Monetary-Fiscal Policy Interactions: Interdependent Policy Rule Coefficients

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Monetary-Fiscal Policy Interactions: Interdependent Policy Rule Coefficients Manuel Gonzalez-Astudillo 2013-58 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Monetary-Fiscal Policy Interactions: Interdependent Policy Rule Coefficients (cid:3) Manuel Gonzalez-Astudillo Federal Reserve Board Washington, D.C., USA manuel.p.gonzalez-astudillo@frb.gov July 16, 2013 Abstract Inthispaper,weformulateandsolveaNewKeynesianmodelwithmonetaryand(cid:12)scal policy rules whose coefficients are time-varying and interdependent. We implement time variation in the policy rules by specifying coefficients that are logistic functions of correlatedlatentfactorsandproposeasolutionmethodthatallowsforthesecharacteristics. The paper uses Bayesian methods to estimate the policy rules with time-varying coefficients, endogeneity, and stochastic volatility in a limited-information framework. Results show that monetary policy switches regime more frequently than (cid:12)scal policy, and that there is a non-negligible degree of interdependence between policies. Policy experiments reveal that contractionary monetary policy lowers in(cid:13)ation in the short run and increases it in the long run. Also, lump-sum taxes affect output and in(cid:13)ation, as the literature on the (cid:12)scal theory of the price level suggests, but the effects are attenuated with respect to a pure (cid:12)scal regime. Keywords: Time-varying policy rule coefficients, Monetary and Fiscal policy interactions, Nonlinear state-space models. JEL Classi(cid:12)cation Numbers: C11, C32, E63. (cid:3)I thank Eric Leeper, Joon Park, Yoosoon Chang, Todd Walker and John Roberts for helpful comments and advice. The views expressed in this paper are solely the responsibility of the author and should not be interpreted as re(cid:13)ecting the views of the Board of Governors of the Federal Reserve System.

1 Introduction The great recession has raised awareness of the role of monetary-(cid:12)scal policy interactions in determining the behavior of economic aggregates. In the United States, for example, reductions in discount and funds rates and even unconventional quantitative easing measures have accompanied (cid:12)scal stimulus policies. Moreover, giventhe high projected (cid:12)scal liabilities in developed as well as in some developing countries, interactions are likely to (cid:12)gure more prominently in determining economic outcomes over the coming decades. The considerations behind monetary-(cid:12)scal policy interactions have been part of the monetary and (cid:12)scal policymaking literature for a long time. Friedman (1948) proposed a scheme of monetary-(cid:12)scal policy interactions to deliver stability in cyclical (cid:13)uctuations. The seminal work of Sargent and Wallace (1981) provides a formal investigation that illustrates how monetary and (cid:12)scal policies interact to determine in(cid:13)ation. The authors show that under (cid:12)scal dominance|whereapartialmonetizationofgovernmentdebtisnecessarytoavoiddefault| the ability of the monetary authority to control in(cid:13)ation disappears. Leeper (1991) examined monetary-(cid:12)scal policy interactions within the dynamic stochastic general equilibrium (DSGE) framework. The work of Canzoneri, Cumby and Diba (2010) surveys the positive and normative aspects of monetary-(cid:12)scal policy interactions in the existing literature. Conventionally, there have been two approaches to how each of the policy authorities respondstodeviationsofitsvariablesofinterestfromtarget. Oneapproachspeci(cid:12)esacentral bank that follows a Taylor (1993) rule under which the nominal interest rate increases more than proportionally when in(cid:13)ation increases. Thus, monetary policy provides the nominal anchor to deliver price-level determinacy. In this approach, the (cid:12)scal authority follows a rule under which (lump-sum) taxes stabilize debt. This (cid:12)rst approach has been referred to as monetary dominance (see Carlstrom and Fuerst, 2000). We will call it a monetary (M) regime. The other approach postulates that the (cid:12)scal authority does not adjust taxes to stabilize debt. In this case, the central bank cannot stabilize the price level. Thus, (cid:12)scal policy provides the nominal anchor for price-level determinacy through expectations about future surpluses, given a level of outstanding nominal liabilities (see Leeper, 1991; Sims, 1994; Woodford, 1994, 1995, 1996; Cochrane, 1998, 2005, 2001). This approach has been referred to as the (cid:12)scal theory of the price level. We will call it a (cid:12)scal (F) regime. When the two approaches are considered together, there are four possible combinations of monetary and (cid:12)scal policy stances that have been referred to as monetary-(cid:12)scal policy interactions. The four combinations are: (i) an M regime; (ii) an F regime; (iii) a regime where no authority provides the nominal anchor and the price level is indeterminate; (iv) a regime where both authorities try to provide the nominal anchor and debt is unbounded. This paper formulates and solves a New Keynesian model that incorporates monetary and (cid:12)scal policy rules whose coefficients are time-varying and interdependent. Time variation and interdependence allow for co-movements in monetary and (cid:12)scal policymaking, thereby introducing a direct channel of interactions. This channel in(cid:13)uences expectations about future monetary and (cid:12)scal policymaking, affecting the dynamics of the variables in equilibrium. In particular, we show that when there are co-movements in monetary and (cid:12)scal policymaking in the direction of stable and determinate equilibria |the M and F regimes| the volatilities of output and in(cid:13)ation are reduced, compared to the case where 1

co-movements in that direction are absent. As Davig and Leeper (2007) emphasize, policymaking is a complicated process of analyzing and interpreting data, receiving advice, and applying judgment. During some periods policymakersmaygivemoreattentiontoin(cid:13)ationordebtstabilization, whileinotherperiods they may give more attention to output stabilization. Furthermore, a substantial empirical literature argues that policy rules have not remained invariant over the past six decades (see Clarida, Gali and Gertler, 2000; Favero and Monacelli, 2003; Lubik and Schorfheide, 2004; Davig and Leeper, 2006; Fernandez-Villaverde, Guerron and Rubio-Ramirez, 2010; Bianchi, 2012). In this paper, policy rule coefficients move across regimes as functions of exogenous latent factors. In particular, the policy rule coefficients are logistic functions of these latent factors. The logistic function allows the policy rule coefficients to move between two regimes |as with a two-state Markov-switching speci(cid:12)cation| with a smooth transition between regimes. We assume that one latent factor in(cid:13)uences the evolution of the coefficients of the monetary policy rule and that a different latent factor in(cid:13)uences the evolution of the coefficients of the (cid:12)scal policy rule. Weintroduce interdependencebetweenmonetary and(cid:12)scal policybyallowingcorrelation between the two latent factors driving the evolution of the monetary and the (cid:12)scal policy rule coefficients. This correlation can be seen as coordination |explicit or implicit| in policymaking. The correlation between the latent factors allows policy interactions through co-movements in the evolution of the policy rule coefficients. Given the logistic speci(cid:12)cation of the policy rule coefficients and the existence of latent factors as additional states, the model is intrinsically nonlinear. We devise a solution method that takes into consideration these nonlinearities. The solution method incorporates agents’ expectations on the joint evolution of the policy rule coefficients. This makes the model appropriate to analyze the impacts of monetary and (cid:12)scal policies in a framework of interactions. We employ a limited-information analysis to estimate the policy rule coefficients using Bayesian methods appropriate for nonlinear state-state models. The results indicate that the policy rule coefficients are far from constant. In particular, the monetary policy rule coefficient on in(cid:13)ation switches more frequently than the (cid:12)scal policy rule coefficient on debt. There is also a degree of policy interdependence: (cid:12)scal policy that focuses on keeping debt under control tends to accompany monetary policy that focuses on keeping in(cid:13)ation under control, and monetary policy that does not increase interest rates actively tends to accompany (cid:12)scal policy that does not pay attention to debt increases. Finally, we calibrate the model to carry out policy analysis. The policy experiments show that contractionary monetary policy can lower in(cid:13)ation, at least in the short-run. The experiments also show that (lump-sum) taxes have effects on output and in(cid:13)ation, as the literature on the (cid:12)scal theory of the price level suggests, but the effects are attenuated with respect to a pure (cid:12)scal regime. 2 Interactions Between Monetary and Fiscal Policy In this section, we introduce the concept of interdependence in policymaking that will be central to the analysis of the subsequent sections. We describe the theoretical framework of 2

monetary-(cid:12)scal policy interactions and the empirical (cid:12)ndings in the literature with respect to policy interactions. Then, we present how we implement interdependence in policymaking through interdependence of the policy rule coefficients. The theoretical framework of monetary-(cid:12)scal policy interactions in a DSGE model was (cid:12)rstintroducedbyLeeper(1991), whoshowshowdifferentcombinationsofthemagnitudesof monetary and (cid:12)scal policy rule coefficients lead to different equilibrium outcomes and local dynamics. Leeper coins the terms \active" and \passive" monetary and (cid:12)scal policies to describe how the central bank adjusts interest rates with respect to in(cid:13)ation deviations from target, and how (cid:12)scal policy adjusts taxes to changes in public debt. A change more than proportionalinnominalinterestrateswithrespecttoin(cid:13)ationdeviationsfromtargetiscalled \active" monetary policy, while a Ricardian view of (cid:12)scal policy, where taxes adjust enough to cover interest payments and to retire debt, is called \passive" (cid:12)scal policy. The alternative scenarios with respect to monetary and (cid:12)scal policies are called \passive" monetary policy and \active" (cid:12)scal policy.1 Leeper (cid:12)nds that the model delivers a bounded unique rational expectations equilibrium as long as monetary policy is active and (cid:12)scal policy is passive |regime M| or if monetary policy is passive and (cid:12)scal policy is active |regime F. Also, the model delivers indeterminacy if both monetary policy and (cid:12)scal policy are passive, and no bounded solution if both are active. Other papers along this line are Sims (1994) and Leith and Wren-Lewis (2000). Davig and Leeper (2006) estimate Markov-switching models of monetary and (cid:12)scal policy ruleswithU.S.data. Theirresultsshowthattherehavebeennumerousswitchesinmonetary and (cid:12)scal policy rule coefficients. In particular, whenever the interest rate rule pays more (less) attention to in(cid:13)ation deviations, less (more) weight is given to output deviations. Also, when the tax rule pays more (less) attention to debt deviations, more (less) weight is given to output deviations |in line with an automatic stabilizers argument. These switches deliver the four regimes of policy interactions described above. Bianchi (2012) conducts a full-information estimation of a Markov-switching DSGE model with policy rule coefficients that switch among three states. The results show that an M regime was in place starting in the 1990s, that an F regime was in place during the 1970s and that a no-bounded-solution regime was in place during the 1980s. We explicitly introduce co-movements between the coefficients of the monetary and the (cid:12)scal policy rules. This setup generalizes the Markov-switching setup in Davig and Leeper (2006) and Bianchi (2012) by introducing interdependence in the evolution of the policy rule coefficients. With this addition, the model allows for a direct channel of interactions in policymaking, so that agents form expectations accordingly, not only in terms of the individual future evolution of policy rule coefficients, but as a framework of joint future policymaking. Below we specify the policy rules, the time variability characteristic of their coefficients, and the way in which we introduce policymaking interdependence. 1Any positive response of taxes to debt constitutes a passive (cid:12)scal policy. If one only wants to consider equilibria with bounded real debt, then real taxes have to respond to real debt deviations with increases higher than the real interest rate. 3

2.1 Policy Rules We specify policy rules with coefficients that are time varying. The time-varying coefficients of a particular policy rule are logistic functions of a latent state. More speci(cid:12)cally, if ϱ is a time-varying coefficient of a policy rule, it has the following functional form: t ϱ (cid:17) ϱ(z ) t t ϱ 1 = ϱ + ; 0 1+exp((cid:0)ϱ (z (cid:0)ϱ )) 2 t 3 where z = (cid:26) z +(cid:24) is a latent factor, 0 (cid:20) (cid:26) (cid:20) 1 and (cid:24) (cid:24) iidN(0;1). t z t(cid:0)1 t z t Under this speci(cid:12)cation, ϱ denotes the lower (upper) bound of ϱ , while ϱ +ϱ denotes 0 t 0 1 its upper (lower) bound (if ϱ < 0). ϱ > 0 is a transition coefficient affecting the speed 1 2 of the transition between the lower and the upper bounds, and ϱ is a location parameter 3 determining the value of z at which ϱ crosses the y-axis. A graph for ϱ(z ) with ϱ = 0:01, t t t 0 ϱ = 0:1, ϱ = 1, and ϱ = 0 is reproduced in Figure 1. 1 2 3 Figure 1: Logistic Function fHzL 0.10 0.08 0.06 0.04 0.02 z -4 -2 2 4 Two approaches have been proposed in the empirical literature to model time-varying policy rule coefficients. One is a Markov-switching speci(cid:12)cation with (cid:12)nite number of states (see Davig and Leeper, 2006; Eo, 2009; Davig and Doh, 2009; Bianchi, 2012). The second is a random-coefficient speci(cid:12)cation (see Kim and Nelson, 2006a; Boivin, 2006; Fernandez- Villaverde et al., 2010). Yksel et al. (2013) offers a survey of the literature on estimation of Taylor rules with time-varying coefficients. An advantage of the Markov-switching speci(cid:12)cationisthatitimpliesboundedcoefficients, whichcouldbeimportantintermsofdeterminacy and relevance of the equilibrium. With respect to the random-coefficient speci(cid:12)cation, an advantage is that it implies smooth transitions between states. The logistic speci(cid:12)cation proposed in this paper generalizes the two approaches: On one hand it allows a policy rule coefficient to switch smoothly from one regime to another, while on the other it allows for a bounded evolution of the coefficient.2 The latent factors can be seen as representing a combination of political and institutional determinants of policymaking. We assume that the latent factors are exogenous with respect to the variables in the model, and that the information that they provide is part of the 2The Markov-switching speci(cid:12)cation is a particular case of the logistic speci(cid:12)cation, when ϱ !1. 2 4

information set of the agents. That is, agents and policymakers share the same information set. This assumption is needed to solve the model, as will be seen. 2.1.1 Monetary Policy Rule Monetary policy takes place by means of an interest rate feedback rule of the form R = R(cid:26)R R (cid:22)(1(cid:0)(cid:26)R)exp("R); t t(cid:0)1 t t where (cid:26) 2 (0;1) indicates the degree of interest rate smoothing, and "R (cid:24) iidN(0;(cid:27)2) is a R t R (cid:22) monetary policy shock. R is the target short-term nominal interest rate. The central bank t sets the interest rate to react to deviations of in(cid:13)ation from target and to the output gap according to ( ) ( ) (cid:5) (cid:11)(cid:25)(z t m) Y (cid:11)y(z t m) (cid:22) t t R = R ; t (cid:5) (cid:22) Y(cid:3) t where (cid:5) = P =P is the gross in(cid:13)ation rate, Y is output, Y(cid:3) is output in the absence of t t t(cid:0)1 t t price rigidities, and R is the steady state nominal interest rate, which is guaranteed to be state independent if the target in(cid:13)ation rate, (cid:5) (cid:22) , is set equal to (cid:5), the steady state in(cid:13)ation.3 The time varying monetary policy rule coefficients are denoted by (cid:11)(cid:25)(zm) for in(cid:13)ation t deviations from steady state, and (cid:11)y(zm) for the output gap. Both are logistic functions of t the monetary policy latent factor, zm. t 2.1.2 Fiscal Policy Rule The(cid:12)scalruleisafeedbackrulefortheratiooflump-sumtaxesnetoftransferstooutput, (cid:28) = T =Y , of the form t t t (cid:28) = (cid:28)(cid:26)(cid:28) (cid:28)(cid:22)(1(cid:0)(cid:26)(cid:28))exp("(cid:28)); t t(cid:0)1 t t where (cid:26) 2 (0;1) indicates the degree of tax rate smoothing, and "(cid:28) (cid:24) iidN(0;(cid:27)2) is a (cid:12)scal (cid:28) t (cid:28) policy shock. (cid:28)(cid:22) is the target level of the ratio of taxes net of transfers to output. The (cid:12)scal t authority sets lump-sum taxes to respond to debt deviations and the output gap according to ( ) ( ) b t(cid:0)1 (cid:13)b(z t f) Y t (cid:13)y(z t f) (cid:28)(cid:22) = (cid:28) ; t (cid:22) b Y(cid:3) t (cid:22) where b = B =(P Y ) denotes the debt-to-output ratio in period t, and b is its target level. t t t t (cid:28) denotes the steady state level of (cid:28) , which is guaranteed to be state independent in the t (cid:22) steady state equilibrium if b is set equal to its steady state value, denoted by b. The time-varying (cid:12)scal policy rule coefficients are (cid:13)b(zf) for debt deviations from steady t state and (cid:13)y(zf) for the output gap. All are logistic functions of the (cid:12)scal policy latent t factor, zf. t 3The target in(cid:13)ation rate is constant to allow the linearization of the model around the steady state conditional on a realization of the latent factor at each period. 5

