PRELIMINARY. August 18, Abstract. Keywords: Fiscal policy, Regime Shifts, Yield curve.

Transcription

1 Non-Linear Eects of Fiscal Policy on the Yield Curve PRELIMINARY Hans Dewachter *, Romain Houssa, Olivier Hubert August 18, 217 Abstract We develop a multi-regimes no-arbitrage term structure model that focuses on the role of scal policy and that incorporates information on switches in the stance of scal policy: a sustainable versus an unsustainable path of government debt. Fiscal policy is deemed unsustainable if current scal policy fails to target the decit that is required to stabilize the debt ratio. The model comprises latent yield curve factors and macroeconomic factors that are not spanned by the yield curve. We apply our macro-nance model to US data for the period 1972 to 211. Results suggest that accounting for the stance of scal policy is important to appraise bond risk premia and the eects of decit shocks on interest rates. Typically, expected excess returns are systematically larger in the unsustainable debt regime as compared to a single regime and a sustainable regime. One standard deviation scal policy shocks identied by recursive identication scheme and sign restrictions increase yields by between 14 and 34 basis points, the increase in bond yields being sensibly larger in the unsustainable regime than the single or sustainable regimes. Variance decompositions indicate that scal shocks matter more for long maturities. Fiscal shocks are also more important in the unsustainable regime. JEL codes: E43, E44, E52, E62, H63 Keywords: Fiscal policy, Regime Shifts, Yield curve. * National Bank of Belgium, University of Leuven University of Namur, University of Leuven and CESifo University of Namur. olivier.hubert@unamur.be 1

2 1 Introduction The theoretical literature has proposed several channels through which scal policy can aect the economy. The dierent theories proposed have reached ambiguous conclusions. On the one hand, conventional theory predicts that an expansionary scal policy creates a shift in Aggregate Demand and a change in the composition of Aggregate Demand, either through a direct increase in goods purchases if the government raises its consumption without changing taxes, or via an improvement in households' disposable income which will increase private consumption and therefore Aggregate Demand. If the government nances its decit by issuing debt it competes with private borrowers for loanable funds. The demand for loanable funds increases and so does the interest rate. This rise in the interest rate crowds out a part of private investment. The economy being Keynesian in the short run, an increased Aggregate Demand raises national income because sticky prices and sticky wages lead to the increased use of the factors of production. Conventional theory also assumes that the economy is classical in the long run. As a consequence, national savings and investment are reduced. A lower stock of capital raises the marginal product of capital, thus raising the interest rate. In turn, local currency appreciates. On the other hand, forward-looking Ricardian households understand that an increase in the stock of debt today induces higher interest payments in the future that would be nanced through higher taxes. These households may thus decide to save in order to support higher future taxes. Thus, the increase in private savings will exactly match the decrease in public savings such that the equilibrium interest rate on the market for loanable funds will remain unchanged. These arguments, however, rest on the assumption that government consumption does not move. If a tax reduction today is met by a decrease in government spending tomorrow, it can still generate wealth eects and would stimulate consumption and reduce national saving. 1 In line with these ambiguous predictions empirical studies have not been able to reach a consensus regarding the eects of scal policy on interest rates. Gale and Orzsag (23) and Barth et al. (1991) provide extensive reviews of empirical results. The latter surveyed 42 empirical studies and report that 17 studies found a predominantly signicant, positive eect of decits on interest 1 In a related literature, Woodford (1996) pioneered the Fiscal Theory of the Price Level. When the increase in debt, that is not is oset but future changes in taxes or spending, generates a wealth eect for households and increase Aggregate Demand. If Aggregate Supply does not change, both goods market and government's budget clearing conditions require that the price level increases enough to match future real debt with its initial value. 2

3 rates, 6 found mixed results and 19 found predominantly insignicant or negative results (Barth et al., 1991). More recent studies suggest a signicant link between decits and interest rates (e.g. Canzoneri, Cumby and Diba (22), Dai and Philippon (25), and Laubach (29)). For instance, Dai and Philippon (25) show that a one percentage point increase in the decit ratio increases the 1-year rate by 25 basis points after three years. Laubach (29) produces similar estimates for the US using projected budget decit. Another strand of the literature has assumed a non-linear relationship between scal policy and bond yields. For instance, Davig and Leeper (27) and Davig and Leeper (211) identify Active and Passive scal policies, in the words of Leeper (1991), and show that the response of the short-rate in a calibrated model diers greatly in sign and magnitude across the two regimes. Dewachter and Toano (212) investigate the link between scal decit and bond yields across scal policy regimes in the USA. Using regression techniques, they show that unsustainable decits are associated with higher bond yields whereas no statistical relationship is found between government decit and bond yields in periods when debt is sustainable. In a related study, Houssa and Hubert (215) use a local projection method to analyze the impact of government decit shocks on bond yields in the G7 economies. They nd that a primary decit shock in the G7 signicantly increases bond yields in the unsustainable scal regime as opposed to a signicant decrease in bond yields in the sustainable scal regime. One important insight from the literature discussed above is that non-linearity is important in uncovering the link between scal policy and interest rates. However, these studies are not able to bind dierent maturities in a consistent manner and this is the reason why models of the term structure are needed. Term structure models are capable of decomposing long rates into expected future short rates and risk premia. Bond risk premia represent the compensations demanded by investors for exposures associated to the fundamental risks. As such, understanding movements in bond risk premia provides valuable information on market participants' valuation of risks. The seminal paper of Ang and Piazzesi (23) extends a latent yield curve term structure model with macro factors such that the relevant state vector for bond pricing consists of both latent and observable factors. In their setup, they impose independence between latent and macro factors. As a result, macro variables determine bond yields, but not the reverse. Ang, Dong and Piazzesi (25) and Diebold and Li (26) allow for bidirectional macro-nance linkages and show 3

