Exchange Rates and Unconventional Monetary Policy Vania Stavrakeva

Jenny Tang

London Business School

Federal Reserve Bank of Boston

This draft: January 19, 2018 Abstract Among policymakers, the folk wisdom prevails that monetary policy lowering interest rates of a country relative to another’s will depreciate the exchange rate. Such beliefs led to accusations against the Federal Reserve of engaging in “currency wars” during the Great Recession. While there is some evidence that this folk wisdom holds prior to the zero lower bound (ZLB) period and during the late-ZLB, the relationship is the opposite over the early-ZLB. Using decomposition of exchange rates that’s disciplined by survey data, we trace the structural breaks primarily to changes in the relationship between monetary policy and currency risk premia. We present a model showing that these results are consistent with a strong signaling channel of monetary policy over the early-ZLB. Over that period, due to higher economic uncertainty and Fed communications indicating weak economic conditions, unconventional US monetary policy that lowered long-term US interest rates was interpreted as a signal of future low economic growth. This led to higher risk aversion and demand for US safe assets, thus decreasing the future required return for holding safe dollar assets and appreciating the dollar.

Emails: [email protected], [email protected]. The views expressed in this paper are those of the authors and do not necessarily represent the views of the Federal Reserve Bank of Boston or the Federal Reserve System. We would like to thank the participants and discussants at the following conferences and seminars: Boston University (seminar), University of Exeter (seminar), London School of Economics (seminar), 2nd Oxford–Federal Reserve Bank of New York Conference, Early Career Women in Finance Conference, Federal Reserve System Committee for International Economic Analysis Meeting at the Federal Reserve Board, CEPR International Macroeconomics and Finance Program Meeting at Cambridge University, Workshop on Financial Determinants of Foreign Exchange Rates at the Bank of England. In particular, we would like to thank Domenico Giannone, Gita Gopinath, Stephen Morris, Ali Ozdagli, Paolo Pesenti, Ricardo Reis, H´el`ene Rey, Kenneth Rogoff, Eric Swanson, and Michael Weber for useful comments.

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Introduction

Monetary policy is one of the most powerful policy tools used to influence the economy. It affects real activity by moving asset prices such as exchange rates, among others. Fluctuations in the relative value of a currency impact the competitiveness of a country and, as a result, its real growth. Therefore, understanding the link between monetary policy and exchange rate movements is of first order importance. The conduct of monetary policy has evolved significantly due to the recent Global Financial Crisis that left many countries constrained by the zero lower bound (ZLB) on short-term nominal interest rates. One might expect these important changes in monetary policy to have had an impact on its transmission to asset prices and, in turn, the real economy. The perception among both financial market participants and policymakers is that lowering interest rates, at any horizon, relative to those in another country, through either conventional or unconventional monetary policy, depreciates a country’s currency.1 In Bernanke (2017), former Federal Reserve Chairman Ben Bernanke, confirms this belief held by policymakers around the world and by himself. In particular, he acknowledges that the accommodative unconventional monetary policy of the US during the Global Financial Crisis was perceived to disadvantage other countries and triggered frequent accusations against the Federal Reserve of engaging in “competitive devaluation” or “currency wars”. However, there is insufficient evidence of whether this perception is supported in the data over the ZLB period. This paper systematically studies how monetary policy affects the nominal exchange rate and how the relationship has changed over time. We disentangle the channels through which monetary policy shocks transmit to the exchange rate and propose a theory that can explain the empirical facts that we document. We perform our core analysis at the quarterly frequency and focus on exchange rates of nine developed-country currencies against the US dollar and on US monetary policy shocks. We look at the relationship between exchange rate changes and changes in relative forward rates of different horizons, conditional on US monetary policy shocks.2 To estimate this conditional comovement, we regress exchange rate changes on relative forward rate changes that have been projected on a set of variables capturing US monetary policy surprises. This set of variables comprises of futures price changes over narrow windows 1

In the financial press, one often sees statements such as “The dollar is almost universally expected to appreciate when US interest rates start rising...” (Anatole Kaletsky, “What a US interest rate rise really means for the dollar,” The Guardian, November 17, 2015. Available at http://www.theguardian.com/ business/2015/nov/17/what-a-us-interest-rate-rise-really-means-for-the-dollar). 2 In contrast to the uncovered interest rate parity (UIRP) literature which studies the relationship between exchange rate changes and lagged interest rate differentials, we focus on the contemporaneous relationship (see Engel (2014) for a recent literature review on the UIRP literature).

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around policy announcements. We capture both conventional and unconventional monetary policy by including both near-term futures on the federal funds rate as well as longer-term futures of 3-month Eurodollar rates and futures on long-term US Treasuries. We focus on monetary-policy-induced fluctuations in forward rates of different horizons in order to isolate the policy effect over different segments of the term structure. Furthermore, we use relative forward rates since other countries’ central bank responses to US monetary policy shocks are an important part of the transmission of these shocks to exchange rates. Other papers have also identified shocks at a monthly or quarterly frequency using highfrequency monetary policy surprises to study the impact of monetary policy on macroeconomic variables and other asset prices. Some of the more prominent papers include Bernanke and Kuttner (2005), Gertler and Karadi (2015) and Nakamura and Steinsson (2017). We identify two important structural breaks in the relationship between exchange rates and relative forward rates. The break dates are informed both by formal structural break date estimation and by the timing of significant changes in monetary policy communication and implementation. In particular, we examine three subsamples representing distinct phases of US monetary policy: the period prior to hitting the ZLB, which was dominated by conventional monetary policy, an early-ZLB period that was mainly characterized by “calendarbased” forward guidance and a late-ZLB period that begins roughly with to the switch to “threshold-based” forward guidance. The concepts of “calendar-based” and “thresholdbased” forward guidance are defined in more detail later on. Prior to the ZLB and during the late-ZLB period, decreases in US forward rates relative to foreign forward rates caused by US monetary policy were associated with a depreciation of the dollar for most forward rate horizons.3 However, during the early-ZLB period, a monetary-policy-induced decrease in US medium- and long-term forward rates relative to those of other countries was associated with a statistically significant and strong appreciation rather than depreciation of the dollar. Thus, during the early-ZLB period, our findings go against the folk wisdom that lowering interest rates relative to another country will depreciate the currency. Given the importance of the results for the policy debate, we shed more light in the second part of the paper on a driver of this changing relationship, both empirically and theoretically. To empirically disentangle the channels through which monetary policy affects exchange rates, we decompose nominal exchange rate changes using a simple accounting identity. This decomposition links exchange rate changes to changing expectations of the entire future 3

This is the relationship that policymakers and financial markets expect. This result is also consistent with the literature that finds, using vector autoregressions (VARs) identified via recursive ordering, that exchange rates depreciate in response to monetary policy easing shocks at a quarterly or monthly frequency prior to the Great Recession (see, for example, Eichenbaum and Evans (1995) and Kim and Roubini (2000)).

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paths of policy rates, excess currency returns, and inflation. We compute these components of exchange rate changes using expectations implied by a vector-autoregressive model (VAR) disciplined with forecast survey data, a method introduced in Stavrakeva and Tang (2018). This method delivers expectations at all horizons by essentially interpolating and extrapolating survey forecasts that are available at only a small number of horizons in a way that’s consistent with a particular data-generating process and actual data.4 While similar forecast-augmented estimation techniques have been used to estimate term premia (see Kim and Wright (2005), Kim and Orphanides (2012), Piazzesi, Salamao, and Schneider (2015), and Crump, Eusepi, and Moench (2016)), they have not been used to decompose exchange rates, to the best of our knowledge, with the exception of Stavrakeva and Tang (2018). Using our VAR-based decomposition, we disentangle the conditional relationship between exchange rate changes and relative forward rate changes driven by US monetary policy shocks and examine the source of the structural break. Most of the structural break can be explained by a change in the relationship between forward rates and expected excess currency returns. In particular, this component played a small or insignificant role in the relationship between exchange rates and relative forward rates of various horizons in both the pre-ZLB and lateZLB periods. However, during the early-ZLB sample, monetary-policy-induced decreases in US medium- and long-term forward rates relative to forward rates of other countries were associated with lower expected future returns from holding the dollar, which is reflected in a contemporaneous appreciation of the dollar. The second most important component of exchange rate changes in accounting for the structural breaks in the overall relationship to relative forward rates is the change in expectations over relative future inflation paths. While over all three subsamples, monetary-policyinduced decreases in US forward rates relative to those of other countries were associated with relatively lower inflation expectations in the US, which strengthened the dollar, this relationship was much stronger and statistically significant during the early-ZLB period. To our knowledge, there are only a few papers that empirically investigate the relationship between exchange rates and US monetary policy over the ZLB period. Kiley (2013), Glick and Leduc (2015), and Swanson (2017) do so at high frequencies (intra-day or daily) while Rogers, Scotti, and Wright (2016) examine exchange rate responses at a monthly frequency. 4

Existing papers that have used a similar decomposition (for example Froot and Ramadorai (2005), Engel and West (2005; 2006; 2010), Engel, Mark, and West (2008), Mark (2009), and Engel (2014; 2016), among others) have calculated expectations based on estimating data-generating processes using only actual (i.e., realized) macroeconomic data. We estimate a data-generating process that is more agnostic in some ways and also incorporate survey-based expectations in the estimation. Stavrakeva and Tang (2018) introduces this methodology and provides details on the model fit including comparisons with an estimation that does not use survey data. A growing literature has shown the importance of using survey-based expectations data both in macroeconomics (see Coibion and Gorodnichenko (2015), Malmendier and Nagel (2016) and Gerko (2017)) and finance (see Malmendier and Nagel (2011) and Adam, Marcet, and Beutel (2017)).

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In contrast to our findings, these papers find that the dollar depreciates in response to accommodative unconventional monetary policy over the ZLB period, where the findings of Kiley (2013) are for a sample which largely coincides with our early-ZLB period.5 To test the robustness of our main result over different frequencies and to relate our results to those in the aforementioned papers, we estimate impulse responses using the local projection method developed by Jord`a (2005).6 More precisely, we regress the change in exchange rates from the day before a policy announcement to m days afterwards directly on high-frequency monetary policy surprises. Since the ZLB period included both forward guidance and quantitative easing (QE), we attempt to understand the type of policy driving our main results using two different approaches. First, we split the sample into announcements featuring important QE announcements versus others. In the early-ZLB period, we find that a positive US monetary policy surprise around non-QE announcements results in a very short-lived dollar appreciation that lasts only about a month while the dollar actually depreciates over the longer term. This is consistent with our main results which are based on quarterly changes. The eventual dollar depreciation is large, statistically significant, and very persistent. In contrast, when we look only at important QE announcements, the response of exchange rates to a positive surprise is broadly consistent with the conventional wisdom of a dollar appreciation. As another approach to disentangling forward guidance from QE, we use the conventional, forward guidance and QE factors estimated by Swanson (2017) using high-frequency monetary policy surprises. We find that over the early-ZLB period, the dollar begins to depreciate in response to contractionary forward guidance shocks about a month after the announcement while QE shocks continue to affect exchange rates in a manner consistent with the conventional wisdom.7 These results, when combined, suggest that the relationship that we find in the data is driven primarily by forward guidance. The fact that the exchange rate response can reverse in the days following a policy shock is not very surprising when one considers the empirical and theoretical literature on “slowmoving” capital in asset markets. Duffie (2010) provides a literature review and examples where asset price dynamics, usually up to a month, can be very different from their longer-term responses due to the slow portfolio rebalancing of various financial participants stemming from inattention and institutional constraints, among other reasons. Finally, we propose a theory that can reconcile the empirical facts that we document. We 5

Another difference is that these papers do not use the same decomposition of the exchange rate to directly pinpoint the channels through which monetary policy affects the exchange rate. 6 A recent example of these methods applied to US policy shocks during the ZLB is Swanson (2017). 7 These results corroborate the fact that the structural break estimation identified the end of 2012 as a break date. This corresponds to an important change in the conduct of forward guidance, but not QE.

