QE in the Fiscal Theory: A Risk-Based View Alexandre Corhay∗

Howard Kung†

June 2017

Gonzalo Morales‡

§

Abstract This paper explores the interactions between yield curve dynamics and nominal government debt maturity operations under fiscal stress in a New Keynesian model with endogenous bond risk premia. Open market debt maturity operations are non-neutral when the slope of the nominal yield curve is nonzero in a fiscally-led policy regime. When the risk profiles of government liabilities differ, rebalancing the maturity structure changes the government cost of capital. In the fiscal theory, changes in discount rates affect inflation through the intertemporal government budget equation. When the yield curve is upward-sloping (downward-sloping), the fiscal discount rate channel implies that shortening the maturity structure dampens (amplifies) the stimulative effects of quantitative easing policies.

Keywords: Fiscal theory of the price level, Government debt, Maturity structure, Inflation, Bond risk premia, Markov-switching DSGE, Nonlinear solution methods, Time-varying risk premia ∗

University of Toronto. [email protected]. London Business School. [email protected] ‡ University of Alberta. [email protected]. § We also thank Ravi Bansal, Marco Bassetto, Frederico Belo, Luigi Bocola, Francesco Bianchi, John Cochrane, Wayne Ferson, Francisco Gomes, Lars Hansen, Christopher Hennessy, Urban Jermann, Tim Landvoight (discussant), Morten Ravn, Lukas Schmid, Andrew Scott, Vania Stavrakeva, Sevin Yeltekin (discussant), and seminar participants at the University of British Columbia, University of Minnesota, University of Southern California, Universidad Carlos III, University College of London, London Business School, Vienna Graduate School of Finance, Society of Economic Dynamics Meetings, CEPR European Summer Symposium, Western Finance Association Meetings, 6th Macro Finance Society Workshop, European Economic Association Meetings, Econometric Society World Congress Meetings, CAPR Workshop on Investment & Production-Based Asset Pricing, and BI-Swedish House of Finance Conference for helpful comments. This paper previously circulated under the title “Government Maturity Structure Twists”. †

1

Introduction

During the recent global financial crisis, central banks, constrained by the zero lower bound (ZLB) on nominal interest rates, conducted open market operations on an unprecedented scale. The series of quantitative easing (QE) operations between 2008 and 2014 reduced the average duration of U.S. government liabilities (including reserve balances) held by the public by over 20%. Fig. 1 illustrates the impact of the QE operations on average maturity.1 These operations dramatically increased both the size and riskiness of the government balance sheet. Meanwhile, during this period, deepening deficits and ballooning debt-to-gdp levels cast doubt on the sustainability of accommodative fiscal policy to support aggressive inflation-targeting monetary policy. This paper investigates how the combination of fiscal stress and a monetary policy stance that weakly responds to inflation – a policy mix that characterizes the fiscal theory – provides a risk-based transmission channel for unconventional monetary policy. In the fiscal theory, government discount rates and surplus innovations affect the price level through a fiscal asset pricing equation that relates the market value of nominal liabilities to the present value of future surpluses. Given that expected bond returns vary by maturity, rebalancing the maturity structure changes marginal financing costs. When accommodative fiscal policy is not possible, discount rate variation requires offsetting changes in the inflation path to satisfy the present value relation, which we refer to as the fiscal discount rate channel. This paper explores the interplay between term structure dynamics and maturity restructuring policies in the presence of the fiscal theory. We illustrate how the effectiveness of operation-twist-type policies depends on the slope of the nominal yield curve, especially in times of fiscal stress. To quantitatively examine role of the fiscal discount rate channel, we build a small-scale New Keynesian model that has several distinguishing features. First, households have recursive preferences (e.g., Epstein and Zin (1989)), which allows the model to generate realistic nominal term premia. Second, the supply of nominal government bonds over various maturities is time-varying and stochastic. Third, the monetary/fiscal policy mix is subject to regime shifts between monetary1

When calculating the average maturity of privately-held public debt we include reserve balances with Federal Reserve Banks. Reserve balances are included since the Federal Reserve started to pay interests on reserves in October 2008 making them, effectively, government debt.

1

and fiscally-led regimes (e.g., Davig and Leeper (2007a), Bianchi and Ilut (2014), and Bianchi and Melosi (2014)). In the monetary-led regime, the monetary authority responds strongly to inflation deviations from target while the fiscal authority accommodates monetary policy by adjusting primary surpluses to stabilize debt. In the fiscally-led regime, the fiscal authority does not respond strongly to changes in debt while the monetary authority accommodates fiscal policy by allowing inflation to adjust in order to stabilize the real value of debt. We show that in the presence of a fiscally-led regime, a nonzero slope of the nominal yield curve implies that debt maturity restructuring affects inflation. To isolate the effects of debt maturity, we consider self-financing shocks to the maturity structure that keep the market value of total government bonds the same at the announcement of the operation (e.g., Maturity Extension Program), but is allowed to adjust freely afterwards through market forces. When the financing costs of bonds vary by maturity, changing the financing mix alters the government cost of capital. In the fiscally-led regime, the price level is determined by the ratio of nominal debt to the present value of surpluses. Thus, variation in the government discount rate changes the fiscal backing, and consequently, the price level and expected inflation adjust to revalue debt in order to satisfy the present value condition. Sticky prices are required for this channel to have real effects. More generally, these results illustrate how accounting for heterogeneity in expected returns across different assets in the fiscal theory contributes to violations of Wallace (1981) neutrality. The slope of the nominal yield curve dictates the effects of the fiscal discount rate channel for maturity restructuring.2 When the yield curve is upward-sloping (downward-sloping), the fiscal discount rate effects from shortening the maturity structure dampens (amplifies) the potential stimulative effects of quantitative easing policies (e.g., from providing short-term liquidity). Taking interest rates as given, increasing the proportion of short-term debt in the maturity structure when the yield curve is upward-sloping implies that the government is refinancing at a lower rate. Lowering the government discount rate puts upward pressure on the real value of debt. In anticipation of this, households increase demand for bonds and decrease demand for consumption goods, which 2

To be precise, the expected excess bond returns across maturities determines the effect of maturity restructuring. However, when yields are persistent, as in the data, then the average slope and expected return spread are approximately proportional to each each other. We discuss this approximation in further detail in Section 2.3.

2

puts downward pressure on the price level. With sticky prices, the fall in the price level generates contractionary pressure in production and output. The opposite results are obtained when the yield curve is downward-sloping. The endogenous yield curve reactions to maturity shocks reinforce the fiscal discount rate channel, however, the effects are attenuated under persistent budget deficits. When the yield curve is flat, the fiscal discount rate channel is neutral even in the presence of a fiscally-led regime since the cost of financing is the same across maturities. In a monetary-led regime without the possibility of regime shifts, the economy is insulated from fiscal disturbances as surplus policy completely offsets changes to the debt burden. Thus, the discount rate channel is also neutral in this case regardless of the slope of the yield curve. However, with regime shifts and rational expectations, the possibility of entering the fiscally-led regime and a nonzero slope is also sufficient for debt maturity changes to impact inflation through the discount rate channel in the monetary-led regime. Since the nominal yield curve provides a key transmission channel for the maturity structure shocks, we also calibrate our model to explain an array of term structure facts. Supply and preference shocks in the model imply sizable inflation risk premia (e.g., a negative consumption-inflation relation) similar to that of Kung (2015). With recursive preferences, these dynamics produce sizable bond risk premia (e.g., Piazzesi and Schneider (2007) and Bansal and Shaliastovich (2013)) and the model matches the average five-year nominal term spread. The model also replicates the persistence in yields and the forecasting ability of the term spread for future inflation. Regime shifts in the monetary/fiscal policy mix generate persistent volatility in macroeconomic fundamentals, such as consumption growth and inflation, which lead to predictability in excess bond returns. We consider an extended version of the model to assess the quantitative importance of the fiscal discount rate channel for operations involving maturity twists, such as QE2. In the extended model, we augment our baseline model with market segmentation and a preference for short-term liquidity demand, two channels that generate expansionary effects from shortening maturity and often cited by policymakers as relevant margins for such actions (e.g., Bernanke (2012)). Also, we start the economy off in the monetary-led regime (with a possibility of entering the fiscally-led regime), in a budget deficit, and at the zero lower bound (ZLB). During the Great Recession, the yield curve was significantly upward-sloping, and we find that the fiscal discount rate channel

3

significantly dampened the stimulative effects of QE arising from the market segmentation and liquidity channels. Indeed, using an estimated process for bond supply and considering a maturity shock of the magnitude of QE2, we find that the fiscal channel dampened the expansionary response of inflation and output by 37% and 52%, respectively, after 10 quarters. Thus, we provide a potential channel for explaining the weak inflation responses at the onset and aftermath of such operations.3

1.1

Related literature

The literature examining how the interactions between monetary and fiscal policy determine the price level begins with Sargent and Wallace (1981) who show that permanent fiscal deficits have to eventually be financed by seignorage when the government only issues real debt. Further, money creation leads to inflation. Building on this paper, the fiscal theory of the price level (FTPL) shows that when the government issues nominal debt and does not provide the necessary fiscal backing, deficits are linked to current and expected inflation through the intertemporal government budget equation, without necessarily relying on seignorage revenues (e.g., Leeper (1991), Sims (1994), Woodford (1994), Woodford (1995), Woodford (2001), Schmitt-Groh´e and Uribe (2000), Cochrane (1999), Bassetto (2002), Bassetto (2008), Cochrane (2005), and Cochrane (2011)). Our paper relates to the literature examining the role of the government maturity structure for policy. Angeletos (2002) and Buera and Nicolini (2004) demonstrate how a portfolio of non-statecontingent real debt of different maturities can replicate the complete markets allocation with statecontingent securities. Lustig, Sleet, and Yeltekin (2008), Leeper and Zhou (2013), and Faraglia, Marcet, Oikonomou, and Scott (2013) analyze optimal maturity structure of nominal debt in DSGE models with distortionary taxes and market incompleteness. Greenwood and Vayanos (2014), Chen, Curdia, and Ferrero (2012), and Guibaud, Nosbusch, and Vayanos (2013) analyze nominal maturity restructuring polices in models with preferred habitats. We differ from these papers by highlighting a distinct but complementary mechanism for maturity structure non-neutralities. Notably, we show how accounting for heterogeneity in risk across nominal bonds in the context of the fiscal theory provides a channel for the maturity structure to affect inflation without market segmentation or distortionary taxation. 3

See, for example, Williams (2014) for a survey on the event-study evidence of QE.