4 that macro factors are much more important when bidirectionality is enforced. This framework has been then successively applied and extended in subsequent studies (e.g. Dewachter and Lyrio, 26). Recently Joslin, Priebschand Singleton (214) have challenged the assumption that macroeconomic factors can be recovered from a collection of bond yields. They assert that this restriction can lead to a misspecied model and biases the estimation of the term premium. They therefore relax what they call the macro-spanning condition: macroeconomic factors do not enter the pricing kernel of bonds, but determine expectations of future short rates. In this paper, we investigate the non-linear impact of scal policy on the yield curve. To achieve this, we propose a regime-dependent term structure model that incorporates the three standard yield curve factors (level, slope and curvature) and three macroeconomic factors (ination, economic activity and government primary decit). We follow Joslin, Priebsch and Singleton (214) and impose that the macro factors are unspanned by the yield curve. Additionally, we distinguish between two economically-grounded scal policy regimes: a sustainable versus an unsustainable path of government debt. The regimes are identied through a regime-switching feedback scal policy rule similar to Favero and Monacelli (25), Dewachter and Toano (212) and Houssa and Hubert (215). This distinction is especially relevant when estimating impulse response functions and risk premia and comparing them to a single-regime case. For this purpose, we employ two approaches for the identication of decit shocks: a recursive identication scheme and sign restrictions. We show that accounting for scal policy regime changes is economically meaningful to understand the dynamics of bond yields and risk premia. Such a claim echoes in both the recursive and sign restrictions identications. Typically, an unexpected increase in the primary decit rst depresses then raises, attens and straightens the yield curve. Responses in the unsustainable regime are also sensibly larger than the single or sustainable regime. A one percentage point increase in the primary decit-to-gdp raises interest rates by between 14 to 34 basis points, depending on the regime considered. If we decompose the total eect on yields into their components, we can see that the bulk of the movement in yields is due to the level and, to a lesser extent, the slope factor. Fiscal policy shocks are mostly important in the unsustainable regime, where its share can be twice as large as in the sustainable regime. Risk premia associated with the unsustainable regime are 4

5 consistently larger than their linear and sustainable counterparts, indicating that investors demand a premium to hold bonds of a government that runs an unsustainable scal policy. The remainder of this paper is structured as follows: Section 2 covers the methodology and describes the features of the model. In particular, we cover the identication of the economic regimes and we present the regime-dependent term structure model with unspanned macro risks. Additionnaly, we explain how the impulse response functions take into account the history of regime switches. Section 3 presents the data collection and treatment for the analysis. Results are found in Section 4 while Section 5 concludes. 2 Methodology 2.1 Fiscal Policy We derive here the scal policy rule and the decit consistent with debt-stabilization that will be useful to indentify the scal policy regimes. We start with the standard debt-accumulation equation 2 : B t = B t 1 + i b t B t 1 + D t (1) where B t is public debt, i t is the average nominal interest rate on bonds and D t is the primary decit. Note that positive values of the primary decit are associated to decits, negative values to surpluses. or, Expressing (1) as a ratio of current GDP yields: B t Y t = ( 1 + i b ) B t 1 t + D t = Y t Y t ( 1 + i b t ) ( 1 + i b t ) (1 + ζ t ) B t 1 Y t 1 + D t Y t (2) b t = (1 + ζ t ) b t 1 + d t (3) 2 Feldstein (1986) uses an even simpler version of the debt-accumulation equation : b t = b b 1 + d t. Obviously, the more elaborate version of Favero and Monacelli (25) is more precise and allows for two ways of stabilization : primary surpluses or restricted decits. 5

6 where ζ t is the growth rate of output between t-1 and t. An economy naturally decreases its debt ratio if the nominal growth rate is positive and larger than the average interest rate paid on debt. Stabilizing the debt ratio implies b t = b t 1 in (3) such that we can express the primary decit required to stabilize the debt ratio as: ( d S ζt i b t t = 1 + ζ t ) b t 1 (4) The above expression can be understood as the decit that the economy can aord, given its output growth and interests paid on its debt. Debt stabilization is achieved either through reduced decits if i b t > ζ t or surpluses if i b t < ζ t. We reproduce the analysis of Dewachter and Toano (212) and Houssa and Hubert (215) to identify scal policy regimes. We assume that scal policy follows a scal rule similar to Favero and Monacelli (25) where the current primary decit-to-gdp ratio d t can be expressed as: ( ) d t = ρ sf t dt ρ sf t dt + σ sf ɛ sf t t (5) d t = c sf t + γ s F t y (y t y t ) + δ sf t d S t where d t 1 is the rst lag of the primary decit ratio and captures inertia in scal policy, dt is the target decit and is composed of a constant c, the output gap (y t y t ) to control for the natural counter-cyclical behavior of scal policy, and the stabilizing decit d S t. The error term ɛ sf t t is assumed to be i.i.d. Each of the coecients is indexed by the superscript s F t that gives the stance of scal policy at time t. For the empirical estimation s F t can take two values. The stance of scal policy and the values of the coecients are uncovered by estimating expression (5) in a First-order Markov- Switching framework with transition matrix P F whose elements p ij gives the probability of being in regime i at time t if the data generating process was in regime j at time t 1. Formally, this gives p ij = Pr [ s F t = i s F t = j ]. The main advantage of Markov-Switching regressions over linear regressions with dummy vari- 6