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explain the structural break in the data using a signaling model of monetary policy that is similar in spirit to Tang (2015). We augment the model by considering currency risk premia determined by an agent with consumption habits in the spirit of Campbell and Cochrane (1999). Given the empirical evidence, we focus on forward guidance as the key source of the signaling channel of monetary policy.8 The intuition of how the signaling channel can deliver the empirical results discussed so far is the following. The central bank is perceived to have better information over the future economic environment. Therefore, forward guidance announcements indicating lower future policy rates can be interpreted as an informative signal of low future growth. If the direct effect of an accommodative forward guidance shock, which leads to higher expected real GDP growth by lowering expected future policy rates, is overshadowed by this signaling channel, then market participants will lower their real GDP growth expectations upon observing an accommodative forward guidance announcements. We test whether the direct effect or the signaling channel dominated in each one of our subsamples by regressing revisions in survey forecasts of future real GDP growth on changes in US forward rates of different horizons caused by US monetary policy shocks. We find that during the early-ZLB period, the behavior of US real GDP growth forecasts indeed reflected, in a highly statistically significant manner, that the signaling channel was stronger than the direct effect. In contrast, the estimates are not statistically significant prior to the ZLB and during the late-ZLB period, consistent with the strength of signaling and direct effects being more balanced. Campbell et al. (2012), Tang (2015), and Nakamura and Steinsson (2017) also provide evidence consistent with accommodative policy revealing central bank private information about worsening economic conditions. This signaling channel theory can match the key empirical findings in the following way. We model currency risk premia using a habit formation framework featuring time-varying risk aversion that is negatively related to real GDP growth. In this setting, lower expected future growth leads to higher expected future risk aversion and, thereby, lower expected future required returns from being long the dollar—a safe haven currency. This is consistent with greater total demand for US safe assets in response to worsened growth expectations, which appreciates the dollar. Likewise, lower US growth expectations will also lower expected future US inflation as long as demand shocks are the main driver of fluctuations. We propose two reasons why the signaling channel might have been stronger over the early-ZLB period. First, the nature of central bank communication changed. Over the 8

Note that our use of the term “signaling” refers to the effect of Delphic forward guidance discussed in Campbell et al. (2012) which is also what Nakamura and Steinsson (2017) refer to as the “information effect”. This is different from the use of “signaling” in the literature examining effects of QE. In that context, it has been used to refer to QE actions signaling a commitment to low future policy rates.

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early-ZLB period, forward guidance in FOMC statements was “calendar-based”, promising low rates until a later date. Most importantly, these statements were qualified with an explanation that the FOMC expected future weak economic conditions would warrant such low rates. In contrast, prior to the ZLB, forward guidance was used less frequently and it was rare for policymakers to give guidance several years into the future as was done during the early-ZLB period.9 During the late-ZLB period, forward guidance announcements contained much less information regarding the economy as the FOMC moved to “threshold-based” guidance, where it was simply stated that policy would remain accommodative as long as the unemployment rate was above 6.5% and inflation was expected to remain below 2.5% in the short run. Such “threshold-based” announcements mainly communicate the Fed’s economic targets and provide less information regarding future economic conditions. Second, another key parameter determining the strength of the signaling channel of monetary policy in our model is the uncertainty about macroeconomic fundamentals relative to uncertainty about exogenous monetary policy shocks. Agents place more weight on the central bank’s announcement when forming expectations over future real GDP growth when this fundamental uncertainty is high. Macroeconomic uncertainty—as measured both by the dispersion of real GDP growth forecasts from the Blue Chip Economic Indicators survey and the estimates of Jurado, Ludvigson, and Ng (2015)—was particularly high over the early-ZLB period, which coincides with the peak of the Great Recession. In contrast, monetary policy uncertainty was lower in the early-ZLB period compared to the pre-ZLB period. Therefore, it is not surprising that the signaling channel of monetary policy was particularly strong over this period, resulting in a dollar appreciation in response to a US policy shock intended to be accommodative. The structure of the paper is as follows. Section 2 presents the main empirical facts regarding the relationship between exchange rates and relative forward rates, conditional on US monetary policy shocks. Section 3 uses our decomposition of exchange rates to disentangle the main results. Section 4 examines the robustness of our main empirical facts at different frequencies and separates the effect of forward guidance from QE. Finally, Section 5 proposes a theory that reconciles all of the empirical findings in the paper and generates additional testable implications that we bring to the data. 9

While conventional monetary policy prior to the ZLB could have had a signaling effect, it is possible that it was less informative about future growth than early-ZLB forward guidance due to the much longer horizon of forward guidance during the ZLB.

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2

Main Empirical Result

In this section, we present the main stylized fact. We examine the relationship between quarterly exchange rate changes and changes in relative forward rates, driven by US monetary policy shocks, at various points along the term structure. We also consider the unconditional relationship in order to gauge whether the relationship driven by monetary policy shocks alone is a good indicator of how these variables comove overall. We start by introducing some notation. The log exchange rate of currency i per US dollar (USD) is denoted by st . Thus, an increase of st corresponds to a depreciation of currency i against the USD. The price of an n-period zero-coupon bond denominated in currency i is Ptn,i . The log per-quarter yield-to-maturity of this same bond is defined as
1 ytn,i = − ln Ptn,i . n The one-year forward rate n periods ahead is given by ftn,i ≡ (n + 4) ytn+4,i − nytn,i .

(1)


The yield on a one-period bond (“short rate”) is also denoted by iit ≡ yt1,i = − ln Pt1,i . Tildes over variables will denote relative quantities defined in terms of country i minus the respective US variable. The frequency that we examine in this paper is quarterly. We study the relationship between exchange rates and forward rates of nine developed countries relative to the US. The countries that we study are: Australia, Canada, Germany/Eurozone, Japan, Norway, New Zealand, Sweden, Switzerland, and the U.K.10 The time period covered is 1990:Q3–2015:Q3, with the start date being determined by availability of long-term zerocoupon yields data. For our baseline results, we estimate panel regressions allowing for currency pair fixed effects and clustering standard errors at the pair level.11 Since yields are an average of forward rates, we focus on forward rates as a way to look more closely at specific segments of the term structure. This is particularly useful for understanding the effects of monetary policy in light of the fact that many unconventional policy actions such as forward guidance or QE have historically had their largest effects on particular regions of the yield curve.12 10 See the Appendix for details on the samples used for each country. The samples differ slightly due to data availability and we also exclude periods where countries had pegged exchange rates. 11 The results are robust to double clustering by currency pair and date. 12 Swanson (2017) finds that the effect of forward guidance typically peaks for yields with a maturity of between one and five years, while QE has its greatest impact on longer maturities, particularly ten years. Greenwood, Hanson, and Vayanos (2015) finds the effects of QE announcements to be the largest for one-year forward rates five and seven years ahead.

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Given that we are interested in how recent unconventional monetary policy has affected the relationship between exchange rates and forward rates, we examine three subsamples representing different phases of US monetary policy. Our pre-ZLB period is 1990:Q3–2008:Q3, our early-ZLB period is 2008:Q4–2012:Q2 and our late-ZLB period is 2012:Q3–2015:Q3.13 These break dates are informed by both the timing of important events as well as formal structural break date estimation procedures. The structural breaks coincide with important changes in the conduct of forward guidance. The first break, 2008:Q4, coincides with the US hitting the ZLB. As a result, 2008:Q4 was also when the Federal Reserve began to heavily use forward guidance with the December 2008 FOMC meeting statement revealing that economic conditions would warrant “exceptionally low levels of the federal funds rate for some time.”14 The second break, 2012:Q3, corresponds with a change in the nature of Federal Reserve forward guidance. More specifically, FOMC statements began moving away from more Delphic forward guidance towards a more Odyssean type of forward guidance, according to the terminology of Campbell et al. (2012). The early-ZLB period is largely characterized by “calendar-based” forward guidance, in which the Federal Reserve specified that it expected economic conditions to warrant keeping the policy rate effectively at zero until a particular future date. Beginning in 2012:Q3, the FOMC statement started introducing “threshold-based” forward guidance where, in 2012:Q4, it specified that policy rates would not lift off from the zero bound until a particular threshold for the unemployment rate was met as long as inflation remained contained.15 As a robustness check, we also use the procedure in Bai and Perron (1998) to find break dates in the relationship between exchange rates and forward rates of different horizons. When we allow for two breaks, 2008:Q4 and 2012:Q3 are closest to the dates most commonly found for longer horizons which are the ones that we are most interested in. See the Appendix for more details on this break date estimation procedure. In addition to forward guidance, the Federal Reserve also used QE over the ZLB. The subsamples chosen based on the different forward guidance regimes are such that the earlyZLB period contains QE1, QE2 and “Operation Twist”while the start of the late-ZLB period roughly coincides with the start of QE3 though it extends past the end of QE3. We find 13

Our results are robust to small changes in the timing of these breaks. Forward guidance was occasionally used prior to that point, particularly during the early-2000s when the federal funds rate was close to 1%, but the period starting in 2008:Q4 was the first time that conventional monetary policy was no longer an available tool. 15 Prior to the introduction of threshold-related language, the September 2012 meeting statement introduced a sentence stating that “a highly accommodative stance of monetary policy will remain appropriate for a considerable time after the economic recovery strengthens [emphasis added].” Some authors have noted this additional phrase as indicating a switch to a more Odyssean, rather than Delphic, style of forward guidance. 14

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results to be generally very similar over the shorter QE1, QE2 and “Operation Twist” subsamples and very different during QE3 while QE3 was quite similar in nature to previous QE programs. This suggests that the important monetary policy structural break for the exchange rate was with respect to the way forward guidance was conducted. We attempt to formally disentangle the effect of the two types of policies in Section 4. Our main results are based on estimating the following regression for one-year forward rates at horizons n ∈ {0, 4, ..., 36}: n ∆st+1 = αf,n + βf,n ∆f˜t+1 + errort+1 ,

(2)

where we use two-stage least squares (2SLS) with a first-stage regression of changes in relative forward rates on high-frequency surprises capturing both conventional and unconventional US monetary policy. More specifically, these surprises are changes of futures prices over a one-hour window around FOMC announcements and important QE announcements. In particular, we use federal funds rate futures expiring 3 months in the future, eurodollar futures expiring 2, 3, and 4 quarters hence, and 2- and 10-year Treasury bond futures expiring in the current quarter.16 There are a number of reasons why we chose to study the effect of US monetary policy on exchange rates using the specification described above. First, one of the important ways in which monetary policy has differed over time is by how much it impacts longterm vs short-term yields or forward rates.17 Therefore, rather than choosing interest rates at particular maturities to act as policy indicators,18 we focus on forward rates at various horizons to more flexibly capture different dimensions of monetary policy. Second, given that this paper focuses on structural breaks in the data, we don’t impose a time-invariant relationship between high-frequency surprises and deep monetary policy shocks which is often assumed in order to estimate a small number of monetary policy factors from high-frequency surprises (see G¨ urkaynak, Sack, and Swanson (2005) and Swanson (2017) for example). Instead, we allow the first-stage regression of forward rate changes on our monetary policy 16

We thank Refet G¨ urkaynak for providing data on federal funds and eurodollar futures. We extend this data and create intra-day changes for Treasury futures using data from Tickdata. 17 For example, the effect of conventional monetary policy on yields depends on the perceived impact of the current policy shock on future policy rates, in other words, the persistence of policy rates. This could have changed over time. The effect of forward guidance depends on the time horizon over which guidance is given. Prior to the ZLB, forward guidance was used mainly to influence expectations about policy decisions likely to be taken in the next couple of meetings. In contrast, forward guidance during the ZLB promised low interest rates for as long as three years, plausibly having an effect much further along the yield curve than pre-ZLB forward guidance. Quantitative easing can also impact different points in the yield curve depending on the maturity of assets being purchased. 18 For example, as in Gertler and Karadi (2015).

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surprises to change over our subsamples.19 Third, the monetary policy of non-US countries can potentially respond to US monetary policy shocks. Therefore, given that the exchange rate is a relative price between two currencies and is affected by the monetary policy in both countries, the more relevant relationship to examine is the one between exchange rate changes and monetary-policy-induced changes in relative forward rate across two countries. Figure 1 plots the slope coefficients from this panel regression along with 90 percent confidence intervals. The regression results are also reported as part of Table 1. In the preZLB sample, we see that an increase in US forward rates relative to foreign forward rates n in response to monetary policy shocks (a negative ∆f˜t+1 ) resulted in an appreciation of the dollar (a positive ∆st+1 ) for most horizons.20 In the early-ZLB sample, we see that for forward rates at horizons eight quarters and above, the relationship flips in a strongly statistically significant way. More specifically, a decrease of US medium- and long-term forward rates relative to foreign forward rates, in response to a US monetary policy shock, is associated with a statistically significant appreciation of the dollar. The result is robust to alternative sets of monetary policy surprises used in the first stage regression. Therefore, instead of stimulating exports, unconventional monetary policy potentially led to lower external demand for US goods over that period. Finally, during the late-ZLB period, the relationship returns to the pattern familiar to policymakers and financial practitioners. We also estimate equation (2) using ordinary least squares (OLS) to obtain the unconditional relationship (see Figure 2 and Table 2). The results are very similar to the 2SLS regressions. The fact that the OLS estimates capture the same structural break and overall pattern implies that US monetary policy shocks are potentially an important driver of the unconditional comovement between exchange rate changes and changes in relative forward rates. In the next section, we examine the forces behind the structural break in the relationship between exchange rates and relative forward rates, conditional on US monetary policy surprise. 19

A cost of this flexibility is that we are unable to isolate different types of monetary policy shocks— conventional, forward guidance, or QE. Instead, what we estimate are relationships conditional on all monetary policy shocks occurring in our sample. For regression results that attempt to disentangle QE from forward guidance, see Section 4. 20 The relationship is the opposite prior to the ZLB when we consider relative forward rates at the short end of the forward rate curve, but this result is not robust to changing the set of monetary policy surprises used in the first stage.

