Assessing the Effects of the Zero-Interest-Rate Policy through the Lens of a Regime-Switching DSGE Model

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Assessing the Effects of the Zero-Interest-Rate Policy through the Lens of a Regime-Switching DSGE Model Han Chen 2014-38 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.

Assessing the E⁄ects of the Zero-Interest-Rate Policy through the Lens of a Regime-Switching DSGE Model Han Chen (cid:3) Abstract Standard dynamic stochastic general equilibrium (DSGE) models assume a Taylor ruleandforecastanincreaseininterestratesimmediatelyafterthe2007-2009economic recession given the predicted output and in(cid:135)ation, contradictory to the extended period of near-zero interest rate policy (ZIRP) conducted by the Federal Reserve. In this paper, I study two methods of modeling the ZIRP in DSGE models: the perfect foresight rational expectations model and the Markov regime-switching model, which I develop in this paper. In this regime-switching model, I assume that, in one regime, the policy follows a Taylor rule, while, in the other regime, it involves a zero interest rate. I also construct the optimal (cid:133)lter to estimate this regime-switching DSGE model with Bayesian methods. I (cid:133)t these modi(cid:133)ed DSGE models to the U.S. data from the third quarter of 1987 to the third quarter of 2010, and then, starting from the fourth quarter of 2010, I simulate the U.S. economy forward with and without the ZIRP intervention. I compare the predicted paths of the macro variables, and I (cid:133)nd that the ZIRP intervention has a signi(cid:133)cant e⁄ect. The estimated regime-switching model I develop implies a substantial stimulative e⁄ect (on average a 0.12% increase in output growth rate and a 0.9% increase in in(cid:135)ation accumulatively over 20 quarters if ZIRP is kept for 6 quarters). The actual path from the fourth quarter of 2010 onward is closer Email: han.chen@frb.gov. Phone: 240-374-2758. I am particularly indebted to my adviser Frank (cid:3) Schorfheide for his help and guidance, as well as my committee member Harold Cole and Frank Diebold. I would like to thank James Nason; several ideas in this paper originated from a conversation with him. I also thank Tao Zha for providing some of his codes and Andrea Ferrero and Andrew Clausen for helpful discussions. The views expressed are those of the author and do not re(cid:135)ect those of the Board of Governors and the Federal Reserve System. 1

to the predicted path derived from the regime-switching model than that generated by the perfect foresight model. The perfect foresight model generates an explosive and spurious rise in in(cid:135)ation. Therefore, the regime-switching model I propose is more appropriate to assess the e⁄ectiveness of the ZIRP, which, I (cid:133)nd, is indeed e⁄ective in stimulating the economy. JEL codes: E43, E44, E52, E58 Keywords: regime switching, zero interest rate policy, unconventional monetary policy 2

1 Introduction In responsze to the 2007-2009 economic recession and the weak recovery that followed, theFederalReservehasbeengivingtheeconomyunprecedentedsupport: besidesthelending facilities and the large-scale asset purchases, the Federal Reserve decided to keep the target range for the federal funds rate at 0 to 1/4 percent.1 Bernanke and Reinhart (2004) refer to the commitment to keep interests low (forward guidance) as "unconventional monetary policy," in contrast to conventional monetary policy, which refers to central banks(cid:146)manipulation of the policy rate (in the United States, the federal funds rate). Standard DSGE models designed to analyze monetary policy and match the macro data well before the crisis must address the challenge of evaluating the Federal Reserve(cid:146)s unconventional policy. StandardDSGEmodels, whichassumeaTaylorrule, oftenpredictsaquickriseofinterest rates immediately after a recession.2 When analyzing the e⁄ects of the policy of keeping the interest rates extremely low for an extended period, the standard approach is to estimate a stochastic model and then conduct a counterfactual analysis using the perfect foresight rational expectations (hereafter PFRE) solution method (Cœrdia and Woodford, 2011).3 This method assumes that agents have perfect foresight of the path of the future shocks and the interest rates, and rational expectations equilibrium can be solved backwards. As a result, the assumption of perfect foresight for policy analysis inherently con(cid:135)icts with the assumption of the stochastic model that is used to (cid:133)t the data. Furthermore, the PFRE model predicts an unrealistic path of macro variables. For example, this model predicts a 1Theforwardguidancehasalwayssuggestedthatthefederalfundsratewouldbelowforsometimeeither date-based or threshold-based. In the January 2014 FOMC meeting, "the Committee also rea¢ rmed its expectationthatthecurrentexceptionallylowtargetrangeforthefederalfundsrateof0to1/4percentwill be appropriate at least as long as the unemployment rate remains above 6-1/2 percent, in(cid:135)ation between one and two years ahead is projected to be no more than a half percentage point above the Committee(cid:146)s 2 percent longer-run goal, and longer-term in(cid:135)ation expectations continue to be well anchored." 2Reifschneider and Williams (2000), Chung et al. (2011), and Del Negro and Schorfheide (2012). 3A detailed description can be found at the online appendix of Chen, Cœrdia and Ferrero (2012). 3