2.2 Interdependence in Policymaking To incorporate direct interactions between policies, we specify the latent factors driving the evolution of the monetary and (cid:12)scal policy rule coefficients as follows: zm = (cid:26) zm +(cid:24)m; (1) t zm t(cid:0)1 t zf = (cid:26) z(cid:28) +(cid:24)f; (2) t zf t(cid:0)1 t where(cid:24)m and(cid:24)f arenormallydistributedwithzeromean, unitvarianceandcorr((cid:24)m;(cid:24)f) = (cid:20). t t t t Noticethatunderthisspeci(cid:12)cation, if(cid:20)isdifferentfromzero, thereexistexplicitinteractions or interdependent changes between monetary and (cid:12)scal policymaking. In this context, policy authorities react more or less strongly to deviations of their feedback variables from target |in(cid:13)ation and the output gap for the monetary authority, and debt and the output gap for the (cid:12)scal authority| depending on the values of the policy rule coefficients as driven by the evolution of the latent factors, zm and zf. These latent t t factors, ultimately, are going to determine the nature of the possible combinations between monetary and (cid:12)scal policymaking. The full speci(cid:12)cation of the monetary and (cid:12)scal policy rule coefficients is as follows: (cid:11)(cid:25) (cid:11)(cid:25)(zm) = (cid:11)(cid:25) + 1 ; (3) t 0 1+exp((cid:0)(cid:11)(cid:25)(zm (cid:0)(cid:11)(cid:25))) 2 t 3 (cid:11)y (cid:11)y(zm) = (cid:11)y + 1 ; (4) t 0 1+exp((cid:0)(cid:11)y(zm (cid:0)(cid:11)y)) 2 t 3 (cid:13)b (cid:13)b(zf) = (cid:13)b + 1 ; (5) t 0 1+exp((cid:0)(cid:13)b(zf (cid:0)(cid:13)b)) 2 t 3 (cid:13)y (cid:13)y(zf) = (cid:13)y + 1 : (6) t 0 1+exp((cid:0)(cid:13)y(zf (cid:0)(cid:13)y)) 2 t 3 A constant-coefficient version of this model delivers an M Regime when (cid:11)(cid:25) and (cid:13)b are sufficientlyhigh|thatis,whenthemonetaryauthorityreactsstronglytoin(cid:13)ationdeviations and the (cid:12)scal authority reacts strongly to debt deviations. The model delivers an F regime when (cid:11)(cid:25) and (cid:13)b are sufficiently low |that is, when the (cid:12)scal authority reacts weakly to debt deviations and the monetary authority reacts weakly to in(cid:13)ation deviations. With timevarying coefficients, we can choose values of the parameters de(cid:12)ning each of the policy rule coefficients (3)-(6) in a way that the model delivers determinate equilibria in the long-run with short-run deviations from these equilibria. Theinteractionswithlong-rundeterminacyoftheequilibriaareonlywellde(cid:12)nedif(cid:20) (cid:21) 0. In that case, at a given point in time, a high value of (cid:11)(cid:25)(zm) |given by a high value of zm| t t is likely to be associated with a high value of (cid:13)b(zf) |given by a high value of zf. Also, at t t a given point in time, a low value of (cid:11)(cid:25)(zm) |given by a low value of zm| is likely to be t t associated with a low value of (cid:13)b(zf) |given by a low value of zf. Then, (cid:20) > 0 tends to t t send the economy towards the M and F regimes. (cid:20) = 0 is the case present in the existing literature, where interactions depend on the values of the policy rule coefficients. 6

3 The Model The economy is populated by a representative household, a (cid:12)nal-goods-producing (cid:12)rm, a continuum of intermediate-goods-producing (cid:12)rms, a monetary authority and a (cid:12)scal authority. The model extends the setup in An and Schorfheide (2007) to incorporate a (cid:12)scal policy rule and time-varying policy rule coefficients. Appendix A details the derivations. 3.1 Households The representative household derives utility from consumption, C ,4 relative to a habit t stock, A , that is given by the level of technology of the economy, and real money balances, t M =P ; and derives disutility from working hours, H . Hence, a representative household t t t chooses consumption, real balances, bond holdings and working hours to maximize ( ) ∑1 (C =A )1(cid:0)(cid:27) H1+φ E (cid:12)t t t +(cid:31) log(M =P )(cid:0)(cid:31) t ; 0 1(cid:0)(cid:27) M t t H 1+φ t=0 where 0 < (cid:12) < 1 is the discount factor, (cid:27) > 0 is the inverse of the elasticity of intertemporal substitution, φ > 0 is the inverse of the Frisch elasticity of labor supply, and (cid:31) > 0 and M (cid:31) > 0 are constants that determine the steady state level of real money balances and hours H worked. The household saves in the form of nominal government bonds, B , that pay a gross t interest rate R each period, and by accumulating money balances that do not pay interests. t It supplies labor services to the (cid:12)rms taking the nominal wage, W , as given; it also receives t its aggregate share on the (cid:12)rms’ nominal pro(cid:12)ts, D , and pays lump-sum taxes, T . Thus, t t the household’s budget constraint is expressed as P C +M +B +P T (cid:20) H W +D +M +R B for t (cid:21) 0; t t t t t t t t t t(cid:0)1 t(cid:0)1 t(cid:0)1 giventheinitialvalueofnominalassetsM +R B ,andwherethetransversalitycondition (cid:0)1 (cid:0)1 (cid:0)1 that rules out Ponzi schemes holds. 3.2 Firms There are two types of producers: perfectly competitive (cid:12)nal goods producers and a continuum of monopolistic intermediate goods producers. 3.2.1 Final Goods Producers Given the composite good price, P , and intermediate goods prices, P (j), for j 2 [0;1], t t producers assemble the intermediate goods, Y (j), to obtain a composite (cid:12)nal good, Y , t t ( ) ∫ (cid:18) 4C t is a composite consumption good given by C t = 0 1 C t (j)(cid:18)(cid:0) (cid:18) 1dj (cid:18)(cid:0)1 and (cid:18) (cid:21) 1. The household chooses C (j) to minimize expenditure on the continuum of goods indexed by j 2 [0;1] which yields j’s t ( ) (cid:0)(cid:18) good demand as C (j) = Pt(j) C , where P is the (cid:12)nal good price at t and P (j) is the price of the t Pt t t t consumption good indexed by j. 7

according to a CES technology, so that (∫ ) 1 (cid:18)t(cid:0)1 (cid:18)t (cid:18) (cid:0) t 1 Y t = Y t (j) (cid:18)t dj ; (7) 0 where (cid:18) 2 [0;1] is the (time varying) price elasticity of demand for each intermediate good. t Here (cid:18) represents a markup, or cost-push, shock in the Phillips curve relationship. This t cost-push shock follows the autoregresive process log(cid:18) = (1(cid:0)(cid:26) )log(cid:18)+(cid:26) log(cid:18) +"(cid:18); t (cid:18) (cid:18) t(cid:0)1 t with (cid:26) 2 (0;1), (cid:18) > 1 and "(cid:18) (cid:24) iid N(0;(cid:27)2). (cid:18) t (cid:18) Finalgoodproducerschoosethedemandofintermediategoods, Y (j), tomaximizepro(cid:12)ts t given by ∫ 1 P Y (cid:0) P (j)Y (j)dj: t t t t 0 Optimization yields the demand function of intermediate good j, ( ) P (j) (cid:0)(cid:18)t t Y (j) = Y : (8) t t P t Combining (8) and (7) yields the expression of the (cid:12)nal good price (∫ ) 1 P = 1 P (j)(1(cid:0)(cid:18)t)dj 1(cid:0)(cid:18)t : t t 0 3.2.2 Intermediate Goods Producers Intermediate goods (cid:12)rms produce type j good according to the linear technology Y (j) = A L (j); (9) t t t where L (j) are hours of work employed by the producer of intermediate good j, and A is t t an exogenous technology shock identical across producers following the stochastic process A = (cid:14)A exp((cid:23) ); t t(cid:0)1 t where (cid:14) is a trend, and (cid:23) is a stochastic component following the process t (cid:23) = (cid:26) (cid:23) +"(cid:23); t (cid:23) t(cid:0)1 t with (cid:26) 2 (0;1) and "(cid:23) (cid:24) iid N(0;(cid:27)2). (cid:23) t (cid:23) Intermediate good producers face an explicit cost of adjusting their price, measured in 8

units of the (cid:12)nished good, and given by ( ) ϕ P (j) 2 t (cid:0)1 Y ; t 2 (cid:5)P (j) t(cid:0)1 where ϕ (cid:21) 0 measures the magnitude of the price adjustment cost, and (cid:5) is the steady state gross in(cid:13)ation rate associated with the (cid:12)nal good. Producers in the intermediate goods sector take wages as given and behave as monopolistic competitors in their goods market, choosing the price for their product taking the demand in (8) as given. Hence, (cid:12)rm j chooses its labor input, L (j), and its price, P (j), to t t maximize [ ] ( ) ∑1 P (j) W ϕ P (j) 2 E MRS t Y (j)(cid:0) t L (j)(cid:0) t (cid:0)1 Y ; (10) 0 0;t t t t P P 2 (cid:5)P (j) t t t(cid:0)1 t=0 where MRS is the household’s marginal rate of substitution between periods 0 and t, 0;t which is given exogenously to the (cid:12)rm. Notice that (cid:12)rm j’s nominal labor cost is given by W Y (j)=A , and its real marginal labor cost is given by = W =P A , which is the same t t t t t t t across (cid:12)rms in the intermediate goods sector. 3.3 Government The government (cid:12)nances purchases of goods, G , with a combination of lump-sum taxes, t T , and money creation, M (cid:0)M , so that the implied process for nominal debt, B , satis(cid:12)es t t t(cid:0)1 t the budget constraint: B +M +P T = P G +M +R B for t (cid:21) 0; (11) t t t t t t t(cid:0)1 t(cid:0)1 t(cid:0)1 given M +R B . Each period, the government demand of the (cid:12)nal good is given by (cid:0)1 (cid:0)1 (cid:0)1 G = (cid:16) Y ; t t t where (cid:16) 2 (0;1) is an exogenous process de(cid:12)ned by the transformation g = 1=(1(cid:0)(cid:16) ) with t t t lng = (1(cid:0)(cid:26) )lng +(cid:26) lng +"g; t g g t(cid:0)1 t where (cid:26) 2 (0;1), g = 1=(1(cid:0)(cid:16)) with (cid:16) being the steady state ratio of government spending g to output, and "g (cid:24) iid N(0;(cid:27)2). t g 3.4 Symmetric Equilibrium In a symmetric equilibrium, all the intermediate-goods-producing (cid:12)rms make identical decisions, the money supply equals money demand, labor supply equals labor demand, and the net supply of government bonds is zero. Hence, the equilibrium conditions for t (cid:21) 0 are 9

given by ( ) ϕ (cid:5) 2 Y = C +G + t (cid:0)1 Y ; (12) t t t t 2 (cid:5) ( ) (cid:5) (cid:5) 0 = 1(cid:0)(cid:18) +(cid:18) (cid:0)ϕ t (cid:0)1 t + t t t (cid:5) (cid:5) ( ) ( ) C =A (cid:27) Y =A (cid:5) (cid:5) +(cid:12)ϕE t t t+1 t+1 t+1 (cid:0)1 t+1 (13) t C =A Y =A (cid:5) (cid:5) ( t+1 t+1) t t C =A (cid:27) A 1 1 = (cid:12)R E t t t ; (14) t t C =A A (cid:5) (t+1) t+1 t+1 t+1 W C (cid:27) t = (cid:31) LφA t ; (15) P H t t A t ( )t ( ) M C (cid:27) R t t t = (cid:31) A ; (16) P M t A R (cid:0)1 t t t with B ;R > 0;A > 0 and M > 0 given. The symmetric equilibrium is comple- (cid:0)1 (cid:0)1 (cid:0)1 (cid:0)1 mented with the monetary and (cid:12)scal policy rules, and the exogenous processes for G ;(cid:18) , t t and A . t 3.5 Frictionless Equilibrium The frictionless equilibrium is given by the above equilibrium with no frictions (ϕ = 0). Aggregate output in the frictionless equilibrium is given by ( ) (cid:18)t (cid:0)1 1=((cid:27)+φ) Y (cid:3) = A (cid:18)t g(cid:27)=((cid:27)+φ): (17) t t (cid:31) t H The above is the potential output over which the output gap in the monetary and (cid:12)scal policy rules is de(cid:12)ned. 3.6 Steady-State Equilibrium Since technology, A , is a non-stationary process, it introduces a stochastic trend in t output, consumption, real money balances, and the real wage. We de(cid:12)ne the stationary variables as: y = Y =A ;c = C =A ;w = W =(A P ) and v = Y =(M =P ). The steady-state t t t t t t t t t t t t t t equilibrium is the stationary equilibrium in the absence of shocks, and is de(cid:12)ned by the following equations: R(cid:12) (cid:5) = ; (18) (cid:14) ( ) (cid:18)(cid:0)1 1=((cid:27)+φ) y = (cid:18) g(cid:27)=((cid:27)+φ) = y (cid:3) ; (19) (cid:31) H 10

( ) (cid:18)(cid:0)1 1=((cid:27)+φ) c = (cid:18) gφ=((cid:27)+φ); (20) (cid:31) H ( ) 1 R = (cid:31) y((cid:27)(cid:0)1)g (cid:0)(cid:27) ; (21) v M R(cid:0)1 ( )( ( )) (cid:12) 1 1 1 1 b = 1(cid:0) (cid:0)(cid:28) (cid:0) 1(cid:0) ; (22) (cid:12) (cid:0)1 g v (cid:5)(cid:14) where b is the steady state level of the debt-to-output ratio, B =(P Y ). t t t 3.7 Log-linearized Model and Solution Method Given that the coefficients of the policy rules are time varying, and the time variation depends on the latent factors, the log-linearization is performed conditioning on the latent factors being at their current values each period. That is the essence of the quasi-linearity of the model. We present the model in log-deviations from the non-stochastic steady state, and show a way to solve it using a method in line with the minimum state variable (MSV) solution approach (McCallum, 1983). Conditioning on a value of the latent factors at period t, zm and zf, the log-linearized t t equations characterizing the economy in equilibrium are (x^ = ln(x =x) denotes the logt t deviation of variable x relative to its non-stochastic steady state, x): t 1 (cid:26) y^ = E y^ (cid:0) (R ^ (cid:0)E (cid:5) ^ )+(1(cid:0)(cid:26) )g^ + (cid:23) (cid:23)^ (23) t t t+1 t t t+1 g t t (cid:27) (cid:27) ((cid:18)(cid:0)1)(φ+(cid:27)) (cid:5) ^ = (cid:12)E (cid:5) ^ + (y^ (cid:0)y^ (cid:3) ) (24) t t t+1 ϕ t t ( ) 1 v^ = (1(cid:0)(cid:27))y^ +(cid:27)g^ + R ^ (25) t t t R(cid:0)1 t ( ) 1 (cid:28) 1 1 1 1 1 ^ b = g^ (cid:0) (cid:28)^ + v^ (cid:0) v^ (cid:0) + ((cid:5) ^ +∆Y ^ )+ (R ^ + ^ b ) (26) t t t t t(cid:0)1 t t t(cid:0)1 t(cid:0)1 bg b bv bv(cid:5)(cid:14) bv(cid:5)(cid:14) (cid:12) (cid:12) 1 (cid:27) (cid:3) ^ y^ = (cid:18) + g^ (27) t (φ+(cid:27))((cid:18)(cid:0)1) t φ+(cid:27) t ( ) R ^ = (cid:26) R ^ +(1(cid:0)(cid:26) ) (cid:11)(cid:25)(zm)(cid:5) ^ +(cid:11)y(zm)(y^ (cid:0)y^ (cid:3) ) +"R (28) t R t(cid:0)1 R t t t t t t ( ) (cid:28)^ = (cid:26) (cid:28)^ +(1(cid:0)(cid:26) ) (cid:13)b(zf) ^ b +(cid:13)y(zf)(y^ (cid:0)y^ (cid:3) ) +"(cid:28); (29) t (cid:28) t(cid:0)1 (cid:28) t t(cid:0)1 t t t t where ∆Y ^ = y^ (cid:0) y^ + (cid:23)^. The exogenous shocks that complete the equilibrium are the t t t(cid:0)1 t government spending shock, the cost-push shock, and the technology shock, given by g^ = (cid:26) g^ +"g (30) t g t(cid:0)1 t (cid:18) ^ = (cid:26) (cid:18) ^ +"(cid:18) (31) t (cid:18) t(cid:0)1 t (cid:23)^ = (cid:26) (cid:23)^ +"(cid:23): (32) t (cid:23) t(cid:0)1 t Tosolvethemodel,let! = [y^;(cid:25)^ ]′,k = [v^; ^ b ;R ^ ;(cid:28)^;∆Y ^ ;y ;y^(cid:3)]′,u = [g^;(cid:18) ^ ;(cid:23)^;"R;"(cid:28)]′, t t t t t t t t t t(cid:0)1 t t t t t t t 11