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3

Exchange Rate and Forward Rates Decomposition

This section decomposes exchange rates and forward rates into easy-to-interpret components. First, we define the expected excess return from taking a long position in one-quarter, riskfree dollar-denominated bonds and a simultaneous short position in one-period, risk-free bonds of currency i. The log expected excess currency return from this trade is defined as σt ≡ i$t − iit + Et ∆st+1 .

(3)

If σt is assumed to be zero, then equation (3) gives the strong form of the uncovered interest rate parity (UIRP) condition, where the currency of country i is expected to depreciate when the interest rate of country i exceeds that of the US. In Section 5, we provide one particular model of σt as a currency risk premium. In our empirical work, we do not impose any restrictions on σt that confine it to capture only risk premia. However, for convenience, we will use the two terms—expected excess currency return and currency risk premia— interchangeably. In reality, the expected excess return may capture numerous additional frictions including the inability of traders to borrow at the risk-free government bond rate, counterparty risk, as well as binding net worth or value-at-risk constraints. Using equation (3), the actual change in the exchange rate can be written as ∆st+1 = iit − i$t + σt + ∆st+1 − Et ∆st+1 .

(4)

The expectational error is assumed to be mean zero and uncorrelated with variables in the information set used to form exchange rate expectations in period t. To further delve into this expectational error, we iterate equation (3) forward to obtain st = −Et

∞ X

[˜ıt+k + σt+k ] + Et lim st+k .

(5)

k→∞

k=0

First-differencing equation (5) and combining the resulting expression with equation (3) implies that the expectational error can be expressed as ∆st+1 − Et ∆st+1 = −

∞ X

(Et+1˜ıt+k+1 − Et˜ıt+k+1 )

(6)

k=0

| −

∞ X k=0

|

{z

}

ϕEH t+1

(Et+1 σt+k+1 − Et σt+k+1 ) + Et+1 lim st+k − Et lim st+k . k→∞ k→∞ | {z } {z } ∆E s t+1,∞

F σt+1

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Equation (6) allows us to express realized exchange rate changes as forward-looking variables which, in addition to the period t interest rate differential and expected excess return, also reflect changes in expectations in: (i) the path of relative short-term rates, ϕEH t+1 , (ii) the F , and (iii) long-run nominal exchange rate levels, s∆E path of excess returns, σt+1 t+1,∞ . We ∆E show in Section 3.1 that if the real exchange rate is stationary, st+1,∞ will reflect changes in expectations over long-run relative price levels or the path of future relative inflation. Combining equations (3) and (6) implies that F ∆E ∆st+1 = ˜ıt − ϕEH t+1 + σt − σt+1 + st+1,∞ .

(7)

Since forward rates capture expectations of 3-month bill rates at various horizons, they are tightly linked to the ϕEH t+1 term. To see this, we can decompose one-year forward rates into an interest rate expectations component and a term premium component as follows: EHtf,n,i ≡ T Ptf,n,i

≡

n+3 X k=n ftn,i

Et iit+k ,

(8)

− EHtf,n,i .

Similarly to the currency risk premium term, σt , we do not make further assumptions that confine the T P term to capture only bond risk premia. Combining equations (7) and (8) implies   f,n n F ∆E gf,n g ∆st+1 = −∆f˜t+1 + ˜ıt − ϕEH + ∆ EH (9) t+1 t+1 + ∆T P t+1 + σt − σt+1 + st+1,∞ . We can now use this expression to further disentangle the results from Section 2. The coefficient βf,n from equation (2) can be rewritten as a scaled covariance that can be further decomposed in the following way:   n ˜ Cov ∆st+1 , ∆ft+1   βˆf,n = (10) n V ar ∆f˜t+1     f,n ˜n g g f,n , ∆f˜n Cov ˜ıt − ϕEH + ∆ EH Cov ∆ T P , ∆ f t+1 t+1 t+1 t+1 t+1     = −1 + + n n V ar ∆f˜t+1 V ar ∆f˜t+1     F n ˜n Cov σt − σt+1 , ∆f˜t+1 Cov s∆E , ∆ f t+1,∞ t+1     + + n n V ar ∆f˜t+1 V ar ∆f˜t+1 In the case of a 2SLS estimate, a similar expression holds with the change in relative forward rates being replaced with the fitted value of the relative forward rate change from 12

the first-stage regression. The first term is equal to −1 since the dependent variable is the negative of the regressor in the OLS case and in the 2SLS case, the fitted values are orthogonal to the residuals from the first-stage regression. Thus, if the sum of the remaining components in equation (9) isn’t strongly correlated with the change in relative forward rates, we would expect a negative relationship between relative changes in forward rates and exchange rate changes in the OLS regression while similar logic applies to the 2SLS specification as well. This is consistent with the conventional wisdom that the dollar appreciates when the US increases policy rates (or expectations of future policy rates) relative to other countries. In the extreme case, if monetary policy affected only the relative forward rates (i.e., the sum of all the other covariances in equation (10) from the 2SLS specification is zero), we can interpret equation (2) as a structural relationship which can be estimated using highfrequency monetary policy surprises as instrumental variables. However, as we show in Section 3.2, the sum of these remaining covariances is usually not zero. Given that each one of the scaled covariances is a univariate regression coefficient from n , we can write βˆf,n regressing the exchange rate change components in equation (9) on ∆f˜t+1 in terms of the following regression coefficients where we now explicitly place the dependent variables in superscripts to avoid ambiguity: EH

g f,n

f,n

F

∆E

g st+1,∞ ˜ıt −ϕ +∆EH t+1 σt −σ ∆T P ∆s βˆf,n = βˆf,nt+1 = −1 + βˆf,n t+1 + βˆf,n t+1 + βˆf,n t+1 + βˆf,n .

(11)

∆s The 2SLS and OLS estimates of βˆf,nt+1 , along with their standard errors, are presented ∆s in Tables 1 and 2. One can easily reject the null hypothesis that βˆf,nt+1 = −1 for the earlyZLB period. As we already pointed out in the previous section, most of the coefficients at the medium and long-end of the forward rate curve are positive instead of negative and statistically significantly different from zero during this subsample. Prior to the ZLB and during the late-ZLB, the results are more mixed where we tend to observe that the contribution of the sum of the final four terms in equation (11) is to make ∆s βˆf,nt+1 more negative than −1. However, more often than not, one cannot reject the null ∆s that βˆf,nt+1 = −1 over these subperiods . ∆s We examine the contributions from each of the four terms for the decomposition of βˆf,nt+1 in Section 3.2. Prior to presenting the decomposition results, we first show how these exchange rate and forward rate components are computed in the next subsection.

3.1

Estimating the Components

To compute the terms in the exchange rate change and forward rate decompositions given by equations (7) and (9), we need interest rate, inflation, and exchange rate expectations at 13

all horizons greater than zero. To obtain estimates of these expectations, we model exchange rates, inflation and short-term interest rates as following a VAR process and estimate this VAR using both actual and survey forecast data. We use the same specification as in Stavrakeva and Tang (2018). To ensure that our VAR-implied expectations capture private sector expectations well, we discipline the VAR using survey forecasts of exchange rates, interest rates, and inflation. This method has primarily been used to fit bond yields (see Kim and Wright (2005), Kim and Orphanides (2012), Piazzesi, Salamao, and Schneider (2015), and Crump, Eusepi, and Moench (2016)), but has not been applied, to our knowledge, to the study of exchange rates, besides in Stavrakeva and Tang (2018). We use survey forecasts obtained from Blue Chip and Consensus Economics.21 We estimate the VAR by maximum likelihood along with additional equations that ensure that VAR-based forecasts remain close to survey forecasts. One can think of this specification as a way to interpolate and extrapolate forecasts at horizons not reported in the surveys. Relative to a standard VAR without the use of survey data, the survey-based VAR forecasts exhibit a number of desirable properties. For instance, the implied 3-month bill rate forecasts have fewer violations of the ZLB and do not predict unrealistically premature lift-off from the ZLB. Our model-implied interest rate expectations are also highly correlated with market-based measures of short-term interest rate surprises computed using futures prices by adapting the method used in Bernanke and Kuttner (2005) to a quarterly frequency. For more details on the fit of this VAR, see Stavrakeva and Tang (2018). With the estimated expectations, we can now obtain the terms in equation (9). The exact expressions are given in Stavrakeva and Tang (2018). One key thing to note is that our estimates imply constant expectations over long-run levels of the real exchange rate. Therefore, the change in expected long-run nominal exchange rates, s∆E t+1,∞ , simply reflects the change in expectations over the future path of relative inflation rates (country i’s inflation relative to US inflation).

3.2

Decomposing the Main Result

We now turn to decomposing our main results using equation (11) and our VAR-implied exchange rate and forward rate components. Figure 3 plots the 2SLS estimates of the regression coefficients in (11) while Figure 4 plots ∆s the corresponding OLS estimates. Estimates of βˆf,nt+1 and the remaining terms in (11) are 21

In Section 5, we introduce a model that helps us interpret the empirical results and, in that section, we will also assume that the survey data from these sources captures well the expectations of the marginal investor in our model.

14

presented, along with their standard errors, in Tables 1 and 2.22 In both the 2SLS and OLS regressions, the most striking result is that the changes we ∆s saw in βˆf,nt+1 at the medium and long ends of the curve in Figures 1 and 2 can be almost entirely explained by the changing relationship between the currency risk premia term and σt −σ F changes in forward rates, that is, by the coefficient βˆf,n t+1 . In results not reported here, F we confirm that the behavior of this coefficient is driven primarily by σt+1 , which captures changes in expectations over the future path of one-period excess returns from being long the three-month US bond and short the three-month bond of country i. The fact that the lagged expected excess return between periods t and t + 1, σt , does not play an important role is not too surprising given that σt is not a function of period t + 1 variables. σt −σ F Prior to the ZLB, βˆf,n t+1 was much smaller and generally insignificantly different from zero. However, in the early-ZLB subsample, the coefficient becomes large and significantly positive. A positive coefficient implies that a decrease in long-horizon US forward rates relative to those of other countries was associated with a decrease in expected future excess returns from being long the dollar and short currency i. In the late-ZLB sample, the coefficients become negative for almost all forward rate horizons, though they are significant only for short-horizon forward rates in the 2SLS regression. σt −σ F In addition to βˆf,n t+1 , we also see larger positive coefficients on the long-run exchange s∆E rate expectations term, βˆ t+1,∞ , in the early-ZLB sample relative to the other two samples. f,n

This implies that over the early-ZLB a decrease of the US forward rates relative to the forward rates of other countries was associated with a lower expected inflation path in the US relative to other countries. The relationship is present in both the OLS and 2SLS specifications. There is no consistent pattern in the contributions of the remaining two terms to the change in the overall coefficient across subsamples. For the most part, the contribution from the nominal rates term is negative, as would be the case if policy rates are persistent. The size of this coefficient differs across subsamples, but not in a way that is consistent with the changes in the overall coefficient. The contribution from the term premium coefficient is always positive and small, with coefficients being slightly smaller for the shortest horizons. Given that the term premia and the expectations hypothesis components of forward rates are generally positively correlated, this last result is not surprising. To summarize, the decomposition presented in Section 3.1 shows that the positive and ∆s statistically significant βˆf,nt+1 estimates for medium and longer-horizons during the earlyZLB period can be attributed predominantly to changes in the relationship between forward rates and expected future currency premia. To a lesser degree, the structural break can be 22

Note that even though the dependent variables are estimated, this does not impact the standard error calculation since we do not have estimated regressors.

15

also explained by a change in the relationship between forward rates and expectations over relative inflation paths.