spurious rise in in(cid:135)ation4. How do we reconcile those con(cid:135)icted assumptions and better predict the distribution of macroeconomic variables? In this work, I propose a Markov regime-switching DSGE model and compare it with the conventional PFREmodel. I propose to model the zero interest rate policy (hereafter ZIRP) by a regime-switching monetary policy rule where, in one regime, the policy rates follow a typical Taylor rule, and, in the other regime, the policy rate is set to zero. Also, I solve this regime-switching DSGE model by using the Farmer, Waggoner, and Zha (2011) minimum state variable solution.5 In addition, I construct the optimal (cid:133)lters in order to estimate this regime-switching DSGE model with Bayesian methods. I (cid:133)t the regime-switching and the standard DSGE models to U.S. data from the third quarter of 1987 to the third quarter of 2010. And then, starting from the fourth quarter of 2010, I simulate the U.S. economy forward under two scenarios: the counterfactual scenario with no policy intervention, and under ZIRP for an extended period using either the regime-switching model or PFRE model. To assess the policy(cid:146)s e⁄ectiveness, I compare the predicted path of the macro variables (output and in(cid:135)ation) with and without the policy intervention. I (cid:133)nd that the ZIRP has a substantial e⁄ect and the predicted path of macro variables generated by the regime-switching model is closer to the actual path, while the PFRE model generates an explosive predicted path. The fundamental di⁄erence between these two types of models is how agents(cid:146)expectations are formulated. In the regime-switching model, at each period agents attach a non-zero probability of exiting the ZIRP regime in the next period despite the Federal Reserve(cid:146)s "extended period" language, because, for example, the simple announcement would be subject 4Carlstrom, Fuerst, and Paustian (2012) also make this observation. 5One can also use the perturbation method developed by Foerster, A., Rubio-Ram(cid:237)rez, J., Waggoner, D. and Zha, T. (2013). The solutions of those two methods are the same for the parameters in front of the lagged state variables and in front of the shocks. I use Liu, Waggoner, and Zha (2011) to get the regime-switching constant. This approach is di⁄erent from the regime-switching constant derived from the perturbation method. 4

to the time inconsistency problem, and is thus not credible. And even after the exit, the agents expect to come back to the ZIRP regime with a non-zero probability in the future. ThePFREassumesthatagentsbelievetheFederalReserve(cid:146)sannouncementandhaveperfect foresightoffutureinterestrates. Iarguethattheregime-switchingmodelismoreappropriate for policy evaluation. Here, I am looking at this extended period of zero interest rates as a policy choice because the central bank could raise interest rates when output starts growing and the economy improving, as predicted by the Taylor rule.6 Carlstrom, Fuerst, and Paustian (2012); Cœrdia and Woodford (2010); and Chen, Cœrdia, and Ferrero (2012) also study the e⁄ects of a transient interest rate peg. Under the assumption of either a deterministic exit or a stochastic exit of the interest rate peg in the previous studies, the policy rate will follow a Taylor rule after the exit and the interest rate peg will never occur again. In the regime-switching model developed in this paper, however, the zero interest rate policy regime is a recurring event. Even at the normal interest rate regime, agents expect to enter a zero interest rate regime in the future with a non-zero probability. Expectations play an important role in the regime-switching model. An alternative way to look at this persistent period of low interest rates is the zero lower bound (ZLB) problem. A persistent shock drives interest rates below zero for an extended period if we keep following a Taylor rule.7 A rapidly growing literature on ZLB considers the zero interest rates as a modeling constraint that has to be considered. Global methods include Judd, Maliar, and Maliar (2011); FernÆndez-Villaverde, Gordon, Guerr(cid:243)n- Quintana, and Rubio-Ram(cid:237)rez (2012); and Aruoba and Schorfheide (2012). There are also a few short cuts for modeling ZLB, such as Braun and K(cid:246)rber (2011); Adam and Billi (2007); Eggertsson and Woodford (2003); and Christiano, Eichenbaum, and Rebelo (2011). Del 6Heretheregime-switchingisexogenouswhileideallyitshouldbeendogenousanddependonthemacroeconomic condition. 7For example a preference shock or a technology shock. 5