" = ["g;"(cid:18);"(cid:23);"R;"(cid:28)]′, z = [zm;zf]′ and rewrite (23)-(32) as t t t t t t t t t 0 = A(z )k +B(zf)k +C(z )! +Du (33) t t t t(cid:0)1 t t t 0 = Gk +JE ! +K! +Mu (34) t t t+1 t t u = Nu +" ; (35) t+1 t t+1 where A;B(zf);C(z );D;G;J;K;M and N are appropriate coefficient matrices shown in t t Appendix B. The proposed solution is given by k = P(z )k +Q(z )u ; (36) t t t(cid:0)1 t t ! = R(z )k +S(z )u ; (37) t t t(cid:0)1 t t where, for F(z ) = fP(z );Q(z );R(z );S(z )g, the i;j (cid:0)th entry is given by t t t t t 1 1 F(z ) = F +F 1+exp((cid:0)F2m(z t m(cid:0)F3m))1+exp((cid:0)F 2f (z t f(cid:0)F 3f )) ; (38) t 0 1 1(cid:0)F exp((cid:0)F2m(z t m(cid:0)F3m)) exp((cid:0)F 2f (z t f(cid:0)F 3f )) 41+exp((cid:0)F2m(z t m(cid:0)F3m))1+exp((cid:0)F 2f (z t f(cid:0)F 3f )) with F 2 [0;1]. This functional form is known as a bivariate logistic function and was 4 introduced by Ali et al. (1978).5 The parameter F allows the model to capture the effect 4 that the dependence between latent factors has on the expectations formations of the agents about the future evolution of the coefficients of the solution. Appendix C illustrates the proceduretoobtaintheparametersofthebivariatelogisticfunctionsbasedonthecoefficients of the structural model. We solve the model using an undetermined coefficients method approach where not only the solution has to be guessed and veri(cid:12)ed, but also the functional form of the coefficients of the solution. Within the logistic speci(cid:12)cation of policy rules, the bivariate logistic function (38) satis(cid:12)es this last requirement. Appendix E shows that the coefficients of the solution indeed follow a bivariate logistic function. 3.8 On Existence, Stability and Uniqueness of the Solution Existence of the solution is guaranteed by using the undetermined coefficients method. Time-varying coefficients pose a difficulty at guaranteeing stability and/or uniqueness of the solution, in particular if one thinks of stability and/or uniqueness holding in each period of time. The method presented (cid:12)nds a solution that is based on the values of the timevarying policy rule coefficients at their limits, or long-run bounds. These limiting coefficient values are chosen to deliver stability and uniqueness of the solution in a constant-coefficient version of the model, offering well-de(cid:12)ned bounds between which the economy evolves and between which agents form expectations. Davig and Leeper (2007) and Farmer et al. (2008) emphasize that stability and uniqueness of Markov-switching rational expectations models 5For identi(cid:12)cation of the latent factors and the coefficients of the policy rules in a estimation setting it is necessary to impose that F >0 and F >0. 2m 2f 12

have to be discussed within a framework of how agents form expectations about the future evolution of policy rule coefficients, and that is the approach taken at deriving the solution of the model in this paper. Here, we take expectations to the realization of future policy rule coefficients as given by realization of logistic functions of the latent factors.6 4 Estimation Strategy This section presents the estimation of the policy rules with time-varying coefficients driven by latent factors as speci(cid:12)ed for the New Keynesian model presented in Section 3. The estimation employs a limited-information estimation approach using Bayesian methods, which allows obtaining the set of parameters characterizing the policy rules, denoted by (cid:2) , y the set of parameters of the latent factors, denoted by (cid:2) , and the latent factors themselves. z 4.1 Time-varying Coefficients, Stochastic Volatility and Endogeneity LetINT denotethedemeanednominalfederalfundsrateinperiodt, TAX thedemeaned t t ratio of federal receipts net of transfers to output in period t, INF the demeaned annual t in(cid:13)ation rate in period t, GAP the output gap in period t and DBT the demeaned average t t debt to output ratio over the last four quarters. The state-space model is composed of the observation equations INT = (cid:26) INT +(1(cid:0)(cid:26) )((cid:11)(cid:25)(zm)INF +(cid:11)y(zm)GAP )+(cid:29)R (39) t R t(cid:0)1 R ( t t t t) t TAX = (cid:26) TAX +(1(cid:0)(cid:26) ) (cid:13)b(zf)DBT +(cid:13)y(zf)GAP +(cid:29)(cid:28); (40) t (cid:28) t(cid:0)1 (cid:28) t t t t t and the transition equations zm = (cid:26) zm +(cid:24)m (41) t zm t(cid:0)1 t zf = (cid:26) z(cid:28) +(cid:24)f: (42) t zf t(cid:0)1 t Assumptions about the distributions of (cid:29)R and (cid:29)(cid:28) are made explicit in the following section. t t 4.1.1 Stochastic Volatility The existence of stochastic volatility in the shocks of policy rules with time-varying coefficients has been documented by Davig and Leeper (2006), Fernandez-Villaverde et al. (2010), Bianchi (2012) and Fernandez-Villaverde et al. (2011b), who (cid:12)nd that not only switches in policy rule coefficients are detectable in estimation, but also a fair amount of 6The debate on the determinacy of the solution of DSGE models with time-varying coefficients has attractedattentionoftheliteraturesinrecentyears,anditisstillanopen(cid:12)eldtofutureresearch(seeDavig and Leeper, 2006, 2007; Fernandez-Villaverde et al., 2010; Farmer et al., 2011; Cho, 2013; Foerster et al., 2013). 13

stochasticvolatility.7 Hence, the distributionof theerrorterms inthe policyrules isspeci(cid:12)ed as (cid:29)R (cid:24) N(0;(cid:27)2 ) and (cid:29)(cid:28) (cid:24) N(0;(cid:27)2 ), where t R;t t (cid:28);t ln(cid:27) = (1(cid:0)(cid:26) )ln(cid:27) +(cid:26) ln(cid:27) +(cid:17) (cid:24)R (43) R;t (cid:27)R R (cid:27)R R;t(cid:0)1 R t ln(cid:27) = (1(cid:0)(cid:26) )ln(cid:27) +(cid:26) ln(cid:27) +(cid:17) (cid:24)(cid:28) (44) (cid:28);t (cid:27)(cid:28) (cid:28) (cid:27)(cid:28) (cid:28);t(cid:0)1 (cid:28) t with (cid:26) 2 [0;1), (cid:26) 2 [0;1), (cid:24)R (cid:24) iidN(0;1) and (cid:24)(cid:28) (cid:24) iidN(0;1). (cid:27)R (cid:27)(cid:28) t t In what follows, let h = [ln(cid:27) ;ln(cid:27) ] and let the set of parameters of the stochastic t R;t (cid:28);t volatility processes be denoted (cid:2) . h Equations (43) and (44) are added to the state-space model (39)-(42) to introduce stochastic volatility to the speci(cid:12)cation of the policy rules with time-varying coefficients. 4.1.2 Endogeneity Since the work of Clarida et al. (2000), the estimation of monetary policy rules with constant coefficients, in particular the Taylor rule, has taken into account the endogeneity that exists between the shocks of the policy rule and in(cid:13)ation and output. The instrument setusedintheir GMMestimationcontainsfour lagsof: in(cid:13)ation, the outputgap, theFederal funds rate, the short-long spread, and commodity price in(cid:13)ation. With respect to (cid:12)scal policy rules with constant coefficients, Li (2009) illustrates the endogeneity/simultaneity problem that arises when estimating a (cid:12)scal policy rule like the one presented in this work. In estimating a (cid:12)scal policy rule that reacts to contemporary debt and the output gap, Claeys (2008) uses a set of instrumental variables in his GMM estimation that contains lags of: the output gap, debt, unit labor costs, growth in labor productivity, NAIRU, a broad money aggregate, a synthetic interest rate of the EURO area, oil price index, and the SEK/DEM exchange rate. In terms of estimating linear equations with time-varying coefficients, either in the conventional random-coefficient or Markov-switching setups, Kim (2006) and Kim (2009) establish a Heckman-type two-stage maximum likelihood estimation technique to deal with the endogeneity problem to yield consistent estimates of the hyper-parameters, as well as to provide correct inferences on the time-varying coefficients. Kim and Nelson (2006b) estimate a random coefficients monetary policy rule for the United States using as the set of instruments four lags of: the Federal funds rate, output gap, in(cid:13)ation, commodity price in(cid:13)ation, and M2 growth. In related work, Bae et al. (2011) estimate a Markov-switching coefficients monetary policy rule for the United States using as the set of instruments three lags of: the Federal funds rate, GDP gap, in(cid:13)ation, commodity price changes, and the spread between the long-term bond rate and the three-month Treasury Bill rate. The set of instruments that we use for both the monetary and the (cid:12)scal policy rules is given by four lags of: in(cid:13)ation, the output gap, government spending as proportion of GDP, M2 growth, and commodity price in(cid:13)ation. In a constant-coefficient version of the policy rules, the GMM estimation obtains the following results with respect to the instrument set: (i) the J test statistics of overidenti(cid:12)cation restrictions for both of the rules do not reject the null hypothesis that the instrument set is appropriate at the 5% level of signi(cid:12)cance; (ii) the 7Sims and Zha (2006), on the other hand, argue that only changes in volatility can be detected in estimation, and not changes in coefficients. 14

exogeneity C test statistics imply that different subsets of instruments are exogenous at the 5% level of signi(cid:12)cance; (iii) the Cragg-Donald test statistics reject the null hypothesis of weak instruments at the 5% level of signi(cid:12)cance for both policy rules. Appendix F describes the setup of the model to correct for endogeneity. Details on the implementation of the Bayesian estimation of (cid:2) , (cid:2) , (cid:2) , fz gT and fh gT appear in Appendix H. y z h s s=0 s s=0 4.2 Data We use quarterly data from 1960:1 to 2008:3. The sample is not extended beyond 2008:3 to avoid having to deal with the zero lower bound (ZLB) of interest rates. It is still possible to estimate the interdependence between monetary and (cid:12)scal policy under the ZLB, but that constitutes a subject of future research. In(cid:13)ation is the percentage change over the last four quarters of the price level given by the GDP price de(cid:13)ator. The nominal interest rate is the quarterly average of the federal funds rate. The output gap is the log difference between real GDP and the Congressional Budget Office’s measure of potential real GDP. M2 growth is the percentage change over the last four quarters of seasonally adjusted M2. Commodity price in(cid:13)ation is the percentage change over the last four quarters of the commodity price index. Government spending is the federal consumption expenditures and gross investment. These variables are obtained from FRED. Lagged debt is the average debt-output ratio over the previousfourquarters, wheredebtistheTreasuryDirectparvalueofgrossmarketablefederal debtheldbythepublic. Taxnetoftransferscorrespondstotheseasonallyadjustedquarterly current receipts of the federal government from which the current transfer payments have been deducted. This variable is obtained from the NIPA Table 3.2. 5 Estimation Results The choice of prior distributions, hyper parameters, means of 5;000 draws from the posteriordistributionaftertrimmingthe(cid:12)rst1;000;000outof2;000;000drawsandthinning every 200th draw, along with 90% con(cid:12)dence sets appear in Table 1.8 In order to keep the estimation relatively simple, we impose two restrictions that do not change the results qualitatively: First, we assume that the output gap coefficients of both policy rules are not time varying, which allows us to focus on capturing the interdependence between monetary and(cid:12)scalpolicymakingintermsoftheco-movementoftheTaylorrulecoefficientofin(cid:13)ation and the tax rule coefficient of lagged debt. Second, we assume that the location parameters of the logistic policy rule coefficients, (cid:11)(cid:25) and (cid:13)b, are zero. 3 3 Figure 2 shows that the estimated model does acceptably well at explaining the observed time series of interest and tax rates. 8The Raftery and Lewis (1992) diagnostic test determines that 606,826 draws from the posterior distribution should be taken to estimate the 50th percentile within 0.01 with a 95% con(cid:12)dence level. It also determines that thinning to achieve an independent chain should occur every 115th draw. 15

Table 1: Results from the Bayesian Estimation Prior Posterior Parameters Density Mean SD Mean 90% Conf. Set (cid:11)(cid:25) Gamma 0.8 0.2 0.57 [0.35, 0.84] 0 (cid:11)(cid:25) Gamma 1.2 0.3 1.06 [0.61, 1.68] 1 (cid:11)(cid:25) Gamma 10 8 5.05 [0.10, 15.72] 2 (cid:11)y Gamma 0.5 0.4 0.25 [0.08, 0.52] (cid:26) Beta 0.9 0.05 0.96 [0.91, 0.99] R (cid:13)b Normal -0.02 0.05 0.00 [-0.06, 0.06] 0 (cid:13)b Gamma 0.1 0.05 0.09 [0.03, 0.20] 1 (cid:13)b Gamma 10 8 4.03 [0.04, 14.70] 2 (cid:13)y Gamma 0.5 0.4 0.76 [0.44, 1.16] (cid:26) Beta 0.9 0.05 0.94 [0.93, 0.96] (cid:28) (cid:26) Beta 0.9 0.05 0.86 [0.74, 0.95] zm (cid:26) Beta 0.9 0.05 0.88 [0.78, 0.95] zf zm Normal 0 2 -0.41 [-2.66, 1.93] 0 zf Normal 0 2 0.05 [-2.26, 2.35] 0 (cid:20) | 0.20 [0.02, 0.47] ln(cid:27) | -2.05 [-2.41, -1.70] R (cid:26) | 0.77 [0.63, 0.90] (cid:27)R (cid:17) | 0.57 [0.39, 0.78] R ln(cid:27) | -2.93 [-5.18, -0.68] R;0 ln(cid:27) | -1.15 [-1.45, -0.89] (cid:28) (cid:26) | 0.48 [0.27, 0.64] (cid:27)(cid:28) (cid:17) | 0.88 [0.65, 1.11] (cid:28) ln(cid:27) | -5.85 [-12.48, 0.59] (cid:28);0 |denotesa(cid:13)atprior. 16

Figure 2: Observed and Predicted Series 20 15 R t R predicted t % 10 5 0 1965 1970 1975 1980 1985 1990 1995 2000 2005 14 12 10 % 8 τ 6 t τ predicted 4 t 2 1965 1970 1975 1980 1985 1990 1995 2000 2005 5.1 Choice of Prior Distributions With respect to the monetary policy rule coefficients, there exist results in the literature about the values that they take in different regimes. Clarida et al. (2000) estimate that the in(cid:13)ation coefficient is 0.83 in the pre-Volcker era and 2.15 in the Volcker-Greenspan era. Lubik and Schorfheide (2004) (cid:12)nd that the coefficient on in(cid:13)ation is estimated at 0.77 or 0.89, depending on the prior used, in the pre-1982 era, and 2.19 in the post-1982 era. Bianchi (2012) (cid:12)nds that the coefficient on in(cid:13)ation is estimated to be 0.94, 1.25, or 1.6 in a three-regime Markov-switching speci(cid:12)cation. Davig and Leeper (2011) (cid:12)nd that the in(cid:13)ation coefficient switches between 0.53 and 1.29. We specify the prior distribution that characterizes the lower bound of the in(cid:13)ation coefficient with a mean of 0.8, and the upper bound with a prior distribution whose mean is 2 (the sum of 0.8 and 1.2 in Table 1). In regard to the (cid:12)scal policy rule coefficients, Bianchi (2012) (cid:12)nds that the coefficient on debt is estimated to be 0.0006, -0.0007, or -0.0036. Davig and Leeper (2011) estimate that the coefficient on debt switches between -0.025 and 0.071. We set the prior mean of the lower bound of the debt coefficient to -0.02, and the mean of the prior distribution that sets the upper bound of the coefficient to 0.08 (the sum of -0.02 and 0.1 in Table 1). The policy rule coefficients on the output gap for both policy rules have 0.5 as the prior mean. The prior distribution speci(cid:12)cation of the transition coefficients of both logistic policy rule coefficients is set to have a mean of 10 with a standard deviation of 8. We recall that the larger the coefficient is, the more rapid the transition between states is. Values of the coefficient greater than 10 imply a speci(cid:12)cation that mimics closely a Markov-switching-like transition. The choice of this prior distribution allows the coefficient to take low or high 17

values. The smoothing coefficients of the policy rules have prior distributions that are standard in terms of the persistence that they represent for interest and tax rates in the data. Acoefficientthatdeservesattentionisthecorrelationcoefficientbetweentheshocksofthe latent factors, which determines the degree of interdependence between monetary and (cid:12)scal policymaking. The prior distribution chosen is uniform on [0,1]. We recall that a positive value of this parameter implies that monetary and (cid:12)scal policy co-move in such a way that they tend to deliver outcomes in the M or the F regime more frequently than outcomes in the indeterminacy or no solution regimes. Hence, by choosing a prior distribution on the nonnegative domain, we are imposing a model with either co-movements towards the determinate regimes or absence of co-movements in that direction, but we are not allowing co-movements in the direction of the indeterminate or unbounded debt regimes. Finally, following Fernandez-Villaverde et al. (2011a), we do not specify prior distributions for the parameters of the stochastic volatility processes. 5.2 Parameter Estimates The estimated parameters of the monetary and (cid:12)scal policy rule coefficients in Table 1 have the expected magnitudes and signs. In particular, for the monetary policy rule, the in(cid:13)ation coefficient takes values in (0.57, 1.63), while the output gap coefficient has a posterior mean of 0.25. With respect to the (cid:12)scal policy rule, the debt coefficient takes on values in (0.001, 0.095), while the output gap coefficient has a posterior mean of 0.76. The speed of transition of the policy rule coefficients is estimated to be higher for the in(cid:13)ation coefficient of the monetary policy rule than for the debt coefficient of the tax rule. This implies that the (cid:12)scal authority has a slower transition across states (in terms of how taxes react with respect to deviations of debt) than the monetary authority does (in terms of how the interest rate reacts with respect to in(cid:13)ation). This result is consistent with the legislative and implementation lags of (cid:12)scal policymaking. The latent factors show relatively high persistence, with the (cid:12)scal policy latent factor being somewhat more persistent than the monetary policy one. This result implies that (cid:12)scal policy regimes are likely to last longer than monetary policy regimes. Finally, thecorrelationbetweenthelatentfactorshasanestimatedposteriormeanof0:20 and a 90% con(cid:12)dence set given by (0.01, 0.47). This result implies that there is a degree of direct interactions between policies. In particular, monetary tightenings to stabilize in(cid:13)ation tend to be accompanied by (cid:12)scal policy that stabilizes debt (M regime), while (cid:12)scal policy that departs from debt stabilization tends to be accompanied by interest rates that react weakly to in(cid:13)ation (F regime). 5.3 Evolution of Policy Rule Coefficients Figure3showsthesmoothedestimatesofthemonetarypolicyrulecoefficientforin(cid:13)ation on the left axis, and of the (cid:12)scal policy rule coefficient for debt on the right axis. Two facts are apparent from this (cid:12)gure: First, the (cid:12)scal policy rule coefficient on debt was low during most of the 1970s, the second half of the 1980s, and the (cid:12)rst half of the 1990s. Second, the monetary policy rule coefficient on in(cid:13)ation was high during the second half of the 1960s 18