4

Other Frequencies

In this section, we link our results to the literature examining the effect of recent unconventional monetary policy on exchange rates at frequencies higher than quarterly. We do so using exercises that are also intended to disentangle whether the surprising empirical result that we find in Section 2 during the early-ZLB is due to QE or to forward guidance. In particular, we estimate impulse responses using the local projection approach of Jord`a (2005), which has recently been used by Swanson (2017) to estimate the effect of conventional and unconventional monetary policy on various asset prices. In particular, we estimate the following x sτ +m − sτ −1 = αm + βm xτ + errorτ,m , (12) where st is the exchange rate measured at a daily frequency and dates indexed by τ are those on which there is a monetary policy announcement. xτ is the intra-day change of a particular future contract’s price around these announcements. In this section, we examine federal funds rate futures expiring 3 months in the future, eurodollar futures expiring one year hence, and 2- and 10-year Treasury bond futures expiring in the current quarter, a representative subset of the surprises used in the first-stage regressions in Sections 2 and 3.2. Our first approach to disentangling QE from forward guidance effects involves estimating regression (12) separately for important QE dates and other dates, which we refer to as QE and non-QE dates, respectively. Prior to the ZLB, estimates using non-QE dates capture the responses to both conventional monetary policy and forward guidance shocks. Over the ZLB, many FOMC announcements contained both statements relevant to QE operations and ones that can be interpreted as forward guidance. We attempt to isolate the effect of forward guidance alone by removing the most important QE announcements from the regression. To the extent that the lack of a statement about QE during a particular announcement could be a surprise to markets, these results should be interpreted as merely suggestive.23 x Figures 5 and 6 plot βm against m for non-QE dates and QE dates, respectively. We find that, prior to the ZLB and during the late-ZLB period, a positive US monetary policy surprise, measured using any of the futures price changes listed above, was associated with an appreciation of the dollar. This is true for both non-QE and QE announcements. This also characterizes the response of exchange rates to important QE announcements during 23

The list of important QE announcements can be found in Appendix C and is collected from existing papers including Rogers, Scotti, and Wright (2014), Wu (2014), and Swanson (2017).

16

the early-ZLB sample. Moreover, these appreciations generally persist up to the maximum horizon that we examine, which is 120 days. This result is consistent with the conventional wisdom. However, during the early-ZLB, when we exclude important QE announcements, a positive US monetary policy surprise was associated with a dollar appreciation that lasted for only about 30 days following the announcement. This appreciation is small and often statistically insignificant. In contrast, after the initial 30 days, the response of the exchange rate reverses. The subsequent dollar depreciation is large, statistically significant, and very persistent. These results suggest that the behavior we find in the early-ZLB subsample is driven by forward guidance rather than QE and that the structural break that we find in the quarterly data regressions is present for a wide range of policy relevant frequencies. The second approach that we use to disentangle QE from forward guidance effects is to use the factors identified in Swanson (2017). Using principle component analysis and identifying restrictions, Swanson (2017) extracts three series from high-frequency monetary policy surprises—LSAP which captures QE, FG which is the forward guidance component and FFR which measures conventional monetary policy.24 Figure 7 presents the coefficients from estimating regression (12) where xτ is now a vector including all relevant factors for a x particular subsample and βm is a vector of coefficients on these factors. We exclude the FFR factor from the regression in the ZLB subperiods, but the results are robust to including all three factors. Note that, for ease of interpretation, we negate the LSAP factor relative to Swanson (2017) so that a positive LSAP factor corresponds to positive surprises in interest rate futures. The results are very similar to what we find by separating policy announcements into QE and non-QE dates. During the early-ZLB sample, positive QE surprises (captured by increases in the LSAP factor) were associated with dollar appreciation. Positive forward guidance surprises were associated with a short-lived dollar appreciation lasting about 30 days that was followed by a large, persistent and statistically significant depreciation. Therefore, we conclude that a structural break in how forward guidance affects the exchange rate is the reason that we observe the structural breaks in our quarterly regressions. In the next section, we propose a theory that would explain this structural break. The changing nature of the exchange rate response as the response horizon m lengthens is consistent with the literature on “slow-moving” capital (see Duffie (2010) for a review of the literature). Similar behavior can be observed in other asset prices. For example, Hanson, Lucca, and Wright (2017) study the relationship between changes in short- and long-term forward rates and also show that the sign flips as one lengthens the time period over which 24

The restrictions are that both forward guidance and QE shocks have no effect on the current federal funds rate while the variance of the LSAP component prior to the ZLB is minimized.

17

the change is considered. In particular, the sign change in their paper occurs somewhere between a month and a quarter. To rationalize their results, they present a model which exogenously assumes that a fraction of traders rebalance infrequently. The theory that we propose in the next section to rationalize our results is that forward guidance had a strong signaling channel regarding economic growth over the early-ZLB. While we do not explicitly introduce slow-moving capital in our model, and, thus, do not match the exchange rate response up to m = 30 days, we believe that the investors who have the resources to interpret monetary policy signals are most likely large institutional investors who are slow to re-optimize their portfolio following a monetary policy announcement.

5

Theoretical Interpretation: Signaling Channel of Monetary Policy

In this section, we present a theory that can reconcile the empirical findings from the previous sections. We explain the structural break in the data using a signaling model of monetary policy that is similar in spirit to Tang (2015). We augment Tang (2015) by modeling currency risk premia using a habit formation model in the spirit of Campbell and Cochrane (1999). The model is partial equilibrium and purposefully kept simple in order to make the signaling effect of monetary policy clear.

5.1

Model

We consider the US to be the home country and assume that real US GDP growth and inflation follow exogenous processes given by πtus = α∆ytus y us ∆ytus = −ν (ius t − πt ) + ε t ,

where α, ν > 0. πtus and ∆ytus denote log US inflation and real GDP growth while ius t is the US net nominal policy rate. Real GDP growth is assumed to be decreasing in a real interest y us rate, ius t − πt , and is subject to i.i.d. mean zero normally-distributed demand shocks, εt . Assuming that α > 0 and not allowing an additional shock to the inflation equation is a way to capture an economy mostly affected by demand shocks. We assume that the policy rate follows a Taylor rule, y us π us i ius t = φ ∆yt + φ πt + εt ,

18

where φy , φπ > 0 and εit is an i.i.d. mean zero normally-distributed monetary policy shock that is uncorrelated with the demand shock. We do not impose a ZLB on the interest rate to preserve the simplicity of the model. However, Andrade et al. (2017) obtain qualitatively similar results in a global game setting with an explicitly-imposed ZLB where the central bank’s policy tool is an announcement about a future ZLB lift-off date.25 Using these three equations, we can express ∆ytus and ius t as functions of exogenous shocks εyt − νεit η + νκ κεyt + ηεit = ius t η + νκ y where κ = φ + φπ α > 0 and η = 1 − να. ∆ytus =

We assume that η > 0, ensuring that a positive interest rate shock indeed increases the nominal rate. That is, we assume that the positive monetary policy shock does not cause large enough contemporaneous falls in inflation and real GDP growth to lead to the nominal interest rate falling due to the endogenous policy reaction to these variables. Next, consider a representative agent located in the US with preferences that allow for time-varying currency risk premia due to time-varying risk aversion following the habit formation literature (see Campbell and Cochrane (1999), for example). Campbell and Cochrane (1999) and the ensuing habit formation literature show that such preferences can match several asset pricing facts. More specifically, assume that the representative agent’s per-period utility function is (C us −X us )1−γ . The consumer can invest in default-free bonds denomigiven by u (Ctus , Xtus ) = t 1−γt us uc (C us ,Xt+1 us ) −πt+1 nated in USD and in the foreign currency. Given that her nominal SDF is β uc (Ct+1 , us ,X us ) e t t the agent’s Euler equation is given by "
 #  us us uc Ct+1 , Xt+1 S us t e−πt+1 (1 + ius (1 + it ) It = 0 (13) E β t )− us us uc (Ct , Xt ) St+1 where St+1 is the level of the nominal exchange rate defined as units of a foreign currency per USD. it is the foreign net nominal policy rate and It is the period t information set of the investor which will be defined below. us us uc (C us ,Xt+1 ) The real stochastic discount factor can be re-written as β uc (Ct+1 = βeγ (∆ρt+1 −∆ct+1 ) , us ,X us ) t t  us    Ct ucc (Ctus ,Xtus ) Ctus where ρt ≡ ln − γ uc (C us ,X us ) = ln C us −X us is the log of the scaled relative risk aversion t t t t coefficient and cus is log US real consumption. In order to obtain analytical results, we t+1 25

The main contribution of Andrade et al. (2017) is theoretical, with a focus on the signaling effects of monetary policy when there is heterogeneity across agents in beliefs. Their framework is richer in some aspects than the one presented here, but they do not model risk premia.

19

us assume that the cross-country debt holdings are negligible, implying that cus t ≈ yt , which would be the limiting case when the economy is approximately closed. This can be a good approximation for large countries such as the US. We also assume that ρt has the following data generating process, which is standard in the habit formation literature, us ρt+1 = θρt − λρt ∆yt+1 , where θ < 1 and λ > 0.

(14)

Assuming that equation (14) holds is an implicit assumption on the functional form of Xtus .26 us , cus Given the assumptions made, πt+1 t+1 and ρt+1 are normally distributed, conditional on time t information. We further assume that the log nominal exchange rate, st+1 , is also conditionally normally distributed and use equation (13) to express the expected excess return of being long the dollar-denominated bond and short the foreign currency-denominated bond as: σt = ius t − it + Et ∆st+1
V ar (∆st+1 |It ) us = − Cov ∆st+1 , −γ∆cus t+1 + γ∆ρt+1 − πt+1 |It . 2 Using the process for real GDP growth, we can write this expected excess return as: σt =


V ar (∆st+1 |It ) us + (γ + α + γλρt ) Cov ∆st+1 , ∆yt+1 |It . 2


We assume that the data-generating process of ∆st+1 is such that Cov ∆st+1 , εyt+1 |It ,

us Cov ∆st+1 , εit+1 |It , and V ar (∆st+1 |It ) are constant, implying that Cov ∆st+1 , ∆yt+1 |It is constant as well. In that case, the only source of time-variation in the expected excess
us |It term. Conditional on dollar appreciation being return is the γλρt Cov ∆st+1 , ∆yt+1
us accompanied by low real GDP growth in the US, i.e., Cov ∆st+1 , ∆yt+1 |It < 0, higher risk aversion in period t is associated with a lower expected excess return from holding the dollar between t and t + 1. For evidence that this is the case in our sample, see Table 3.27

5.2

The Effect of Forward Guidance

To keep the analysis as simple as possible, we assume that at time t + 1, the central bank knows the state of the economy and the monetary policy surprise in period t+h for h ≥ 2. In In the habit formation literature, it is the norm to specify a data generating process for ρt or ρ1t instead of Xtus (see Campbell and Cochrane (1999) and the discussion in Brandt and Wang (2003)). 27 The only key assumption that we need regarding the exchange rate is that the dollar is a good hedge. The partial equilibrium nature of the model allows us not to take a stand on whether markets are complete or incomplete, thus making it easier to identify the properties of exchange rates that are crucial for matching the empirical findings with our theory. 26

20

other words, the central bank can observe εit+h and εyt+h .28 We consider a forward guidance announcement to be the central bank’s truthful expectation of ius t+h . Given that there is no persistence in the variables affecting the policy rate, this is equivalent to the central bank announcing the actual policy rate h periods from now. Denote by at+1 the announcement in period t + 1. Given the assumptions made, at+1 = ius t+h . Assume that the change in the one-period relative forward rate, defined as the non-US rate minus the US rate, prevailing between periods t + h and t + h + 1 and caused by the F 29 ˆσt −σt+1 and . Then, our estimates of β announcement at+1 , is approximately equal to −ius t+h f,n F s∆E ∂ (σt −σt+1 ∂s∆E ) t+1,∞ t+1,∞ ˆ βf,n in Section 3.2 correspond to the derivatives − ∂at+1 and − ∂at+1 in the model.30 To compute these derivatives within the model, we assume that the agent’s time t information set contains current and past values of shocks and announcements, It ≡ {at , εy,t , εi,t }. First, we derive the posterior expectation of future real GDP growth which, as we will show, is the main driver of the changes in expectations over future currency risk premia and over the relative inflation paths in response to monetary policy shocks. The agent’s expectation of ∆yt+h involves a signal extraction problem. Since the future policy rate is a function of both future monetary policy and demand shocks, actual realizations of both shocks are not completely revealed by the forward guidance announcement. However, the agent uses this announcement to extract information about εyt+h and εit+h , which then informs her expectation about ∆yt+h . Using the posterior expectations of the two shocks in t + h, which are presented in the Appendix, one can show that V ar(εyt+h )

E [∆yt+h |It+1 ] = Kat+1

− νη (εit+h ) = at+1 , V ar(εy ) κ2 V ar εit+h + η 2 ( t+h ) κ V ar

(15)

where E [∆yt+h |It ] = 0 due to the shocks being i.i.d. V ar(εy ) Note that when V ar εit+h = 0, the agent believes that the forward guidance announcement ( t+h ) η i is driven only by a future exogenous monetary policy shock, i.e., at+1 = ius t+h = η+νκ εt+h . In this case, the effect of the announcement on GDP growth expectations is given by − νη < 0, capturing only the direct effect of the future interest rate shock on expected real GDP growth, where a negative interest rate surprise improves GDP growth expectations. 28

To obtain our results, it’s sufficient for the agents who trade these assets to believe that the central bank has superior information about εit+h and εyt+h . We choose to allow the central bank to perfectly observe these shocks for ease of exposition. 29 For this assumption to hold, the movement of the other country’s forward rate and the relative term premia of both forward rates in response to US forward guidance should be zero. These assumptions can be relaxed and are made primarily for tractability. 30 With i.i.d. shocks, the expectation for it+h before the announcement is zero. Thus, derivatives with respect to the announcement itself are equivalent to those with respect to the monetary policy surprise.