Negro and Schorfheide (2012) describe how to impose zero interest rates via unanticipated or anticipated monetary policy shocks in a DSGE model. Those models focus on modeling howthe economy enters an extended period of zero interest rates, while the regime-switching model developed here is more of a model of exiting the zero interest rates. The rest of the paper proceeds as follows. Section 2 presents a standard DSGE model. Section3discusseshowtomodelZIRPwitharegime-switchingmonetarypolicyandwiththe PFRE. Section4 describes the estimation of the regime-switching model, some basic analysis of parameterestimates, anevaluationof thee⁄ects of theZIRP, andthecomparisonbetween the regime-switching model and the PFRE model. Finally, section 5 summarizes the main (cid:133)ndings. 2 Models Except for the monetary policy rule, the model used here is a standard medium-scale DSGE model (Christiano et al., 2005; Smets and Wouters, 2007): Households maximize utility; monopolistic competitive (cid:133)rms hire the labor to produce intermediate goods; and competitive (cid:133)nal-goods-producing (cid:133)rms package intermediate goods into a homogeneous consumption good. Finally, the government sets monetary and (cid:133)scal policy. To simplify the analysis, I abstract from capital. 2.1 Households The representative household(cid:146)s objective function is 1 (cid:27) Et 1 (cid:12)sb t+s2(cid:16) C Z 1 t t + + s s (cid:17)(cid:27) (cid:0) (cid:0) 1 L + 1 t+ + s (cid:23) (cid:23)3 ; s=0 (cid:0) X 6 7 4 5 6

where (cid:12) is the discount factor, b is a preference shock, C is consumption (relative to t t productivity Z , as in An and Schorfheide (2007) to ensure the existence of a balanced t growth path with constant relative risk aversion preferences (cid:27)), L is labor supply, and (cid:23) is t the inverse of the Frisch elasticity of labor supply. The time t budget constraint for a household is P C +B R B +W L + T ; (2.1) t t t t 1 t 1 t t t t (cid:20) (cid:0) (cid:0) P (cid:0) where, B ; are one-period securities purchased at time t that pay a nominal return, R ; at t t time t+1, P is the price of the (cid:133)nal consumption good, W is the competitive wage, are t t t P the pro(cid:133)ts distributed by the intermediate goods producers, and T are lump-sum taxes. t Let(cid:4)P representtheLagrangemultiplierfor(2.1). TheEulerequationfortheshort-term t bonds is (cid:4)p t = (cid:12)Et [(cid:4)p t+1 R t ]: 2.2 Final Goods Producers The (cid:133)nal good, Y ; is a composite made of a continuum of intermediate goods indexed t by i (0;1) 2 1 1 1+(cid:21)f Y t = Y t (i)1+(cid:21)fdi : (2.2) (cid:20)Z0 (cid:21) The (cid:133)nal goods producers buy the intermediate goods on the market, package to Y , and t sell it to consumers. These (cid:133)rms maximize pro(cid:133)ts in a perfectly competitive environment. Their problem is: 1 max P Y P (i)Y (i)di; Yt(i) t t (cid:0) 0 t t (2.3) s.t. Y t = R 0 1 Y t (i)1+ 1 (cid:21)fdi 1+(cid:21)f : h i R 7

From the (cid:133)rst order conditions: 1+(cid:21)f P t (i) (cid:0) (cid:21)f Y (i) = Y : t t P (cid:18) t (cid:19) Combining this condition with the zero pro(cid:133)t condition, I obtain the expression for the price of the composite (cid:133)nal good: 1 1 (cid:0) (cid:21)f P t = P t (i)(cid:0)(cid:21)fdi : (2.4) (cid:20)Z0 (cid:21) 2.3 Intermediate goods producers Intermediate goods producer i uses the following technology: Y (i) = Z L ; (2.5) t t t where Z is the technology, and L is labor input. t t Prices are sticky (cid:224) la Calvo (1983). Speci(cid:133)cally, each (cid:133)rm can readjust prices with a probability 1 (cid:16) in each period. For those (cid:133)rms that cannot adjust prices, P (i) will p t (cid:0) increase at the steady state rate of in(cid:135)ation (cid:25). For those (cid:133)rms that can adjust prices, the ~ problem is to choose a price level, P (i); that maximizes the sum of the expected discounted t pro(cid:133)ts in all states of the future where the (cid:133)rm cannot adjust the price: 1+(cid:21)f 1 s P ~ t (i)(cid:5)s P ~ t (i)(cid:5)s (cid:0) (cid:21)f m P~ t a (i x ) Et s=0 (cid:12)(cid:16) p (cid:4) t+s P t+s ! (cid:0) w z;t+s ! P t+s ! Y z;t+s ; X(cid:0) (cid:1) where (cid:4) = (cid:4)p P Z , w = wt+s , and Y = Yt+s: t+s t+s t+s t+s z;t+s Pt+sZt+s z;t+s Zt+s The (cid:133)rst order condition for the (cid:133)rm is 8