4

Our paper is most closely related to Cochrane (2001) who also considers the maturity structure in the fiscal theory. In a partial equilibrium setting with a constant interest rate, Cochrane illustrates how restructuring the face value of nominal government debt alters the timing of inflation. In contrast, we focus on market value restructuring operations (i.e., holding the market value of debt constant initially), and highlight how a nonzero yield curve slope is essential for such open market procedures to affect inflation through the fiscal discount rate channel. Furthermore, we also quantitatively evaluate our mechanism in a Dynamic Stochastic General Equilibrium (DSGE) framework, with a calibrated stochastic process for the average duration of public debt, to quantitatively examine effects of maturity restructuring policies. The Markov-switching Dynamic Stochastic General Equilibrium (DSGE) framework builds on Davig and Leeper (2007a), Davig and Leeper (2007b), Farmer, Waggoner, and Zha (2009), Bianchi and Ilut (2014), and Bianchi and Melosi (2014). We differ from these papers in that we focus on how the interaction between risk composition dynamics and the fiscal policy regime (or expectations of entering this regime) propagates maturity restructuring shocks. Our paper connects to the theoretical literature studying the effects of unconventional monetary policy and transmission channels. C´ urdia and Woodford (2010), Gertler and Karadi (2011), Ara´ ujo, Schommer, and Woodford (2015), and Williamson (2016) analyze the role of financial frictions for central bank purchases of risky assets. Correia, Farhi, Nicolini, and Teles (2013) demonstrate how distortionary tax policy can deliver an economic stimulus when monetary policy is constrained at the zero lower bound. Reis (2015) explores QE operations in an environment with an exogenous fiscal limit, default risk, and a fiscally-led regime. Gomes, Jermann, and Schmid (2014) illustrate how incorporating nominal corporate debt in a DSGE framework provides an important source of monetary non-neutrality. We offer an alternative transmission channel for thinking about unconventional monetary policy that relies on the interaction between the fiscal theory and bond risk premia to break Wallace neutrality. More broadly, this paper relates to general equilibrium models that link policy to risk premia. For example, Rudebusch and Swanson (2012), Palomino (2012), Dew-Becker (2014) Campbell, Pflueger, and Viceira (2014), and Kung (2015) link asset prices to monetary policy. Croce, Kung, Nguyen, and Schmid (2012), Pastor and Veronesi (2012), Gomes, Michaelides, and Polkovnichenko

5

(2013), and Belo, Gala, and Li (2013) look at fiscal policy and asset prices. The paper is organized as follows. Section 2 provides a simple partial equilibrium model to qualitatively illustrate the basic mechanisms. Section 3 presents the quantitative model. Section 4 studies the implications of the quantitative model. Section 5 describes an extended model and considers a policy experiment relating to QE2. Section 6 concludes.

2

Simple Example

In this section, we propose a simple partial equilibrium model to illustrate how the risk composition of the government portfolio affects inflation. Using approximate analytical solutions, we explicitly show how the impact of changing debt maturity on inflation depends on the slope of the yield curve. These concepts are integrated into a general equilibrium model in the next section.

2.1

Government Budget Equation

The government finances nominal surpluses by issuing one- and two-period nominal debt. The flow government budget constraint at time t is therefore given by:

p1q

p1q

p2q

p1q

p1q

p2q

p2q

Bf,t ` Qt Bf,t “ Qt Bf,t`1 ` Qt Bf,t`1 ` St ,

(1)

pnq

where Bf,t is the nominal face value of debt issued by the treasury with a maturity of n periods, Qnt is the corresponding nominal bond price, and St ” Tt ´ Gt is the nominal primary surplus. The relative supply of government bonds (portfolio weights) is assumed to be exogenous and constant. In particular, we define the relative supply of the one-period bond, in terms of market values, as: p1q

p1q

Qt Bf,t`1 p1q

p1q

p2q

p2q

Qt Bf,t`1 ` Qt Bf,t`1 which we assume to be constant in this example.

6

” wt “ w,

(2)

2.2

Monetary/Fiscal Policy Mix

The monetary/fiscal policy mix is permanently characterized by a fiscally-led policy regime (without the possibility of regime shifts). We consider a particular policy mix that allows for analytical tractability. Fiscal policy sets the real surplus independently of debt. We assume that st ” St {Pt , follows an exogenous stochastic process:

st “ p1 ´ ρs q¯ s ` ρs st´1 ` σs s,t ,

(3)

where s,t „ iid N p0, 1q. Monetary policy sets the short-term nominal interest rate independently of inflation. Specifically, the nominal short rate is determined by an exogenously specified nominal stochastic factor, Mt , via the Euler equation: p1q

logpRt`1 q “ ´ log pEt rMt`1 sq .

(4)

The log nominal stochastic discount factor is assumed to follow an ARMA(1,1):

´ logpMt q “ p1 ´ ρqδ ´ ρ logpMt´1 q ` t ` θt´1 ,

(5)

where t „ iid N p0, σ 2 q. This specification of the pricing kernel allows for various average slopes of the nominal yield curve (e.g., Backus and Zin (1994)). The real rate is assumed to be constant and given by the parameter, Rreal ą 0. To close model, we assume that the Fisher equation holds:

p1q

logpRt`1 q “ logpRreal q ` Et rlogpΠt`1 qs,

where Πt is gross inflation.

7

(6)

2.3

Bond Pricing

The nominal price of a n-period nominal zero-coupon bond can be written recursively using the Euler equation: ” ı pn´1q “ Et Mt`1 Qt`1 ,

pnq

Qt p1q

where Qp0q ” 1 and Qt

(7)

p1q

” 1{Rt`1 . The corresponding yield-to-maturity for the n-period bond is

defined as:

pnq

yt

1 pnq ” ´ log Qt . n

(8)

When yields are persistent, there is tight relation between the yield spread and the corresponding return spread:

p2q

p1q

p2q

ErlogpRt q ´ logpRt qs « 2 ¨ Eryt p2q

where Rt

2.4

p2q

p1q

p1q

” Qt {Qt´1 and Rt

p1q

´ yt s,

(9)

p1q

” 1{Qt´1 .4

Fiscal Discount Rate Channel

We can rewrite the flow government budget constraint in terms of market values of debt and returns:

p1q

p1q

Rt Bt piq

where Bt

piq

piq

p2q

” Qt´1 Bf,t , Rt

p2q

p2q

` Rt Bt

p1q

p2q

p1q

” Qt {Qt´1 , Rt

p1q

p2q

“ Bt`1 ` Bt`1 ` St , p1q

(10)

p1q

” 1{Qt´1 , Bt ” Bt

` Bt , Rtg ” wRt p2q

p1q

` p1 ´

p2q

wqRt . Iterating Eq. (10) forward and imposing the transversality condition, we obtain the present value formula relating the real value of government liabilities at t “ 0 to the present value of future 4

p2q

p2q

p1q

p1q

p1q

p2q

Note that logpRt q “ 2yt´1 ´yt and logpRt q “ yt´1 . Thus, if yields are persistent, then ErlogpRt q´ p1q p2q p1q p2q p1q logpRt qs « Er2yt ´ 2yt s “ 2 ¨ Eryt ´ yt s.

8

surpluses (see Appendix A for the derivation): fi

» b1 ”

8 ÿ

B1 – “ E1 śi P0 i“0

s ) fl . ! i`1 g R {Π j`1 j“0 j`1

(11)

In the fiscally-led regime, the price level today, P0 , is determined by the intertemporal government budget equation (Eq. (11)). Any current or expected fiscal disturbances directly affect the price level. The slope of the nominal yield curve determines the relative cost of debt financing across maturities. When the yield curve is not flat, rebalancing the maturity structure changes the government cost of capital, Rg , which directly affects the price level through Eq. (11). Importantly, the sign of the effect on the price level depends on the sign of the slope of the nominal yield curve. Suppose the yield curve is downward-sloping, then shortening the maturity structure increases Rg . To satisfy the intertemporal government budget equation, the increase in discount rates is reflected by a devaluation of the debt portfolio through inflation (increases in P0 ). The effects are reversed when the yield curve is upward-sloping, and neutral when the yield curve is flat. If we add sticky prices (and the slope is nonzero), rebalancing the maturity structure will also have real effects, so that the fiscal discount rate channel violates Wallace neutrality. Fig. 2 illustrates comparative statics from shortening the maturity structure (i.e., increasing w) on the government discount rate (top figure) and inflation (bottom figure). We show these comparative statics for parameterizations of the SDF where the nominal yield curve is upwardsloping (solid line), flat (line with circles), downward-sloping (dashed line). Consistent with the intuition above, the sign of the slope determines the sign of the responses to changes in maturity restructuring. These comparative statics can be obtained analytically by using a log-linear approximation for the definition of the return on the government bond portfolio. As derived in Appendix B, the impact of shortening maturity on expected inflation is negatively related to the slope of the nominal yield curve: ” ı rt dE Π dω

p2q



p1q

´2 ¨ Eryt ´ yt s . 1 ´ θ1

9

(12)

While we focus primarily on maturity restructuring operations, the fiscal discount rate channel also applies to other types of asset purchases by the government. Accounting for differences in expected returns across financing instruments, regardless of maturity, in the fiscal theory provides a direct channel for changes in the government portfolio to impact inflation. In Appendix C we provide a simple example illustrating the impact from rebalancing a portfolio with defaultable and nondefaultable nominal debt.

3

Quantitative Model

This section presents the quantitative general equilibrium model that integrates the concepts from the simple example above.

3.1

Households

The representative household is assumed to have Epstein-Zin preferences over streams of consumption Ct and labor Lt : ´ ¯ 1 1´θ 1´θ Ut “ u pCt , Lt q ` βt Et rUt`1 s ,

where θ “ 1 ´

1´γ 1´1{ψ ,

(13)

γ is the coefficient of risk aversion, and ψ is the elasticity of intertemporal

substitution. Time preference shocks are modeled as:

logpβt q “ p1 ´ ρβ q logpβq ` ρβ logpβt´1 q ` σβ β ,

(14)

where β is a standard normal shock.5 The utility kernel is time-separable: 1 1´ ψ

u pCt , Lt q “

Ct 1´

1 ψ

5

1 1´ ψ

´ χ0 Nt

Lt1`χ , 1`χ

(15)

This specification is used in Fern´ andez-Villaverde, Gordon, Guerr´on-Quintana, and Rubio-Ramirez (2015). Microfoundations for such preference shocks can be obtained in models with hetergeneous agents, as show in Gollier and Zeckhauser (2005).

10

where χ is the inverse of the Frisch labor supply elasticity. We scale the second term in the utility kernel, up¨q, by the exogenous trend component in productivity, Nt , to ensure that utility for leisure does not become trivially small along the balanced growth path. The budget constraint of the household is

Pt Ct ` Bt`1 “ Pt Dt ` Wt Lt ` Rt Bt ´ Tt ,

(16)

where Pt is the aggregate price level, Bt is the nominal market value of a portfolio of government bonds, Dt represents the aggregate payout received from the intermediate firms, Rt is the gross nominal interest rate on the bond portfolio, Wt is the nominal competitive wage, and Tt are nominal lump sum taxes raised by the government. The household chooses sequences of Ct , Lt , and Bt to maximize lifetime utility subject to the budget constraint. The corresponding first-order conditions are listed in Appendix D

3.2

Firms

Production in our economy is comprised of two sectors: the final goods sector and the intermediate goods sector.