21

The signaling channel appears when increasing in

V ar(εyt+h ) V ar(εit+h )

V ar(εyt+h ) V ar(εit+h )

. For a sufficiently high

> 0. Given our parameterization, K is

V ar(εyt+h ) V ar(εit+h )

(i.e., a sufficiently strong signaling

channel), K can become positive, meaning that an announcement of a lower future policy rate can lower expectations over future real GDP growth. V ar(εy ) < νη More generally, if V ar εt+h , then the direct channel dominates (K < 0), and if κ ( it+h ) the opposite is true, the signaling channel dominates (K > 0). This result is intuitive as large real GDP growth uncertainty (i.e., larger variance of the prior distribution of the real GDP growth shock) implies that the agent will place more weight on the public signal of real GDP growth, at+1 . In this paper, when we say that the signaling channel is strong, we implicitly mean that the signaling channel is strong enough to dominate the direct effect of interest rate movements on real GDP growth, implying that an announcement of a lower future policy rate is associated with a lower expected future real GDP growth. That is, in our terminology, a strong signaling channel of forward guidance implies that the response of a GDP forecast revision to the announcement is positive, i.e., K > 0. We test the hypothesis that the signaling channel was strong over the early ZLB but not strong enough prior to and after that period by estimating K for the three subperiods. We regress revisions in survey forecasts of real GDP growth 4 quarters from now on changes in US forward rates, conditional on monetary policy shocks.31 Indeed, we find a very strong and statistically significant signaling effect of monetary policy over the early-ZLB and no such effect prior to and after that period. Table 4 presents the regression results. Next, we derive above-mentioned derivatives of our exchange rate components with ret+h |It+1 ] . We start spect to the announcement and show that they are tightly linked to ∂E[∆y ∂at+1 F with the one for σt − σt+1 . First, note that since σt contains information up to only t, F F ∂ (σt −σt+1 ) ∂σt+1 − ∂at+1 = ∂at+1 . In the Appendix, we show that F ∂ σt − σt+1 − ∂at+1




∞

F X ∂ ∂σt+1 = = (E [ρt+k+1 |It+1 ] − E [ρt+k+1 |It ]) γλσ s,y ∂at+1 ∂at+1 k=0

= −

∂E [∆yt+h |It+1 ] γλ2 σ s,y E [ρt+h−1 |It+1 ] . ∂at+1 1−θ | {z } K


us where σ s,y denotes the constant value of Cov ∆st+1 , ∆yt+1 |It and in the data σ s,y < 0. We further assume that the realizations of shocks in our sample are such that expected future log relative risk aversion is always nonnegative, i.e., that E [ρt+h−1 |It+1 ] ≥ 0.32 In that case, 31

More precisely, the forecast revision is the change between the lagged 4-quarter-ahead forecast and the current 3-quarter ahead forecast, thus keeping the forecasted quarter fixed. 32 This amounts to assuming a sufficiently low λ, high θ and positive initial value, ρ0 .

22

if the signaling channel is strong, K > 0, as it appears to be the case during the early-ZLB period, a negative forward guidance shock lowers the expectations of future growth. This, in turn, increases the expectations of future risk aversion and lowers the expected excess return σt −σ F from being long the dollar. Thus, our finding that βˆf,n t+1 > 0 during this subsample is consistent with the signaling channel being strong over this period. Another related testable implication that emerges is that expectations of risk aversion should have increased in response to announcements of lower future policy rates. We test that by regressing quarterly changes in the VIX on changes in US forward rates due to US monetary policy surprises. The results are given in Table 5. We find that, consistent with a strong signaling channel, in the early ZLB, a fall in US forward rates induced by monetary policy shocks led to a strongly significant increase in the VIX.33 This brings additional evidence in support of the fact that it is time variation in currency risk aversion, due to the signaling channel of US monetary policy, that contributed to the appreciation of the dollar during the early-ZLB in response to accommodative forward guidance announcements. Finally, we derive the effect of the monetary policy surprise on the long-run nominal exchange rate component, the second term of the exchange rate decomposition that contributed
P∞ us i us i to the structural break. Since s∆E t+1,∞ = k=1 E[πt+k − πt+k |It+1 ] − E[πt+k − πt+k |It ] for some foreign country i, if we assume that US monetary policy shocks do not affect expectations regarding inflation in other countries, then ∞  us   us 
∂s∆E ∂ X ∂E [∆yt+h |It+1 ] t+1,∞ − = E πt+k |It+1 − E πt+k . |It = α ∂at+1 ∂at+1 k=1 ∂at+1 | {z } K

Once again, understanding the effect of the forward guidance surprise on real GDP growth expectations is sufficient to understanding the second key derivative in the model. Since we assumed that the economy is driven primarily by demand shocks, lower real GDP growth is associated with lower inflation. The result that a negative forward guidance announcement leads to lower expectations of future real GDP growth and inflation, when the signaling ∂s∆E channel is strong, follows directly. Thus, our finding that − ∂at+1,∞ > 0 over the early-ZLB t+1 subsample is again consistent with a strong signaling channel over that period. In summary, we show that the hypothesis of a strong signaling channel over the earlyZLB, when combined with preferences featuring habit formation, can explain the relationship between exchange rate changes or its components and relative forward rates, conditional on 33

The VIX can be perceived as a proxy of today’s risk aversion while the model with exogenous output implies that forward guidance affects expectations of future risk aversion. However, if we were to relax the assumption that output is exogenous, the real GDP growth today will be a function of expected real GDP growth in the future through the Euler equation, implying that today’s risk aversion will also change due to forward guidance.

23

monetary policy shocks, that we observe in the data. Moreover, the model predictions are also consistent with other empirical facts that we document, such as accommodative forward guidance policy over the early-ZLB leading to downward revisions of US GDP growth forecasts and to higher VIX. Our results raise the question of why the signaling channel was so much stronger than the direct channel of monetary policy over the early-ZLB period. We can use the model to V ar(εy ) is try to answer this question. In the model, the signaling channel is strong when V ar εt+h ( it+h ) sufficiently high. Evidence that this ratio was particularly high during the early-ZLB can be found by examining average values of macroeconomic and monetary policy uncertainty measures over the three subsamples, which are presented in Table 6. The macroeconomic uncertainty measures we examine are the 12-month-ahead macroeconomic uncertainty estimated by Jurado, Ludvigson, and Ng (2015) and dispersion in 4-quarter-ahead US real GDP forecasts from Blue Chip Economic Indicators. The monetary policy uncertainty measure that we examine is the monetary policy subcomponent of the Baker, Bloom, and Davis (2016) index. Note that both measures of macroeconomic uncertainty were higher in the early-ZLB subsample versus other subsamples. Furthermore, monetary policy uncertainty actually declines slightly in the early-ZLB period and further still in the late-ZLB period.34 V ar(εy ) This is consistent with V ar εit+h being particularly high in the early-ZLB period relative to ( t+h ) both of the other two subsamples. Another reason why the signaling channel was weaker during the late-ZLB could be that the FOMC’s switch away from ”calendar-based” towards ”threshold-based” forward guidance reduced the amount of information about future economic conditions contained in its communications. In the model, ”threshold-based” forward guidance can be interpreted as announcements about the functional form and coefficients of the policy rate rule rather than future interest rates themselves. Such announcements provide no signal regarding the future state of the economy. This could explain why we observe stronger signaling channel over the early-ZLB relative to the late-ZLB. At the same time, forward guidance was rarely used prior to the ZLB and it is possible that conventional monetary policy was, on average, less informative or that the direct effect of monetary policy was stronger prior to the ZLB. This would explain why we don’t detect a relatively strong signaling channel, on average, prior to the ZLB. 34

Note that this measure of monetary policy uncertainty would capture uncertainty both about the exogenous monetary policy shock as well as the systematic component of monetary policy, which is a function of the demand shock in our model. However, the divergence of macroeconomic and monetary policy uncertainty during the early-ZLB subsample suggests that uncertainty about the monetary policy shock likely declined.

24

5.3

Discussion of the Model

In reality, central banks make announcements regarding a whole path of future policy rates. In our model, this can be captured by allowing each Fed announcement to be about all  us for some h ≥ 1. Note that in , ..., i policy rates up to some date t + h, at+1 = ius t+1 t+h the current setting with the central bank having perfect information about future shocks, the new information revealed in at+1 relative to the information in at will be only it+h . So revealing at+1 still leads to an update in the agent’s belief only about t + h variables. Thus, the results above continue to hold with a slight difference that our empirical results are now ∂ (σt −σ F ) ∂s∆E estimates of the derivatives, − ∂ius t+1 and − ∂it+1,∞ . us t+h t+h If we instead allow the central bank to receive noisy signals in each period about future shocks, then each at+1 will reveal the central bank’s evolving posterior beliefs about the     us CB  CB path of future rates, i.e., at+1 = E ius |I , ..., E it+h |It+1 . In this setting, revealing t+2 t+1 at+1 leads to an update in the agent’s belief about all variables at times t + 2 through t + h. However, the signal extraction problem works in an analogous way with the agent updating beliefs about time t + k variables by splitting the observed announcement update,    us CB  CB E ius |I − E it+k |It , into updates in beliefs about the demand shock, monetary t+k t+1 policy shock, and shocks to the central bank’s information. Thus, derivatives of GDP forecast F and s∆E revisions, σt − σt+1 t+1,∞ with respect to changes in forward rates caused by central bank policy announcements will be qualitatively similar with a sufficiently strong signaling effect producing derivatives of the signs discussed above.

6

Conclusion

In this paper, we revisit an old but important question—what is the effect of monetary policy on the nominal exchange rate? With recent changes in the conduct of monetary policy, disentangling the effect of monetary policy has become more challenging. By using conventional and unconventional monetary policy, a central bank can affect both the short and long ends of the yield curve. For that reason, in this paper, we consider a broader range of interest rates targeted by the central bank when examining the relationship between monetary policy and exchange rates. We find that, during the ZLB, monetary policy that moved the medium and long ends of the yield curve indeed had an impact on exchange rates. Over the early-ZLB period, a policy-induced decrease of medium- and long-term US forward rates relative to those of other countries led to a contemporaneous appreciation of the dollar while the opposite was true over the late-ZLB period. This fact can also be found by directly estimating exchange rate responses to monetary policy shocks using high-frequency data and

25

can be attributed to forward guidance rather than to QE. This implies that forward guidance actions which tried to stimulate the economy by lowering yields over the early-ZLB period, while the economy was in the depths of the Great Recession, had the unintended consequence of suppressing US external demand by causing the dollar to appreciate. We present a model showing that this surprising result is consistent with a particularly strong signaling effect of monetary policy during the early-ZLB period. This strong signaling effect leads to policy-induced falls in US forward rates being interpreted as signals of worsening economic conditions. This, in turn, raises agents’ expected risk aversion which lowers the required excess return from being long the dollar (the hedge currency) and short the foreign currency, which is consistent with what we find in the data. As a result, lowering US forward rates relative to those of other countries leads to the dollar appreciating rather than depreciating. Another channel through which a strong signaling effect of monetary policy contributes to the structural breaks we document is through inflation expectations. In particular, we find that policy-led falls in US forward rates relative to those of other countries lowered inflation expectations more strongly during the early-ZLB period, as markets interpreted the lower rates as signaling weaker-than-anticipated future demand. Lower inflation expectations in the US led to expectations that the dollar will appreciate over the long run, which, due to the fact that the exchange rate is a forward-looking variable, translates into a stronger dollar today.

26

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31

Figures and Tables Figure 1: Fixed Effect Panel 2SLS Estimates For Exchange Rate Changes Regressed on Relative Forward Rate Changes

-20

-10

0

10

20

30

Drelfwd, IV: FF4 ed2-ed4 fut2y fut10y

0

4

8

12

1990:Q2 - 2008:Q3

16 20 Horizon

24

2008:Q4 - 2012:Q2

28

32

36

2012:Q3 - 2015:Q3

Note: 90% confidence intervals based on standard errors clustered by country pair. Surprises used: Price changes in a 1-hour window around FOMC and QE announcements of federal funds rate futures expiring 3 months hence, eurodollar futures expiring 2, 3, and 4 quarters hence, and 2- and 10-year Treasury bond futures expiring in the current quarter.

