stata Journal, 15, 1075-1097.
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Canova, F. and L. Sala. 2009. Back to Square One: Identification Issues in DSGE Models. Journal of Monetary Economics, 56, 431-449.
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Choi, W. 1999. Estimating the Discount Rate Policy Reaction Function of the Monetary Authority. Journal of Applied Econometrics, 14, 379-401.
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Clarida, R., J. Gali, and M. Gertler. 2000. Monetary Policy Rules and Macroeconomic Stability: Evidence and Some Theory. Quarterly Journal of Economics, 115, 147-180.
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Coibion, O. and Y. Gorodnichenko. 2012. Why are Target Interest Rate Changes So Persistent? American Economic Journal: Macroeconomics, 4, 126-162. Consolo, A. and C. Favero. 2009. Monetary Policy Inertia: More a Fiction than a Fact? Journal of Monetary Economics, 49, 1161-1187. Cragg, J. and S. Donald. 1993. Testing Identifiability and Specification in Instrumental Variable Models. Econometric Theory, 9, 222-240. Dueker, M. 1999. Measuring Monetary Policy Inertia in Target Fed Funds Rate Changes. Federal Reserve Bank of St. Louis Review, 81, 3-10. 28
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