3.2.1

Final Goods

A representative firm produces the final consumption goods Yt in a perfectly competitive market. The firm uses a continuum of differentiated intermediate goods Xit as input in a constant elasticity of substitution (CES) production technology:

ˆż 1 Yt “ 0

ν´1 ν

Xi,t

ν ˙ ν´1

,

(17)

where ν is the elasticity of substitution between intermediate goods. The profit maximization problem of the final goods firm yields the following isoelastic demand schedule with price elasticity

11

ν: ˆ Xi,t “ Yt

Pi,t Pt

˙´ν ,

(18)

where Pt is the nominal price of the final goods and Pi,t is the nominal price of the intermediate goods i.

3.2.2

Intermediate Goods

The intermediate goods sector is characterized by a continuum of monopolistic firms. Each intermediate goods firm produces Xi,t using labor, Li,t :

Xi,t “ Zt Li,t ,

(19)

where Zt represents an aggregate productivity shock common across firms, and consists of transitory and permanent components (e.g., Croce (2014) and Kung and Schmid (2015)):

logpZt q “ at ` nt ,

(20)

where at is the cyclical component and nt is the trend component. The evolution of these components follow:

at “ ρa at´1 ` σa at , ∆nt “ µ ` ρn ∆nt´1 ` σn nt ,

(21) (22)

where at and nt are correlated standard normal shocks with a contemporaneous correlation equal to ρan . The intermediate firms face a cost of adjusting the nominal price `a la Rotemberg (1982), measured in terms of the final good as φR 2

ˆ

Pi,t ´1 Πss Pi,t´1

12

˙2 Yt ,

(23)

where Πss ě 1 is the steady-state inflation rate and φR dictates the magnitude of the costs. The source of funds constraint is

Pt Di,t

φR “ Pi,t Xi,t ´ Wt Li,t ´ Pt 2

ˆ

˙2 Pi,t ´ 1 Yt , Πss Pi,t´1

(24)

where Di,t is the real dividend paid by the firm. The objective of the firm is to maximize sharepiq

holder’s value Vt

“ V piq p¨q taking the pricing kernel, Mt , the competitive nominal wage, Wt , and

the vector of aggregate state variables, Υt “ pPt , Zt , Yt q as given:

V piq pPi,t´1 ; Υt q “ max

Pi,t ,Li,t

! ” ı) Di,t ` Et Mt`1 V piq pPi,t ; Υt`1 q ,

(25)

subject to Eqs. (18) and (24). The corresponding first-order conditions are listed in Appendix D.

3.2.3

Government and Bond Supply

The government issues nominal bonds across infinitely many maturities with the face value declining geometrically. We assume that in each period the government retires outstanding debt and issues new liabilities to satisfy a particular maturity structure, λt . In particular, the amount of debt (in face value) outstanding at time t that matures at time t ` n is given by:

pnq

p1q

, Bf,t`1 “ Bf,t`1 λn´1 t

(26)

p1q

where Bf,t`1 is the amount of one-period bond outstanding, and λt is a stochastic process that determines the duration of the government bond portfolio, determined at time t´1.6 The evolution of logpλt`1 q ” xt`1 is given by:

x ` ρx xt ` σx xt , xt`1 “ p1 ´ ρx qs

(27)

s determines the average duration of debt in the steady-state and xt is a standard normal where x shock. 6

p1q

To be precise, total face value is Bf,t`1 {p1 ´ λt q.

13

The maturity shocks are initially self-financing, in the sense that the total market value of debt is unaffected upon announcement, but bond prices and market prices adjust immediately afterwards when the restructuring policy is implemented through open market operations. Appendix F describes the intraperiod timing of the maturity shocks and Appendix G illustrates the mapping between λt and the portfolio weights on the government bond portfolio. The flow budget constraint is:

p1q

p1q

Bf,t r1 ` λt Qt pλt qs “ Bf,t`1 Qt pλt q ` St ,

(28)

where St denotes the nominal value of primary surpluses and Qt pλt q is the value of the bond portfolio per unit of total supply with maturity structure λt . Qt is calculated as:

Qt pλt q “

8 ÿ

pn`1q

λnt Qt

,

(29)

n“0 pnq

where Qt

is the nominal price of a n-period zero coupon bond as defined in Eq. (7).

For parsimony, we assume that the government only levies lump-sum taxes and government expenditures are excluded. Thus, the primary surplus equals lump-sum taxes. Denoting the total p1q

market value of public debt by Bt “ Bf,t Qt´1 and scaling the budget constraint by nominal output Pt Yt , we get:

bt`1 “

Rtg bt ´ st , Πt ∆Yt

(30)

where bt`1 ” Bt`1 {pPt Yt q, st ” St {pPt Yt q, and Rtg is the nominal gross interest paid on the portfolio of government debt.7

3.2.4

Monetary and Fiscal Rules

The central bank follows an interest rate feedback rule: ˜ ln 7

p1q

Rt`1 Rp1q

¸

ˆ “ ρπ,ζt ln

Appendix G shows how to obtain Rtg from Eq. (28).

14

Πt Π

˙ ,

(31)

p1q

where Rt`1 is the gross one-period nominal interest rate, and Πt is inflation. Note that the coefficient ρπ,ζt is indexed by ζt , which determines the policy mix at time t. The fiscal authority adjusts the primary surplus-to-GDP ratio, st ” St {pPt Yt q, according to the following rule:

st ´ s “ ρs pst´1 ´ sq ` p1 ´ ρs q δb,ζt pbt ´ bq ` σs st .

(32)

The coefficient δb,ζt is also indexed ζt and is therefore depends on the policy mix at time t.

3.2.5

Monetary/Fiscal Policy Mix

Leeper (1991) distinguishes four policy regions in a model with fixed policy parameters. Two of the parameter regions admit a unique bounded solution for inflation. One of the determinacy regions is what Leeper refers to as the Active Monetary/Passive Fiscal (AM/PF) regime, which is the familiar textbook case (e.g., Woodford (2003) and Gal´ı (2015)). The Taylor principle is satisfied ˘´1 ` ´ 1). In this (ρπ ą 1) and the fiscal authority adjusts taxes to stabilize debt (δb ą β∆Y 1´1{ψ policy mix, monetary policy determines inflation while fiscal policy passively provides the fiscal backing to accommodate the inflation targeting objectives of the monetary authority. We refer to this regime as the monetary-led regime. The other determinacy region is the Passive Monetary/Active Fiscal (PM/AF) regime. The ` ˘´1 ´ 1), but instead the fiscal authority is not committed to stabilizing debt (δb ă β∆Y 1´1{ψ monetary authority passively accommodates fiscal policy (ρπ ă 1) by allowing the price level to adjust (to satisfy the government budget constraint). In this setting, fiscal policy determines inflation while monetary policy stabilizes debt and anchors expected inflation. Importantly, in this regime, fiscal disturbances, including non-distortionary taxation, have a direct impact on the price level via the government budget constraint because households know that changes in taxes will not be offset by future tax changes.8 We refer to this regime as the fiscally-led regime. When both the fiscal and monetary authorities are active, no stationary equilibrium exists. When both authorities are passive, there are multiple equilibria. In our regime-switching specifica8

In this regime, the government budget constraint is an equilibrium condition rather than a constraint that has to hold for any price path.

15

tion, we assume that the policy mix alternates between monetary- and fiscally-led regimes according to a two-state Markov chain with the following transition matrix: ¨ ˚ M“˝

˛ pM M 1 ´ pM M

1 ´ pF F ‹ ‚, pF F

where pij ” P rpζt`1 “ i|ζt “ jq and M denotes the monetary-led regime and F denotes the fiscally-led regime.

4

Results

This section presents the results from the quantitative model. We begin with a description of the calibration of the model followed by a quantitative analysis. The model is solved using a global projection method that is outlined in Appendix E.

4.1

Calibration

Table 1 presents the quarterly calibration. Panel A reports the values for the preference parameters. The elasticity of intertemporal substitution ψ is set to 1.35 and the coefficient of relative risk aversion γ is set to 10.0, which are within the standard values of the long-run risks literature (e.g., Bansal and Yaron (2004)). The mean of the time discount factor, β, is calibrated to 0.993 to match the average return on the government bond portfolio. The parameters driving the persistence and volatility of the time preference shocks, ρβ and σβ , are set to 0.8 and 0.25%, respectively, following Fern´andez-Villaverde, Gordon, Guerr´ on-Quintana, and Rubio-Ramirez (2015). The parameter χ is calibrated to 0.25, which implies a Frisch elasticity of labor of 4 (e.g., King and Rebelo (1999)). Panel B reports the calibration of the parameters relating to production and price-setting. The price elasticity of demand ν is set to 2. The price adjustment cost parameter φR is set to 7.5.9 The mean growth rate of productivity µ is set to obtain a mean annualized output growth rate of 2%. The parameters dictating the cyclical dynamics of productivity, ρa and σa , are set to be consistent 9

For example, in a log-linear approximation, the parameter φR can be mapped directly to a parameter that governs the average price duration in a Calvo pricing framework. In this calibration, φR “ 7.5 corresponds to an average price duration of 3.3 quarters.

16

with the standard deviation and persistence of realized consumption growth. The parameters governing the dynamics of the trend component of productivity, ρn and σn , are calibrated to be consistent with the expected consumption growth dynamics from Bansal and Yaron (2004). Panel C reports the calibration of the policy rule parameters. We set the steady-state debt-toGDP ratio to match the empirical average. The persistence and volatility parameters, ρs and σs , are chosen to match primary surplus dynamics. The surplus rule parameter determining the degree of debt smoothing, δb , is set to 0.05 and 0.00 in the monetary and fiscally-led regimes, respectively. The interest rate rule parameter governing the degree of inflation smoothing, ρπ , is set to 1.5 and 0.5 in the monetary- and fiscally-led regimes, respectively. The calibration of these policy parameters, conditional on regime, are consistent with structural estimation evidence from Bianchi and Ilut (2014). Steady-state inflation Πss is set to 1.005 to match average inflation. Following Bianchi and Melosi (2014), we assume that the transition matrix governing the dynamics of the policy/mix is symmetric, pM M “ pF F ” p, and is equal to 0.9875, implying that the economy stays on average for 20 years in a given regime. Overall, the model produces realistic macroeconomic dynamics and bond risk premia, as evidenced in the summary statistics reported in Panel A of Table 2. We use this framework to quantitatively examine the effects of maturity restructuring through the lens of the fiscal theory. To this end, we calibrate the bond supply process to capture salient features of the maturity structure dynamics (reported in Panel D of Table 1). The steady state maturity structure x ¯ is set to match average duration, while the standard deviation and persistence of the stochastic process driving bond duration dynamics, xt , are calibrated to match the empirical counterparts.