Figure 2: Fixed Effect Panel OLS Estimates For Exchange Rate Changes Regressed on Relative Forward Rate Changes

-15

-10

-5

0

5

10

Drelfwd, OLS

0

4

8

1990:Q3 - 2008:Q3

12

16 20 Horizon

24

2008:Q4 - 2012:Q2

28

32

36

2012:Q3 - 2015:Q3

Note: 90% confidence intervals based on standard errors clustered by country pair.

32

Figure 3: Fixed Effect Panel 2SLS Estimates For Exchange Rate Change Components Regressed on Relative Forward Rate Changes Drelfwd, IV: FF4 ed2-ed4 fut2y fut10y

0

4

8

12

16

20

24

28

32

36

10 -20

-10

0

10 0 -10 -20

-20

-10

0

10

20

2012:Q3 - 2015:Q3

20

2008:Q4 - 2012:Q2

20

1990:Q2 - 2008:Q3

0

Other Interest Rates

4

8

12

16

20

24

28

Term Premium

32

36

0

4

LR Exch Rate

8

12

16

20

24

28

32

36

Excess Returns

Note: Bars with darker shading represent estimates that are significant at the 10% level based on standard errors clustered by country pair. Surprises used: Price changes in a 1-hour window around FOMC and QE announcements of federal funds rate futures expiring 3 months hence, eurodollar futures expiring 2, 3, and 4 quarters hence, and 2- and 10-year Treasury bond futures expiring in the current quarter..

Figure 4: Fixed Effect Panel OLS Estimates For Exchange Rate Change Components Regressed on Relative Forward Rate Changes Drelfwd, OLS

0

4

8

12

16

20

24

28

32

36

Other Interest Rates

5 -10

-5

0

5 0 -5 -10

-10

-5

0

5

10

2012:Q3 - 2015:Q3

10

2008:Q4 - 2012:Q2

10

1990:Q3 - 2008:Q3

0

4

8

12

16

Term Premium

20

24

28

32

36

LR Exch Rate

0

4

8

12

16

20

24

28

32

36

Excess Returns

Note: Bars with darker shading represent estimates that are significant at the 10% level based on standard errors clustered by country pair.

33

1-Year Ahead Eurodollar Future

3-Month Ahead Fed Funds Future

Figure 5: Daily Impulse Responses of Exchange Rates to Monetary Policy Surprises on Non-QE Dates

Current 2-year Treasury Future

09/13/12 - 10/28/15

0 -0.5 -1 0

30

60

90

120

1

2

6

0.5

0

4

0

-2

2

-0.5

-4 0

30

60

90

120

0 0

1

2

0.5

0

0

-2

30

60

90

120

0

30

60

90

120

0

30

60

90

120

0

30

60

90

120

4

2

-0.5

-4 0

Current 10-year Treasury Future

12/16/08 - 08/31/12

02/08/90 - 10/29/08

0.5

30

60

90

120

0 0

30

60

90

120

1

2

3

0.5

0

2

0

-2

1

-0.5

-4 0

30

60

90

120

0 0

30

60

90

120

Note: 90% confidence intervals based on standard errors clustered by country pair.

34

Figure 6: Daily Impulse Responses of Exchange Rates to Monetary Policy Surprises on Important QE Dates 12/16/08 - 08/31/12

1-Year Ahead Eurodollar Future

2 1

10

0

0

-1

-10

-2

-20

Current 2-year Treasury Future

0

30

60

90

120

4

10

2

5

0

0

-2

0

30

60

90

120

0

30

60

90

120

0

30

60

90

120

-5 0

Current 10-year Treasury Future

09/13/12 - 10/28/15

20

30

60

90

120

1.5

4

1

2

0.5

0

0

-2 0

30

60

90

120

Note: 90% confidence intervals based on standard errors clustered by country pair.

35

Figure 7: Daily Impulse Responses of Exchange Rates to Monetary Policy Surprises on Swanson (2017) Factors 02/08/90 - 10/29/08

0.4

12/16/08 - 08/31/12

09/13/12 - 10/28/15

FFR

0.2 0 -0.2 0

30

60

90

120

1

2

2

1

FG

0.5

1

0 0 -0.5

-2 0

30

60

90

120

2

LSAP

0

-1

-1 0

30

60

90

120

1.5

30

60

90

120

0

30

60

90

120

1

1

1

0

0.5

0.5 0

0

0

-1

-0.5 0

30

60

90

120

-0.5 0

30

60

90

120

Note: 90% confidence intervals based on standard errors clustered by country pair. For ease of interpretation, we negate the LSAP factor relative to Swanson (2017) so that a positive LSAP factor corresponds to positive surprises in interest rate futures.

36

Table 1: Fixed Effect Panel 2SLS Estimates For Exchange Rate Change Components Regressed on Relative Forward Rate Changes n=

0

4

8

12

16

20

24

28

32

36

1990:Q3 - 2008:Q3 1.43 (0.95)

2.52 (0.95)

∗∗∗

2.13 (0.77)

1.18 (0.89)

-0.56 (1.16)

-1.50 (1.16)

-2.59∗∗ (1.25)

-2.41 -4.74∗∗∗ -6.77∗∗∗ (1.51) (1.44) (1.89)

-1.36 (0.89)

-2.30∗∗∗ -1.55∗∗ (0.89) (0.79)

-1.16 (0.97)

-1.35 (1.11)

-0.39 (1.55)

-0.39 (1.65)

-0.60 (1.56)

0.30 (1.70)

0.50 (1.86)

g ∆T P t+1

-0.00 (0.03)

0.60∗∗∗ 0.74∗∗∗ 0.83∗∗∗ (0.07) (0.04) (0.07)

0.84∗∗∗ (0.08)

0.93∗∗∗ (0.10)

0.89∗∗∗ (0.09)

0.91∗∗∗ (0.08)

0.87∗∗∗ (0.08)

0.85∗∗∗ (0.08)

s∆E t+1,∞

0.44 (0.28)

1.38∗∗∗ 1.20∗∗∗ 1.30∗∗∗ (0.45) (0.41) (0.46)

1.60∗∗∗ (0.50)

1.45∗∗ (0.65)

1.64∗∗ (0.72)

1.72∗∗ (0.80)

1.47∗∗ (0.59)

1.60∗∗ (0.64)

F σt − σt+1

3.35∗∗∗ (1.02)

3.84∗∗∗ 2.73∗∗∗ (1.12) (1.00)

-0.65 (1.56)

-2.49 (2.04)

-3.74∗ (2.17)

-3.43 -6.38∗∗∗ -8.73∗∗∗ (2.14) (2.41) (3.08)

∆st+1 f,n

g ˜ıt − ϕEH t+1 + ∆EH t+1 f,n

∗∗∗

1.20 (1.30)

2008:Q4 - 2012:Q2 ∗∗∗

∆st+1 f,n

g ˜ıt − ϕEH t+1 + ∆EH t+1 f,n

g ∆T P t+1

∗

-10.68 (2.13)

4.40 (2.32)

15.98∗∗ 11.62∗∗∗ 10.19∗∗∗ 9.87∗∗∗ 10.50∗∗∗ 11.60∗∗∗ 11.71∗∗∗ 12.69∗∗∗ (7.54) (4.35) (3.32) (3.03) (3.11) (3.33) (4.02) (4.22)

3.03 (3.46)

-2.40 (1.83)

-5.91∗∗ -4.63∗∗∗ -4.03∗∗∗ -3.90∗∗∗ -4.07∗∗∗ -4.39∗∗∗ -4.40∗∗ (2.76) (1.53) (1.22) (1.17) (1.30) (1.51) (1.73)

-0.16 (0.17)

0.49∗∗∗ 0.93∗∗∗ 0.66∗∗∗ (0.11) (0.26) (0.13)

0.55∗∗∗ (0.11)

0.49∗∗∗ (0.13)

0.46∗∗∗ (0.16)

0.46∗∗∗ (0.17)

0.54∗∗∗ (0.15)

0.62∗∗∗ (0.12)

3.95∗∗∗ (0.78)

3.78∗∗∗ (0.75)

3.99∗∗∗ (0.83)

4.36∗∗∗ (0.96)

4.48∗∗∗ (1.07)

4.71∗∗∗ (1.10)

-4.59∗∗ (1.85)

s∆E t+1,∞

-4.87∗∗∗ (1.60)

2.39∗∗ (1.02)

6.75∗∗∗ 4.64∗∗∗ (2.10) (1.02)

F σt − σt+1

-7.68∗∗ (3.20)

4.92∗ (2.54)

15.20∗∗ 11.95∗∗∗ 10.73∗∗∗ 10.49∗∗∗ 11.12∗∗∗ 12.16∗∗∗ 12.10∗∗∗ 12.95∗∗∗ (6.56) (4.10) (3.18) (2.92) (3.01) (3.24) (3.95) (4.17) 2012:Q3 - 2015:Q3

∗∗∗

-18.84 (2.81)

-12.83 (2.30)

-3.88∗ (2.06)

-1.59 (2.25)

-1.53 (2.73)

-2.01 (3.33)

-1.41 (3.45)

-1.55 (2.63)

-2.40 (2.12)

-2.35 (1.58)

0.41 (1.61)

0.75 (2.59)

-0.79 (0.83)

-1.14∗ (0.69)

-1.61 (1.50)

-2.13 (2.54)

-2.38 (3.40)

-2.10 (3.08)

-1.87 (2.91)

-1.47 (2.49)

g ∆T P t+1

0.35∗∗∗ (0.10)

0.46∗∗∗ 0.90∗∗∗ 0.84∗∗∗ (0.12) (0.07) (0.05)

0.75∗∗∗ (0.12)

0.70∗∗∗ (0.22)

0.72∗∗ (0.31)

0.79∗∗∗ (0.28)

0.85∗∗∗ (0.26)

0.90∗∗∗ (0.20)

s∆E t+1,∞

-1.62∗ (0.84)

-0.48 (0.55)

1.00∗∗∗ 1.20∗∗∗ (0.27) (0.36)

1.44∗∗∗ (0.50)

1.69∗∗ (0.66)

1.65∗∗ (0.73)

1.20∗ (0.66)

0.77 (0.66)

0.38 (0.59)

-1.11 (3.02)

-1.27 (4.03)

-0.40 (4.78)

-0.45 (4.14)

-1.15 (3.69)

-1.16 (3.02)

∆st+1 f,n

g ˜ıt − ϕEH t+1 + ∆EH t+1 f,n

F σt − σt+1

∗∗∗

-16.98∗∗∗ -12.57∗∗∗ -4.00 (3.34) (3.95) (2.50)

-1.49 (2.35)

Note: Each cell of this table gives the slope coefficient from regressing the variable at the left on the change in the n ). Standard errors clustered by country pair are in parentheses. one-year relative forward rate n quarters hence (∆f˜t+1 Constants are included in the regression, but omitted from this table. Surprises used: Price changes in a 1-hour window around FOMC and QE announcements of federal funds rate futures expiring 3 months hence, eurodollar futures expiring 2, 3, and 4 quarters hence, and 2- and 10-year Treasury bond futures expiring in the current quarter.