1 ~ 1 s 1 (cid:5)s (cid:0)(cid:21)f 0 = P t (i)Et (cid:12)(cid:16) p (cid:4) t+s (cid:21) P Y z;t+s (2.6) s=0 f (cid:18) t+s(cid:19) X(cid:0) (cid:1) 1+(cid:21)f 1 s 1+(cid:21) f (cid:5)s (cid:0) (cid:21)f (cid:0) Et (cid:12)(cid:16) p (cid:4) t+s (cid:21) w z;t+s P Y z;t+s : s=0 f (cid:18) t+s(cid:19) X(cid:0) (cid:1) Notethatall(cid:133)rmsthatcanreadjustpricesfacethisidenticalproblem. Iwillonlyconsider the symmetric equilibrium in which all (cid:133)rms that can readjust prices will choose the same price, so I can drop the i index. From 2.4 it follows that: P t = (1 (cid:16) p )P ~ t (cid:0)(cid:21) 1 f +(cid:16) p [(cid:5)P t 1 ](cid:0)(cid:21) 1 f (cid:0) (cid:21)f : (2.7) (cid:0) (cid:0) (cid:20) (cid:21) So 1 P ~ t (cid:0)(cid:21)f (cid:5) (cid:0)(cid:21) 1 f 1 = (1 (cid:16) ) +(cid:16) : (cid:0) p P p (cid:5) t! (cid:20) t(cid:21) 2.4 Government Policies The monetary policy assumption is taken from Chen et al. (2012). The central bank followsaconventionalfeedbackinterestraterulesimilartoTaylor(1993), modi(cid:133)edtoinclude interest rate smoothing (Clarida et al., 2000) and to use the growth rate of output instead of the output gap (Justiniano et al., 2011): 1 (cid:26) R R (cid:26) m (cid:5) (cid:30) (cid:25) Y =Y (cid:30) y (cid:0) m t = t (cid:0) 1 t t t (cid:0) 4 e(cid:15)m;t; (2.8) R R (cid:5) e4(cid:13) " # (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) where (cid:5) P =P is the in(cid:135)ation rate, (cid:26) (0;1), (cid:30) > 1, (cid:30) 0; and (cid:15) is an i.i.d. t t t 1 m (cid:25) y m;t (cid:17) (cid:0) 2 (cid:21) innovation.8 In the section (3.1), I will elaborate how to modify the monetary policy rule to 8Chen et al. (2012) use the output growth in the Taylor rule, instead of the output gap, to avoid the complication of solving and estimating the system by characterizing the (cid:135)exible price equilibrium. In 9

assess ZIRP. The standard government budget constraint is as follows: B = R B +P G T : (2.9) t t 1;t t 1 t t t (cid:0) (cid:0) (cid:0) The left-hand side of expression (2.9) is the market value, in nominal terms, of the total amount of bonds issued by the government at time t. The right-hand side is the total de(cid:133)cit at time t, that is, market value plus interest payment of the bonds maturing in that period plus spending G net of taxes. t 2.5 Exogenous Processes The model is supposed to be (cid:133)tted to data on output, in(cid:135)ation, and nominal interest rates. There are four structural shocks in total. The logarithm of the technology follows a random walk with drift: lnZ = (cid:13) +lnZ +z ; t t 1 t (cid:0) where the shock z follows a (cid:133)rst order autoregressive process (AR(1)): t z t = (cid:26) z z t +(cid:15) z;t ;(cid:15) z;t (cid:24) N 0;(cid:27)2 "z : (cid:0) (cid:1) The shock to the discount factor (cid:12) (intertemporal preference shifter) is also assumed to follow an AR(1) process: lnb t = (cid:26) b lnb t (cid:0) 1 +(cid:15) b;t ;(cid:15) b;t (cid:24) N 0;(cid:27)2 "b : practice, real GDP growth relative to trend is often cited as one o(cid:0)f the m(cid:1)ain indicators of real activity for the conduct of monetary policy. 10

The government spending is assumed to be an exogenous process: lng t = (cid:26) g lng t (cid:0) 1 +(cid:15) g;t ;(cid:15) g;t (cid:24) N 0;(cid:27)2 "g : (cid:16) (cid:17) The monetary policy shock (cid:15) is an independent and identically distributed shock. m;t 3 Zero Interest Rate Policy In this section, I describe two methods of studying the e⁄ects of ZIRP in DSGE models: the regime-switching method and the PFRE method. I am going to consider a regimeswitching model where, in one regime, the policy rate follows a typical Taylor rule, and, in the other regime, it involves ZIRP. Although the Markov regime switching is imposed to the monetary policy rule before loglinearizing the system, and thus the method begins from (cid:133)rst principles rather than adding Markov-switching parameters after linearizing the model, the model is still a linear model. Simplifying assumptions in my model may miss some nonlinear interactions between the zero interest rates and the policy functions of the agents, however, I substantially gain tractability. I also construct the optimal (cid:133)lter and likelihood function so that I can (cid:133)t this model to the macro data including the recent time where the interest rates are maintained near zero for an extended period. This regime-switching model can not only explain the interest rate data, but it also provides a plausible explanation for exiting the zero interest rate policy. It is more of a model for exiting the ZIRP than a model for entering the ZIRP. This regime-switching model o⁄ers a tool to conduct forecasts and counterfactual analysis. The other approach to assessing the ZIRP, PFRE, cannot explain the recent episodes of near-zero interest rates. It only asks the counterfactual questions such as what are the e⁄ects on the macro variables if the interest rates are pegged at zero for an extended period, and agents have perfect knowledge of this policy experiment? Now I de(cid:133)ne 11