4.2

Yield Curve

The dynamics of the nominal yield curve provide the key transmission channel – both qualitatively and quantitatively – for maturity restructuring operations in the fiscal theory. In this section, we show that the model endogenously generates realistic term structure implications. Panel A of Table 3 reports the mean, standard deviation, and first autocorrelation of nominal and real yields for maturities of one quarter to five years. The model closely matches the mean and volatility of the 5-year minus 1-quarter nominal term spread. The preference shocks help to produce an upward-sloping

17

real term structure of interest rates, while the cyclical shocks to productivity and preference shocks generate significant inflation risk premia.10 The persistent shocks to time preferences and expected consumption growth are reflected in persistent yield dynamics, as in the empirical counterparts. The volatility of nominal yields falls short of the empirical targets, however Kung (2015) shows that a richer DSGE model that incorporates exogenous volatility and monetary policy shocks fits second moments better. Panel B illustrates that the model can reproduce the well-established empirical fact that the slope of the nominal yield curve forecasts future inflation at business cycle frequencies. The interest rate rule plays an important role in these forecasting regressions. Suppose that inflation falls persistently today, then the monetary authority responds by lowering the short rate. A temporary fall in the short rate steepens the slope of the yield curve. The responsiveness of the interest rate rule to inflation deviations controls the degree of predictability in the inflation forecasting regressions. The regime changes generate endogenous variation in macroeconomic volatility and is a source of time-varying bond risk premia. Panel B of Table 2 reports macroeconomic dynamics conditional on being in the monetary and fiscally-led regimes, respectively. The volatility of consumption growth, labor hours, and inflation is significantly lower in the monetary-led regime relative to the fiscally-led regime. The economy is primarily insulated from fiscal disturbances in the monetaryled regime due to the Ricardian tax/surplus policy characterizing this regime.11 In contrast, fiscal disturbances directly impact inflation and expected inflation through the intertemporal government budget equation in the fiscally-led regime. Panel A of Table 2 illustrates that the policy regime shifts generate a significant degree of consumption and inflation heteroskedasticity. With recursive preferences, the time-varying inflation and consumption volatility lead to predictable excess nominal bond returns. Indeed, Table 4 shows that excess bond returns are forecastable by a single factor, such as the forward premium (Fama and Bliss (1987)) or a linear combination of forward rates (Cochrane and Piazzesi (2005)), as in the data. 10

Albuquerque, Eichenbaum, Luo, and Rebelo (2016) also document how preference shocks can help generate an upward-sloping real yield curve in a real endowment economy. 11 Due to the regime shifts and rational expectations, fiscal disturbances are not perfectly neutral in the monetary-led regime.

18

Tables 2, 3, and 4 collectively demonstrate that the model provides a reasonable account of salient features of nominal bond yields and macroeconomic dynamics. In the following sections, we use this framework to quantitatively explore the interactions between the term structure and maturity operations in the fiscal theory.

4.3

Maturity Structure Shocks

As described in the simple model from Section 2, the combination of a fiscally-led regime and a non-zero slope implies that rebalancing the maturity structure affects inflation. We again can derive a similar present value relation in the quantitative model from Section 3 by iterating forward the government budget equation (Eq. (28)): « ff 8 ÿ Bt st`i bt ” , “ Et śi g Pt j“0 Rt`j { pΠt`j ∆Yt`j q i“0

(33)

which is derived in Appendix A. In the fiscally-led regime, Eq. (33) is an equilibrium asset pricing condition that determines the price level. This equation is similar to the present value relation derived in the simple model except that there is now debt issued up to infinitely many maturities and that the equation accounts for trend growth. When the expected returns of bonds vary by maturity, changing the financing mix alters the government cost of capital, Rg , which leads to an adjustment in inflation (and expected inflation) to satisfy Eq. (33). The addition of sticky prices in the benchmark model implies that the fiscal discount rate channel has real effects and therefore violates Wallace neutrality. We illustrate that the direction of these effects is determined by the sign of the expected return spread between long and short maturity bonds. As discussed in Section 2.3, when yields are persistent (like in the data and the model), the average return spread and the yield spread are approximately proportional. Thus, for ease of exposition, we state the subsequent results in terms of the yield curve slope.

4.3.1

Conditional on Different Slopes

Fig. 4 plots impulse response functions, conditional on staying in the fiscally-led regime, to a shock that corresponds to an open market operation that reduces average maturity by 0.18 years, a

19

similar magnitude as in QE2. We illustrate the effects of the maturity shock for when the yield curve is upward-sloping (solid line), flat (line with circles), and downward-sloping (dashed line). Since the average slope in the model is positive, for the downward-sloping case, we look at maturity structure shocks conditional on periods when the slope is negative. For the flat yield curve scenario, we assume that the monetary authority implements an interest rate peg. In the positive slope case, shortening the maturity structure implies that the government is refinancing at a lower rate. In the fiscally-led regime, a persistent decline in the government discount rate requires that inflation falls persistently to revalue nominal debt obligations so that the intertemporal government equation is satisfied. A heuristic interpretation of the discount rateinflation link is as follows. A fall in the government discount rate puts upward pressure on the real value of debt. Households, in anticipation of the appreciation in their debt portfolio, increase demand for debt and decrease demand for consumption goods. The fall in aggregate demand leads to a decline in the price level. Without sticky prices, the fall in prices will be sufficient to leave households content with their original consumption allocation. However, with sticky prices, the fall in prices is sluggish, so that prices are temporarily too high relative to the flexible price case, which depresses production (output) and increases the real rate. Thus, when the yield curve is upward-sloping, the fiscal channel highlights a potential “cost” of QE operations. In the negative slope case, shortening the maturity structure gives the opposite effects compared to when the slope is positive. In particular, reducing maturity in this case means that the government is refinancing at a higher rate. In the fiscally-led regime, an increase in the discount rate requires a devaluation in the real value of the bond portfolio via higher inflation to satisfy the intertemporal government budget constraint. With sticky prices, the increase in inflation stimulates an expansion in output. Finally, when the yield curve is flat, our fiscal channel is neutral even in the fiscally-led regime, as the financing costs are the same across maturities. These results illustrate how monetary policy is important for open market operations even when it is “passive”. Overall, we highlight the importance of the yield curve for maturity restructuring in the fiscal theory. The endogenous yield curve responses to the maturity structure shocks provide a feedback channel that amplifies the fiscal channel. For example, the fall in inflation (in the upward-sloping case) leads to a decline in the short rate due to the interest rate rule. A temporary fall in the short

20

rate steepens the slope of the nominal yield curve, which further deepens the fall in the government discount rate. Furthermore, the fall in the overall level of the yields from falling inflation further depresses the nominal bond portfolio return. A similar logic for the amplification effect also applies for the downward-sloping case.

4.3.2

Conditional on Different Regimes

Due to the recurrent regime shifts and rational expectations, maturity restructuring also has nonneutral effects in the monetary-led regime when the yield curve is nonzero. Fig. 5 displays impulse response functions, conditional on staying in the fiscally-led (dashed line) and the monetary-led (solid line) regimes for the relevant period, to a shock that reduces the average maturity of the government bond portfolio by 0.18 years. We show these plots for the upward-sloping case. Without regime shifts, changes in fiscal discount rates are neutral in the monetary-led regime due to offsetting Ricardian tax policy. However, the possibility of entering the fiscally-led regime propagates the restructuring effects, through agent’s expectations, to the monetary-led regime. Indeed, the responses in the monetary-led regime are qualitatively similar to the reactions in the fiscally-led regime. However, since the unconditional probability of changing regimes is small, the responses of macroeconomic quantities in the monetary-led regime are smaller. For example, output drops by 14.5 basis points in the fiscally-led regime compared to 0.7 basis points in the monetary-led regime.

4.4

Market Timing Policies

Fig. 6 plots impulse response functions from a shock that lengthens average maturity, conditional on staying in the fiscally-led regime, for various slopes as in Fig. 4. The effects of lengthening maturity are the opposite to shortening maturity. When the yield curve is upward-sloping (downwardsloping), lengthening maturity is expansionary (contractionary). These results suggest that there might be a role for market timing maturity operations based on the slope of the nominal yield curve. Consider modifying the maturity restructuring process so it that depends directly on the slope

21

of the nominal yield curve: ´ xt “ ρx xt´1 ` φ

¯ yt5Y ´ yt1Q ` σx x,t .

(34)

A positive coefficient (φ ą 0) implies that the government lengthens the maturity structure when yield curve is upward-sloping and shortens it when the yield curve is downward-sloping. A negative coefficient implies the opposite policy. Fig. 7 plots the comparative statics for varying φ from -1 to 1. More positive values of φ smooth macroeconomic fluctuations, reduce risk premia, and improve welfare through the fiscal discount rate channel. In contrast, more negative values of φ increase consumption and inflation volatility, and, in turn, decrease welfare. Positive values of φ shorten the maturity structure when the yield curve is downward sloping, which stimulates the economy and generates fiscal inflation exactly during low growth states. Using similar logic, negative values for φ deepen recessions and increase deflationary pressure.

5

Extended Model and Policy Experiment

In this section we investigate the quantitative importance of the fiscal discount rate channel in the context of the quantitative easing operations during the Great Recession. To enrich the analysis, we augment the quantitative model from Section 3 with a zero lower bound (ZLB) constraint on nominal interest rates, market segmentation, and short-term liquidity demand shocks. The economy is hit with large negative surplus shocks to replicate the sizable budget deficits during this period. The extended model is recalibrated to fit the macroeconomic and term structure dynamics from Tables 2, 3, and 4. We take a conservative approach in evaluating the fiscal channel by analyzing the effects in the monetary-led regime. At the onset and during the aftermath of the Great Recession, short-term nominal interest rates were near zero, and therefore, the monetary authority was unable to prevent deflationary/contractionary pressure by lowering interest rates using conventional measures. Consequently, policymakers resorted to unconventional monetary policy, such as the maturity twist operations (e.g., the Maturity Extension Program and QE2). Market segmentation and providing short-term

22

liquidity are often-cited motivations by policymakers for implementing the maturity restructuring operations (e.g., Bernanke (2012)). Indeed, absent the fiscal channel, these channels imply that shortening maturity on the scale of QE2 would have a powerful inflationary and expansionary effect. As outlined in the previous section, the impact of the fiscal channel depends on the slope of the nominal yield curve. Given that the slope was large and positive during the Great Recession, the fiscal channel provides an opposing force that weakens the stimulative effects of QE. In the following sections, we introduce each new model ingredient in isolation and then discuss their role in the transmission of maturity shocks. At the end of this section, we integrate all of the new ingredients together for counterfactual analysis, where we discuss tradeoffs for QE2-type operations and quantitatively evaluate the significance of the fiscal discount rate channel.

5.1

ZLB

We incorporate a ZLB constraint in the extended model: ˜ ln

p1q

Rt`1 Rp1q

¸

" ˆ ˙* Πt “ max 0, ρπ,ζt ln . Π

(35)

As in Aruoba, Cuba-Borda, and Schorfheide (2013), Fern´andez-Villaverde, Gordon, Guerr´ onQuintana, and Rubio-Ramirez (2015), and Gust, L´opez-Salido, and Smith (2012), we use global projection methods and rational expectations, as in the previous model analysis, to approximate the policy functions with the ZLB constraint. Fig. 8 plots impulse response functions for a shock that reduces average maturity by 0.18 years (e.g., QE2), conditional on being in the monetary-led regime, at the ZLB (solid line) and away from the ZLB (dashed line). We consider the benchmark case where the yield curve is, on average, upward-sloping. For the case where the ZLB binds, large time preference shocks are used to send the economy to the ZLB for 10 quarters. The response functions plotted for the ZLB case are deviations from a counterfactual economy where the maturity shock does not occur. The impact of macroeconomic shocks, including maturity restructuring, are amplified at the ZLB because the monetary authority loses the ability to offset such disturbances through inflation smoothing.