Figure 3: Evolution of Policy Rule Coefficients and NBER Recession Periods απ (zm) t 1.25 0.07 0.06 γb απ 1 0.05 0.04 γb(zf) t 0.75 11996655 11997700 11997755 11998800 11998855 11999900 11999955 22000000 22000055 and the (cid:12)rst half of the 1970s, during the (cid:12)rst half of the 1980s, and during the (cid:12)rst half of the 2000s. In the next paragraphs, we discuss the evolution of each of the policy rule coefficients to compare the econometric results with the narrative about monetary and (cid:12)scal policymaking and with similar results in the literature. The evolution of the monetary policy rule coefficient on in(cid:13)ation reveals that the Federal Reserve conducted a hawkish monetary policy during the second half of 1960s, as Davig and Leeper (2011), Bianchi (2010), Eo (2009) and Fernandez-Villaverde et al. (2010) (cid:12)nd, with a tendency to ease it towards the end of the decade. During the mid 1970s the interest rate did not react strongly to in(cid:13)ation. Most studies in the literature (cid:12)nd a passive monetary policy during this decade, except Boivin (2006), who (cid:12)nds that monetary policy was tight during the (cid:12)rst half of the 1970s. At the end of the 1970s monetary policy switched rapidly to a hawkish regime. This switch in policymaking is also found by Davig and Doh (2009), Eo (2009) and Bianchi (2010) who (cid:12)nd, based on estimations of Markov-switching policy rule coefficients, that the active monetary policy periods started around, or a little earlier than, the mid 1980s. The graph shows that monetary policy continued to be relatively hawkish during most of the 1980s. Starting in the late 1980s, the estimates of the model suggest a relatively weak response to in(cid:13)ation, which lasted through much of the 1990s. A study that (cid:12)nds a similar result is Fernandez-Villaverde et al. (2010).9 In 2000s, monetary policy increased its reaction to in(cid:13)ation until 2005 and then the strength of the reaction declined. The empirical evidence is divided with respect to this result: On one hand Eo (2009), Davig and Doh (2009) and Bianchi (2010) (cid:12)nd that monetary policy was actively (cid:12)ghting in(cid:13)ation, while on another Fernandez-Villaverde et al. (2010) and Davig and Leeper (2011) (cid:12)nd the 9KimandNelson(2006a)(cid:12)ndthattheircon(cid:12)denceintervalstartsincludingthepassivemonetarypolicy region at the beginning of the 1990s. 19

opposite. For a narrative perspective on the results, we rely on the history of the monetary policy of the Federal Reserve as told by Hetzel (2008). He compares Fed Chairman William Martin to Fed Chairman Paul Volcker and Fed Chairman Alan Greenspan in that Martin believed that raising short-term interest rates in an expansion was a way to preempt in(cid:13)ation. That allows to understand the behavior of monetary policy, according to the graph, during the second half of the 1960s. Hetzel also mentions the weak reaction of interest rates to in(cid:13)ation during the 1970s due to the focus of the central bank to promote employment and the belief that in(cid:13)ation was a nonmonetary phenomenon. According to Hetzel, the 1980s saw the commitment of the Federal Reserve to money targets allowing the FOMC to raise interest rates by whatever extent necessary to lower in(cid:13)ation. The results show that the reaction coefficient of the interest rate to in(cid:13)ation was indeed relatively high. After tightening monetary policy at the end of the 1980s to counteract concerns about in(cid:13)ation, the results show a decline in the reaction of the federal funds rate to in(cid:13)ation at the beginning of the 1990s to help with the recovery after the recession. With in(cid:13)ation declining during this decade and interest rates remaining nearly (cid:13)at, except for the rise between 1993 and 1995 due to the \1994 in(cid:13)ation scare" (see Hetzel, 2008), the reaction coefficient on in(cid:13)ation declines until the events of the Asian and Russian crises. During 1998 the Federal Reserve moved the interest rate in the same direction as declining in(cid:13)ation, causing a spike in the evolution of the in(cid:13)ation coefficient. In the 2000s, with declining in(cid:13)ation and interest rates between 2001 and the beginning of 2003, the reaction coefficient to in(cid:13)ation rises rapidly to reach a peak in the second quarter of 2003 when concerns about interest rates being close to the zero-bound arose for the (cid:12)rst time. With in(cid:13)ation stable and interest rates rising during 2004, the reaction coefficient declines. Then the coefficient rises at the end of the sample, between 2005 and 2008, re(cid:13)ecting interest rates that risen more than what in(cid:13)ation risen during the period. For the debt policy rule coefficient, the results show a coefficient responding weakly to lagged debt during the 1970s, a result mostly in the same lines as those in Davig and Leeper (2006), Bianchi (2012), and Favero and Monacelli (2003). The results also show a low debt coefficient during the second half of the 1980s and the (cid:12)rst half of the 1990s, a result that is compatible with the (cid:12)ndings in Bianchi (2012). In terms of the narrative, the tax reductions in the revenue act of 1964 can help understand the decline of the coefficient during the 1960s, while the revenue and expenditure control act of 1968 and the tax reform act of 1969 can help explain the rise of the coefficient at the end of the decade. The subsequent tax reforms of 1971, 1975 and 1976 help explain the decline in the debt coefficient during the 1970s. The rise of the coefficient observed at the end of the decade has to do with the fact that the debt-GDP ratio rises between 1975 and 1980, inclusive, and the net revenue-GDP ratio rose in that period. That relationship breaks up during the 1980s due to the tax reforms of the decade, in particular the recovery tax act of 1981. During the 1990s, the coefficient rises rapidly, most likely due to the changes implemented in the budget reconciliation act of 1993. The taxpayer relief act of 1997 can help understand the decline in the coefficient at the end of the decade. The decline of the debt coefficient in the 2000s can be attributed to the tax relief acts of 2001 and 2003, and the subsequent stimulus acts of 2004 and 2008. In a historical perspective, the evolution of the estimated policy rule coefficients suggest 20

Figure 4: Evolution of Estimated Stochastic Volatilities 1.5 1 σ % R,t 0.5 0 1965 1970 1975 1980 1985 1990 1995 2000 2005 3 2 σ % τ,t 1 0 1965 1970 1975 1980 1985 1990 1995 2000 2005 that policy leaned towards a (cid:12)scal regime during some part of the 1970s, a result emphasized by Bianchi (2012). On the other hand, the estimates favored a monetary regime during the secondhalfofthe1960s,the(cid:12)rsthalfofthe1980s,andthe(cid:12)rsthalfofthe2000s. Themacroeconomic model allows short-run dynamics that visit temporarily all the regimes, including the indeterminate and the no-bounded-solution regimes, but the long-run expectations of a mixed regime imply dynamics that tend to move the economy between the two determinate regimes. 5.4 Evolution of Estimated Stochastic Volatilities Figure 4 shows the evolution of the estimated stochastic volatilities for the interest and tax rate rules. The estimation shows that the volatility of interest rates was signi(cid:12)cantly higherduringthebeginningofthe1980sduetotheimportantchangeinconductingmonetary policy. Also, volatility is somewhat lower starting in the 1990s, a phenomenon referred to as the great moderation (see Stock and Watson, 2002), and stays low with a small peak at the end of the sample period when the great recession hit. On the other hand, (cid:12)scal policy volatility shows spikes in the mid 1970s, the (cid:12)rst half of the 2000s, and the beginning of the great recession. All these events are associated with some kind of (cid:12)scal stimulus. It is worth notingthatthepersistenceofthevolatilityofmonetarypolicymaking, givenbytheestimated coefficient (cid:26) , is higher than the persistence of the volatility of (cid:12)scal policymaking, given (cid:27)R by the estimated coefficient (cid:26) , as can be seen in Table 1. (cid:27)(cid:28) 6 Monetary and Fiscal Policy Analysis This section presents the results of monetary and (cid:12)scal policy experiments through impulse-response functions. The section also presents the effect of policy interdependence in 21

reducing the volatility of output and in(cid:13)ation. 6.1 Nonlinear Impulse Response Analysis Weperformpolicyexperimentswiththemodelsolvedusingtheproposedsolutionmethod andcalibratedaccordingtotheestimatedparametersofthepolicyrulecoefficients. Sincethe estimation is not constrained to deliver a continuous solution to the macroeconomic model as presented in Appendix E, we calibrate the model with coefficients that are consistent with continuity of the solution. In particular, we calibrate the lower bound of the monetary policy rule coefficient on in(cid:13)ation, (cid:11)(cid:25), to be 0.85 instead of 0.57. The (cid:12)scal policy rule coefficients, 0 (cid:13)b and (cid:13)b, are calibrated to 0.03 and 0.07 instead of 0 and 0.09, respectively. 0 0 For the markup in steady state and the price adjustment cost parameter, Keen and Wang (2007) show that, given a steady state markup and a fraction of (cid:12)rms that re-optimize each period, there is a corresponding value for the price adjustment cost parameter. We set the markup to 20% and the fraction of re-optimizing (cid:12)rms to 25% each period (a (cid:12)rm reoptimizes every 12 months). These values correspond, approximately, to (cid:18) = 6 and ϕ = 60. Other parameters that need to be speci(cid:12)ed are: (cid:14), the steady state gross quarterly rate of output growth, which is set to 1:0081, the average over the sample period that implies a steady state annual growth rate of approximately 3.25%; (cid:5), the steady state gross quarterly rate of price in(cid:13)ation, which is set to 1:0084, the average over the sample period that implies a steady state annual in(cid:13)ation of approximately 3.4%; b, the steady state level of debt to output, which is set to 0.3354, the average over the sample period; (cid:16), the ratio of government spending to output is set to 0.081, also the average over the sample period; 1=vb, the ratio of outstanding money balances to debt, is set to 0.2 following Kim (2003); (cid:12) is set to 0.99; (cid:27) is set to 1; and φ is set to 1, so that the Frisch elasticity of labor supply is unity. We perform two policy experiments. First, a one-time i.i.d shock on the interest rate. Second, a one-time i.i.d shock on the tax-to-output ratio. The responses to these shocks are calculated under three scenarios: starting at regime M and staying there forever, starting at regime F and staying there forever, and a Mixed regime. IntheMixedregime,thelatentfactorsstartattheirmeans,whichwesettozeroaccording to the con(cid:12)dence sets in Table 1. These zero values imply corresponding values for the initial monetary and (cid:12)scal policy rule coefficients ((cid:11)(cid:25)(0) = 1:38 and (cid:13)b(0) = 0:065). Over the response horizon, the policy rule coefficients evolve according to their logistic speci(cid:12)cations and the joint evolution of the latent factors. To compute the responses in the Mixed regime, we take the average of the responses for given realizations of the policy rule coefficients. Note that in the Mixed regime the initial values of the policy rule coefficients imply that the economy is in regime M. However, because agents expect regime F to take place for some time in the future, the dynamics of the Mixed regime are not those of a pure M regime, but of a regime that incorporates the features of both regimes M and F. 6.1.1 A Monetary Contraction Figure 5 shows the responses of some of the variables of the model to a one-time 25 basis point i.i.d. shock on the interest rate. 22

Figure 5: Response to a 25 bp Increase in the Interest Rate Interest Rate Tax Rate 0.4 0.2 0.2 0.1 0 0 0 10 20 30 40 0 10 20 30 40 Inflation Debt/Output 0 1.5 1 −0.5 0.5 −1 0 0 10 20 30 40 0 10 20 30 40 Output Gap PV Surplus (% change) 6 0 4 −0.5 2 −1 −1.5 0 0 10 20 30 40 0 10 20 30 40 PV Seigniorage (% change) 1 F Regime 0 M Regime −1 Mixed Regime −2 0 10 20 30 40 23

Figure 6: Response to a 100 bp Increase in the Tax Rate Interest Rate Tax Rate 0 1 −0.05 0.5 −0.1 0 0 10 20 30 40 0 10 20 30 40 Inflation Debt/Output 0 0.5 −0.2 0 −0.4 −0.5 −0.6 −1 0 10 20 30 40 0 10 20 30 40 Output Gap PV Surplus (% change) 0 2 −0.2 0 −0.4 −2 −0.6 −0.8 −4 0 10 20 30 40 0 10 20 30 40 PV Seigniorage (% change) 0 F Regime −0.5 M Regime Mixed Regime −1 0 10 20 30 40 24

The effects of a monetary contraction in a New Keynesian model in regime M are well known: an open market operation that sells debt to households, and that is expected to be corrected in the future via higher taxes, does not change household wealth and only increases the nominal interest rate. With sticky prices the real interest rate increases, reducing consumption, output and in(cid:13)ation. As the nominal interest rate decreases, as well as the real interest rate, the output gap returns to zero and in(cid:13)ation to its steady state. Taxes (cid:12)rst decrease due to feedback from the output gap and then increase due to their Ricardian response to lagged debt, which is now higher. Given the persistence of taxes, the present value of expected future surpluses increases and slowly returns to steady state, while the present value of expected future seignorage decreases on impact. InregimeF,anopenmarketoperationthatsellsdebttohouseholds,whichisnotexpected to be corrected in the future via higher taxes, increases household’s wealth. The increase in wealth attenuates the liquidity-induced contraction in demand. In(cid:13)ation drops on impact given that the liquidity effect dominates the wealth effect.10 The real interest rate increases on impact, causing output to fall. Note that a \price puzzle", in the sense of Sims (1992), is present. Over the response horizon, in(cid:13)ation increases due to some unpleasant monetarist arithmetic: Taxes are not expected to increase to pay back the debt and some monetization of the de(cid:12)cit is expected. Taxes decrease on impact, reacting mainly to the output gap, and then increase above steady state due to the initial increase in debt, since the debt coefficient of the (cid:12)scal policy rule is small but positive. Given the persistence of taxes, the tax increase above steady state makes the present value of expected future surpluses increase on impact. In the Mixed regime, agents compute their responses based on their expectation of the evolutionoffutureregimes. Asaresult, theeffectsofamonetarycontractiononin(cid:13)ationand output are between the two previously described regimes. Agents’ expectations incorporate the behavior of the economy under both regimes M and F. Thus, there is lower in(cid:13)ation on impact and there is a wealth effect that translates into marginally higher in(cid:13)ation in the long run. Output also falls on impact due to higher real interest rates. These results mean that, as long as the economy does not start at one of the limiting regimes and stays there forever, some of the wealth effect derived from regime F will in(cid:13)uence the economy’s response to a monetary contraction. That is, a monetary contraction decreases in(cid:13)ation on impact, but the long-run effect is an increase of in(cid:13)ation above steady state. Sims (2011) describes this feature of monetary policy as \stepping on a rake". The impact response of in(cid:13)ation and output to a 25 basis points increase in the interest rate is shown in Figure 7 for a combination of values of (cid:11)(cid:25)(zm) and (cid:13)b(zf) for zm = zf in [- 10,10]. The responses of in(cid:13)ation and output range from -1.03% and -1.37%, respectively, in regime M (zm = zf = 10), to -0.15% and -0.42%, respectively, in regime F (zm = zf = (cid:0)10). In between these limits there exists a continuum of impact responses given by the possible combinationsof the monetaryand (cid:12)scal policy rulecoefficients. Hence, the size of theimpact response of in(cid:13)ation and output to a monetary policy shock will depend on the state of the 10Todeterminethedirectionoftheinitialresponseofin(cid:13)ationinregimeF,weusethefollowingexpression: ∑1 1 1 R(cid:0)1 b(cid:0)1(cid:5) ∆Y =E 0 MRS 0;t (s t +m t ); 0 0 t=0 where s and m denote the primary surplus and seignorage, respectively, in output terms. t t 25