the regime-switching model more precisely . 3.1 Regime-Switching Policy Rule Consider a regime-switching policy rule where, in one regime, the federal funds rate follows a Taylor rule while, in the other regime, it sets the interest rates to zero. I will use the Farmer, Waggoner, and Zha (2011) minimum state variable solution method to solve this regime-switching model, and the estimation strategy will be described in section 4. The policy rule is (cid:25) r ’ (cid:25) (Kt) Y =Y ’ y (Kt) (1 (cid:0) (cid:26) R (Kt)) R t = (R t(cid:3) (K t ))1 (cid:0) (cid:26) R (Kt) R ( t K ) t e4(cid:13) t (cid:0) 4 R t (cid:26) R 1 (Kt) exp(" R;t ); " (cid:18) t(cid:3) t (cid:19) (cid:18) (cid:19) # (cid:0) (3.1) where all the parameters denoted by (K ) are regime-dependent. R (K ) are the desired t t(cid:3) t regime-dependent target nominal interest rates. Let K = 1 denote the normal regime and t K = 2 denote the ZIRP regime. For example, I can set R (K = 1) = R = 1:005; which t t(cid:3) t 1(cid:3) corresponds to a target 2% annual interest rate at the normal regime and set R (K = 2) = t(cid:3) t R = 1:0005; which corresponds to a target 20 basis points annual interest rate at the second 2(cid:3) regime. To study the ZIRP, I set R = 1; 2(cid:3) (cid:26) (K = 2) = 0; R t ’ (K = 2) = 0; (cid:25) t ’ (K = 2) = 0; and y t (cid:27) (K = 2) = 0: "R;t t I de(cid:133)ne the ergodic mean of the logarithm of the steady state interest rates as 12

(cid:22) (cid:22) log(R) = (cid:21) log(R )+(cid:21) log(R ); 1 1(cid:3) 2 2(cid:3) (cid:22) (cid:22) where (cid:21) and (cid:21) are the ergodic probabilities. 1 2 Divide 3.1 by its ergodic mean, R; and thus: R R t = R R t(cid:3) (1 (cid:0) (cid:26) R (Kt))(1 (cid:0) ’ (cid:25) (Kt)) (cid:25) (cid:25) t ’ (cid:25) (Kt) Y t = e Y 4(cid:13) t (cid:0) 4 ’ y (Kt) (1 (cid:0) (cid:26) R (Kt)) R R t (cid:0) 1 (cid:26) R (Kt) exp(" R;t ): " # (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:16) (cid:17) (3.2) Loglinearize 3.2 and thus: i=3 R ^ = (cid:26) (K )R ^ +(1 (cid:26) (K )) ’ (K )(cid:25)^ +’ (K ) y^ 9 y^ + z t R t t 1 R t (cid:25) t t y t z;t z;t 4 t i (cid:0) (cid:0) " (cid:0) (cid:0) (cid:0) !# i=0 X ^ +" +(1 (cid:26) (K ))(1 ’ (K ))R (K ); (3.3) R;t (cid:0) R t (cid:0) (cid:25) t t(cid:3) t wherethelasttermrepresentsaregime-switchingconstant. Iamgoingtoapplythetrickused by Liu, Waggoner, and Zha (2011). They solve a system where the only regime-switching coe¢ cient is the constant. I can rewrite 3.3 as i=3 ^ ^ R = (cid:26) (K )R +(1 (cid:26) (K )) ’ (K )(cid:25)^ +’ (K ) y^ y^ + z +" t R t t 1 R t (cid:25) t t y t t t 4 t i R;t (cid:0) (cid:0) " (cid:0) (cid:0) (cid:0) !# i=0 X R R 1 2 +(1 (cid:26) (K ))(1 ’ (K )) log ; log ^e ; (cid:0) R t (cid:0) (cid:25) t R R s;t (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) where ^e = e (cid:22)e ; and (cid:22)e is the ergodic probability. e is de(cid:133)ned as: s;t s;t s s s;t (cid:0) 1 e = St=1 ; s;t 2 3 1 St=2 6 7 4 5 with 1 s = j = 1 if s = j; and 0 otherwise. As shown in Hamilton (1994), the random t t f g 13