7 ables is that the identication of regimes is determined by the data and not a priori by the researcher. Another advantage is that Markov-Switching regressions give a probability rather than a /1 distinction. The ratio 1 gives the average duration of regime i, enabling the researcher to p F ii gain knowledge about the probable regime several periods in the future. This reaction function allows the government to react dierently to debt developments depending on the value of the stabilizing decit. Following the taxonomy of Leeper (1991), we dene passive (or sustainable) scal policy if the Government aims at stabilizing the debt-to-gdp ratio whereas active (or unsustainable) scal policy targets macroeconomic variables, irrespective of the debt-to-gdp dynamics. Empirically, it corresponds to the question whether the target decit d t aims towards the stabilizing decit d S t in the long run. That is, if ρ < 1 (the relation is nonexplosive), c = and δ = 1. We identify the Active regime as a regime where δ <. A coecient δ < implies that government increases the decit when a reduced decit or a surplus is required. In the empirical estimation, we remove the cyclical component of the raw stabilizing decit d S t series implied by equation (4) using the Hodrick-Prescott lter setting a value of λ = 16 for quarterly observations so as to obtain a smooth long-run trend for the stabilizing decit. Doing so, we stress that debt sustainability is a long-run goal and actual scal policy is allowed to deviate from it in the short-run. In this respect, this is a small deviation of the rule set forth by Favero and Monacelli (25). 2.2 A Term Structure Model with Unspanned Macro Risks In this section, we lay out the setup of our model. The notations follow Bauer and Rudebusch (215) as a matter of easiness for the comparison. Yields are collected in the vector Y t which contains rates for J dierent maturities. The set of variables that characterize bond yields is denoted Z t. The matrix Z t contains both yield factors and macro factors. We denote the M macro factors by M t. Joslin, Singleton and Zhu (211) have proved that there exists a rotation of L < J latent factors that is observationally equivalent to observed yields portfolios, provided zero measurement error is assumed for the latent factors. For the yields portfolios, we are free to choose any combination of yields such that P t = W Y t where W is a full-rank weighting matrix that transforms observed yields into portfolios of yields. For our 7

8 analysis, we take W as the loadings coming from the eigenvector-eigenvalue decomposition of the variance-covariance matrix of observed yields. We therefore select the rst L principal components [ ] such that Z t = M t, Pt L. Our model thus contains N = M + L observable risk factors. We depart from the usual view predominant in term structure modeling that all relevant information about the state of the economy is embedded in the yield curve, what Joslin, Priebsch and Singleton (214) have called the maco-spanning condition. As a consequence, variation in macro variables is not perfectly correlated with the yield curve. Evidence suggests that there is indeed unspanned macroeconomic variation (see Joslin, Priebsch and Singleton (214) and Duee (22), Duee (211), Duee (212), Cooper (29), Ludvigson and Ng (29)). 3 Any no-arbitrage term structure model is fully described by three elements: a time series representation of the risk factors under the real-world and risk-neutral probability measures as well as an equation that links the short-rate to the priced risk factors. First, the risk factors under the real-world probability measure P follow a Vector Autoregression (VAR) of order 1 of the form: Z t = µ + ΦZ t 1 + ɛ t, ɛ t i.i.d. N (, Σ N ) (6) where Z t represents the new information that market participants obtain at time t. The parameters µ represent the constant terms, Φ is the feedback matrix that stacks the coecients of the regressions together, ɛ t is the vector of residuals, and Σ is the variance-covariance matrix of the residuals. In more details, eq. (6) gives: Z t = M t = µ M P L t µ P + Φ MM Φ PM Φ MP Φ PP Z t 1 + ɛ t = M t = µ M Pt L µ P + Φ MM Φ PM Φ MP Φ PP M t 1 + ɛ M t ɛ Pt P L t 1 (7) 3 See also Annex E for an application of the several tests of the macro-spanning condition set forth by the literature. 8

9 and Σ = E[ɛ t, ɛ t ] = Σ MM Σ PM Σ MP Σ PP. (8) Second, for unspanned macro risk term structure models, it is important to make a distinction between the total set of risk factors and the priced risk factors. The whole set of risk factors is denoted by Z t and includes both macro and yield curve risk factors while the priced factors only include the yield curve risk factors. The priced risk factors under the risk-neutral probability measure Q follow a VAR that is independent of the macro factors because macro risk factors are not spanned by the yield curve: P L t = µ Q P + ΦQ PP PL t 1 + ɛ Q P,t, ΣQ L ɛq P,t i.i.d. N (, I L ) (9) The only way macro factors enter the model is as additional predictors in the VAR in Eq. (6). The macro factors will therefore aect real-world expectations of future yields. The bottom-left corner of expression (7) is crucial as it determines the eects of macro variables on expectations of yields. If this part of the feedback matrix Φ is restricted to zero, macro variables completely drop out of the model. They can neither aect bond pricing under the risk-neutral probability measure nor the real-world factors dynamics. In that case, we end up with a yields-only model, where only P L t are the risk factors. Third, the one-period interest rate is an ane function of the priced risk factors (i.e. the risk factors that enter the bond pricing equation) and is given by: r t = ρ + ρ 1P L t. (1) As opposed to spanned macro risks, the one-period interest rate only loads on the rst L principal components and not on the macroeconomic factors. Assuming no-arbitrage, there exists a risk-neutral probability measure Q that can be used to price government bonds. The stochastic discount factor m t+1 denes the change of probability measure between P and Q-measures is exponentially ane and takes the form: 9