Figure 7: Response on Impact to a 25 bp Increase in the Interest Rate Response of Inflation −0.2 −0.4 −0.6 −0.8 −1 1.8 1.6 1.4 1.2 1 0.1 0.08 0.06 0.04 γb απ Response of Output −0.6 −0.8 −1 −1.2 1.8 1.6 1.4 1.2 1 0.1 0.08 0.06 0.04 γb απ monetary and (cid:12)scal policy rule coefficients at the time of policy implementation. 6.1.2 A Fiscal Contraction - Tax increase Figure 6 shows the responses of some of the variables of the model to a one-time 100 basis point i.i.d shock on the tax-to-output ratio. In regime M, a tax increase that retires debt does not have any effect on output or in(cid:13)ation due to the Ricardian nature of the equilibrium: higher taxes today are expected to be fully compensated with lower taxes in the future, with no wealth effect on households. Wealth is unchanged because, on impact, the present value of expected future surpluses and seignorage are unchanged. Taxes decrease after impact in reaction to the reduction in debt and, given the persistence of taxes, remain below steady state at long horizons. Debt falls, anticipating the decrease in the present value of expected future surpluses. There are no changes in seignorage. AtaxincreaseinregimeFreduceswealthsincetheincreaseisnotexpectedtobereversed in the future. The decrease in wealth reduces demand for goods and, hence, in(cid:13)ation. Since the nominal interest rate responds passively to changes in in(cid:13)ation, the real interest rate increases, reducing consumption and output. The present value of expected future surpluses increases due to persistently higher taxes that will not be compensated for in the future. The present value of expected future seignorage declines on impact due to lower in(cid:13)ation and interest rates. Overall, the change in the expected present value of government net receipts implies that debt initially increases before returning to steady state. In the Mixed regime, as with the monetary contraction, agents’ behavior is based on their expectation of the evolution of future regimes. As a result, the effects of a (cid:12)scal contraction are between the two previously described regimes. That is, a change in taxes has effects on in(cid:13)ation and output due to the wealth effect that emerges because agents do not expect the 26

Figure 8: Response on Impact to a 100 bp Increase in the Tax Rate Response of Inflation 0 −0.2 −0.4 1.8 1.6 −0.6 1.4 1.2 1 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 απ γb Response of Output 0 −0.2 −0.4 1.8 1.6 −0.6 1.4 1.2 −0.8 1 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 απ γb initial increase in taxes to be completely reversed in the future. However, the effects are attenuated with respect to a pure F regime. The impact response of in(cid:13)ation and output to a 100 basis point increase in the tax-tooutputratioisshowninFigure8foracombinationofvaluesof(cid:11)(cid:25)(zm)and(cid:13)b(zf)forzm = zf in [-10,10]. The responses of in(cid:13)ation and output range from 0% and 0%, respectively, in regime M (zm = zf = 10), to -0.71% and -0.78%, respectively, in regime F (zm = zf = (cid:0)10). In between these limits there exists a continuum of impact responses dependent on the combinations of monetary and (cid:12)scal policy rule coefficients. Hence, the magnitude of the impact responses of in(cid:13)ation and output to a (cid:12)scal policy shock will depend on the state of the monetary and (cid:12)scal policy rule coefficients at the time of policy implementation. 6.2 Policy Interdependence: Its Effects In this section we analyze the effect of interdependence in monetary and (cid:12)scal policy making as measured by a positive correlation coefficient between the latent factors, (cid:20). In Section 5.2, we showed that the estimated posterior mean of the correlation coefficient between the latent factors is 0.2. As mentioned previously, this value implies that monetary tightenings to stabilize in(cid:13)ation tend to be accompanied by (cid:12)scal policy that stabilizes debt (regime M) while (cid:12)scal policy that deviates from debt stabilization tends to be accompanied by a loose monetary policy that contributes to keep debt stable (regime F). The existing literature on Markov-switching policy rule coefficients assumes that independent states drive the switches between regimes of the coefficients of each policy rule. In that sense, there are not explicit interactions between monetary and (cid:12)scal policymaking. In the context of the current paper, that would be analogous to the case where (cid:20) = 0. To examine the effect of policy interdependence ((cid:20) > 0), we solve and simulate the model for different values of the correlation coefficient between the latent factors. We simulate the 27

Figure 9: Interdependence and Unconditional Volatilities of Output and In- (cid:13)ation 1 σ y σ π 0.995 0.99 0.985 0.98 0.975 0.97 0 0.2 0.4 0.6 0.8 1 κ model for 1,000 periods 1,000 times, including the evolution of latent factors, and take the standard deviation of output and in(cid:13)ation.11 Figure 9 illustrates the results of the exercise where the unconditional standard deviations have been normalized to one when (cid:20) = 0, which is the case of absence of coordination. As can be seen from the (cid:12)gure, the stronger the association between monetary and (cid:12)scal policymaking, as measured by the degree of synchronization of changes in the coefficients of the policy rules, the smaller the unconditional standard deviation of the series. The results show that going from a scenario of independent switching in the coefficients of the policy rules to a scenario of relatively low interdependence, as measured by (cid:20) = 0:25, would result in a decrease of between 1% and 2% in the unconditional standard deviation of in(cid:13)ation or output. To understand this result, recall that the only two determinate equilibria arise in regimes M and F, while the other two equilibria imply either an indeterminacy or a solution with unbounded debt. Under the indeterminate equilibrium, the presence of sunspot shocks may imply a higher volatility of in(cid:13)ation, whereas under the unbounded debt equilibrium there is not a stationary solution to the macroeconomic model. Hence, policy co-movements in the direction of the M and F regimes would help stabilize output and in(cid:13)ation with respect to the case of no policy co-movements or co-movements away from determinate regimes. 7 Concluding Remarks In this paper, we formulated and solved a New Keynesian model with time-varying policy rule coefficients that allows for interdependence between monetary and (cid:12)scal policymaking. This speci(cid:12)cation permits analyzing the effects of monetary and (cid:12)scal policy in an environment where agents expect the economy to evolve between times where the nominal anchor is provided by monetary policymaking and times where the nominal anchor is provided by 11The standard deviation of the shocks is kept (cid:12)xed across simulations. 28

(cid:12)scal policymaking. We estimated the policy rules, and obtained results showing that there have been (cid:13)uctuations in the monetary and (cid:12)scal policy rule coefficients. Additionally, policymaking shows important persistence, with (cid:12)scal policy being somewhat more persistent than monetary policy. The results also show that there is a degree of policy interdependence, given by a positive correlation coefficient between the latent factors that drive the evolution of policy rule coefficients. More speci(cid:12)cally, (cid:12)scal policy that focuses on keeping debt under control tends to accompany monetary policy that focuses on keeping in(cid:13)ation under control, and monetary policy that keeps interest rates with a low reaction to in(cid:13)ation tends to accompany (cid:12)scal policy that departs from debt stabilization. Policyexperimentsshowthatwhenagentsexpecttheeconomytoevolvebetweenaregime of monetary price determination and a regime of (cid:12)scal price determination, contractionary monetary policy lowers in(cid:13)ation in the short run and increases it in the long run. The experiments reveal that (lump-sum) taxes have effects on output and in(cid:13)ation, as the literature on the (cid:12)scal theory of the price level suggests, but effects are attenuated relative to a pure (cid:12)scal regime. 29

Appendix A Model Setup The representative household solves the following problem: ( ) ∑1 (C =A )1(cid:0)(cid:27) ( ) H1+φ max E (cid:12)t t t +(cid:31) log Md=P (cid:0)(cid:31) t fCt;Mt=Pt;Bt g1 t=0 0 t=0 1(cid:0)(cid:27) M t t H 1+φ subject to Md B T W D Md B C + t + t + t (cid:20) H t + t + t(cid:0)1 +R t(cid:0)1 for t (cid:21) 0; t t t(cid:0)1 P P P P P P P t t t t t t t M +R B (cid:0)1 (cid:0)1 (cid:0)1 given; P (cid:0)1 M +B t t lim MRS = 0; 0;t t!1 P t where MRS denotes the marginal rate of substitution between period 0 and period t. The 0;t necessary (cid:12)rst order conditions are: ( ) 1 C (cid:0)(cid:27) C : t (cid:0)(cid:21) = 0 (45) t t A A t t W H : (cid:0)(cid:31) Hφ +(cid:21) t = 0 (46) t H t t P ( ) t Md Md (cid:0)1 P t : (cid:31) t (cid:0)(cid:21) +(cid:12)E (cid:21) t = 0 (47) M t t t+1 P P P t t t+1 (cid:21) (cid:21) B : (cid:0) t +(cid:12)R E t+1 = 0; (48) t t t P P t t+1 Md B T W D Md B (cid:21) : C + t + t + t (cid:0)H t (cid:0) t (cid:0) t(cid:0)1 (cid:0)R t(cid:0)1 = 0 (49) t t t t(cid:0)1 P P P P P P P t t t t t t t where (cid:21) is the Lagrange multiplier associated to the budget constraint at time t. t From (45) and (48), ( ) C =A (cid:27) A P 1 = (cid:12)R E t t t t : (50) t t C =A A P t+1 t+1 t+1 t+1 From (45) and (46) ( ) C (cid:27) W (cid:31) HφA t = t : (51) H t t A P t t 30

From (45), (47) and (48), ( ) ( ) M C (cid:27) R t t t = (cid:31) A : (52) P M t A R (cid:0)1 t t t Pro(cid:12)ts of intermediate (cid:12)rm j are given by ( ) D (j) P (j) W ϕ P (j) 2 t = t Y (j)(cid:0) t L (j)(cid:0) t (cid:0)1 Y : (53) t t t P P P 2 (cid:5)P (j) t t t t(cid:0)1 Substituting (8) and (9) in (53) yields [ ] ( ) ( ) ( ) D (j) P (j) 1(cid:0)(cid:18)t P (j) (cid:0)(cid:18)t ϕ P (j) 2 t = t (cid:0) t (cid:0) t (cid:0)1 Y : t t P P P 2 (cid:5)P (j) t t t t(cid:0)1 Then, intermediate (cid:12)rm j chooses P (j) to maximize t [ ] ( ) ( ) ( ) ∑1 P (j) 1(cid:0)(cid:18)t P (j) (cid:0)(cid:18)t ϕ P (j) 2 E MRS t (cid:0) t (cid:0) t (cid:0)1 Y : t t;t+k t t P P 2 (cid:5)P (j) t t t(cid:0)1 k=0 The (cid:12)rst order condition to this maximization problem is [ ] ( ) ( ) ( ) P (j) (cid:0)(cid:18)t 1 P (j) (cid:0)(cid:18)t (cid:0)1 P (j) 1 0 = (cid:21) Y (1(cid:0)(cid:18) ) t +(cid:18) t t (cid:0)ϕ t (cid:0)1 + t t t t P P P P (cid:5)P (cid:5)P (j) t t t t t(cid:0)1 t(cid:0)1 ( ) P (j) P (j) +(cid:12)(cid:21) Y ϕ t+1 (cid:0)1 t+1 : (54) t+1 t+1 (cid:5)P (cid:5)P (j)2 t t In a symmetric equilibrium, P (j) = P , L (j) = L and Y (j) = Y , hence t t t t t t [ ] ( ) D (j) D ϕ P 2 t = t = 1(cid:0) (cid:0) t (cid:0)1 Y : (55) t t P P 2 (cid:5)P t t t(cid:0)1 In equilibrium, Md = Ms = M , B = 0 and H = L . Then, (11), (55) and (49) imply t t t t t t (12). In the symmetric equilibrium, substituting (45) into (54) yields (13). Finally, the symmetric equilibrium yields (15), (52) is (16), and (50) is (14). Before proceeding to the log-linearization of the model, it is convenient to write (11) in terms of nominal output. The resulting expression is 1 1 1 1 1 1 1 b = 1(cid:0) (cid:0)(cid:28) (cid:0) + +R b ; (56) t t t(cid:0)1 t(cid:0)1 g v v (cid:5) ∆Y (cid:5) ∆Y t t t(cid:0)1 t t t t where ∆Y = Y =Y . t t t(cid:0)1 31

To obtain (17), start with (15) and recall the de(cid:12)nition of = W =P A to get T t t t ( ) C (cid:27) = (cid:31) LφA t : t H t t A t In the symmetric equilibrium, from (9), Y = A L , and without price rigidities, = (cid:18)t (cid:0)1 t t t t (cid:18)t and C = Y =g , hence t t t ( ) (cid:18) (cid:0)1 Y(cid:3) φ+(cid:27) t = (cid:31) t g (cid:0)(cid:27): (cid:18) H A t t t Once the variables have been de-trended by dividing them by A , the absence of shocks t yields (18) from (14), (21) from (16), (19) from combining (12) and (17), (20) from (19) and c = y =g , and (22) from (56) with ∆y = (cid:14). t t t B Matrices for Solving the Model The matrices in system (33)-(35) are given by [ ] [ ] 0 0 (cid:0)1=(cid:27) 0 0 0 0 1 1=(cid:27) G = ; J = ; 0 0 0 0 0 0 (cid:0)((cid:18)(cid:0)1)(φ+(cid:27))=ϕ 0 (cid:12) [ ] [ ] (cid:0)1 0 1(cid:0)(cid:26) 0 (cid:26) =(cid:27) 0 0 K = ; M = g (cid:23) ; ((cid:18)(cid:0)1)(φ+(cid:27))=ϕ (cid:0)1 0 0 0 0 0 2 3 (cid:0)1 0 1=(R(cid:0)1) 0 0 0 0 6 61=vb (cid:0)1 0 (cid:0)(cid:28)=b (cid:0)(1=bv(cid:5)(cid:14) +1=(cid:12)) 0 0 7 7 6 6 0 0 (cid:0)1 0 0 0 (cid:0)(1(cid:0)(cid:26) R )(cid:11)y(z t m) 7 7 6 7 A(z ) = 6 0 0 0 (cid:0)1 0 0 (cid:0)(1(cid:0)(cid:26) )(cid:13)y(zf)7; t 6 (cid:28) t 7 6 0 0 0 0 (cid:0)1 (cid:0)1 0 7 4 5 0 0 0 0 0 (cid:0)1 0 0 0 0 0 0 0 (cid:0)1 2 3 0 0 0 0 0 0 0 6 6 (cid:0)1=v(cid:5)(cid:14)b 1=(cid:12) 1=(cid:12) 0 0 0 0 7 7 6 7 6 0 0 (cid:26) R 0 0 0 07 B(zf) = 6 6 0 (1(cid:0)(cid:26) )(cid:13)b(zf) 0 (cid:26) 0 0 0 7 7; t 6 (cid:28) t (cid:28) 7 6 0 0 0 0 0 0 07 4 5 0 0 0 0 1 1 0 0 0 0 0 0 0 0 2 3 1(cid:0)(cid:27) 0 6 6 0 (cid:0)(1=bv(cid:5)(cid:14) +1=(cid:12)) 7 7 6 6(1(cid:0)(cid:26) R )(cid:11)y(z t m) (1(cid:0)(cid:26) R )(cid:11)(cid:25)(z t m) 7 7 6 7 C(z ) = 6(1(cid:0)(cid:26) )(cid:13)y(zf) 0 7; t 6 (cid:28) t 7 6 1 0 7 4 5 0 0 0 0 32

2 3 (cid:27) 0 0 0 0 6 7 6 1=bg 0 0 0 07 6 7 6 0 0 0 1 07 6 7 D = 6 0 0 0 0 17; 6 7 6 0 0 1 0 07 4 5 0 0 0 0 0 (cid:27)=(φ+(cid:27))+1=(φ+(cid:27)) 0 0 0 0 2 3 (cid:26) 0 0 0 0 g 6 7 60 (cid:26) (cid:18) 0 0 07 6 7 N = 60 0 (cid:26) 0 07 (cid:23) 4 5 0 0 0 0 0 0 0 0 0 0 C Obtaining the Coefficients of the Logistic Functions Each of the elements in the matrices P(z );Q(z );R(z );S(z ) have the bivariate logistic t t t t functional form (38), which is reproduced here for convenience and is illustrated in Figure 10: 1 1 F(z ) = F +F 1+exp((cid:0)F2m(z t m(cid:0)F3m))1+exp((cid:0)F 2f (z t f(cid:0)F 3f )) : t 0 1 1(cid:0)F exp((cid:0)F2m(z t m(cid:0)F3m)) exp((cid:0)F 2f (z t f(cid:0)F 3f )) 41+exp((cid:0)F2m(z t m(cid:0)F3m))1+exp((cid:0)F 2f (z t f(cid:0)F 3f )) Figure 10: Bivariate Logistic Function F(zm,zf) −6 −6 −4 −4 −2 −2 0 zf 0 2 2 zm 4 4 6 6 To avoid computational costs, we (cid:12)nd the solutions to the coefficients in R(z ) and S(z ) t t only, solving for in(cid:13)ation and output, leaving the full solution of the model to be accounted for the structural equations of the state variables (25)-(32). Given the structure of the vector of shocks, S(z ) is a matrix with 10 distinct elements. Given the structure of the vector of t state variables, R(z ) is a matrix with 10 distinct elements as well, although it has 14 entries. t 33