23

5.2

Persistent Deficits

Fig. 9 explores the impact of persistent budget deficits on maturity restructuring policies. This figure displays impulse response functions, conditional on starting in the fiscally-led regime and an upward-sloping nominal yield curve, to a shock that reduces average maturity by 0.18 years under positive surpluses around the steady-state level as in the benchmark (solid line), a negative surplus shock calibrated to match the budget deficits during the Great Recession (dotted line), and a severe deficit shock that is three times the magnitude of the deficit during the Great Recession (dashed line).12 The responses under persistent deficits are deviations from a counterfactual economy where the maturity shock does not occur. Restructuring under a deficit attenuates the fiscal discount rate effects on inflation. In the case where the yield curve is upward-sloping, shortening the maturity lowers the discount rate persistently. For the negative surpluses today and in the immediate future, lowering the discount rate has a negative effect on the present value of the fiscal backing. Since the surplus process is mean-reverting, surpluses will be positive at some point in the future, and a lower discount rate applied to these cash flows has a positive effect on the present value.13 Hence, if the deficit today is particularly severe or persistent, the discount rate effects from the deficit component can potentially dominate. Quantitatively, when the magnitude of the deficit is calibrated to the recent U.S. data or even three times larger, we have similar responses as the positive surplus steadystate case (benchmark), albeit somewhat weakened. For the responses to be reversed (i.e., negative component dominates), requires a deficit that is more than an order of magnitude larger than the current deficit.

5.3

Market Segmentation and Liquidity Demand

While this paper highlights a potential cost of QE through the fiscal discount rate channel, for our policy experiment, we also account for some of the key proposed benefits of shortening maturity during the Great Recession – lowering long-term borrowing costs and meeting short-term liquidity needs. These additional channels allow for a richer analysis of policy tradeoffs from maturity 12 During the onset of crisis, the surplus dropped sharply, which corresponds to around a negative 3.5-sigma shock to the surplus process. In the severe deficit case, the shock is set to be three times the magnitude of the deficit shock at the onset of the crisis. 13 This is implicitly assuming that the maturity structure shock is more persistent than the deficit.

24

restructuring considered in the next section. To this end, we incorporate market segmentation via transaction costs for bonds of different maturities (e.g., Bansal and Coleman (1996)) and shortterm liquidity demand shocks (e.g., Krishnamurthy and Vissing-Jorgensen (2012)) in the extended model. The transaction costs captures, in reduced-form, a preferred habitat motive (e.g., Vayanos and Vila (2009)). These costs imply that operations shortening debt maturity flattens the yield curve by reducing the risk premium of bonds at a specific maturity. More specifically, the household pnq

now pays an additional fee eκt

´ 1 for each dollar of bond of maturity n purchased. The budget

constraint of the representative household is modified in the following way:

Pt Ct `

8 ÿ

pnq

pnq

pnq

eκt Qt Bf,t`1 “ Pt Dt ` Wt Lt `

8 ÿ

pn´1q

Qt

pnq

Bf,t ´ St ,

(36)

i“1

i“1

p1q

pnq

where Bf,t`1 is the face value of a bond with maturity n, and is equal to Bf,t`1 λtn´1 . Solving the household problem, the nominal price of a n-maturity bond is « pnq Qt

“ Et

pn´1q

Mt`1 Qt`1 Πt`1 eκpnq t

ff .

(37)

pnq

Note that the existence of transaction costs gives rise to a liquidity premium that depends on κt . Following the literature on preferred habitats, it is assumed that the liquidity premium on a bond of specific maturity depends on its total supply: » pnq

κt

¨

“ κ ¯ pnq ` κ1 –log ˝

pnq

Bf,t`1 p1q

˛

¨

‚´ log ˝

Bf,t`1

“ κ ¯ pnq ´ κ1 pn ´ 1qxt ,

pnq

˛fi

p1q

‚fl

Bf

(38)

Bf

(39)

where κ ¯ pnq ą 0 is the steady state transaction cost, and κ ¯ pnq ě κ ¯ pn´1q so that longer-term bonds are relatively more costly to trade than shorter-term ones. For parsimony, we assume that κ ¯ pnq grows linearly with maturity so that,

pnq

κt

“ pn ´ 1q pκ0 ´ κ1 xt q ,

25

(40)

where κ0 ě 0 drives the average slope the yield curve, and κ1 ě 0 measures the elasticity of transaction costs to a change in the relative supply of bonds. Demand for short-term liquidity is captured by replacing consumption Ct with a “consumption” composite, Ct‹ , that captures, in a reduced form a preference for short-term debt:14 Ct‹ “ Ct Vt , logpVt q “ %xt ,

(41) (42)

where % ď 0 captures preference for short-term debt. Fig. 10 shows impulse response functions from shortening maturity with market segmentation and short-term liquidity demand. To isolate the effects of these two new transmission channels, we shut down the fiscal discount rate channel by assuming that the economy is permanently in the monetary-led regime without the possibility of regime changes. The market segmentation parameters are calibrated so that a QE2-type shock reduces the five-year minus one quarter nominal yield spread, on average, by around 15 basis points on impact to be consistent with empirical estimates from Krishnamurthy and Vissing-Jorgensen (2011), while still matching the average yields implied by the benchmark model (Table 3). The liquidity demand parameter is calibrated to match the 5-year inflation expectation reactions to QE2 of 3 basis points inferred from inflation swaps. With market segmentation and a short-term liquidity demand, shortening maturity pushes down long-term rates while stimulating economic activity.

5.4

Policy Experiment

In this section, we combine all of the new ingredients into the quantitative model from Section 3 to evaluate the significance of the fiscal channel in the context of QE2. We start the economy off in the monetary-led regime (and allow for the policymakers to switch to the fiscally-led regime in the future), at the ZLB, and in a sizable budget deficit calibrated to the data around this period. In 14 One can think of this specification as having preference for the aggregate supply of short term relative to the supply ” of any longer-term debt. To seeı this, note that for any maturity j ą 1, we can define p1q pjq p1q pjq 1 ¯ “ %xmt , logpVt q “ % logpBf,t`1 {Bf,t`1 q ´ logpBf {Bf q , which simplifies to logpVt q “ %1 pj ´ 1q logpλt {λq

where % “ %1 pj ´ 1q.

26

Fig. 11 we evaluate the impact of a QE2-type shock on the economy in the extended model (dashed line) and in an alternative economy (solid line) that shuts down the fiscal channel (i.e., policy mix is permanently characterized by the monetary-led regime without the possibility of regime changes). These responses are deviations from a counterfactual economy where the maturity shock does not occur. As discussed above, the market segmentation and liquidity preference parameters are calibrated to match the responses of the yield spread and inflation to QE2 in the extended model. Given that the yield curve is sharply upward-sloping during this period, which is replicated in the model, the fiscal discount rate channel significantly dampens the expansionary effects of QE2 arising from liquidity demand and market segmentation. Shortening maturity when the slope is positive puts downward pressure on government discount rates. When there is a possibility of entering the fiscally-led regime, the downward pressure in discount rates imparts a downward force on inflation through the intertemporal government budget equation presented in Eq. 33. Comparing the impulse response functions, we find that the fiscal discount rate channel dampens inflation responses by 14% after 5 quarters and 37% after 10 quarters. Similarly, the fiscal channel dampens output responses by 9% after 5 quarters and 52% after 10 quarters. The increasing effects by horizon reflect that the cumulative probability of switching to the fiscally-led regime increases. In addition, the fiscal channel also partially offsets the flattening of the yield curve from market segmentation. Interpreting these results, accounting for the fiscal discount rate channel provides a potential explanation for why strong responses in inflation were not observed after the QE operations.

6

Conclusion

This paper explores the interactions between yield curve dynamics and nominal government debt maturity operations through the fiscal discount rate channel. Open market debt maturity operations are non-neutral when the slope of the nominal yield curve is nonzero in the fiscal theory. When the risk profile of bonds varies by maturity, rebalancing the maturity structure affects the cost of government financing. Changes in government discount rates directly affect inflation through the

27

intertemporal government budget equation. With sticky prices, the fiscal discount rate channel has real effects, and therefore breaks Wallace neutrality. When the yield curve is upward-sloping (downward-sloping) the effects of maturity restructuring implied by the fiscal channel are contractionary (expansionary). The effects are neutral when the slope is zero. In the nonzero slope cases, the discount rate effects are magnified when the likelihood of entering the fiscally-led regime is higher or the magnitude of the nominal yield curve slope is larger. Thus, we highlight a novel risk-based transmission channel for unconventional monetary policy that depends on the state of bond yields and expectations regarding the monetary/fiscal policy mix. We show that the power of maturity restructuring operations implemented during, and in the aftermath, of the Great Recession are attenuated during periods of fiscal stress and positive term premia. We quantify the fiscal discount rate channel in a small scale New Keynesian model with a stochastic maturity structure and policy regime changes between monetary-led and fiscally-led regimes. Calibrating the model to explain salient features of the term structure of interest rates, macroeconomic dynamics, and bond supply data we find that changes in fiscal discount rates arising from maturity restructuring shocks have sizable effects on inflation and output. In the policy experiment, we examine policy tradeoffs in a counterfactual economy and demonstrate that the fiscal channel significantly dampened the stimulative effects of QE2. Overall, we provide a potential channel for explaining the weak inflation responses following the quantitative easing operations.