10 m t+1 = exp ( r t λ tλ t 2 λ tɛ t+1 ) (11) where the scaled market prices of risk are given by: [, where λ = λ PL ] LxL [, λ 1 = λ t = (Σ PP ) 1 (λ + λ 1 Z t ) (12) λ PL 1 LxL λm 1 LxM ]. This specication of the prices of risk corresponds to the essentially-ane class of Duee (22) such that the prices of risk have a constant component λ and a time-varying component λ 1. They measure the additional expected return required per unit of risk in each of the shocks in ɛ t (Bauer and Rudebusch (215)). Notice that there are as many rows as there are priced risk factors, and as many columns are there are risk factors. Since macro factors are not priced factors, there are only L rows in λ and λ 1 but M + L columns. As noted by JPS, the market prices of risk are an ane function of the whole set of risk factors Z t despite the fact that the only priced risks are the P L t. It follows that agents are sensitive to broader information than just yield curve factors. Equations (1) to (12) imply that the log price p (n) t by an ane function that links the log bond price to the priced risk factors: of an n-period bond at time t is determined p (n) t ( ) = A n + B np t L (13) The no-arbitrage condition is written as: ( exp p (n+1) t ) [ ( = E t exp (m t+1 ) exp p (n) t+1 )] (14) and states that longer bonds are risk-adjusted expected shorter bonds. To obtain no-arbitrage loadings A n and B n one needs to solve the following dierence equations: A n+1 = A n + B nµ Q B nσ PP Σ PP B n + A 1 B n+1 = B nφ Q PP + B 1 (15) with initial conditions A 1 = and B 1 = ρ 1 so as to satisfy r t = p (1) t. 1

11 Once we have solved the dierence equations for bond prices, yields can be computed as: y (n) t where a n = A n /n and b n = B n /n for n = 1, 2, 3,... = p(n) t n = a n + b npt L (16) 2.3 Market Prices of Risk The market prices of risk transform the risk factors dynamics under P into the risk factors dynamics under Q and vice-versa. More specically, µ Q P = µ P Σ PP λ (17) Φ Q P = Φ PP Σ PP λ 1 (18) Scaled market prices of risk are thus given by: Σ PP λ Σ PP λ 1 = µ P µ Q P = [Φ PP Φ PM ] [ ] (19) Φ Q PP LxM 2.4 Excess Returns Recall that the price of a bond of maturity n at time t is given by: P (n) t = exp ( A n B n Pt L ) (2) and the Expected Excess Returns are computed as: ( rx (n) t = E t [log P (n 1) t+1 ) ( log P (n) t )] r t (21) [ = E t An 1 B n 1 Pt+1 L ( A n B n Pt L )] rt (22) Notice that only P L t+1 is unknown at time t. Reworking expression (22) yields: 11

12 rx (n) t = (A n A n 1 ) + ( B n P L t B n 1 E t+1 [ P L t ]) rt (23) = (A n A n 1 ) + ( B n P L t B n 1 P L t [Φ PM Φ PP ] ) r t (24) 2.5 Regime-dependence of the MTSM In the subsequent empirical application we allow the dynamics under the risk-neutral Q and realworld P measures to be regime-dependent. In this sub-section, we precise how the regime dependence feature enters the empirical estimation. We restrict the bottom-right corner of Φ to be the same across regimes because the regime-switching feature should only matter for the expectations of future yields. These restrictions are consistent with the unspanned macro risks attribute of the model: yields factors matter for the characterization of the current state of the economy while both yields factors and macro factors determine the expectations of the state of the economy. Eq. (7) and (8) become: k Z t = M t = P L t µ M µ P k + Φk MM Φ k PM Φk MP Φ PP Z t 1 + ɛ k t = M t = Pt L µ M µ P + Φk MM Φ k PM Φk MP Φ PP M t 1 + ɛk M t Pt 1 L ɛ k Pt L (25) and Σ k = Σk MM Σ k PM Σk MP Σ PP (26) where the superscript k stands for the regime one considers. We see that despite the identical bottom-right corner of the Φ matrix, the complete Φ is regime-dependent. As a consequence, the variance-covariance matrix is also regime-dependent. Notice that the bottom-right of Σ k is especially important as it enters the no-arbitrage model and thus determines the no-arbitrage loadings A, B n, a, b n as well as the market prices of risk and 12

13 excess returns. Typically, as Σ PP does not depend on the regime considered, it enters the recursive dierential equations in (15) equally for both regimes. Consequently, the short-rate loadings in (1) are also unaected. The largest dierence in bond risk premia will presumably come from the market prices of risk in expression (19) because Φ k MP can greatly dier. In the same way, most of the dierences between the responses of the dependent variables will be due to dierent parameter values in Φ k and Σ k. We decide to identify regimes with a scal reaction function rather than within the no-arbitrage framework because we want to have a clear economic interpretation of the regimes we identify. We estimate the VAR by Maximum Likelihood where stationarity of both regimes is imposed. The stationarity criterion states that the modulus of the largest eigenvalue should be strictly inferior to one and that the second largest eigenvalue be les than the largest eigenvalue of the single regime. In order to overcome the diculty to reach the global optimum, we start the optimization procedure from a hundred sets of random parameter values that respect the stationarity criterion centered around the OLS coecients. We then select the best parameter values combination based on the value of the log-likelihood of the VAR. 2.6 Impulse Response Functions In order to identify economic shocks, one needs to transform the estimated residuals ɛ t into structural shocks v t = SRɛ t, where S is a unique decomposition of the variance-covariance matrix Σ and R is a rotation matrix. For the recursive identication strategy, S is the Cholesky decomposition of the variance-covariance matrix, and R is the identity matrix. With this ordering, we follow Afonso and Martins (212) and, to some extent, Ang and Piazzesi (23). It stands to reason, just like Afonso and Martins (212) claim, that nancial variables may react contemporaneously to shocks to macro variables, while the opposite may not be true. The macro block is therefore ordered rst. Within this block, the scal variable is in the last position because scal policy can directly be aected by ination and output developments through automatic stabilizers, whereas scal policy can only aect ination and output with a delay that is due to policy implementation lag. For the Sign Restrictions identication scheme, the matrix R is replaced by a rotation matrix 13