Therefore, in total there are 20(cid:2)6 = 120 coefficients of the logistic functions to be found to obtain a solution. C.1 Finding F and F 0 1 Notice that the time-varying policy rule coefficients have the following limiting combinations (bounds): zm ! +1 zm ! (cid:0)1 t t (cid:11)(cid:25)(zm) = (cid:11)(cid:25) +(cid:11)(cid:25) (cid:11)y(zm) = (cid:11)y +(cid:11)y zf ! +1 t 0 1 t 0 1 t (cid:13)b(zf) = (cid:13)b +(cid:13)b (cid:13)y(zf) = (cid:13)y +(cid:13)y t 0 1 t 0 1 (cid:11)(cid:25)(zm) = (cid:11)(cid:25) (cid:11)y(zm) = (cid:11)y zf ! (cid:0)1 t 0 t 0 t (cid:13)b(zf) = (cid:13)b (cid:13)y(zf) = (cid:13)y t 0 t 0 Also, notice that F(z ) has the following limiting expressions: t zm ! +1 zm ! (cid:0)1 t t zf ! +1 F(z ) = F +F t t 0 1 zf ! (cid:0)1 F(z ) = F t t 0 To obtain F and F , we solve the constant-coefficient versions of the model two times 0 1 using Uhlig (1998), at the two different limiting combinations of the latent factors zm and t zf. We solve for the two bounds making sure that the solutions deliver determinacy of the t equilibria at the bounds. C.2 Finding F ;F ;F ;F 2m 2f 3m 3f Substitute (36)-(37) in (33)-(35) to obtain [ ] A(z t )P(z t )+C(z t )R(z t )+B(z t f) k t(cid:0)1 +[A(z t )Q(z t )+C(z t )S(z t )+D]u t = 0; {[ ] } {[ ] } JR(cid:22)(z t )+G P(z t )+KR(z t ) k t(cid:0)1 + JR(cid:22)(z t )+G Q(z t )+JS(cid:22)(z t )N+KS(z t )+M u t = 0; where R (cid:22) (z ) (cid:17) E R(z ) and S (cid:22) (z ) (cid:17) E S(z ). By the undetermined coefficients method, t t t+1 t t t+1 we have A(z )P(z )+C(z )R(z )+B(zf) = 0 t t t t t A(z )Q(z )+C(z )S(z )+D = 0 [ t t ] t t (cid:22) JR(z )+G P(z )+KR(z ) = 0 [ ] t t t (cid:22) (cid:22) JR(z )+G Q(z )+JS(z )N+KS(z )+M = 0: t t t t Then, solving for R(z ) and S(z ), we have t t [K+[G+JR(cid:22)(z )][(cid:0)A(z )] (cid:0)1C(z )]R(z ) = (cid:0)[G+JR(cid:22)(z )][(cid:0)A(z )] (cid:0)1B(zf) t t t t t t t [K+[G+JR(cid:22)(z )][(cid:0)A(z )] (cid:0)1C(z )]S(z ) = (cid:0)JS(cid:22)(z )N(cid:0)M(cid:0)[G+JR(cid:22)(z )][(cid:0)A(z )] (cid:0)1D: t t t t t t t 34

Let T(z ) (cid:17) K+[G+JR (cid:22) (z )][(cid:0)A(z )] (cid:0)1C(z ) (57) t t t t U(z ) (cid:17) (cid:0)[G+JR (cid:22) (z )][(cid:0)A(z )] (cid:0)1B(zf) (58) t t t t V(z ) (cid:17) (cid:0)JS (cid:22) (z )N(cid:0)M(cid:0)[G+JR (cid:22) (z )][(cid:0)A(z )] (cid:0)1D: (59) t t t t Then, using the relevant elements of R(z ) and S(z ), we can write t t [ ][ ] [ ] T(z ) 0 R(z ) U(z ) t t = t : (60) 0 T(z ) S(z ) V(z ) t t t Given F , F and F , F(z ) takes the following expressions that include F , F , F 0 1 4 t 2m 2f 3m and F : 3f F (cid:15) lim F(0;zf) = F + 1 zf!1 t 0 1+exp(F 2m F 3m ) t F (cid:15) lim F(zm;0) = F + 1 z t m!1 t 0 1+exp(F 2f F 3f ) @ (cid:15) F(zm;0)j = g (F ;F ;F ;F ;F ;F ) @zm t z t m=0 m 1 2m 3m 2f 3f 4 t @ (cid:15) F(zm;0)j = g (F ;F ;F ;F ;F ;F ), @zf t z t f=0 f 1 2m 3m 2f 3f 4 t where g ((cid:1)) and g ((cid:1)) are nonlinear functions of their respective arguments. m f Then, to (cid:12)nd F ;F ;F and F we solve the following system of equations for the 2m 2f 3m 3f relevant elements of R(z ) and S(z ): t t 2 32 3 2 3 lim T(0;zf) 0 lim R(0;zf) lim U(0;zf) 6 4z t f!1 t 7 5 6 4z t f!1 t 7 5 = 6 4z t f!1 t 7 5; 0 lim T(0;zf) lim S(0;zf) lim V(0;zf) t t t zf!1 zf!1 zf!1 2 t 32t 3 t2 3 lim T(zm;0) 0 lim R(zm;0) lim U(zm;0) 4zm!1 t 54zm!1 t 5 4zm!1 t 5 t t = t ; 0 lim T(zm;0) lim S(zm;0) lim V(zm;0) zm!1 t zm!1 t zm!1 t t t t 2 3 @ T(zm;0)j 0 [ ] 6 4 @z t m t z t m=0 7 5 R(0;0) + @ S(0;0) 0 T(zm;0)j @zm t z t m=0 2 t 3 [ ] @ R(zm;0)j + T(0;0) 0 6 4 @z t m t z t m=07 5 = 0 T(0;0) @ S(zm;0)j @zm t z t m=0 t 35

2 3 @ U(zm;0)j = 6 4 @z t m t z t m=07 5; @ V(zm;0)j @zm t z t m=0 2 t (cid:12) 3 @ (cid:12) T(0;zf)(cid:12) 0 [ ] 6 6 4 @z t f 0 t z t f=0 @ T(0;zf) (cid:12) (cid:12) (cid:12) 7 7 5 R S( ( 0 0 ; ; 0 0 ) ) + @zf t zf=0 2 t (cid:12) t 3 @ (cid:12) [ ] R(0;zf)(cid:12) + T(0 0 ;0) T(0 0 ;0) 6 6 4 @z @ t f S(0;zf t ) (cid:12) (cid:12) (cid:12) z t f=0 7 7 5 = @zf t zf=0 2 (cid:12) 3 t t @ (cid:12) U(0;zf)(cid:12) = 6 6 4 @ @ z t f t (cid:12) (cid:12) z t f=0 7 7 5; V(0;zf)(cid:12) ; @zf t zf=0 t t where 2 3 (cid:0)1 0 1=(R(cid:0)1) 0 0 0 0 6 61=vb (cid:0)1 0 (cid:0)(cid:28)=b (cid:0)(1=v(cid:5)(cid:14)+1=(cid:12)) 0 ( 0 ) 7 7 6 6 6 0 0 (cid:0)1 0 0 0 (cid:0)(1(cid:0)(cid:26) R ) (cid:11) 0 y+ 1+exp (cid:11) ( y 1 (cid:11)y(cid:11)y) 7 7 7 z t f l ! im +1 A(0;z t f)=6 6 6 6 0 0 0 0 0 0 (cid:0) 0 1 (cid:0) 0 1 (cid:0) 0 1 (cid:0)(1(cid:0)(cid:26) (cid:28) ) 0 ((cid:13) 0 y+(cid:13) 1 y) 2 3 7 7 7 7 4 0 0 0 0 0 (cid:0)1 0 5 0 0 0 0 0 0 (cid:0)1 2 3 (cid:0)1 0 1=(R(cid:0)1) 0 0 0 0 6 61=vb (cid:0)1 0 (cid:0)(cid:28)=b (cid:0)(1=v(cid:5)(cid:14)+1=(cid:12)) 0 0 7 7 z t m l ! im +1 A(z t m;0)= 6 6 6 6 6 6 6 0 0 0 0 0 0 (cid:0) 0 0 1 (cid:0) 0 0 1 (cid:0) 0 0 1 (cid:0) 0 0 1 (cid:0)(1(cid:0) (cid:0)( (cid:26) 1 (cid:28) ) (cid:0) ((cid:26) (cid:13) R 0 y ) 0 + ((cid:11) 1 0 y + + exp (cid:11) (cid:13) ( 1 y 1 y (cid:13) ) 2 y(cid:13) 3 y) ) 7 7 7 7 7 7 7 4 0 0 0 0 0 (cid:0)1 0 5 0 0 0 0 0 0 (cid:0)1 2 3 1(cid:0)(cid:27) 0 6 6 ( 0 ) (cid:0)((1=v(cid:5)(cid:14)+1=(cid:12)) ) 7 7 6 6 6 (1(cid:0)(cid:26) R ) (cid:11) 0 y+ 1+exp (cid:11) ( y 1 (cid:11)y(cid:11)y) (1(cid:0)(cid:26) R ) (cid:11) 0 (cid:25)+ 1+exp (cid:11) ( (cid:25) 1 (cid:11)(cid:25)(cid:11)(cid:25)) 7 7 7 zf l ! im +1 C(0;z t f)=6 6 6 (1(cid:0)(cid:26) (cid:28) )((cid:13) 0 y+(cid:13) 1 y) 2 3 0 2 3 7 7 7 ; t 6 1 0 7 4 5 0 0 0 0 2 3 1(cid:0)(cid:27) 0 6 6 0 (cid:0)(1=v(cid:5)(cid:14)+1=(cid:12)) 7 7 lim C(z t m;0)= 6 6 6 6 6 (1(cid:0) ( (cid:26) 1 (cid:28) (cid:0) ) ((cid:26) (cid:13) R 0 y ) + ((cid:11) 1 0 y + + exp (cid:11) (cid:13) ( 1 y 1 y (cid:13) ) y(cid:13)y) ) (1(cid:0)(cid:26) R )( 0 (cid:11) 0 (cid:25)+(cid:11) 1 (cid:25)) 7 7 7 7 7 ; zf!+1 6 2 3 7 t 6 1 0 7 4 5 0 0 0 0 36

2 3 0 0 0 0 0 0 0 6 6 (cid:0)1=v(cid:5)(cid:14) 1=(cid:12) 1=(cid:12) 0 0 0 0 7 7 6 7 6 0 ( 0 ) (cid:26) R 0 0 0 07 6 7 B(0)=6 6 0 (1(cid:0)(cid:26) (cid:28) ) (cid:13) 0 b+ 1+exp (cid:13) ( 1 b (cid:13)b(cid:13)b) 0 (cid:26) (cid:28) 0 0 07 7 ; 6 2 3 7 6 0 0 0 0 0 0 07 4 5 0 0 0 0 1 1 0 0 0 0 0 0 0 0 2 3 0 0 0 0 0 0 0 6 6 (cid:0)1=v(cid:5)(cid:14) 1=(cid:12) 1=(cid:12) 0 0 0 0 7 7 6 7 6 0 0( ) (cid:26) R 0 0 0 07 z t f l ! im +1 B(z t f)= 6 6 6 6 0 0 (1(cid:0)(cid:26) (cid:28) ) 0 (cid:13) 0 b+(cid:13) 1 b 0 0 (cid:26) 0 (cid:28) 0 0 0 0 0 0 7 7 7 7 : 4 5 0 0 0 0 1 1 0 0 0 0 0 0 0 0 2 3 0 0 0 0 0 0 0 60 0 0 0 0 0 0 7 6 7 @z @ t m A(z t m;0)j z t m=0 = 6 6 6 6 6 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (cid:0)(1(cid:0)(cid:26) R )(cid:11) (1 0 0 y 1 + (cid:11) e y 2 x e p x ( p (cid:11) ( y 2 (cid:11) (cid:11) y 2 y 3 (cid:11) ) y 3 )2 )7 7 7 7 7 7 4 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 3 0 0 6 0 0 7 6 7 @z @ t m C(z t m;0)j z t m=0 = 6 6 6 6 6 6 (1(cid:0)(cid:26) R )(cid:11) (1 y 1 + 0 0 (cid:11) e y 2 x e p x ( p (cid:11) ( y 2 (cid:11) (cid:11) y 2 y 3 (cid:11) ) y 3 )2 ) (1(cid:0)(cid:26) R )(cid:11) (1 (cid:25) 1 + 0 0 (cid:11) e (cid:25) 2 x e p x ((cid:11) p( (cid:25) 2 (cid:11) (cid:11) (cid:25) 2 (cid:25) 3 (cid:11) ) (cid:25) 3 )2 )7 7 7 7 7 7 4 5 0 0 0 0 2 3 0 0 0 0 0 0 0 60 0 0 0 0 0 7 6 7 @ @ zf B(z t f) (cid:12) (cid:12) (cid:12) zf=0 = 6 6 6 6 6 0 0 (1(cid:0)(cid:26) (cid:28) ) ( (cid:13) 1 1 + 0 (cid:13) e 2 x e p x ( p (cid:13) ( 2 (cid:13) (cid:13) 2 3 (cid:13) ) 3 ) ) 2 0 0 0 0 0 0 0 0 0 0 7 7 7 7 7 : t t 60 0 0 0 0 0 07 4 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C.3 Finding F 4 Given F ;F ;F ;F ;F and F , we have the following expression that includes F : 0 1 2m 2f 3m 3f 4 1 1 F(0) = F +F 1+exp(F2mF3m)1+exp(F 2f F 3f ) 0 1 1(cid:0)F4 exp(F2mF3m) exp(F 2f F 3f ) 1+exp(F2mF3m)1+exp(F 2f F 3f ) Then, to (cid:12)nd F we solve the following system of equations: 4 [ ][ ] [ ] T(0) 0 R(0) U(0) = : 0 T(0) S(0) V(0) 37

(cid:22) (cid:22) D Computation of R(z ) and S(z ) t t To obtain E R(z ) and E S(z ) we perform a second order Taylor expansion to t t+1 t t+1 F(z ) = F(zm ;zf ) t+1 t+1 t+1 = F((cid:26) zm +(cid:24)m ;(cid:26) zf +(cid:24)f ) zm t t+1 zf t t+1 around (cid:24)m = (cid:24)f = 0 as follows: t+1 t+1 F(zm ;zf ) (cid:25) F((cid:26) zm;(cid:26) zf)+F ((cid:26) zm;(cid:26) zf)(cid:24)m +F ((cid:26) zm;(cid:26) zf)(cid:24)f + t+1 t+1 zm t zf t m zm t zf t t+1 f zm t zf t t+1 +0:5F ((cid:26) zm;(cid:26) zf)((cid:24)m )2 +0:5F ((cid:26) zm;(cid:26) zf)((cid:24)f )2+ mm zm t zf t t+1 ff zm t zf t t+1 +F ((cid:26) zm;(cid:26) zf)(cid:24)m (cid:24)f ; mf zm t zf t t+1 t+1 where F ((cid:1)) and F ((cid:1)) denote the partial derivative of F((cid:1)) with respect to (cid:24)i and the pari ij tial derivative of F((cid:1)) with respect to (cid:24)i (cid:12)rst and then with respect to (cid:24)j, for j = m;f, respectively. Then, taking expectation conditional to zm and zf, we get t t ( ) E F(zm ;zf )jzm;zf (cid:25) F((cid:26) zm;(cid:26) zf)+ t+1 t+1 t t zm t zf t +0:5F ((cid:26) zm;(cid:26) zf)+0:5F ((cid:26) zm;(cid:26) zf)+ mm zm t zf t ff zm t zf t +F ((cid:26) zm;(cid:26) zf)(cid:20): mf zm t zf t Notice that the expectation of the coefficient matrices involves (cid:20), the correlation coefficient between the latent factors that drive the evolution of the monetary and (cid:12)scal policy rule coefficients. When taking expectations, the agents of the model incorporate the degree of interdependence in policymaking. E Verifying the Guessed Functional Form Having obtained the coefficients of the logistic functions that characterize the solution, it is necessary to verify that the guessed functional forms for R(z ) and S(z ) are indeed t t logistic. Recall the system of equations in (60): [ ][ ] [ ] T(z ) 0 R(z ) U(z ) t t = t : 0 T(z ) S(z ) V(z ) t t t Notice that if both zm and zf were white noise processes, then both R (cid:22) (z ) and S (cid:22) (z ) t t t t would be constant matrices. In that case, if the coefficients on the output gap of both policy rules were constant, T((cid:1)) in (57) would be a function of zm only through C(zm), U((cid:1)) in (58) t t would be a function of zf only through B(zf), and V((cid:1)) would be constant. This implies t t that some of the elements of R((cid:1)) are bivariate logistic functions, and that all the elements of S((cid:1)) are univariate logistic functions that depend on zm. In particular, only the entries t in the second column of R((cid:1)), those that relate the behavior of in(cid:13)ation and the output 38