28

Appendix A Present Value Relations The flow budget constraint of the government for the simple model Eq. (10) can be written as ˆ ˙ Bt Πt Bt`1 “ g ` st , Pt´1 Rt Pt p1q

where Bt “ Bt p1q

p1q Bt

p2q

Rt Bt ` Rt we have:

p2q

` Bt

p2q Bt

Bt

(43)

is the total nominal value of government debt, st is the real surplus, and Rtg “

is the nominal gross interest rate paid on the portfolio of government debt. In real terms

bt “

Πt pbt`1 ` st q. Rtg

Leading Eq. (44) for one period and taking the conditional expectation at time t,  „  „ Πt`1 Πt`1 bt`2 ` Et st`1 . Et rbt`1 s “ Et g g Rt`1 Rt`1

(44)

(45)

Iterating forward on Et rbt`1 s using the law of iterated expectations and assuming a transversality condition on real debt, we have « ff 8 ÿ Pt`i 1 Et rbt`1 s “ Et st`i . (46) śi g Pt i“1 j“1 Rt`j Using Eq. (44) and Eq. (46) together, the present value of government surpluses is ff « 8 ÿ st`i bt “ Et ( . śi g i“0 j“0 Rt`j {Πt`j

(47)

Evaluating Eq. (47) at time t “ 0 yields Eq. (11). The present value relation for the benchmark model can be calculated in a similar way. Leading Eq. (30) for one period and taking the conditional expectation at time t we get: „  „  Πt`1 ∆Yt`1 Πt`1 ∆Yt`1 Et rbt`1 s “ Et bt`2 ` Et st`1 . (48) g g Rt`1 Rt`1 Iterating forward on Et rbt`1 s using the law of iterated expectations and assuming a transversality condition on real debt, ff « 8 ÿ 1 Pt`i Yt`i st`i . (49) Et rbt`1 s “ Et śi g Pt Yt i“1 j“1 Rt`j Simplifying, we get that the present value of government surpluses is « ff 8 ÿ st`i . bt “ Et śi ´1 ´1 g i“0 j“0 Πt`j ∆Yt`j Rt`j

29

(50)

Appendix B Simple Model: Approximate Analytical Solution The real return of the government portfolio is: Rtg Πt



bt`1 ` st , bt

(51)

where lowercase variables denote real variables. Define log x ” x r, and take logs of Eq. (51): rtg ´ Π rt R

log pbt`1 ` st q ´ log pbt q , ˙ ˆ st r ´ rbt . “ bt`1 ` log 1 ` bt`1



(52) (53)

Since surpluses can be negative, substitute st “ τt ´gt (τt and gt are real taxes and government expenditures, respectively) into Eq. (52) and rearrange: ˆ ˙ τt ´ gt g r r r Rt ´ Πt “ bt`1 ` log 1 ` ´ rbt , (54) bt`1 ´ ´ ¯ ´ ¯¯ “ rbt`1 ´ rbt ` log 1 ` exp τrt ´ rbt`1 ´ exp grt ´ rbt`1 . (55) Following Berndt, Lustig, and Yeltekin (2012), we log-linearize the last term in Eq. (55) with respect to the log tax-to-debt and the log government expenditures-to-debt ratios around the steady-state: ¯¯ ¯ ´ ´ ´ ¯ ´ ¯¯ ´ ´ log 1 ` exp τrt ´ rbt`1 ´ exp grt ´ rbt`1 » log 1 ` exp τr ´ rb ´ exp gr ´ rb ¯ ´ ¯¯ ´ ´ exp τr ´ rb ¯ τrt ´ rbt`1 ´ τr ´ rb ´ ¯ ´ ` 1 ` exp τr ´ rb ´ exp gr ´ rb ¯ ´ ´ ¯¯ ´ exp gr ´ rb ¯ grt ´ rbt`1 ´ gr ´ rb . ¯ ´ ´ (56) ´ 1 ` exp τr ´ rb ´ exp gr ´ rb Collecting constant terms and rearranging: ¯¯ ´ ¯ ¯ ´ ´ ´ log 1 ` exp τrt ´ rbt`1 ´ exp grt ´ rbt`1 » θ0 ` p1 ´ θ1 q µτ τrt ´ µg grt ´ rbt`1 ,

(57)

where µτ

and θ0

´ ¯ ´ ¯ exp τr ´ rb exp gr ´ rb 1 ´ ¯ ´ ¯ , µg “ ´ ¯ ´ ¯ , θ1 “ ´ ¯ ´ ¯, “ r r r r r exp τr ´ b ´ exp gr ´ b exp τr ´ b ´ exp gr ´ b 1 ` exp τr ´ b ´ exp gr ´ rb ´ ¯ ´ ´ ¯ ´ ´ ¯ ¯¯ exp τr ´ rb ´ ¯ ´ ¯ τr ´ rb “ log 1 ` exp τr ´ rb ´ exp gr ´ rb ´ 1 ` exp τr ´ rb ´ exp gr ´ rb ´ ¯ ´ ¯ exp gr ´ rb ´ ¯ ´ ¯ gr ´ rb . ` 1 ` exp τr ´ rb ´ exp gr ´ rb

Therefore Eq. (55) can be written as: rtg ´ Π r t « θ1rbt`1 ´ rbt ` θ0 ` p1 ´ θ1 qpµτ τrt ´ µg grt q. R

30

(58)

Iterating Eq. (58) forward and imposing the transversality condition, the present value relation at time 0 is: rbt “

8 ¯ ´ ÿ θ0 r t`j . rg ` Π ` θ1j p1 ´ θ1 qpµτ τrt`j ´ µg grt`j q ´ R t`j 1 ´ θ1 j“0

(59)

taking unconditional expectations: ” ı E rbt “

8 ´ ” ı ” ı¯ ÿ θ0 rg r t`j . ` E Π ` θ1j p1 ´ θ1 qE rµτ τrt`j ´ µg grt`j s ´ E R t`j 1 ´ θ1 j“0

(60)

´ ” ı ” ı ¯ r t`j`1 “ E R rp1q rreal in Eq. (60) we get: Using the unconditional Fisher equation E Π ´ R t`j`1 ” ı θ0 rtg ` Π rt ` E p1 ´ θ1 qpµτ τrt ´ µg grt q ´ R 1 ´ θ1 8 ´ ” ı ” ı ¯ ÿ rg rp1q rreal . ` θ1j p1 ´ θ1 qE rpµτ τrt`j ´ µg grt`j qs ´ E R t`j ` E Rt`j ´ R ” ı E rbt “

(61)

j“1

Since the unconditional expectation is the same for all j, we can simplify the expression: ” ı ” ı rg rp1q ´ R rreal ” ı ” ı E R E R t`j t`j τ r g r s E rµ ´ µ θ τ t`j g t`j 0 r t ` θ1 ` p1 ´ θ1 q ´ `E Π , (62) E rbt “ 1 ´ θ1 1 ´ θ1 1 ´ θ1 1 ´ θ1 ” ı ” ı rg rp1q ´ R rreal ” ı E R E R t`j t`j θ0 r “ ` E rµτ τrt`j ´ µg grt`j s ´ ` E Πt ` θ 1 . (63) 1 ´ θ1 1 ´ θ1 1 ´ θ1 ” ı rt : Solving for E Π

” ı rt E Π “



´

” ı E rbt ´

θ0 ´ E rµτ τrt`j ´ µg grt`j s ` 1 ´ θ1

” ı θ0 ´ E rµτ τrt`j ´ µg grt`j s ` E rbt ´ 1 ´ θ1 ” ı rp1q ´ R rreal E R t`j θ1 . 1 ´ θ1

” ı rg E R t`j

” ı rp1q ´ R rreal E R t`j

´ θ1 1 ´ θ1 1 ´ θ1 ” ı p1q r rp2q E ωR t`j ` p1 ´ ωqRt`j

,

(64)

1 ´ θ1 (65)

rtg for ω R rtp1q ` p1 ´ ωqR rtp2q .15 Reordering the previous equation we have: In Eq. (65) we substituted R ” ı ” ı rp2q ´ ωpR rp2q ´ R rp1q q rp1q ´ R rreal ” ı ” ı E R E R t`j t`j t`j t`j θ0 r r ´ E rµτ τrt`j ´ µg grt`j s ` ´ θ1 . (66) E Πt “ E bt ´ 1 ´ θ1 1 ´ θ1 1 ´ θ1 Finally, we can calculate the derivative of expected inflation with respect to the weight of the 1-period bond ω: ” ı ” ı rt rp2q ´ R rp1q dE Π ´E R t`j t`j “ (67) dω 1 ´ θ1 p2q

“ 15

p1q

´2 ¨ Eryt ´ yt s 1 ´ θ1

˜

˜ We used the fact that eR ´ 1 « R.

31

(68)

where the second equality uses the approximation from Eq. (9).

Appendix C Other Asset Classes p1q

Assume now that the government issues one-period riskless nominal debt Bt nominal debt Btd . The government flow budget constraint becomes: p1q

p1q

Rt Bt

pdq

` Rt Btd

and one-period defaultable

p1q

(69)

1{Qt´1 ,

pdq

(70)

 Mt`1 ´κd e , Πt`1

(71)

d “ Bt`1 ` Bt`1 ` St .

The return on the defaultable bond is: pdq

Rt



pdq

where the price, Qt , is given by the Euler equation: „ pdq Qt

“ Et

and κd ą 0 is a parameter driving the proportion of debt that defaults each period. The government discount rate is given by: pgq

Rt



p1q

p1 ´ wd qRt

pdq

` wd Rt .

(72)

p1q

pdq

Given default risk, we can show that ErRt s ą ErRt s. Therefore, increasing the financing weight on risky debt will increase the government discount rate and generate inflation. These comparative statics are illustrated in Fig. 3.

Appendix D Equilibrium piq

p1q

The equilibrium is given by the sequence tYt , Ct , Ut , Lt , Wt , Λt , bt , Mt , Zt , at , ∆nt , βt , st , xt , λt , wt , Rt , Πt , Qt , Rtg u8 t“0 determined by: 1. Household’s first-order conditions: „  Mt`1 p1q 1 “ Et Rt`1 Πt`1 „  Mt`1 r1 ` λt Qt`1 pλt qs Qt pλt q “ Et Πt`1 ¨ ˛´θ ˆ ˙´ ψ1 Ut`1 Ct`1 ‚ Mt`1 “ βt ˝ “ ‰ 1 Ct E U 1´θ 1´θ t

(73) (74)

(75)

t`1

1

Wt “

χ0 Lχt Ctψ

(76)

2. Household’s utility: 1 1´ ψ

C Ut “ t 1´

1 ψ

1 1´ ψ

´ χ 0 Nt

` ˘ 1 L1`χ t 1´θ 1´θ ` β Et rUt`1 s 1`χ

32

(77)

3. Intermediate firm’s first-order conditions: ˆ ˙ ˆ ˙ Wt 1 1 Zt “ 1´ Zt ` Λt Pt ν ν Yt ˙ ˙  ˆ „ ˆ Πt Yt Yt`1 Πt`1 Πt Πt`1 Λt “ φR ´1 ´ Et Mt`1 φR ´1 Πss Πss Πss Πss 4. Government policy: ˜ p1q ¸ ˆ ˙ Rt`1 Πt “ ρ ln ln π,ζ t p1q Π R

(79)

(80)

st ´ s “ ρs pst´1 ´ sq ` p1 ´ ρs q δb,%t pbt ´ bq ` σs st bt`1 “

(78)

Rtg

bt ´ s t Πt ∆Yt 1 ` λt Qt pλt q Rtg “ Qt´1 pλt q

(81) (82) (83)

5. Output: Yt “ Zt Lt

(84)

6. Market clearing: φR Yt “ Ct ` 2

ˆ

˙2 Πt ´ 1 Yt Πss

(85)

7. Stochastic processes: logpZt q “ at ` nt

(86)

at “ ρa at´1 ` σa at

(87)

∆nt “ µ ` ρn ∆nt´1 ` σn nt

(88)

logpβt q “ p1 ´ ρβ q logpβq ` ρβ logpβt´1 q ` σβ βt

(89)

xt “ p1 ´ ρx q¯ x ` ρx xt´1 ` σx xt

(90)