14 that preserves orthogonality of the structural shocks but rotates them such that the responses of some of the variables respect certain conditions set a priori and grounded in economic theory. Notice that the identication is not unique as with the recursive identication. One should therefore collect rotations that are consistent with the restrictions established a priori. To do so, an orthogonal rotation matrix is randomly selected from the uniform distribution. If the draw respects the criteria, it is stored whereas if it fails to respect the criteria, it is discarded. We later summarize the collection of accepted draws by its median value and a condence interval around the median. We can thus rewrite Equation (7) as: Z t = M t = µ M P L t µ P + Φ MM Φ PM Φ MP Φ PP Z t 1 + SRɛ t = M t = µ M P L t µ P + Φ MM Φ PM Φ MP Φ PP M t 1 + SR ɛ M t P L t 1 ɛ Pt = M t = µ M Pt L µ P + Φ MM Φ PM Φ MP Φ PP M t 1 + ν M t ν Pt P L t 1 (27) Following Hamilton (1994), Expresssion (27) can be written as a MA ( ) process such that: Z t = µ + v t + Ψ 1 v t 1 + +Ψ 2 v t 2 + = Ψ (L) v t (28) where Ψ h = Φ h and L stands for the lag operator. Consequently, element (i, j) of the matrices Ψ h can be interpreted as the impulse response function of variable i to a shock in variable j at time t + h since Z t+h v t = Ψ h. We compute the Impulse Response Functions in a way that preserves the information contained in the transition matrix P F. Impulse Response Functions of shock j are essentially the dierence between a forecast of Z t h periods ahead where a shock occured in variable j and a forecast where no shock occured. When forecasting the VAR, we need to take into account that the coecients of the VAR dier by regimes and that the one period forecast between h 1 and h will depend on the coecients of regime 1 or regime 2 according to the transition matrix P F. Therefore, a forecast h 14

15 periods ahead should take into account the history of switches from one regime to the other. The transition matrix P F is extremely informative about the likelihood of occurrence of regime k h periods ahead. If there are no absorbing regimes (i.e. a regime in which the data-generating process is locked), the diagonal elements of ( P ) F h will tend to the long-term likelihood of occurrence of the regimes. Let us call H the number of periods necessary for the transition matrix P F to converge to its long-term likelihood of occurrence of each regime. We compute 2 H possible histories of switches (i.e. at each horizon h = 1,..., H one can observe a switch or a non-switch) for h H and compute the IRFs accordingly. We then weight each of the paths by its likelihood of occurrence such that highly unlikely histories carry little weight in the nal value of the IRF. For h > H we consider that the regime occuring at horizon H will perdure indenitely. 4 3 Data We use the same yields dataset as Joslin, Priebsch, and Singleton (214) (JPS henceforth). They construct zero-coupon bonds from individual CRSP Treasury coupon bonds. They lter out bonds that are illiquid or have embedded options. They then apply the same Fama-Bliss bootstrap method as CRSP (see Fama and Bliss (1987)). The nal dataset ranges from the rst quarter of 1972 to the last quarter of 211. To express monthly yields at a quarterly frequency we take the last observation of the quarter. The maturities used are 6 months, 1 through 5 years, 7 years and 1 years. The principal components are extracted by decomposing the variance-covariance matrix of the eight standardized observed maturities into eigenvalues and eigenvectors as in Litterman and Scheikman (1991). Consistent with previous estimates (e.g. Cochrane and Piazzesi (25), Cochrane and Piazzesi (29) ) the rst three principal components explain 97.9, 1.9 and.14% of the total variance of observed yields, respectively. The level factor evenly loads on all maturities, the slope factor loads negatively on short maturities and positively on long maturities while the 4 Because of computational limits, the highest H achievable is 19 periods. However, choosing H = 1 or H = 15 makes little dierence. To circumvent this physical limitation, we can compute the histories of switches by simulation. We draw from the uniform distribution at each horizon h and compare the value of the draw to the probability of switching to another regime. If the value of the draw is smaller than the probability of switching, the data-generating process remains in the regime it was in and switches otherwise. Taking 1 2 H draws, where H is the maximum horizon of the computed IRFs, and averaging gives similar results as the analytical method. 15