37

Table 2: Fixed Effect Panel OLS Estimates For Exchange Rate Change Components Regressed on Relative Forward Rate Changes n=

0

4

8

12

16

20

24

28

32

36

-0.44 (0.24)

-0.65∗∗ (0.26)

1990:Q3 - 2008:Q3 -1.59∗∗∗ -2.03∗∗ (0.41) (0.71)

∆st+1

-2.31∗∗ (0.75)

-2.45∗∗ -2.67∗∗∗ -2.09∗∗∗ -1.77∗∗∗ -0.98∗∗∗ (0.77) (0.77) (0.62) (0.45) (0.28)

f,n

∗∗∗ g -3.67∗∗∗ -3.81∗∗∗ -4.17∗∗∗ -3.88∗∗∗ -3.61∗∗∗ -3.40∗∗∗ -2.64∗∗∗ -1.51∗∗ -1.90∗∗∗ ˜ıt − ϕEH t+1 + ∆EH t+1 -3.54 (0.61) (0.59) (0.51) (0.48) (0.39) (0.39) (0.43) (0.37) (0.64) (0.34) f,n

g ∆T P t+1

0.14∗∗∗ (0.03)

0.32∗∗∗ (0.04)

0.48∗∗∗ (0.04)

0.59∗∗∗ (0.03)

0.67∗∗∗ (0.03)

0.73∗∗∗ (0.03)

0.78∗∗∗ (0.03)

0.86∗∗∗ (0.04)

0.92∗∗∗ (0.04)

0.93∗∗∗ (0.03)

s∆E t+1,∞

1.04∗∗∗ (0.23)

1.06∗∗∗ (0.14)

1.21∗∗∗ (0.12)

1.53∗∗∗ (0.17)

1.59∗∗∗ (0.12)

1.71∗∗∗ (0.16)

1.82∗∗∗ (0.20)

1.55∗∗∗ (0.26)

1.08∗∗ (0.44)

1.19∗∗∗ (0.24)

F σt − σt+1

1.76∗ (0.80)

1.26 (0.97)

0.81 (0.92)

0.60 (0.84)

-0.04 (0.74)

0.08 (0.57)

0.02 (0.43)

0.25 (0.34)

0.06 (0.26)

0.13 (0.29)

∗

∗∗

3.93∗ (1.72)

4.57∗∗ (1.72)

4.98∗∗ (1.62)

3.69 (2.31)

4.13∗∗ (1.74)

2008:Q4 - 2012:Q2 ∆st+1

-5.93 (2.63)

-4.63 (1.65)

-2.33 (2.70)

-4.57∗∗ -4.78∗∗∗ -4.32∗∗∗ -3.88∗∗∗ -3.60∗∗∗ -3.38∗∗∗ -3.13∗∗∗ -2.72∗∗∗ -2.65∗∗∗ (1.59) (0.87) (0.69) (0.72) (0.78) (0.80) (0.77) (0.66) (0.61)

g ∆T P t+1

0.29∗ (0.15)

0.33∗∗∗ (0.06)

0.49∗∗∗ (0.04)

0.55∗∗∗ (0.06)

0.58∗∗∗ (0.07)

0.60∗∗∗ (0.08)

0.63∗∗∗ (0.09)

0.68∗∗∗ (0.08)

0.76∗∗∗ (0.06)

0.80∗∗∗ (0.05)

s∆E t+1,∞

-0.50 (1.09)

1.16∗∗ (0.40)

2.25∗∗∗ (0.28)

2.61∗∗∗ (0.33)

2.64∗∗∗ (0.30)

2.61∗∗∗ (0.26)

2.53∗∗∗ (0.25)

2.40∗∗∗ (0.25)

2.13∗∗∗ (0.25)

2.08∗∗∗ (0.26)

F σt − σt+1

-2.38 (4.17)

-0.55 (2.90)

1.32 (1.91)

3.11∗ (1.62)

4.42∗∗ (1.70)

5.32∗∗ (1.81)

5.79∗∗ (1.84)

6.03∗∗∗ (1.73)

4.52 (2.45)

4.89∗∗ (1.80)

g f,n ˜ıt − ϕEH t+1 + ∆EH t+1 f,n

-1.72 (1.25)

0.95 (1.40)

2.76 (1.61)

2012:Q3 - 2015:Q3 -7.68 (3.17)

-5.89 (1.60)

-4.95 (1.39)

-4.56∗∗ (1.37)

-4.55∗∗ (1.40)

-4.66∗∗ -4.15∗∗∗ -4.32∗∗∗ -3.96∗∗∗ -4.21∗∗∗ (1.50) (1.13) (1.07) (0.85) (0.76)

∗∗ g f,n ˜ıt − ϕEH t+1 + ∆EH t+1 -1.15 (0.48)

-0.89 (0.52)

-1.02∗ (0.47)

-1.18∗∗ (0.41)

-1.40∗∗ (0.44)

-1.70∗∗ (0.55)

-1.78∗∗ (0.57)

-1.85∗∗ (0.61)

-1.65∗∗ (0.59)

-1.53∗∗ (0.61)

∆st+1

f,n

∗∗

∗∗∗

∗∗∗

g ∆T P t+1

0.70∗∗∗ (0.14)

0.69∗∗∗ (0.08)

0.83∗∗∗ (0.05)

0.85∗∗∗ (0.05)

0.83∗∗∗ (0.07)

0.82∗∗∗ (0.07)

0.83∗∗∗ (0.07)

0.86∗∗∗ (0.06)

0.88∗∗∗ (0.06)

0.90∗∗∗ (0.05)

s∆E t+1,∞

0.15 (0.34)

0.26 (0.14)

0.53∗∗∗ (0.14)

0.76∗∗∗ (0.18)

0.96∗∗∗ (0.22)

1.07∗∗∗ (0.26)

1.04∗∗∗ (0.27)

0.97∗∗ (0.29)

0.89∗∗ (0.29)

0.75∗∗ (0.30)

F σt − σt+1

-6.37∗ (2.73)

-4.94∗∗∗ -4.29∗∗ (1.30) (1.24)

-3.99∗∗ (1.27)

-3.93∗∗ (1.29)

-3.86∗∗ (1.31)

-3.24∗∗ -3.30∗∗∗ -3.08∗∗∗ -3.33∗∗∗ (0.99) (0.92) (0.78) (0.78)

Note: Each cell of this table gives the slope coefficient from regressing the variable at the left on the change in the one-year n ). Standard errors clustered by country pair are in parentheses. Constants relative forward rate n quarters hence (∆f˜t+1 are included in the regression, but omitted from this table.

38

Table 3: Regressions of Actual or Forecasted Exchange Rate Change on US GDP Growth Actual Quarterly Growth

Forecasted 3M Growth

Forecasted 12M Growth

US GDP Growth

-0.65∗ (0.32)

-0.79∗ (0.42)

-0.99∗∗∗ (0.18)

# of Observations

909

918

918

Note: Each column of this table gives estimates from an OLS regression of an actual or forecasted exchange rate change on the corresponding actual or forecasted US GDP growth growth rate (both annualized) over the 1990:Q2–2015:Q3 sample period. Standard errors clustered by country pair are in parentheses. Constants are included in the regression, but omitted from this table.

Table 4: 2SLS Regression of US GDP Forecast Revisions on US Forward Rate Changes n=

0

4

8

12

16

20

24

28

32

36

1990:Q2 - 2008:Q3 n,U S ∆ft+1

# of Observations

0.03 (0.12)

-0.15 (0.20)

-0.28 (0.29)

-0.38 (0.33)

-0.48 (0.36)

-0.58 (0.37)

-0.65∗ (0.37)

-0.68∗ (0.36)

-0.66∗ (0.34)

-0.62∗ (0.33)

71

71

71

71

71

71

71

71

71

71

2008:Q4 - 2012:Q2 n,U S ∆ft+1

# of Observations

0.64∗∗∗ 0.67∗∗∗ 0.55∗∗∗ 0.47∗∗∗ 0.43∗∗∗ 0.42∗∗∗ 0.43∗∗∗ 0.45∗∗∗ 0.47∗∗∗ 0.48∗∗∗ (0.09) (0.10) (0.14) (0.13) (0.13) (0.12) (0.12) (0.13) (0.13) (0.13) 15

15

15

15

15

15

15

15

15

15

2012:Q3 - 2015:Q3 n,U S ∆ft+1

# of Observations

-0.65 (0.43)

-0.22 (0.15)

-0.07 (0.07)

-0.06 (0.05)

-0.06 (0.04)

-0.06 (0.04)

-0.06 (0.05)

-0.05 (0.05)

-0.05 (0.06)

-0.05 (0.07)

13

13

13

13

13

13

13

13

13

13

Note: Each cell of this table gives the slope coefficient from regressing the revision in the Blue Chip Economic Indicators 4-quarter ahead GDP forecast change on the change in the one-year US forward rate n quarters n ). Heteroskedasticity-robust standard errors are in parentheses. Constants are included in the hence (∆ft+1 regression, but omitted from this table. Surprises used: Price changes in a 1-hour window around FOMC and QE announcements of federal funds rate futures expiring 3 months hence, eurodollar futures expiring 2, 3, and 4 quarters hence, and 2- and 10-year Treasury bond futures expiring in the current quarter.

39

Table 5: 2SLS Regression of Changes in the VIX on US Forward Rate Changes n=

0

4

8

12

16

20

24

28

32

36

1990:Q2 - 2008:Q3 n,U S ∆ft+1

# of Observations

3.47 (4.17)

-0.38 (5.34)

-3.70 (4.24)

-5.03 (4.38)

-5.72 (4.76)

-6.00 (5.18)

-5.91 (5.47)

-5.62 (5.56)

-5.35 (5.52)

-5.24 (5.52)

71

71

71

71

71

71

71

71

71

71

2008:Q4 - 2012:Q2 n,U S ∆ft+1

# of Observations

-0.67 (3.34)

-9.69 (6.62)

15

15

-10.81∗ -9.99∗∗ -9.40∗∗∗ -9.20∗∗∗ -9.23∗∗∗ -9.34∗∗∗ -9.39∗∗ -9.25∗∗ (5.65) (4.32) (3.63) (3.33) (3.27) (3.41) (3.69) (4.04) 15

15

15

15

15

15

15

15

2012:Q3 - 2015:Q3 n,U S ∆ft+1

# of Observations

∗∗

∗∗∗

43.82 11.87 (19.77) (3.36) 13

13

0.83 (4.29)

-0.74 (3.39)

-1.30 (2.90)

-1.69 (2.61)

-2.07 (2.45)

-2.49 (2.41)

-2.96 (2.47)

-3.47 (2.60)

13

13

13

13

13

13

13

13

Note: Each cell of this table gives the slope coefficient from regressing change in the VIX index on the change in the n ). Heteroskedasticity-robust standard errors are in parentheses. one-year US forward rate n quarters hence (∆ft+1 Constants are included in the regression, but omitted from this table. Surprises used: Price changes in a 1-hour window around FOMC and QE announcements of federal funds rate futures expiring 3 months hence, eurodollar futures expiring 2, 3, and 4 quarters hence, and 2- and 10-year Treasury bond futures expiring in the current quarter.

Table 6: Subsample means of uncertainty measures

JLN Macro Uncertainty GDP Forecast Dispersion BBD Monetary Policy Uncertainty

1990:Q2-2008:Q3 2008:Q4-2012:Q2 2012:Q3-2015:Q3 -0.04 0.80 -0.68 0.03 0.90 -1.23 0.11 -0.05 -0.58

Note: The JLN macro uncertainty measure is the 12-month ahead measure of macroeconomic uncertainty estimated by Jurado, Ludvigson, and Ng (2015). GDP forecast dispersion is the 25th-75th percentile range of 4-quarter-ahead US real GDP forecasts from Blue Chip Economic Indicators. BBD monetary policy uncertainty is the monetary policy subcomponent of the Baker, Bloom, and Davis (2016) policy uncertainty index. All three measures are standardized over the full 1990:Q1–2015:Q3 sample to facilitate interpretation.

40

Appendix A

Data Description • Exchange Rates: End-of-quarter exchange rates are obtained using daily data from Global Financial Data. • Short-term rates: End-of-quarter three-month bill rates were obtained from the following sources: – Australia, Canada, New Zealand, Norway, Sweden, Switzerland, United Kingdom, and United States: Central bank data obtained through Haver Analytics. – Germany: Reuters data obtained through Haver Analytics. German three-month bill rates are replaced with three-month EONIA OIS swap rates starting in 1999:Q1. – Japan: Bloomberg • Zero-Coupon Yields: End-of-quarter zero-coupon yields were obtained from the following sources: – Canada, Germany, Sweden, Switzerland, and United Kingdom: Central banks – Norway: Data from Wright (2011) extended with data from the BIS – Australia, New Zealand: Data from Wright (2011) extended with data from central banks – Japan: Bloomberg. – United States: G¨ urkaynak, Sack, and Wright (2007) • Output Gap and Current Account-to-GDP ratio: All macro data are from the OECD Main Economic Indicators and Economic Outlook databases. The GDP gap is computed using the OECD’s annual estimates of potential GDP, which were log-linearly interpolated to the quarterly frequency. German data are replaced with euro-area data starting in 1999:Q1. • Market-Based Interest Rate Surprises and Expected Changes: These are computed using prices of futures on three-month interest rates on the last trading day of each quarter. These expectations refer to the three-month rates on each contract’s last trading day, which typically falls within the second-to-last week of each quarter. When computing the surprises and expected changes in these interest rates, the actual rate used is the underlying rate of each futures contract. The futures data are all obtained from Bloomberg and are based on the following underlying rates: – Australia: Australian 90-day bank accepted bills – Canada: Canadian three-month bankers’ acceptance A-1

– – – – –

Switzerland: three-month Euroswiss Germany/EU: ICE three-month Euribor Norway: three-month NIBOR New Zealand: New Zealand 90-day bank accepted bills Sweden: three-month Swedish T-bill (1992:Q4–2007:Q4); three-month STIBOR (2008:Q1-present) – United Kingdom: three-month Sterling LIBOR – United States: three-month Eurodollar

Data Sample Ranges Australia Canada Germany Japan New Zealand Norway Sweden Switzerland United Kingdom United States