λt “ logpxt q

(91)

Appendix E Numerical Procedure The model is solved using a global method following Maliar and Maliar (2015) and Judd, Maliar, Maliar, and Valero (2014). A subset of the policy functions are approximated by piece-wise ordinary polynomials of the state variables. The state variables are: ` ˘ St “ at , βt , st´1 , bt´1 , xt , ∆nt , st , Qt´1 , (92) where at is the transitory productivity shock, βt is the preference shock, st´1 is the government surplus, bt´1 is the total value of government debt, xt is the stochastic process driving the maturity structure, ∆nt is the permanent productivity shock, st is the innovation to the government’s surplus, and Qt´1 is the price of the government bond portfolio. The approximated policy functions are: ` ˘ G “ Ft , EMt$ , EUt , Qt , (93)

33

where ˆ Ft “

˙ Πt Πt ´1 , Πss Πss

(94)

EMt$ is the expected nominal discount factor, EUt is the expected household utility, Qt is the price of the government bond portfolio. Let L be a policy function, the policy function is approximated as: Lp “ 1F pF ` 1M pM ,

(95)

where 1i is an indicator function equal to one in regime i and zero otherwise, and pi is a polynomial. Note that a different polynomial is used in each regime as indicated by the F and M subscript. The model is solved by finding the set of polynomial coefficients Θ that minimizes the mean squared residuals for the approximated decision rules over a fixed grid. For each point j on the grid the residuals tRjk u4k“1 are calculated as: Rj1 Rj2 Rj3 Rj4

φR Ftj Ytj ´ Etj rMt`1 φR Ft`1 Yt`1 s ´ Λt ,  „ Mt`1 ´ EMt$,j , “ Etj Πt`1 “

“ EUtj ´ Etj rUt`1 s , „  Mt`1 p1 ` λt Qt`1 q ´ Qjt . “ Etj Πt`1

(96) (97) (98) (99)

Rj1 is calculated using the first order condition of the firms in the intermediate goods sector, Rj2 is calculated using the expected value of the nominal discount factor, Rj3 is computed using the value function equation. Finally, Rj4 is computed using the Euler equation for the portfolio of government bonds. The grid determining the state space is calculated in 5 steps. First, for each of the regimes the model is solved using a second-order perturbation approximation to obtain an initial guess for the set of coefficients Θ. Second, the regime-specific model is simulated and principal components of the state variables are calculated. Third, an auxiliary grid on the principal components space is calculated using the Smolyak algorithm. Fourth, the regime-specific grid on the state variables space is calculated by performing a linear transformation of the auxiliary grid calculated in the previous step. Lastly, the final grid is calculated as the union of the two single regime grids (fiscally-led regime and monetary-led regime). The Smolyak algorithm is used for the auxiliary grid because it is a highly efficient method to calculate a sparse grid in a hypercube. The drawback of the Smolyak algorithm is that the points are not chosen to maximize the number of points on the region of the state space where the ergodic distribution of the model is located. The Smolyak algorithm is improved by adapting it to the characteristics of the model using the principal components transformation. Second-order standard ordinary polynomials are used for each of the regime-specific polynomials. The minimization is done using a numerical optimizer. To improve the speed of the code, monomial integration is used. The mean square error is of the order of 10´6 .

Appendix F Intraperiod Timing Within each period t, there is a morning (beginning of period t) and an evening (end of period t). 1. During the morning, the following events unfold concurrently: • financial and goods markets open • consumption, labor, and production decisions are made • productivity, monetary, preference, and surplus shocks are realized

34

• open market operations are conducted to implement the maturity structure policy, λt , announced in the evening of the previous period • government determines total amount debt outstanding Bt`1 2. The financial and goods markets close before the evening begins. 3. During the evening: • announcement of the maturity structure policy (to be implemented next period), λt`1 , such that total market value of debt this period would be unchanged, given the prices determined in the morning

Appendix G Portfolio Return The government flow budget constraint is: p1q

p1q

Bf,t r1 ` λt Qt pλt qs “ Bf,t`1 Qt pλt q ` St .

(100)

Given the timing assumption defined in Appendix F, the face value of 1-period bonds at the end of period p1q t ´ 1, Bf,t , is calculated so that the market value of debt with the old maturity structure, λt´1 , is equal to the market value of debt with the new maturity structure, λt . Thus, we have: p1q

Bt “ Bf,t Qt´1 pλt q .

(101)

Using Eq. (101), we rewrite Eq. (100) in terms of market values: Bt

1 ` λt Qt pλt q “ Bt`1 ` St , Qt´1 pλt q

(102)

where 1 ` λt Qt pλt q “ Rtg Qt´1 pλt q

(103)

is the return of the portfolio of government’s bonds held by the public. We rewrite the return of the portfolio of government bonds as a weighted average of the returns for the different bonds. Given Eq. (29) we can express Eq. (103) as: Rtg “

1 ` λt

ř8

pn`1q

λnt Qt Qt´1 pλt q n“0

.

(104)

Rearranging terms we get: Rtg “

Since

1 p1q Qt´1

p1q

“ Rt

pn`1q

and

Qt

pn`2q

Qt´1

1 p1q

Qt´1

pn`2q

“ Rt

Rtg “

p1q pn`2q 8 pn`1q ÿ Qt´1 Q Q ` λt λnt tpn`2q t´1 . Qt´1 Qt´1 Q n“0

(105)

t´1

(@n ě 0) we get:

p1q pn`2q 8 ÿ Qt´1 p1q Q pn`2q Rt ` λt λnt t´1 Rt . Qt´1 Q t´1 n“0

35

(106)

From the previous equation we see that the bonds’ weights are: p1q

p1q

wt



Qt´1 , Qt´1



λtj´1

(107) pjq

pjq

wt

Qt´1 @j ą 1. Qt´1

(108)

We can check than the weights sum up to one. The sum of all the weights is given by: 8 ÿ

pjq

wt



j“1

p1q pn`1q 8 ÿ Qt´1 Q ` λnt t´1 . Qt´1 n“1 Qt´1

(109)

Rearranging terms we get: 8 ÿ

p1q

pjq wt



Qt´1 `

ř8 n“1

Qt´1

j“1

pn`1q

λnt Qt´1

.

(110)

We can see from Eq. (29) that the numerator in the previous equation is equal to Qt´1 . Finally, we get: 8 ÿ j“1

pjq

wt



Qt´1 “ 1. Qt´1

(111)

Appendix H Data We obtain quarterly data for consumption and output from the Bureau of Economic Analysis (BEA). Consumption is measured as real personal consumption expenditures (DPCERX1A020NBEA). Output is measured as real gross domestic product (GDPC1). Inflation is computed by taking the log return on the Consumer Price Index for All Urban Consumers (CPIAUCSL), obtained from the Bureau of Labor Statistics (BLS). Monthly yield data are from CRSP. Nominal yield data for maturities of 4, 8, 12, 16, and 20 quarters are from the CRSP Fama-Bliss discount bond file. The one-quarter nominal yield is from the the CRSP Fama risk-free rate file. Finally, we obtain the maturity distribution of privately-held Treasury marketable securities from Table FD-5 of the Treasury Bulletin. We supplement the information on the Treasury Bulletin by including reserve balances with Federal Reserve Banks from the Federal Reserve H.4.1.

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41

Table 1: Calibration Parameter

Description

Model

β ρβ σβ ψ γ χ

Mean of the time discount factor Persistence of βt Volatility of preference shock Elasticity of intertemporal substitution Risk aversion Inverse of the Frisch labor supply elasticity

0.993 0.8 0.25% 1.35 10 0.25

B. Production ν φR µ ρa σa ρn σn

Price elasticity for intermediate goods Magnitude of price adjustment costs Unconditional mean growth rate Persistence of at Volatility of transitory shock at Persistence of ∆nt Volatility of permanent shock nt

2 7.5 0.5% 0.95 0.75% 0.99 0.015%

C. Policy ρs σs δb pM {F q ρπ pM {F q p

Persistence of government surpluses Volatility of government surpluses st Sensitivity of taxes to debt Sensitivity of interest rate to inflation Switching probability

0.900 0.125% 0.05 / 0.00 1.5 / 0.5 0.9875

D. Bond Supply ¯b ex¯ ρx σx

Steady state Debt-to-GDP ratio Steady state maturity structure Persistence of xt Volatility of xt

A. Preferences

0.5 0.950 0.900 0.128%

This table reports the parameter values used in the quarterly calibration of the model. The table is divided into four categories: Preferences, Production, Policy, and Bond Supply parameters.

42

Table 2: Summary Statistics Panel A: Unconditional Moments

I. Means ` ˘ E y p5Y q ´ y p1Qq (in %)

Data

Model

1.02

1.04

1.42 1.64 0.92 0.84 0.97

1.49 1.20 0.67 1.26 0.46

-0.56 0.25 0.73 0.40 0.26

-0.14 0.08 0.84 0.34 0.24

II. Standard deviations σp∆cq (in %) σpπq (in %) σp∆lq{σp∆yq σpV ol∆c,t,t`4 q (in %) σpV olπ,t,t`4 q (in %) III. Correlations corrpπ, ∆cq AC1p∆cq AC1pπq AC1pV ol∆c,t,t`4 q AC1pV olπ,t,t`4 q

Panel B: Conditional moments

I. Means ` ˘ E y p5Y q ´ y p1Qq (in %)

Monetary

Fiscal

1.40

0.67

1.30 0.68 0.13

1.77 1.49 1.01

-0.29

-0.10

II. Standard deviations σp∆cq (in %) σpπq (in %) σp∆lq{σp∆yq III. Correlations corrpπ, ∆cq

Panel A presents the unconditional means, standard deviations, and correlations of key variables. Panel B presents summary statistics from the model conditional on being in the monetary and fiscal regimes. The series for realized volatility are computed following Beeler and Campbell (2012) and Bansal, Kiku, and Yaron (2012). First, a series is fitted to an AR(1): wt “ a0 ` a1 wt´1 ` ut . Then, annual (four-quarter) realized ř4´1 volatility is computed as V olw,t,t`4 “ j“0 “ |ut`j |. The quantitative model is calibrated at a quarterly frequency and the reported statistics are annualized.