gap with the evolution of lagged debt, depend on both zm and zf; all the other elements t t are univariate logistic functions dependent on zm. Hence, in the absence of persistence in t the latent factors, it is easy to show that the solution matrices follow indeed bivariate (or univariate) logistic functions. It can be shown that the same argument holds if the policy rule coefficients on the output gap are time varying. In any case, however, and in particular when the latent factors are persistent, the solution has to be such that in (60) we do not ‘divide by zero’. The issue has to do with the fact that since the transition between states is continuous, as opposed to the Markov-switching setup of policy rule coefficients, for some values of z there may exist a discontinuity in the t solution. To see this, consider the simple Fisherian model: R = E (cid:25) +u t t t+1 t R = (cid:11)(z )(cid:25) t t t u = (cid:26) u +" ; " (cid:24) iid N(0;(cid:27)); j(cid:26) j < 1 t+1 u t t+1 t u z = (cid:26) z +(cid:24) ; (cid:24) (cid:24) iid N(0;1); 0 (cid:20) (cid:26) (cid:20) 1 t+1 z t t+1 t z z ?u 8s;t t s (cid:11) 1 (cid:11)(z ) = (cid:11) + ; t 0 1+exp((cid:0)(cid:11) z ) 2 t where the notation is the same as in the new Keynesian model, since this is a particular version of the model presented in the paper. We guess that the fundamental solution is given by (cid:25) = a(z )u ; t t t where a 1 a(z ) = a + : t 0 1+exp((cid:0)a (z (cid:0)a )) 2 t 3 Solving for a(z ), we have t (cid:26) E a(z )+1 u t t+1 = a(z ): t (cid:11)(z ) t Performing a (cid:12)rst-order Taylor expansion to a(z ) around (cid:24) = 0, we get t+1 t+1 a 1 a(cid:22)(z ) = a((cid:26) z ) = a + t z t 0 1+exp((cid:0)a ((cid:26) z (cid:0)a )) 2 z t 3 If (cid:26) = 1, it is easy to see that z 1 a(z ) = : t (cid:11)(z )(cid:0)(cid:26) t u Hence, for the solution to be logistic |and continuous, we need (cid:11)(z ) > (cid:26) 8z , which is t u t the same as (cid:11) > (cid:26) , given the monotonicity of (cid:11)(z ). 0 u t To derive the analogous condition in the more general case of the model presented in the 39

paper, rewrite (60) for S(z ) ignoring the last two terms of (59), as follows: t T(z )S(z ) = (cid:0)JS (cid:22) (z )N; t t t which can be rewritten as ( ) ( ) (I (cid:10)T(z ))vec(S(z )) = (cid:0) N ⊤ (cid:10)J vec S (cid:22) (z ) : 5 t t t Hence, a sufficient condition for the continuity of the solution i(s that ea)ch of the elements of (cid:0)(I (cid:10)T((cid:0)1)) is greater than the corresponding element of N⊤ (cid:10)J . 5 F Endogeneity Setup Let GSP be the demeaned government spending to output ratio in period t, M2G the t t annual rate of growth of M2 in period t, and CMP commodity price in(cid:13)ation. In order to t account for the existence of endogeneity, the observation equations (39)-(40) of the statespace model have to be modi(cid:12)ed by introducing a system of simultaneous equations. To that end, let y = INT ; 1;t t y = TAX ; 2;t t x = [INF ;GAP ]; 1;t t t x = [DBT ;GAP ]; 2;t t(cid:0)1 t (cid:11)(zm) = [(cid:11)(cid:25)(zm);(cid:11)y(zm)] ′ ; t t t (cid:13)(zf) = [(cid:13)b(zf);(cid:13)y(zf)] ′ ; t t t w = [fINF g4 ;fGAP g4 ;fGSP g4 ;fM2G g4 ;fCMP g4 ]; 1;t t(cid:0)s s=1 t(cid:0)s s=1 t(cid:0)s s=1 t(cid:0)s s=1 t(cid:0)s s=1 w = [DBT ;fINF g4 ;fGAP g4 ;fGSP g4 ;fM2G g4 ;fCMP g4 ]: 2;t t(cid:0)1 t(cid:0)s s=1 t(cid:0)s s=1 t(cid:0)s s=1 t(cid:0)s s=1 t(cid:0)s s=1 Hence, conditional on z , the state-space model (39)-(44) can be divided into two models: t one for the interest rate equation, and another for the tax rate equation. The observation equations of each of the models are: y = (cid:26) y +(1(cid:0)(cid:26) )x (cid:11)(zm)+(cid:29)R (61) 1;t R 1;t(cid:0)1 R 1;t t t x = w (cid:5) +v ; (62) 1;t 1;t 1 1;t and y = (cid:26) y +(1(cid:0)(cid:26) )x (cid:13)(zf)+(cid:29)(cid:28) (63) 2;t (cid:28) 2;t(cid:0)1 (cid:28) 2;t t t x = w (cid:5) +v : (64) 2;t 2;t 2 2;t Here, (cid:5) and (cid:5) are conformable parameter matrices, and v (cid:24) iid N(0;(cid:9) ) for i = 1;2. 1 2 i;t i We introduce endogeneity in (61)-(62) and (63)-(64) by specifying (cid:29)R(cid:3) = v (cid:14) +eR; t 1;t 1 t 40

(cid:29)(cid:28)(cid:3) = v (cid:14) +e(cid:28); t 2;t 2 t where (cid:29)j = (cid:27) (cid:29)j(cid:3) for j = R;(cid:28), and t j;t t eRjy ;v (cid:24) iid N(0;1(cid:0)(cid:14) ′ (cid:9) (cid:14) ); t 1;t(cid:0)1 1;t 1 1 1 e(cid:28)jy ;v (cid:24) iid N(0;1(cid:0)(cid:14) ′ (cid:9) (cid:14) ): t 2;t(cid:0)1 2;t 2 2 2 Let y = [y ;y ]′ and Y = fy gt . Appendix G shows how to obtain the conditional t 1;t 2;t t s s=1 likelihood function of Y . T G The Likelihood Function of the Model with Endogeneity, Time-Varying Coefficients and Stochastic Volatility Notice that the joint density function of y and v , conditional on y , for i = 1;2, and i;t i;t i;t(cid:0)1 the latent factors and stochastic volatility (these last two omitted from the density functions below) can be written as p(y ;v jy ;(cid:2) ;(cid:2) ) = p (y jy ;v ;(cid:2) ;(cid:2) )p (v j(cid:2) ) i;t i;t i;t(cid:0)1 yi vi y i;t i;t(cid:0)1 i;t yi vi v i;t vi = p (y jy ;x ;v ;(cid:2) ;(cid:2) )p (x jw ;(cid:2) ); y i;t i;t(cid:0)1 i;t i;t yi xi x i;t i;t xi where (cid:2) for i = 1;2 is the set of parameters f(cid:14) ;(cid:5) ;(cid:9) g. Adding the conditionality on vi i i i latent factors and stochastic volatility, ( ( ) ) y 1;t jy 1;t(cid:0)1 ;x 1;t ;v 1;t ;z t m;(cid:27) R;t ;(cid:2) y1 ;(cid:2) x1 (cid:24)N ( (cid:26) R y 1;t(cid:0)1 +(1(cid:0)(cid:26) R )x 1;t (cid:11)(z t m)+(cid:27) R;t v 1;t (cid:14) 1 +eR t ;1(cid:0)(cid:14) 1 ′ (cid:9) 1) (cid:14) 1 y 2;t jy 2;t(cid:0)1 ;x 2;t ;v 2;t ;z t f;(cid:27) (cid:28);t ;(cid:2) y2 ;(cid:2) x2 (cid:24)N (cid:26) (cid:28) y 2;t(cid:0)1 +(1(cid:0)(cid:26) (cid:28) )x 2;t (cid:13)(z t f)+(cid:27) (cid:28);t (v 2;t (cid:14) 2 +e(cid:28) t );1(cid:0)(cid:14) 2 ′ (cid:9) 2 (cid:14) 2 x jw ;(cid:2) (cid:24)N(w (cid:5) ;(cid:9) ); for i=1;2: i;t i;t xi i;t i i Let Y = fy gt , for i = 1;2 and let X and V be de(cid:12)ned in a similar fashion. Let i;t i;s s=1 i;t i;t Z = fzkgt for k = m;f and let H = f(cid:27) gt for j = R;(cid:28) . Then, the conditional k;t s s=1 j;t j;s s=1 log-likelihoodfunctionsofY givenY , X , V , Z , H , andofY givenY , 1;T 1;T(cid:0)1 1;t 1;t m;t R;t 2;T 2;T(cid:0)1 X , V , Z , H are, respectively, 2;t 2;t f;t (cid:28);t ( ) ∑T ( ) ^ ^ L (cid:2) ((cid:2) ) = l (cid:2) ((cid:2) ) ; 1;T y1 x1 1;t y1 x1 t=1 ( ) ∑T ( ) ^ ^ L (cid:2) ((cid:2) ) = l (cid:2) ((cid:2) ) ; 2;T y2 x2 2;t y2 x2 t=1 where ∑T (cid:2) ^ = max log(p (x jw ;(cid:2) )) xi (cid:2)xi t=1 x i;t i;t xi 41

is the maximum likelihood estimator of (cid:2) for i = 1;2, and xi ( ) ( ( )) l (cid:2) ((cid:2) ^ ) = log p y jy ;x ;v ;zm;(cid:27) ;(cid:2) ;(cid:2) ^ ; 1;t ( y1 x1 ) ( y ( 1;t 1;t(cid:0)1 1;t 1;t t R;t y1 x )1 ) l (cid:2) ((cid:2) ^ ) = log p y jy ;x ;v ;zf;(cid:27) ;(cid:2) ;(cid:2) ^ : 2;t y2 x2 y 2;t 2;t(cid:0)1 2;t 2;t t (cid:28);t y2 x2 This is the two-stage conditional maximum likelihood estimation procedure suggested by Vuong (1984). Let X = [2 X , and let V be de(cid:12)ned in a similar way. Let Z = fz gt t i=1 i;t t t s s=1 and H = fh gt . The log-likelihood function of Y given Y ;X ;V ;Z ;H is t s s=1 T T(cid:0)1 T T T T ( ) ∑T ( ) ^ ^ L (cid:2) ((cid:2) ) = l (cid:2) ((cid:2) ) ; T y x t y x t=1 where ( ) ( ) ( ) ^ ^ ^ l (cid:2) ((cid:2) ) = l (cid:2) ((cid:2) ) +l (cid:2) ((cid:2) ) ; t y x 1;t y1 x1 2;t y2 x2 and (cid:2) ^ = (cid:2) ^ [(cid:2) ^ , (cid:2) = (cid:2) [(cid:2) . x x1 x2 y y1 y2 H Bayesian Estimation The method presented here extends the estimation method implemented in Geweke and Tanizaki(2001)byallowingendogeneity. Letp (y jX ;V ;Z ;H ;(cid:2) )denotetheconditional y t t t t t y density of y given X , V , Z , H and (cid:2) . Let p (z jz ;(cid:2) ) denote the conditional density t t t t t y z t t(cid:0)1 z of z given z and (cid:2) . Let p (h jh ;(cid:2) ) denote the conditional density of h given h t t(cid:0)1 z h t t(cid:0)1 h t t(cid:0)1 and (cid:2) . De(cid:12)ne Z(cid:3) = fz gT and H(cid:3) = fh gT . Under this setup, the joint density h t+1 s s=t+1 t+1 s s=t+1 of Z , H and Y given X , V , (cid:2) , (cid:2) and (cid:2) is given by T T T T t z h y p(ZT;HT;YT jXT;VT;(cid:2)z;(cid:2) h ;(cid:2)y)=pz(ZT jXT;VT;(cid:2)z)p h (HT jXT;VT;ZT;(cid:2) h )py(YT jXT;VT;ZT;HT;(cid:2)y) =pz(ZT j(cid:2)z)p h (HT j(cid:2) h )py(YT jXT;VT;ZT;HT;(cid:2)y); where the last equality follows from the Markov property of fz gt and fh gt . Then, if s s=0 s s=0 z and h are assumed to be stochastic, 0 0 ∏T p (Z j(cid:2) ) = p (z j(cid:2) ) p (z jz ;(cid:2) ); z T z z 0 z z t t(cid:0)1 z t=1 ∏T p (H j(cid:2) ) = p (h j(cid:2) ) p (h jh ;(cid:2) ); h T h h 0 h h t t(cid:0)1 h t=1 ∏T p (Y jX ;V ;Z ;H ;(cid:2) ) = p (y jX ;V ;Z ;H ;(cid:2) ); y T T T T T y y t t t t t y t=1 p(z jZ ;Z (cid:3) ;Y ;X ;V ;H ;(cid:2) ;(cid:2) ;(cid:2) ) / (65) t t(cid:0)1 t+1 T T T T y z h 42

{ p (y jX ;V ;Z ;H ;(cid:2) )p (z jz ;(cid:2) )p (z jz ;(cid:2) ); if t (cid:20) T (cid:0)1 y t t t t t y z t t(cid:0)1 z z t+1 t z p (y jX ;V ;Z ;H ;(cid:2) )p (z jz ;(cid:2) ); if t = T; y t t t t t y z t t(cid:0)1 z p(h jH ;H (cid:3) ;Y ;X ;V ;Z ;(cid:2) ;(cid:2) ;(cid:2) ) / (66) { t t(cid:0)1 t+1 T T T T y z h p (y jX ;V ;Z ;H ;(cid:2) )p (h jh ;(cid:2) )p (h jh ;(cid:2) ); if t (cid:20) T (cid:0)1 y t t t t t y h t t(cid:0)1 h h t+1 t h p (y jX ;V ;Z ;H ;(cid:2) )p (h jh ;(cid:2) ); if t = T; y t t t t t y h t t(cid:0)1 h p((cid:2) jY ;X ;V ;Z ;H ;(cid:2) ;(cid:2) ) / p (Y jX ;Z ;H ;(cid:2) )p ((cid:2) ); (67) y T T T T T z h y T T T T y (cid:2)y y p((cid:2) jY ;X ;V ;Z ;(cid:2) ;(cid:2) ) / p (Z j(cid:2) )p ((cid:2) ); (68) z T T T T y h z T z (cid:2)z z p((cid:2) jY ;X ;V ;H ;(cid:2) ;(cid:2) ) / p (H j(cid:2) )p ((cid:2) ); (69) h T T T T y z h T h (cid:2) h h where p ((cid:2) ), p ((cid:2) ) and p ((cid:2) ) are the prior densities of (cid:2) , (cid:2) and (cid:2) , respectively. (cid:2)y y (cid:2)z z (cid:2) h h y z h From the posterior densities (67)-(69), the smoothing random draws are generated as follows: Step 0. Take appropriate initial values for (cid:2) , (cid:2) , fz gT , (cid:2) and fh gT , and (cid:12)x (cid:5) ^ , y z t t=0 h t t=0 i (cid:9) ^ for i=1,2.12 i Step 1. Generate a random draw of z from p(z jZ ;Z(cid:3) ;Y ;X ;V ;H ;(cid:2) ;(cid:2) ;(cid:2) ) t t t(cid:0)1 t+1 T T T T y z h for t = 1;2;:::;T. Draw z(i) using the normal proposal density t z(i) (cid:24) N(z(i(cid:0)1);cKz) t t t where Kz is the (cid:12)ltered variance-covariance matrix of the random coefficients of a t random-coefficient speci(cid:12)cation of the policy rules, and c is a proper scale coefficient. The algorithm accepts z(i) with probability r: t 8 ( ) 9 < p z(i)jZ(i(cid:0)1);Z (cid:3)(i(cid:0)1);Y ;X ;V ;H(i(cid:0)1);(cid:2)(i(cid:0)1);(cid:2)(i(cid:0)1);(cid:2)(i(cid:0)1) = ( t t(cid:0)1 t+1 T T T T y z h ) r = min ;1 : : p z(i(cid:0)1)jZ(i(cid:0)1);Z (cid:3)(i(cid:0)1);Y ;X ;V ;H(i(cid:0)1);(cid:2)(i(cid:0)1);(cid:2)(i(cid:0)1);(cid:2)(i(cid:0)1) ; t t(cid:0)1 t+1 T T T T y z h Step 2. Generate a random draw of (cid:2) from p((cid:2) jY ;X ;V ;Z ;H ;(cid:2) ;(cid:2) ). y y T T T T T z h (a) Partition (cid:2) = (cid:2) [(cid:2) [(cid:2) , where (cid:2) = f(cid:11)(cid:25);(cid:11)(cid:25);(cid:11)(cid:25);(cid:11)(cid:25), (cid:11)y;(cid:11)y;(cid:11)y;(cid:11)yg, y1 y1:1 y1:2 y1:3 y1:1 0 1 2 3 0 1 2 3 (cid:2) = (cid:26) and (cid:2) = (cid:14) . Partition (cid:2) = (cid:2) [ (cid:2) [ (cid:2) , where (cid:2) = y1:2 R y1:3 1 y2 y2:1 y2:2 y2:3 y2:1 f(cid:13)b;(cid:13)b;(cid:13)b;(cid:13)b;(cid:13)y;(cid:13)y;(cid:13)y;(cid:13)yg, (cid:2) = (cid:26) and (cid:2) = (cid:14) . 0 1 2 3 0 1 2 3 y2:2 (cid:28) y2:3 2 12The initial latent factors, fz gT , are smoothed estimates of a random coefficients model of the policy t t=0 ruleswithconstantvolatility. Theinitialvolatilityprocesses,fh gT ,aresmoothedestimatesofastochastic t t=0 volatility model of the policy rules with constant coefficients. The initial values of the parameters of the policyrules,(cid:2) ,areobtainedfromthemaximizationofthelikelihoodfunctiongiventheinitialprocessesfor y the latent factors and stochastic volatilities. The initial values for (cid:2) come from a least-squares regression z ofcurrentagainstlaggedinitiallatentfactors. Theinitialvaluesfor(cid:2) comefromaleast-squaresregression h of current against lagged initial log stochastic volatilities. (cid:5)^ and (cid:9)^ for i=1;2 are the least squares estimates of the parameters in equations (62) and (64), respeci i tively. 43