vector e follows an AR(1) process: s;t e = Pe +(cid:23) ; (3.4) s;t s;t 1 t (cid:0) where P is the transition matrix of the Markov switching process, and the innovation vector has the property that Et 1 (cid:23) t = 0. In the steady state, (cid:23) t = 0 so that 3.4 de(cid:133)nes the ergodic (cid:0) probabilities for the Markov process (cid:22)e . Schorfheide (2005) also proposes an algorithm to s solve DSGE models with a regime-switching constant in the policy rule. One can prove that Schorfheide (2005) and Liu, Waggoner, and Zha (2011) give rise to the same solution.10 By adding two extra variables e , I can use Farmer, Waggoner, and Zha(cid:146)s (2011) minis;t mum state variable solution to solve this regime-switching model. The solution of the model can be represented by Z = G (K )Z +R (K )" t t t t 1 t t t (cid:0) Z G G Z R R " 1;t 11 12 1;t 1 11 12 1;t = (cid:0) + ; 2 3 2 32 3 2 32 3 Z 0 P Z 0 1 " 2;t 2;t 1 2;t 6 7 6 76 (cid:0) 7 6 76 7 4 5 4 54 5 4 54 5 where I can partition the variables Z and the shocks " into two parts, respectively. Z t t 2;t are [e^ (1) e^ (2)], " are [v v ], Z are the rest of the states, and " are the structural t t 0 2;t 1;t 2;t 0 1;t 1;t shocks of the DSGE models. I de(cid:133)ne C(K ) = G Z +R " : t 12 2;t 12 2;t Notice that C(K ) are the regime-dependent constants. Finally I can rewrite the system t as follows with regime-switching coe¢ cients: 10See the appendix for proof. 14

Z = C(K )+G (K )Z +R (K )" : t t t t t 1 t t t (cid:0) 3.2 Model ZIRP by the PFRE The solution method of the PFRE model can be found in Cœrdia and Woodford (2011). For a detailed description of the algorithm and an application, please refer to Chen et al. (2012) and its companion online appendix.11 The basic idea is that agents have perfect foresight of the path of future interest rates and of all shocks until an arbitrary point in time. From this point forward all the shocks are zero, and the solution method is standard, such as in Sims (2002). The system can be solved backwards from this point. The following is a very simple example to illustrate the solution method. Consider the equilibrium system: y^ t = Et [y^ t+1 ] (cid:27) (cid:0) 1(^{ t Et [(cid:25)^ t+1 ]) (cid:0) (cid:0) (cid:25)^ t = (cid:12)Et [(cid:25)^ t+1 ]+(cid:20)y^ t ^{ = (cid:30) (cid:25)^ +(cid:23) for t > K (cid:23) = 0 t (cid:25) t t t = 0 for t = 1;:::K 1;K: (cid:0) The solution for t > K is: y^ t y(cid:23) 2 3 2 3 (cid:25)^ = (cid:23) : t (cid:25)(cid:23) t 6 7 6 7 6 ^{ 7 6 7 6 t 7 6 i(cid:23) 7 6 7 6 7 4 5 4 5 11The appendix can be found at http://onlinelibrary.wiley.com/doi/10.1111/j.1468- 0297.2012.02549.x/suppinfo 15