16 curvature factor is U-shaped and thus loads positively on short and long maturities but negatively on medium maturities. These results are standard in the literature (see, for instance, Ang and Piazzesi (23)). Note, however, that the loadings have been rescaled according to the following rules as in JPS such that the scores of the principal components have the same scale: 56 P C new 1 = P C new 2 = P C new 3 = P C1 old Σ 8 i=1 P Cold 1 P C2 old P C2 old (1y) P C2 old(6m) P C3 old P C3 old (1y) 2P C3 old (2y) + P C3 old (6m) Economic activity is captured by the CBO output gap calculated as the log dierence between actual nominal GDP and nominal potential GDP. Ination is the yearly growth rate of Consumer Price Index less Energy and Food prices. Bauer and Rudebusch (215) stress that the use of core CPI ination is supported by the statements of monetary policymakers. Typically, very volatile series do not greatly inuence ination expectations. To estimate the scal feedback rule 5 we compute the primary decit-to-gdp ratio as total government expenditures minus total taxes and interest payments, divided by nominal GDP. A positive value indicates decit while a negative value indicates surplus. The debt ratio is calculated as total outstanding debt divided by nominal GDP. g t is given as the quarterly growth rate of nominal GDP and the implied interest rate on debt i b t is computed as quarterly interest payments divided by the outstanding stock of debt one period before. See Appendix (B) for details on the 5 Since principal components are, by construction, independent from each other, a rescaling of one of the principal components has no impact on the others. 6 We also used an alternative measure for yield curve factors, as in Afonso and Martins (212), results are robust to this specication: Level t = Slope t = Curvature t = [ y (6m) t [ y (1y) t [ y (1y) t + y (2y) t y (6m) t 2 y (2y) t ] + y (1y) t /3 ] ] + y (6m) t 16

17 computation of the data. 4 Empirical Results 4.1 Fiscal Policy Regimes Panel (a) of Table 1 provides some interesting features. First, scal policy is quite persistent, as indicates the autoregressive coecient. Second, scal policy behaves in a counter-cyclical manner as expected. Notice also that the stabilizing decit does not matter for the conduct of scal policy in the single-regime case. This apparent lack of importance may be due to a succession of heterogenous regimes that cancel each other out. We test this hypothesis by allowing the coecients of the regression to switch across regimes. The t is signicantly improved when scal policy is allowed to switch from a sustainable to an unsustainable regime and vice-versa 7. Moreover, the non-stabilizing regime is characterized by a negative δ coecient. Such negative coecient implies that the government increases its decit when larger surpluses are needed. We also see that scal policy reacts much stronger to economic developments than in the stabilizing regime. [Insert Table 1 here] We estimate Eq. (5) from 1949 until 215. Doing so, we use the maximum information available, even though we only keep the identied regimes from 1972 to 211 in the rest of the analysis. The literature on scal preferences shifts is extensive and dierent scal policy reaction functions have been proposed (see, for example, Bohn (1998) ; Davig and Leeper (27), Davig and Leeper (211) ; Favero and Monacelli (25) ; Afonso and Toano (213)). In line with Dewachter and Toano (212), we nd that scal policy has been predominantly passive except for four shortlived episodes of unsustainable scal policy. Those episodes match documented discretionary scal policy decisions. The rst episode is matched with the 1973 recession that saw a drop in tax revenue while the second episode corresponds to the President Ford's tax cuts following the oil shock and the 2-years long recession of The year 1982 sees President Reagan's tax cut and military buildup. US government has generally run positive primary decits throughout the 7 The Log-likelihood ratio test-statistic between Regime-switching and Single-regime is equal to and the critical value at 1% signicance level is

18 sample with the exception of the nineties that saw a strong buildup of surpluses. This trend was put to an halt and even reversed with the successive tax cuts of the Bush administration in 22 and 23. The scal stimulus packages of 28 and 29 (Economic Stimulus Act ($152 billions) and the American Recovery and Reinvestment Act ($83 billions) complete the set of unsustainable scal episodes. Those periods correspond to extreme events: either large increases in spending, or sharp decreases in government revenue. [Insert Figure 1 here] Our estimates of unsustainable scal policy compare well against the scal policy rule developed by Davig and Leeper (27) and Davig and Leeper (211). They estimate a regime-switching scal policy rule for the United States between 1949 and 24 where the government adjusts taxes as a function of government debt, output, and the level of government spending. The authors estimate the following rule: τ t = γ sf t + γ sf t b b t 1 + γ sf t y (y t y t ) + γ sf t g g t + στ sf ɛ sf t t (29) where τ is tax revenues less transfer payments, b t is the debt held by the public divided by GDP, (y t y t ) is the output gap and g t is current government purchases. The superscript s F t stands for the scal policy regime. Notice that the variance is regime-dependent. Removing transfers from tax revenues partly removes the natural movement of tax revenue due to automatic stabilizers. The identication of sustainable and unsustainable scal policy hinges on the sign of the response of tax revenues to the lagged value of debt. A positive co-movement indicates an unsustainable scal policy while a negative sign indicates a sustainable scal policy. Davig and Leeper (27), Davig and Leeper (211) identify an unsustainable scal policy in , , 1975, , and These periods broadly correspond to the periods we uncover. There are two main dierences between Eq. (5) and (29). The rst is that Davig and Leeper consider that the principal scal policy instrument are taxes and that they react to developments in government spending. Our rule, on the other hand, considers the nal outcome of developments in both taxes and government spending and removes the eect of borrowing on the budget. Secondly, 8 Additional information can be found in Appendix (C). 18