43

Table 3: Term Structure of Interest Rates Panel A: Unconditional Moments Maturity 3Y 4Y

1Q

1Y

2Y

5Y

5Y - 1Q

Mean (Model) (in %) Mean (Data) (in %)

4.57 4.58

5.06 4.96

5.21 5.16

5.34 5.34

5.47 5.49

5.61 5.60

1.04 1.02

Std (Model) (in %) Std (Data) (in %)

1.54 3.12

1.10 3.12

0.89 3.11

0.77 3.03

0.66 2.97

0.59 2.90

0.50 1.00

AC1 (Model) AC1 (Data)

0.82 0.95

0.86 0.95

0.88 0.96

0.90 0.96

0.91 0.97

0.92 0.97

0.78 0.75

2.25 0.69 0.91

2.37 0.59 0.94

2.48 0.51 0.95

2.56 0.46 0.96

2.61 0.43 0.96

2.65 0.41 0.96

0.40 0.37 0.80

I. Nominal yields

II. Real yields Mean (Model) (in %) Std (Model) (in %) AC1 (Model)

Panel B: Inflation Forecasts Data

β S.E. R2

1

4

-1.328 0.227 0.180

-1.030 0.315 0.157

Model Horizon (in Quarters) 8 1 -0.649 0.330 0.071

-0.606 0.040 0.623

4

8

-0.490 0.061 0.469

-0.385 0.071 0.333

Panel A presents the mean, standard deviation, and first autocorrelation of the one-quarter, one-year, twoyear, three-year, four-year, and five-year yields and the 5-year minus one-quarter spread for nominal and real yields. Panel B presents inflation forecasts for horizons of one, four, and eight quarters using the five-year p5q nominal yield spread. The n-quarter regressions, n1 pxt,t`1 ` ¨ ¨ ¨ ` xt`n´1,t`n q “ α ` βpyt ´ y p1Qq q ` t`1 , are estimated using overlapping quarterly data and Newey-West standard errors are used to correct for heteroscedasticity. The model is calibrated at a quarterly frequency and the moments are annualized.

44

Table 4: Bond Return Predictability Data 2

3

Model 4

Maturity (in years) 5 2 3

4

5

A. Fama-Bliss β pnq S.E. R2

1.076 1.476 1.689 1.150 0.239 0.321 0.407 0.619 0.175 0.190 0.185 0.068

0.408 0.439 0.461 0.483 0.100 0.104 0.102 0.100 0.098 0.112 0.124 0.134

0.455 0.862 1.229 1.449 0.027 0.014 0.011 0.030 0.379 0.415 0.446 0.421

0.571 0.926 1.161 1.343 0.072 0.065 0.016 0.121 0.091 0.120 0.145 0.152

B. Cochrane-Piazzesi β pnq S.E. R2

This table presents forecasts of one-year excess returns on bonds of maturities of two to five years. Panel pnq A reports forecasts of excess bond returns using the forward spread (i.e., Fama-Bliss regressions): rxt`1 “ pnq p1q pnq α ` βpft ´ yt q ` t`1 . Panel B reports forecasts of excess bond returns using the Cochrane-Piazzesi ř5 pnq factor. First, the factor is obtained by running the regression: 14 n“2 rxt`1 “ γ 1 ft ` t`1 , where γ 1 ft ” p1q p2q p5q γ0 ` γ1 yt ` γ2 ft ` ¨ ¨ ¨ ` γ5 ft . Second, use the factor γ 1 ft obtained in the previous regression to forecast pnq pnq bond excess returns of maturity n: rxt`1 “ bn pγ 1 ft q ` t`1 . The forecasting regressions use overlapping quarterly data and Newey-West standard errors are used to correct for heteroscedasticity.

45

Figure 1: Average Maturity of Public Debt 5.4 5.2 average maturity (years)

5

QE1



4.8 4.6 4.4

QE2



4.2

MEP



4

QE3



3.8 3.6 3.4

2006

2008

2010 time (years)

2012

This figure plots the average maturity structure of government debt held by the public from Q1-2005 to Q3-2013.

Figure 2: Comparative Statics: Shortening Maturity

E[rg]

Slope>0 Slope=0 Slope<0

0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

E[π]

w

0

0.2

0.4

w

This figure plots the average effect from increasing the portfolio weight on short-term debt, w, on the government expected discount rate ErRg s (top figure) and expected inflation Erπs (bottom figure) for when the average yield curve is upward-sloping (solid line), downward-sloping (dashed line), and flat (line with circles) in the simple model.

46

E[rg]

Figure 3: Comparative Statics: Defaultable Debt

0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

E[π]

wd

0

0.2

0.4

wd

This figure plots the average effect from increasing the portfolio weight on risky debt, wd , on the expected government discount rate ErRg s (top figure) and expected inflation Erπs (bottom figure) in the simple model.

-0.05

Slope>0 Slope=0 Slope<0

-0.1 -0.15 0

10

0.05

0.05

0

0

-0.05

-0.05

20

0

10

20

0

output

0

0 -0.02

-0.02

10

20

10

20

0.1

0.02

0.02

r(1)

yield spread

E[r g ]

maturity

Figure 4: Maturity Restructuring with Different Slopes

0 -0.1

-0.04 0

10

quarters

20

0

10

quarters

20

0

quarters

This figure plots conditional impulse response functions for the government discount rate, inflation, the 5-year minus 1-quarter nominal yield spread, the 1-quarter nominal yield, and output to a decrease in the average maturity of government debt (similar in magnitude as QE2) in the fiscal regime. Results are reported for when the yield curve is upward-sloping (solid line), flat (line with circles), and downward-sloping (dashed line). The units of the y-axis are annualized percentage deviations from the steady-state. The units for average maturity are in years.

47

Figure 5: Maturity Restructuring in Different Regimes Monetary Fiscal

-0.1

E[r g ]

maturity

0 -0.05

-0.15

-0.02

-0.02

-0.04

-0.04

-0.06

-0.06 -0.08

10

20

0

10

20

0

output

0.02

10

20

10

20

0

0 0.03

r(1)

yield spread

0

-0.02

0.01

-0.05 -0.1

-0.04

0 0

10

20

0

quarters

10

20

0

quarters

quarters

This figure plots the conditional impulse response functions for the government discount rate, inflation, the 5-year minus 1-quarter nominal yield spread, the 1-quarter nominal yield, and output to a decrease in the average maturity of government debt (similar magnitude as QE2) in the monetary regime (solid line) and fiscal regime (dashed line). The units of the y-axis are annualized percentage deviations from the steadystate. The units for average maturity are in years.

0.15

Slope>0 Slope=0 Slope<0

0.1

E[r g ]

maturity

Figure 6: Lengthening Maturity 0.05

0.05

0

0

0.05

-0.05

-0.05 10

20

0

10

20

0

10

20

10

20

0.04

0 -0.02

0.02

output

0.02

r(1)

yield spread

0

0 -0.02

0

10

quarters

20

0.1 0 -0.1

0

10

quarters

20

0

quarters

This figure plots conditional impulse response functions of the government discount rate, inflation, the 5-year minus 1-quarter nominal yield spread, the 1-quarter nominal yield, and output to an increase in the average maturity of government debt in the fiscal regime. Results are reported for when the yield curve is upward sloping (solid line), flat (line with circles), and downward sloping (dashed line). The units of the y-axis are annualized percentage deviations from the steady-state. The units for average maturity are in years.

48

Figure 7: Market Timing Policies

6.5687 6.5603 -1

0

1

1.0991

2.093

std( )

yield spread

welfare

6.5771

1.04 0.9808 -1

1.2 0.32

0

1

-1

0

1

This figure plots comparative statics for the welfare, average yield spread, and the standard deviation of inflation when the market timing sensitivity to the yield spread varies.

Figure 8: Maturity Restructuring at the ZLB -0.08 -0.1 -0.12 -0.14 -0.16 -0.18

away ZLB at ZLB

-5

-0.2

-10 -0.4

0

5

10

-15 0

0.02

10

-0.5 -1

0

5

quarters

10

0

output

0.04

5

5

10

5

10

10-3

0

r(1)

yield spread

E[r g ]

maturity

10-3

0

5

quarters

10

-0.02 -0.04 -0.06 -0.08 0

quarters

This figure plots the conditional impulse response functions of the government discount rate, inflation, the 5-year minus 1-quarter nominal yield spread, the 1-quarter nominal yield, and output to a decrease in the average maturity of government debt (similar magnitude as QE2) when in the monetary regime. The solid line represents the response at the zero lower bound and the dashed line represents the response away from the zero lower bound. The units of the y-axis are annualized percentage deviations from a counterfactual economy where the decrease in average maturity does not occur. The units for average maturity are in years.

49

Figure 9: Maturity Restructuring with Persistent Deficits

-0.1 -0.15

-0.04 -0.06

0

10

20

0

0.04

10

20

0

0

r(1)

yield spread

0 -0.02 -0.04 -0.06 -0.08

-0.02

0.02 0

10

20

10

20

0

output

maturity

Positive surplus Less severe deficit More severe deficit

E[r g ]

0 -0.05

-0.02

-0.05 -0.1

-0.04 0

10

20

0

quarters

10

20

0

quarters

quarters

This figure plots the conditional impulse response functions of the government discount rate, inflation, the 5-year minus 1-quarter nominal yield spread, the 1-quarter nominal yield, and output to a decrease in the average maturity of government debt (similar magnitude as QE2) when the economy starts initially in the fiscal regime. The solid line represents the response under positive surpluses. The dotted line represents the response under a deficit shock (-3.5 sigma) calibrated to the data during the Great Recession (2008-2010). The dashed line represents a severe deficit shock that is three times the magnitude of the deficit during the Great Recession. The units of the y-axis are annualized percentage deviations from a counterfactual economy where the decrease in average maturity does not occur. The units for average maturity are in years.

-0.05

E[r g ]

maturity

Figure 10: Maturity Restructuring with Market Segmentation and Liquidity Demand

-0.1

0.04

-0.2

-0.15

0.03

-0.4 10

20

0

10

20

0

output

-0.1

10

20

10

20

0.2

0.07 -0.05

r(1)

yield spread

0

0.06 0.05

0.1

-0.15 0

10

quarters

20

0

10

quarters

20

0

quarters

This figure plots the impulse response functions of the government discount rate, inflation, the 5-year minus 1-quarter nominal yield spread, the 1-quarter nominal yield, and output to a decrease in the average maturity of government debt (similar magnitude as QE2) in the monetary regime (without regime shifts) with market segmentation. The units of the y-axis are annualized percentage deviations from the steady-state. The units for average maturity are in years.

50

-0.05 -0.1

E[r g ]

maturity

Figure 11: Policy Experiment

No switching Switching

-0.15

0.07 0.06 0.05 0.04 0.03

-0.2 -0.4

10

20

0

-0.1

10

20

-0.15

0

10

20

10

20

0.2

output

0.04

r(1)

yield spread

0

0.02

0.1

0 0

10

quarters

20

0

10

quarters

20

0

quarters

This figure plots the conditional impulse response functions of the government discount rate, inflation, the 5-year minus 1-quarter nominal yield spread, the 1-quarter nominal yield, and output to a decrease in the average maturity of government debt (similar magnitude as QE2) when the economy starts initially in the monetary regime, at the zero lower bound, and under a deficit shock of magnitude ´3.5σs . These plots come from the extended model that also incorporates market segmentation and a short-term liquidity demand. The dashed line represents the response from the extended model that allows for policymakers to switch to the fiscally-led regime and the solid line represents the response from an alternative economy where the policy mix is permanently characterized by the monetary regime without the possibility of regime changes. The units of the y-axis are annualized percentage deviations from a counterfactual economy where the decrease in average maturity does not occur. The units for average maturity are in years.

51

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