The system can be broken into the forward-looking and the backward-looking parts. The forward-looking part is: y^ t 1 (cid:27) (cid:0) 1 Et [y^ t+1 ] 1 (cid:27) (cid:0) 1 0 2 3 = ^{ 2 32 3 2 3 t 0 (cid:12) Et [(cid:25)^ t+1 ] (cid:20) 0 1 6 7 6 76 7 6 (cid:0) 7 6 (cid:25)^ 7 6 t 7 4 54 5 4 5 6 7 4 5 and the backward-looking part is: y^ t 2 3 [0 1 (cid:30) ] ^{ = (cid:23) : (cid:0) (cid:25) t t 6 7 6 (cid:25)^ 7 6 t 7 6 7 4 5 At t = K, plug in the solution to the forward-looking part and thus: y^ t 1 (cid:27) (cid:0) 1 y(cid:23) Et (cid:23) t+1 1 (cid:27) (cid:0) 1 0 2 3 = ^{ : 2 32 3 2 3 t 0 (cid:12) (cid:25)(cid:23) Et (cid:23) t+1 (cid:20) 0 1 6 7 6 76 7 6 (cid:0) 7 6 (cid:25)^ 7 6 t 7 4 54 5 4 5 6 7 4 5 Combine this part with the backward-looking part and thus: 0 1 (cid:30) y^ 0 (cid:23) (cid:25) t t (cid:0) 2 32 3 2 3 2 3 1 (cid:27) (cid:0) 1 0 ^{ t = y(cid:23) +(cid:27) (cid:0) 1 (cid:25)(cid:23) Et (cid:23) t+1 + 0 : 6 76 7 6 7 6 7 6 6 (cid:20) 0 1 7 7 6 6 (cid:25)^ t 7 7 6 6 (cid:0) (cid:12) (cid:25)(cid:23) Et (cid:23) t (cid:1) +1 7 7 6 6 0 7 7 6 (cid:0) 76 7 6 7 6 7 4 54 5 4 5 4 5 We can solve this system by inverting a matrix. The solution is: 1 (cid:0) y^ 0 1 (cid:30) 0 (cid:23) t (cid:25) t (cid:0) 2 3 2 3 02 3 2 31 ^{ t = 1 (cid:27) (cid:0) 1 0 y(cid:23) +(cid:27) (cid:0) 1 (cid:25)(cid:23) Et (cid:23) t+1 + 0 : 6 7 6 7 B6 7 6 7C 6 6 (cid:25)^ t 7 7 6 6 (cid:20) 0 1 7 7 B B 6 6 (cid:0) (cid:12) (cid:25)(cid:23) Et (cid:23) t (cid:1) +1 7 7 6 6 0 7 7 C C 6 7 6 (cid:0) 7 B6 7 6 7C 4 5 4 5 @4 5 4 5A 16

We can iterate backwards until the (cid:133)rst period. 4 Empirical Analysis In this section, I compare two approaches to modeling ZIRP in DSGE models. I estimate the DSGE model that either incorporates a regime-switching monetary policy as 3.1 or a typical Taylor rule as 2.8. I extract the (cid:133)ltered states of those estimated DSGE models, and then, starting from the fourth quarter of 2010, I simulate the U.S. economy forward under two scenarios: no intervention and ZIRP for an extended period.12 I compare the predicted path of macro variables generated from the di⁄erent models. When I evaluate ZIRP in the DSGE model with the regular Taylor rule, the PFRE method is used to simulate the economy. I will only explicate the estimation strategy of the regime-switching DSGE model. The description of the estimation procedure of the other non-regime-switching model was omitted here. The Bayesian estimation methods for a linearized DSGE model with constant coe¢ cients can be found, for example by An and Schorfheide (2007). Bayesian estimation combines prior information on the parameters with the likelihood function of the model to form the posterior distribution. In the regime-switching model, the optimal (cid:133)lter is no longer the Kalman Filter. I will (cid:133)rst illustrate the optimal (cid:133)lter and the likelihood function for this regime-switching model, and then describe data, show estimation results, and make comparisons of simulation results. 12Under both scenarios, there are no macro shocks. 17

4.1 Optimal Filter and Likelihood Function The regime-switching model is complicated because we have to keep track of the long history of the distribution of the states, and the number of the states grows exponentially.13 Fortunately, in my application, when updating the states probability, the distribution of the states at each time is degenerated because I observe the interest rates and thus deduce whether or not the economy is at the ZIRP regime in that period. This practice makes inference more e¢ cient. In this New Keynesian economy, the states are denoted by S and the observables are t denotedbyy . LetK denotetheMarkovregime-switchingstatesand(cid:21) denotetheprobabilt t t ity in the ZIRP regime (K = 2) at time t, thus K = 1, the normal regime, has probability t t ^ 1 (cid:21) . Let R denote the log deviation of the regime-switching interest rates from their t t (cid:0) ergodic mean. Its density function can be written as: 1 P R ^ t = (cid:21) 1 tf Rt=1 g (1 (cid:21) t )f t R ^ t f Rt>1 g; (cid:0) (cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17) where f R ^ is the conditional density, conditional on at the normal state.14 That is t t (cid:16) (cid:17) ^ ^ P R R > 1 = f R : t t t t j (cid:16) (cid:17) (cid:16) (cid:17) De(cid:133)ne the Dirac function as 0 if x = x~ (cid:14) (x) = 6 and (cid:14) (x)dx = 1: x~ x~ (cid:26) if x = x~ Z 1 Using the Dirac function, I can express the density of the interest rates as 13Evenwithatwo-stateMarkovregime-switchingprocess, attimet, thenumberofstatesis2t:Traditionally, we can follow the approximation approaches by Kim and Nelson (1999). 14R represents gross interest rates. t 18