(b) Generate a random draw of (cid:2)(i) sequentially, as follows: y1 i. Generate a random draw of a transformation of (cid:2)(i) , (cid:2) ~(i) , using the normal y1:1 y1:1 proposal density (cid:2) ~(i) (cid:24) N((cid:2) ~(i(cid:0)1);cS ); y1:1 y1:1 1:1 where S is the variance-covariance matrix of the maximum likelihood esti- 1:1 ~ mator of (cid:2) given the initial latent factors and stochastic volatilities, and y1:1 c is a scale coefficient. The algorithm accepts (cid:2)i with probability r: y1:1 8 ( ) 9 r=min < p ( (cid:2)( y i 1 ) :1 [(cid:2)y (i 1 (cid:0) :2 1)[(cid:2)( y i 1 (cid:0) :3 1)jY1;T;X1;T;V1;T;Z( T i);H( T i(cid:0)1);(cid:2)( z i(cid:0)1);(cid:2)( h i(cid:0)1) );1 = : : p (cid:2)y (i 1 (cid:0) :1 1)[(cid:2)y (i 1 (cid:0) :2 1)[(cid:2)( y i 1 (cid:0) :3 1)jY1;T;X1;T;V1;T;Z( T i);H( T i(cid:0)1);(cid:2)( z i(cid:0)1);(cid:2)( h i(cid:0)1) ; ii. Generate a random draw of a transformation of (cid:2)(i) , (cid:2) ~(i) , using the normal y1:2 y1:2 proposal density (cid:2) ~(i) (cid:24) N((cid:2) ~(i) ;cS ) y1:2 y1:2 1:2 where S is the variance-covariance matrix of the maximum likelihood esti- 1:2 ~ mator of (cid:2) given the initial latent factors and stochastic volatilities, and y1:2 c is a scale coefficient. The algorithm accepts (cid:2)(i) with probability r: y1:2 8 ( ) 9 r=min < p ( (cid:2)( y i 1 ) :1 [(cid:2)( y i 1 ) :2 [(cid:2)y (i 1 (cid:0) :3 1)jY1;T;X1;T;V1;T;Z( T i);H( T i(cid:0)1);(cid:2)( z i(cid:0)1);(cid:2)( h i(cid:0)1) );1 = : : p (cid:2)( y i 1 ) :1 [(cid:2)y (i 1 (cid:0) :2 1)[(cid:2)y (i 1 (cid:0) :3 1)jY1;T;X1;T;V1;T;Z( T i);H( T i(cid:0)1);(cid:2)( z i(cid:0)1);(cid:2)( h i(cid:0)1) ; iii. Generate a random draw of (cid:2)(i) using the normal proposal density y1:3 (cid:2)(i) (cid:24) N((cid:2) ^ ;S ) y1:3 y1:3 1:3 ^ where (cid:2) is the least squares estimator of (cid:2) using the residuals of least y1:3 y1:3 squares estimations of (61) and (62), and S is its least squares estimated 1:3 variance-convariance matrix. The algorithm accepts (cid:2)(i) with probability r: y1:3 8 ( ) 9 r=min < p ( (cid:2)( y i 1 ) :1 [(cid:2)( y i 1 ) :2 [(cid:2)( y i 1 ) :3 jY1;T;X1;T;V1;T;Z( T i);H( T i(cid:0)1);(cid:2)( z i(cid:0)1);(cid:2)( h i(cid:0)1) ) g((cid:2)( y i 1 (cid:0) :3 1)) ;1 = ; : p (cid:2)( y i 1 ) :1 [(cid:2)( y i 1 ) :2 [(cid:2)y (i 1 (cid:0) :3 1)jY1;T;X1;T;V1;T;Z( T i);H( T i(cid:0)1);(cid:2)( z i(cid:0)1);(cid:2)( h i(cid:0)1) g((cid:2)( y i 1 ) :3 ) ; where g((cid:1)) is the proposal density. (c) Generate a random draw of (cid:2)(i) sequentially, as follows: y2 i. Generate a random draw of a transformation of (cid:2)(i) , (cid:2) ~(i) , using the normal y2:1 y2:1 proposal density (cid:2) ~(i) (cid:24) N((cid:2) ~(i(cid:0)1);cS ); y2:1 y2:1 2:1 where S is the variance-covariance matrix of the maximum likelihood esti- 2:1 ~ mator of (cid:2) given the initial latent factors and stochastic volatilities, and y2:1 c is a scale coefficient. The algorithm accepts (cid:2)(i) with probability r: y2:1 8 ( ) 9 r=min < p ( (cid:2)( y i 2 ) :1 [(cid:2)y (i 2 (cid:0) :2 1)[(cid:2)( y i 2 (cid:0) :3 1)jY2;T;X2;T;V2;T;Z( T i);H( T i(cid:0)1);(cid:2)( z i(cid:0)1);(cid:2)( h i(cid:0)1) );1 = : : p (cid:2)y (i 2 (cid:0) :1 1)[(cid:2)y (i 2 (cid:0) :2 1)[(cid:2)( y i 2 (cid:0) :3 1)jY2;T;X2;T;V2;T;Z( T i);H( T i(cid:0)1);(cid:2)( z i(cid:0)1);(cid:2)( h i(cid:0)1) ; 44

ii. Generate a random draw of a transformation of (cid:2)(i) , (cid:2) ~(i) , using the normal y2:2 y2:2 proposal density (cid:2) ~(i) (cid:24) N((cid:2) ~(i) ;cS ) y2:2 y2:2 2:2 where S is the variance-covariance matrix of the maximum likelihood esti- 2:2 ~ mator of (cid:2) given the initial latent factors and stochastic volatilities, and y2:2 c is a scale coefficient. The algorithm accepts (cid:2)(i) with probability r: y2:2 8 ( ) 9 r=min < p ( (cid:2)( y i 2 ) :1 [(cid:2)( y i 2 ) :2 [(cid:2)y (i 2 (cid:0) :3 1)jY2;T;X2;T;V2;T;Z( T i);H( T i(cid:0)1);(cid:2)( z i(cid:0)1);(cid:2)( h i(cid:0)1) );1 = : : p (cid:2)( y i 2 ) :1 [(cid:2)y (i 2 (cid:0) :2 1)[(cid:2)y (i 2 (cid:0) :3 1)jY2;T;X2;T;V2;T;Z( T i);H( T i(cid:0)1);(cid:2)( z i(cid:0)1);(cid:2)( h i(cid:0)1) ; iii. Generate a random draw of (cid:2)(i) using the normal proposal density y2:3 (cid:2)(i) (cid:24) N((cid:2) ^ ;S ) y2:3 y2:3 2:3 ^ where (cid:2) is the least squares estimator of (cid:2) using the residuals of least y2:3 y2:3 squares estimations of (63) and (64), and S is its least squares estimated 2:3 variance-convariance matrix. The algorithm accepts (cid:2)(i) with probability r: y2:3 8 ( ) 9 r=min < p ( (cid:2)( y i 2 ) :1 [(cid:2)( y i 2 ) :2 [(cid:2)( y i 2 ) :3 jY2;T;X2;T;V2;T;Z( T i);H( T i(cid:0)1);(cid:2)( z i(cid:0)1);(cid:2)( h i(cid:0)1) ) g((cid:2)( y i 2 (cid:0) :3 1)) ;1 = ; : p (cid:2)( y i 2 ) :1 [(cid:2)( y i 2 ) :2 [(cid:2)y (i 2 (cid:0) :3 1)jY2;T;X2;T;V2;T;Z( T i);H( T i(cid:0)1);(cid:2)( z i(cid:0)1);(cid:2)( h i(cid:0)1) g((cid:2)( y i 2 ) :3 ) ; where g((cid:1)) is the proposal density. Step 3. Generate a random draw of h from p(h jH ;H(cid:3) ;Y ;X ;V ;Z ;(cid:2) ;(cid:2) ;(cid:2) ) t t t(cid:0)1 t+1 T T T T y z h for t = 1;2;:::;T. Draw h(i) using the normal proposal density t h(i) (cid:24) N(h(i(cid:0)1);cKh) t t t where Kh is the (cid:12)ltered variance-covariance matrix of the volatility of a constantt coefficient speci(cid:12)cation of the policy rules, and c is a proper scale coefficient. The algorithm accepts h(i) with probability r: t 8 ( ) 9 < p h(i)jH(i(cid:0)1);H (cid:3)(i(cid:0)1);Y ;X ;V ;Z(i);(cid:2)(i);(cid:2)(i(cid:0)1);(cid:2)(i(cid:0)1) = ( t t(cid:0)1 t+1 T T T T y z h ) r = min ;1 : : p h(i(cid:0)1)jH(i(cid:0)1);H (cid:3)(i(cid:0)1);Y ;X ;V ;Z(i);(cid:2)(i);(cid:2)(i(cid:0)1);(cid:2)(i(cid:0)1) ; t t(cid:0)1 t+1 T T T T y z h Step 4. Generate a random draw of (cid:2) from p((cid:2) jY ;X ;V ;Z ;(cid:2) ;(cid:2) ) sequentially, z z T T T T y h as follows: (a) Partition (cid:2) = (cid:2) [(cid:2) where (cid:2) = f(cid:26) ;(cid:26) g and (cid:2) = f(cid:20)g. z z1 z2 z1 zm zf z2 (b) Generatearandomdrawf(cid:26)(i);(cid:26)(i)gusingtwoindependentBetaproposaldensities zm zf (expressed in terms of means and standard deviations) (cid:26)(i) (cid:24) Beta((cid:26)^ ;(cid:27)^ ) zm zm (cid:26)^ zm (cid:26)(i) (cid:24) Beta((cid:26)^ ;(cid:27)^ ); zf zm (cid:26)^ zf 45

where (cid:26)^ is the ordinary least squares estimate of (cid:26) in (41) using fzm(i)gT , zm zm t t=1 and (cid:27)^ is its standard error. The same applies for (cid:26)^ and (cid:27)^ , which come (cid:26)^ zm zf (cid:26)^ zf from the estimation of (42) using fzf(i)gT . The algorithm accepts f(cid:26)(i);(cid:26)(i)g t t=1 zm zf with probability r: 8 ( ) 9 r = min < p ( (cid:2) z (i 1 ) [(cid:2)( z i 2 (cid:0)1)jY T ;X T ;V T ;Z( T i);(cid:2) y ;(cid:2) h ) g((cid:26)( z i m (cid:0)1);(cid:26)( z i f (cid:0)1)) ;1 = : : p (cid:2)(i(cid:0)1) [(cid:2)(i(cid:0)1)jY ;X ;V ;Z(i);(cid:2) ;(cid:2) g((cid:26)(i);(cid:26)(i)) ; z1 z2 T T T T y h zm zf (c) Generate a random draw (cid:20)(i) using a four-parameter Beta proposal density with range on [-1,1] (expressed in terms of mean and standard deviation) p (cid:20)(i) (cid:24) TransformedBeta((cid:20)^;(1(cid:0)(cid:20)^2)= n(cid:0)1); where (cid:20)^ is the correlation coefficient between the residuals of equations (41) and (42)using(cid:26)(i) and(cid:26)(i) asestimatedcoefficientswherecorresponds. Thealgorithm zm zf accepts (cid:20)(i) with probability r: 8 ( ) 9 < p ( (cid:2)( z i 1 ) [(cid:2) z (i 2 )jY T ;X T ;V T ;Z( T i);(cid:2) y ;(cid:2) h ) g((cid:20)(i(cid:0)1)) = r = min ;1 : : p (cid:2)(i) [(cid:2)(i(cid:0)1)jY ;X ;V ;Z(i);(cid:2) ;(cid:2) g((cid:20)(i)) ; z1 z2 T T T T y h Step 5. Generate a random draw of (cid:2) from p((cid:2) jY ;X ;V ;H ;(cid:2) ;(cid:2) ). h h T T T T y z (a) Partition (cid:2) = (cid:2) [(cid:2) where (cid:2) = fln(cid:27) ;(cid:26) ;ln(cid:27) ;(cid:26) g and (cid:2) = f(cid:17) ;(cid:17) g. h h1 h2 h1 R (cid:27)R (cid:28) (cid:27)(cid:28) h2 R (cid:28) (b) Generate a random draw fln(cid:27)(i);(cid:26)(i);ln(cid:27)(i);(cid:26)(i)g using the independent proposal R (cid:27)R (cid:28) (cid:27)(cid:28) densities (expressed in terms of means and standard deviations) (cid:26)(i) (cid:24) Beta((cid:26)^ ;(cid:27)^ ) (cid:27)R (cid:27)R (cid:26)^(cid:27)R (cid:26)(i) (cid:24) Beta((cid:26)^ ;(cid:27)^ ) (cid:27)(cid:28) (cid:27)(cid:28) (cid:26)^(cid:27)(cid:28) c(i) (cid:24) N(c^ ;(cid:27)^ ) R R cR c(i) (cid:24) N(c^ ;(cid:27)^ ); (cid:28) (cid:28) c(cid:28) with ln(cid:27)(i) = c(i)=(1 (cid:0) (cid:26)(i)) and ln(cid:27)(i) = c(i)=(1 (cid:0) (cid:26)(i)), where c^ ;(cid:26)^ are the least squ R ares es R timates o (cid:27) f R (1 (cid:0) (cid:26) )l (cid:28) n(cid:27) a (cid:28) nd (cid:26) in (cid:27)(cid:28) (43), respe R ctiv (cid:27) e R ly, using (cid:27)R R (cid:27)R fln(cid:27)(i)gT , and where c^ ;(cid:26)^ are obtained similarly from (44) using fln(cid:27)(i)gT . R;t t=0 (cid:28) (cid:27)(cid:28) (cid:28);t t=0 The algorithm accepts fln(cid:27)(i);(cid:26)(i);ln(cid:27)(i);(cid:26)(i)g with probability r: R (cid:27)R (cid:28) (cid:27)(cid:28) 8 ( ) 9 r =min < p ( (cid:2)( h i 1 )[(cid:2)( h i 2 (cid:0)1)jY T ;X T ;V T ;H( T i);(cid:2) y ;(cid:2) z ) g(ln(cid:27) R (i(cid:0)1);(cid:26)( (cid:27) i R (cid:0)1);ln(cid:27) (cid:28) (i(cid:0)1);(cid:26)( (cid:27) i (cid:28) (cid:0)1)) ;1 = : : p (cid:2)( h i 1 (cid:0)1)[(cid:2)( h i 2 (cid:0)1)jY T ;X T ;V T ;H( T i);(cid:2) y ;(cid:2) z g(ln(cid:27) R (i);(cid:26)( (cid:27) i R );ln(cid:27) (cid:28) (i);(cid:26)( (cid:27) i (cid:28) )) ; 46

(c) Generate a random draw f(cid:17)(i);(cid:17)(i)g using the inverted gamma proposal densities R (cid:28) (cid:17)(i) (cid:24) IG((cid:24) ~′R(i) (cid:24) ~R(i);df) R (cid:17)(i) (cid:24) IG((cid:24) ~′(cid:28)(i) (cid:24) ~(cid:28)(i);df); (cid:28) where (cid:24) ~R(i) = ln(cid:27)(i) (cid:0) (1 (cid:0) (cid:26)(i))ln(cid:27)(i) (cid:0) (cid:26)(i) ln(cid:27)(i) and (cid:24) ~(cid:28)(i) = ln(cid:27)(i) (cid:0) (1 (cid:0) t R;t (cid:27)R R (cid:27)R R;t(cid:0)1 t (cid:28);t (cid:26)(i))ln(cid:27)(i) (cid:0) (cid:26)(i)ln(cid:27)(i) are residuals, and (cid:24) ~R(i) and (cid:24) ~(cid:28)(i) are vectors that stack (cid:27)(cid:28) (cid:28) (cid:27)(cid:28) (cid:28);t(cid:0)1 the respective residuals. The algorithm accepts f(cid:17)(i);(cid:17)(i)g with probability r: R (cid:28) 8 ( ) 9 r = min < p ( (cid:2)( h i 1 ) [(cid:2)( h i 2 )jY T ;X T ;V T ;H( T i);(cid:2) y ;(cid:2) z ) g((cid:17) R (i(cid:0)1);(cid:17) (cid:28) (i(cid:0)1)) ;1 = : : p (cid:2)( h i 1 ) [(cid:2)( h i 2 (cid:0)1)jY T ;X T ;V T ;H( T i);(cid:2) y ;(cid:2) z g((cid:17) R (i);(cid:17) (cid:28) (i)) ; Step 6. Repeat steps 1-4 N times to obtain N random draws of Z , H , (cid:2) , (cid:2) and T T y z (cid:2) . h In steps (ii)-(vi) the random draws of Z, H, (cid:2) , (cid:2) and (cid:2) are updated. This sampling y z h method is referred to as the Gibbs sampler. To generate the random draws of z , h for t t t = 1;2;:::;T, (cid:2) , (cid:2) and (cid:2) , we use the Metropolis-Hastings (M-H) algorithm. That is, y z h the Gibbs sampler and the M-H algorithm are combined to obtain the smoothing random draws from the state-space model. 47

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