19 recall that d S t = (ζt ib t) (1+ζ b t) t 1. The multiplying factor in front of the lagged value of the debt is time-varying and oscillates around 1, whereas it is xed to 1 in the rule of Davig and Leeper. In some sense, we relax this embedded assumption in Davig and Leeper's rule. Figure 2 contains the six risk factors used in the rest of the analysis. The level factor broadly follows the average of the yield curve. Except from the period , the level factor exhibits a downward trend, with some cyclical movements associated with business cycles and recessions. The slope factor is much more volatile than the level factor, positive on average but with some negative values. Extreme negative values are found in 198 at the beginning of the monetary experiment of Paul Volcker, the then-chairman of the Federal Reserve. Harvey (1986) has stressed the importance of the slope factor to predict upcoming recessions: an inverted yield curve (a negative slope coecient) accurately forecasts recessions two to six quarters ahead. Two alternative denitions of the slope factors (the dierence between the 1-years and 3-months yields or the dierence between the 1-years and Federal Funds Rate) are even published in the Financial Stress Index of the St. Louis Fed and the Index of Leading Economic Indicators of the Conference Board. The curvature factor is quite choppy and reaches its maximum at around the same time the slope factor reaches its minimum. High values correspond to convex yield curves and are found at, or around, times of economic recessions. Core CPI ination has experienced two high peaks: 1975 and 198. These two peaks correspond to the aftermath of the rst oil shock and the Iran-Iraq war, respectively. After 1982, core ination has remained constant for over a decade before entering a smooth decline since 199 onwards. The output gap variable exhibits a clear cyclical pattern, however the amplitude and the regularity of the cycles are not constant across time. [Insert Figure 2 here] 4.2 Risk Factor Dynamics We estimate Equation (6) for each regime separately. In order to enforce the same parameter values for Φ PP, we interact the constant and the six risk factors M t and P L t with a dummy variable that takes the value 1 if the probability that scal policy is unsustainable is larger than.5. An alternative would be to use the probabilities as such. We choose the rst option because the probabilities 19

20 are, in general, either close to 1 or close to, making the distinction dummy variable/probabilities nil. Secondly, interpretation is clearer since the other regime does not contaminate the rst in the way model parameters are estimated. We estimate the VAR by Maximum Likelihood and impose stationarity of both regimes such that the largest modulus of the eigenvalues of Φ k is strictly inferior to 1 and that the second largest modulus be less than the largest modulus of Φ in the single regime. Importantly, we restrict the bottom-right corner of the feedback matrix Φ to be the same across regimes in order to be consistent with the unspanned macro factors feature of the model and to save on the number of degrees of freedom. Log-Likelihood ratio tests indicate that the t of the dependent variables between the single- and multi-regimes cases is statistically signicant. Restricting the bottom-right corner to be identical across regimes is, however, rejected by the Log-likelihood ratio test. Nevertheless, we maintain this restriction for two reasons. First, we save on the number of degrees of freedom. The restriction imposed reduces the number of parameters to be estimated from 126 to 111. Second, the restriction limits the degree with which regimes can dier. If results are dierent across regimes with the restriction enforced, they will dier even more strongly when every parameter of the model is free to change. The restriction ensures that any dierence in the results is the consequence of the sole macro factors. 4.3 Risk Premia - Excess Returns Table 6 provides information about the scaled market prices of risk. The price of risk gives the compensation required by bondholders to be exposed to this risk. The prices of risk should be understood as the sensitivity of risk premia to the exposure of shocks. A positive value in column i implies a positive co-movement of the risk premium with the exposure to risk i. Conversely, a negative value should be considered as an hedging premium bondholders are ready to pay in order to be exposed to such risk. First, although JPS impose some zero-restrictions on the prices of risk motivated by improvement in Information Criteria, we obtain the same signs of the co-movements between the exposures to Level, Slope, Curvature shocks and the expected excess returns on the yields portfolios Level, Slope and Curvature as JPS, whether we consider the single, sustainable or unsustainable regime. 2

21 Exposures to Decit shocks have ambiguous eects on the Level, Slope and Curvature portfolios depending on the regime one considers. Indeed, while the sign and magnitude of the pro-cyclicality of the risk premium associated to the Curvature portfolio induced by exposure to Decit shocks is the same for the three cases, we can see that the sensitivity to Decit shocks is markedly larger in the unsustainable regime for the Slope portfolio. It is also worth nothing that the exposure to Decit shocks has positive co-movement with the Level risk premium in the unsustainable regime, as compared to a negative co-movement for the single and sustainable regime. [Insert Table 6 here] We present the one-period excess returns for a selection of maturities in Figure 3. Similarly to the IRFs, we consider that expectations h periods ahead take into account the history of switches until h. With one-period forecast and two regimes, this amounts to: [ L, k=i Pt = µ k=i P + [Φ MP, Φ PP ] k=i [ M t 1, Pt 1 L ] ] [ p ii + µ k=j P + [Φ MP, Φ PP ] k=j [ M t 1, Pt 1 L ] ] p ij, where the asterisk denotes a forecast, i is the regime considered and j the other regime. The estimated risk premia broadly correspond to risk premia described in Cochrane and Piazzesi (25) and Cochrane and Piazzesi (29) with a clear business-cycle pattern. Strong negative excess returns match with the start of the monetary experiment of It is clear that unsustainable scal policy consistently implies higher risk premia than in the sustainable or linear case, although the dierence is less pronounced for long maturities. This dierence is particularly large in the second half of the nineties. It is worth noting, however, that this pattern vanishes from 21 onward where we see that risk premia associated with the sustainable regime are larger than in its unsustainable counterpart for short maturities. We conjecture that this feature is due to the accommodative monetary policy that was taking place at the time. Notice also that risk premia associated with the unsustainable scal policy regime experience a sharp decrease in periods of unsustainable scal policy while sustainable scal policy risk premia increase during those periods. (3) 21