B

1989:Q4 1992:Q2 1991:Q2 1992:Q3 1990:Q1 1989:Q4 1992:Q4 1992:Q1 1992:Q4 1989:Q4

– – – – – – – – – –

2015:Q4 2015:Q4 2015:Q4 2015:Q4 2015:Q1 2015:Q4 2015:Q4 2011:Q2 2015:Q4 2015:Q1

Break Date Estimation

To estimate break dates, we follow the procedure of Bai and Perron (1998) using OLS estimation of equation (2). Though our main interest is in the two-stage least squares estimate, Perron and Yamamoto (2015) argues that estimation of the break dates using OLS is in general more precise. The procedure involves searching over a grid of possible break dates, for a predefined number of breaks, to find the set that minimizes the regression’s sum of squared residuals (SSR). We do this estimation for 1, 2, and 3 breaks and set a minimum subsample length of 10 quarters, which corresponds to roughly 10% of our sample. Table A-1 presents the optimal break dates for each horizon considered while the dashed lines in Figure A-1 plot the resulting SSRs relative to the SSR achieved with no break. Note that for most horizons, including the longer ones which we are mainly interested in, the largest improvement in SSRs is achieved with 2 breaks. The set of 2 break dates that occurs most commonly, particularly for longer horizons, is 2008:Q4 and 2012:Q2. For A-2

Table A-1: Break dates that minimize sum of squared residuals Horizons

1 Breaks

0 4 8 12 16 20 24 28 32 36

2001:Q3 2008:Q1 2012:Q4 2012:Q4 2013:Q1 2013:Q1 2013:Q1 2013:Q1 2013:Q1 2012:Q4

2 Breaks 2002:Q2, 2002:Q2, 2002:Q2, 2002:Q2, 2008:Q4, 2008:Q4, 2008:Q4, 2008:Q4, 2001:Q3, 2002:Q1,

3 Breaks

2005:Q1 2005:Q1 2005:Q1 2005:Q1 2012:Q2 2012:Q2 2012:Q2 2012:Q2 2012:Q4 2012:Q4

1995:Q2, 1995:Q2, 1995:Q2, 2002:Q2, 2001:Q3, 2006:Q2, 2001:Q3, 2001:Q3, 2001:Q3, 1997:Q1,

2002:Q2, 2001:Q3, 2001:Q3, 2005:Q1, 2008:Q4, 2008:Q4, 2008:Q4, 2008:Q4, 2008:Q4, 2001:Q3,

2005:Q1 2005:Q1 2005:Q1 2012:Q4 2012:Q2 2012:Q2 2012:Q2 2012:Q2 2012:Q4 2012:Q4

Note: Break dates given are the start dates of subsamples. the longest horizons, 2012:Q4 also occurs as a break date. Given these results, we choose 2008:Q4 and 2012:Q3 to be the two break dates we use for all horizons. The red solid line in Figure A-1: Sums of squared residuals (SSRs) relative to no break case 1

0.96

0.92

0.88

0.84

0.8 0

4

8

12 1 break

2013:Q1

16

20 2 breaks

2008:Q4 and 2012:Q3

24

28

32

36

3 breaks 2001:Q3, 2008:Q4, and 2012:Q3

Note: Finer lines with x’s are the SSRs relative to the case of no breaks for the optimal 1, 2, or 3 break dates for each horizon (shown in Table A-1). Thicker solid lines are the relative SSRs for each horizon at the break dates shown in the legend.

A-3

Figure A-1 plots the relative SSRs obtained when we apply these two breaks to all horizons. Note that for horizons equal to or above 12 quarters, the SSR achieved using these two breaks is very close to the ones achieved using the optimal horizon-specific breaks shown in Table A-1. Figure A-1 also plots the relative SSRs for the most commonly found single break and set of 3 breaks across horizons. When we allow for a third break in 2001:Q3 in our regressions, the coefficient estimates from the first two subsamples are very similar, particularly for longer horizons.

C

Important QE Dates

The following list of dates is used to define our QE and non-QE samples in Section 4. The dates are collected from a number of papers including Rogers, Scotti, and Wright (2014), Wu (2014), and Swanson (2017). 11/25/2008 Initial large-scale-asset-purchase announcement 12/1/2008 Bernanke states Treasuries may be purchased 12/16/2008 The FOMC indicated that “it stands ready to expand its purchases of agency debt and mortgage-backed securities as conditions warrant. The Committee is also evaluating the potential benefits of purchasing longer-term Treasury securities.” 1/28/2009 FOMC Statement 3/18/2009 FOMC announces it will purchase $750B of mortgage-backed securities, $300B of longer-term Treasuries, and $100B of agency debt (a.k.a. “QE1”) 8/12/2009 The FOMC eliminated the “up to” phrase in its intended purchase amount of Treasury securities. It also stated that it would “slow the pace of these transactions and anticipates that the full amount will be purchased by the end of October. 9/23/2009 The FOMC eliminated the “up to” phrase in its intended purchase amount of the MBS, as well as its plan to “slow the pace of these purchases in order to promote a smooth transition in markets and anticipates that they will be executed by the end of the first quarter of 2010.” 11/4/2009 The FOMC clarified that the intended purchase amount of agency debt would be $175 billion, instead of the previously announced “up to $200 billion” A-4

8/10/2010

8/27/2010 9/21/2010 10/15/2010 11/3/2010 8/26/2011 9/21/2011

6/20/2012

9/13/2012 12/12/2012 5/22/2013 6/19/2013 12/18/2013

The FOMC announced that it “will keep constant the Federal Reserve’s holdings of securities at their current level by reinvesting principal payments from agency debt and agency mortgage-backed securities in longer-term Treasury securities. The Committee will continue to roll over the Federal Reserve’s holdings of Treasury securities as they mature.” Bernanke Speech at Jackson Hole FOMC Statement Bernanke Speech at the Boston Fed FOMC announces it will purchase an additional $600B of longer-term Treasuries (a.k.a. “QE2”) Bernanke Speech at Jackson Hole FOMC announces it will sell $400B of short-term Treasuries and use the proceeds to buy $400B of long-term Treasuries (a.k.a. “Operation Twist”) The FOMC announced its intention “to continue through the end of the year its program to extend the average maturity of its holdings of securities.” FOMC announces it will purchase $40B of mortgage-backed securities per month for the indefinite future. FOMC announces it will purchase $45B of longer-term Treasuries per month for the indefinite future. Bernanke Testimony (“Taper Tantrum”) FOMC Statement FOMC announces it will start to taper its purchases of longer-term Treasuries and mortgage-backed securities to paces of $40B and $35B per month, respectively.

A-5

D

Model: Additional Derivations

The central bank’s signal can be decomposed as y i at+1 = ius t+h = at+h + at+h κεyt+h y at+h = η + νκ ηεit+h ait+h = η + νκ

Note that ayt+h and ait+h are both mean zero and i.i.d. normal. Thus, the posterior mean of the two shocks is given by    η + νκ  y E εyt+h |at+1 , εy,t+1 , εi,t+1 = E at+h |at+1 , εy,t+1 , εi,t+1 κ  η + νκ  y E at+h |at+1 since ayt+h is i.i.d. = κ
V ar ayt+h |at+1 at+h η + νκ

= κ V ar ayt+h |at+1 + V ar ait+h |at+1
κ (η + νκ) V art εyt+h at+1

= 2 κ V art εyt+h + η 2 V art εit+h Similarly,    η + νκ  i E at+h |at+1 E εit+h |at+1 , εy,t+1 , εi,t+1 = η
V ar ait+1 |at+1 at+1 η + νκ

= η V ar ayt+1 |at+1 + V ar ait+1 |at+1
η (η + νκ) V art εit+h at+1

= 2 κ V art εyt+h + η 2 V art εit+h The posterior distribution of GDP growth is then given by      us t+1 y,t+1 i,t+1  E εyt+h |at+1 , εy,t+1 , εi,t+1 − νE εit+h |at+1 , εy,t+1 , εi,t+1 E ∆yt+h |a , ε ,ε = η + νκ = Kat+1 V ar(εy ) − νη κ V ar εt+h ( it+h ) . where K = V ar(εyt+h ) 2 2 κ V ar εi +η ( t+h ) Note that due to the i.i.d. nature of shocks, the only information relevant for time t + h

A-6

outcomes is the central bank’s announcement at+1 . That is,  us t+1 y,t+1 i,t+1   us  E ∆yt+h |a , ε ,ε = E ∆yt+h |at+1 . We now relate the terms from our exchange rate decomposition to this GDP growth expectation. First, note that, using our previous assumptions about the stationarity of V art (∆st+1 )
us , the expected excess returns term is the following in our model and Covt ∆st+1 , ∆yt+1 F σt+1

=

∞ X

(E (σt+k+1 |It+1 ) − E (σt+k+1 |It ))

k=0

=

∞ X


   E ρt+k+1 |at+1 , εy,t+1 , εi,t+1 − E ρt+k+1 |at , εy,t , εi,t γλσ s,y

k=0


us where σ s,y denotes the constant value of Covt ∆st+1 , ∆yt+1 . Next, we relate the update in expectations regarding ρt+k+1 to beliefs about GDP growth.
us Since ρt+1 = θ − λ∆yt+1 ρt , we have
us ρt+k+1 = Πk+1 θ − λ∆y t+i ρt+1 i=2
us = Πk+1 i=1 θ − λ∆yt+i ρt so that   
t+1 y,t+1 i,t+1  us E ρt+k+1 |at+1 , εy,t+1 , εi,t+1 = E Πk+1 ,ε ,ε ρt+1 i=2 θ − λ∆yt+i |a  
us t+1 y,t+1 i,t+1 = Πk+1 ,ε ,ε ρt+1 . i=2 θ − λE ∆yt+i |a us due to ∆yt+i being i.i.d. Then, since at+1 = {a1−h , ..., at+1 } = {i0 , ..., it+h } (  us 
  Πk+1 ρt+1 if k ≤ h − 1 i=2 θ − λE ∆yt+i |at−h+i+1 t+1 y,t+1 i,t+1  us 
E ρt+k+1 |a , ε ,ε = k+1−h h θ Πi=2 θ − λE ∆yt+i |at−h+i+1 ρt+1 if k > h − 1

Similarly, (   E ρt+k+1 |at , εy,t , εi,t =

 us 
Πk+1 ρt if k ≤ h − 2 i=1 θ − λE ∆yt+i |at−h+i+1  us 
k+2−h h−1 θ Πi=1 θ − λE ∆yt+i |at−h+i+1 ρt if k > h − 2

A-7

Thus, the update in expectations of ρt+k+1 between time t and t + 1 is     E ρt+k+1 |at+1 , εy,t+1 , εi,t+1 − E ρt+k+1 |at , εy,t , εi,t   us 
  us 
 Πk+1 ρt+1 − θ − λE ∆yt+1 |at−h+2 ρt if k ≤ h − 2  i=2 θ − λE ∆yt+i |at−h+i+1   
  
 us 
 h−1 us |a us |a = θk+1−h Πi=2 θ − λE ∆yt+i θ − λE ∆yt+h t+1 ρt+1 − θ θ − λE ∆yt+1 |at−h+2 ρt t−h+i+1   if k ≥ h − 1

so that the derivative w.r.t. the announcement is    ∂   E ρt+k+1 |at+1 , εy,t+1 , εi,t+1 − E ρt+k+1 |at , εy,t , εi,t ∂at+1 ( 0 if k ≤ h − 2 = us |a 
 ∂E ∆y [ ] t+1 t+h us −θk+1−h λ ρt+1 if k ≥ h − 1 Πh−1 i=2 θ − λE ∆yt+i |at−h+i+1 ∂at+1 ( 0 if k ≤ h − 2 = . us |a ∂E ∆y [ ] t+1 t+h t+1 y,t+1 i,t+1 −θk+1−h λ E [ρ |a , ε , ε ] if k ≥ h − 1 t+h−1 ∂at+1 Then, we have ∞ F X    
∂σt+1 ∂ = E ρt+k+1 |at+1 , εy,t+1 , εi,t+1 − E ρt+k+1 |at , εy,t , εi,t γλσ s,y ∂at+1 ∂at+1 k=0  us   ∂E ∆yt+h |at+1 γλ2 σ s,y  = − E ρt+h−1 |at+1 , εy,t+1 , εi,t+1 ∂at+1 1−θ ∞   X 2 s,y t+1 y,t+1 i,t+1 = −Kγλ σ E ρt+h−1 |a , ε ,ε θk+1−h k=h−1 2 s,y

= −K

  γλ σ E ρt+h−1 |at+1 , εy,t+1 , εi,t+1 . 1−θ

Assuming that E [ρt+h−1 |at , εy,t+1 , εi,t+1 ] ≥ 0—which is possible with a sufficiently low λ, high θ and positive initial value, ρ0 —and as long as σ s,y ≤ 0, then this derivative is always positively proportional to K though its magnitude can be time-varying.

A-8