22 [Insert Figure 3 here] 4.4 Impulse Response Functions Cholesky Identication Scheme Figure 4 reports the impulse response functions following a one standard deviation unexpected shock on the primary decit. Panels (a) through (d) give the responses of yields from 1-year, 3-years, 5-years and 1-years. The responses clearly depend on the maturity. Increased decits initially decrease interest rates but lead to higher yields after two years, irrespective of the regime considered. A notable exception, however, is the response of the one-year bond yield which sees its yield increase from horizon in the unsustainable case. The ten-years bond also decreases more in the unsustainable regime than in the single or sustainable regimes. If we decompose these responses according to their components (panels (h) to (j)), we can see that the most marked dierence relies in the response of the slope factor and, to some extent, the curvature factor. Taken together, these results suggest that the yield curve attens after a decit shock. If the slope of the yield curve is of any indication about the likelihood of an upcoming recession, this would suggest that recessions follow increases in decits. Evidence of such claim can be found in panel (f) where the response of output in the unsustainable regime is more negative than in the other regimes. Taken together, these results indicate snon-ricardian eects that are consistent with the new- Keynesian view. Our results are broadly in line with Dai and Philippon (25) who, despite a dierent identication strategy, report that a shock of 1% to the decit increases the 1-years bond by about basis points. Laubach (29) uses the projected decit/gdp and reach the same number. Gale and Orszag (23) survey the literature and report similar numbers. [Insert Figure 4 here] Sign Restrictions An alternative identication strategy is to impose a priori the sign of the response of some variables of the VAR. Those restrictions are dictated by economic theory. A positive aggregate demand shock 22

23 raises both output and the price level, wheras a positive aggregate supply shock raises output but reduces the price level. A positive decit shock raises the primary decit to GDP ratio as well as Ination and Output. One advantage of the sign restrictions is that shocks identied through sign restrictions do not suer from doubts about the timing and exogeneity of variables included in the VAR. [Insert Table 7 here] Figure 5 presents the Impulse Responses from a one standard deviation decit shock identied through sign restrictions together with their corresponding 66% condence bands based on 1 accepted draws. Panel (j) presents the response of the primary decit to a decit shock. The persistence of the shock diers greatly across regimes. The single regime returns to zero after 35 quarters, while the sustainable regime remains signicantly higher than its pre-shock level even after 4 quarters. The decit ratio in the unsustainable regime, on the other hand, quickly returns to its pre-shock level. In panel (i) one can observe that output quickly returns to its pre-shock level after the four quarters of the restrictions and the response even turns negative after two years. Similarly to the decit variable, the response in the linear regime returns to zero quicker than its sustainable counterpart, but the eect of decit on output is short-lived in the unsustainable regime. Ination, on the other hand, shows a very dierent story. The response to scal policy shock is undoubtebly positive, and almost twice as strong in the unsustainable regime than the other two cases after 12 quarters. The persistence of the increase in ination is also larger in the unsustainable regime. Fiscal policy raises interest rates with a delay of 2 to 8 quarters. The eects on the yield curve can be read in panels (a) to (d) and shows a clear dominance of the level eect in the linear regime and, to a lesser extent, in the sustainable regime. The unsustainable regime is peculiar in this respect because short-term interest rates raise more sharply than their linear or sustainable counterparts whereas the response of longer term interest rates is indistinguishable from the linear or sustainable regime. This indicates a more important slope and curvature eect, as can be seen in panels (f) and (g). [Insert Figure 5 here] 23

24 4.5 Forecast Error Variance Decomposition Cholesky Identication scheme Tables 8 and 9 present the variance decomposition for the Cholesky identication scheme at horizons of 1, 4, 2 and 4 quarters. Variance decomposition gives the share of the response that can be attributed to each shock, at a specic horizon. In general, the bulk of the variance is explained by the latent factors Level, Slope and Curvature. This holds for dierent horizons, regimes and maturities.yield curve shocks explain between 1/3 and 3/4 of the variance of yields. The share of the yield curve shocks decreases with maturity and the horizon of the variance decomposition, however. Most of the variance in yields is in fact due to the Level. Conversely, the share of the variance explained by macro factors increases with maturity and the horizon of the FEVD. Within the macro block, it is Output shocks that mostly drive yields variance, ranging from 1/1 to 1/2, depending on the horizon and the regime considered. The importance of Decit shocks is fairly limited, with shares of variance of yields that never reach 5 percent. The distinction across regimes brings important results. First, the importance of the three latent factors generally decreases by half in the unsustainable regime as compared to either the single or the sustainable regime. At the same time, it seems that Ination becomes a more important driver of bond yields, almost doubling in importance compared to the single regime case. Output shocks gain in importance as well, but less than Ination. Second, although the absolute value remains low, the importance of decit shocks is larger for long maturities. Third, it is also worth noting that while decit shocks have a stable share across horizons in the single regime, this share increases with horizon in the unsustainable case but decreases in the sustainable case. Taken together, these three main results highlight the importance of macro factors for long-term investors and stress the importance of regimes for bond yields variance. Ang and Piazzesi (23)estimate that their macro factors explain up to 85% of the total variance of impulse response functions for short bonds and 5% for long bonds. Our results are closer, however, to Dai and Philippon (25) who nd that scal policy shocks explain up to 12% of the variance in yields from 1 quarter to 1-years at a 4 quarters horizon. [Insert Table 8 here] 24