^ ^ P R = (cid:21) (cid:14) (R )+(1 (cid:21) )f R : t t 1 t t t t (cid:0) (cid:16) (cid:17) (cid:16) (cid:17) The transition equations are S (K ) = C(K )+G (K )S (K )+R (K )" ; t t t t t t 1 t 1 t t t (cid:0) (cid:0) where all the coe¢ cients are regime-dependent and the measurement equations are (no measurement error): y (K ) = TS (K ): t t t t (cid:22) Let (cid:21) denote the ergodic probability of the Markov chain and (cid:6) denote the state-dependent k variance-covariance matrix of the structural shocks: (cid:6) = E[" " K = k]: k t 0tj t The algorithm of the optimal (cid:133)lter is as follows: Initializing at time t = 1;the mean of the states: (cid:15) S(cid:22) = (cid:21) (cid:22) (I G(K = 1)) 1C(K = 1)+ 1 (cid:21) (cid:22) (I G(K = 2)) 1C(K = 2); 1 1 t (cid:0) t 1 t (cid:0) t (cid:0) (cid:0) (cid:0) (cid:0) (cid:1) and the variance, (cid:22) (cid:22) (cid:22) P = (cid:21) X + 1 (cid:21) X ; 1 1 1 1 2 (cid:0) (cid:0) (cid:1) 19

where X and X solve the discrete Lyapunov matrix equations: 1 2 G(K = 1)X G(K = 1) X +R(K = 1)(cid:6) R(K = 1) = 0 t 1 t 0 1 t 1 t (cid:0) and G(K = 2)X G(K = 2) X +R(K = 2)(cid:6) R(K = 2) = 0; t 2 t 0 2 t 2 t (cid:0) respectively. Forecasting t+1 given t (cid:15) (cid:150)Transition equation P S ;K Yt;(cid:18) t+1 t+1 j = (cid:0) P (S ;K S ; (cid:1) K )P S ;K Yt;(cid:18) d(S ;K ) t+1 t+1 t t t t t t j j Z (cid:0) (cid:1) = P S ;K R ^ ;S ;K P R ^ ;K S ;K P S ;K Yt;(cid:18) d(S ;K ) t+1; R^ t+1 t+1 j t+1 t t t+1 t+1 j t t t t j t t (cid:0) Z (cid:16) (cid:17) (cid:16) (cid:17) (cid:0) (cid:1) = P S K ;S ;K P R ^ K ;S ;K P (K S ;K )P S ;K Yt;(cid:18) d(S ;K ) t+1; R^ t+1j t+1 t t t+1 j t+1 t t t+1 j t t t t j t t (cid:0) Z (cid:16) (cid:17) (cid:16) (cid:17) (cid:0) (cid:1) = P S K = 2;S ;K (cid:14) R ^ = 0 P (K = 2 S ;K )P S ;K Yt;(cid:18) d(S ;K ) t+1; R^ t+1j t+1 t t 0 t+1 t+1 j t t t t j t t (cid:0) Z (cid:16) (cid:17) (cid:16) (cid:17) (cid:0) (cid:1) + P (S K = 1;S ;K )P (K = 1 S ;K )P S ;K Yt;(cid:18) d(S ;K ); t+1 t+1 t t t+1 t t t t t t j j j Z (cid:0) (cid:1) where S denotes all of the states excluding the interest rates. Since the density t+1; R^ t+1 (cid:0) of the regime K ; conditional on the last period states and regime, P (K S ;K ), is t+1 t+1 t t j discrete, I can break the integral into two parts when it is in a ZIRP regime, and when it is in the normal regime. Notice that when it is in the ZIRP regime, I do not need to track the distribution of interest rates because it is degenerate. 20

(cid:150)Measurement equation = likelihood function (cid:15) ) P y Yt;(cid:18) t+1 j = (cid:0) P y S (cid:1) ;K Yt;(cid:18) P S K ;Yt;(cid:18) P K Yt;(cid:18) dS dK t+1 t+1 t+1; t+1 t+1 t+1 t+1 t+1 j j j Z (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) = P K = 1 Yt;(cid:18) P y S ;K Yt;(cid:18) P S K Yt;(cid:18) dS t+1 t+1 t+1 t+1; t+1 t+1; t+1 j j j Z (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) +P K = 2 Yt;(cid:18) P y S ;K Yt;(cid:18) P S K Yt;(cid:18) dS : t+1 j t+1 Rt+1j t+1 t+1; t+1 j t+1; t+1 (cid:0) Z (cid:16) (cid:17) (cid:0) (cid:1) (cid:0) (cid:1) b Updating (cid:15) (cid:150)Updating states P S ;K Yt+1;(cid:18) t+1 t+1 j P (cid:0) y S ;K Yt(cid:1) ;(cid:18) P S ;K Yt;(cid:18) t+1 t+1 t+1; t+1 t+1 / j j P (cid:0) y S ;K Yt;(cid:18) (cid:1) P (cid:0) S K ;Yt;(cid:18) (cid:1) P K Yt;(cid:18) t+1 t+1 t+1; t+1 t+1 t+1 / j j j P (cid:0) y S ;K Yt;(cid:18) (cid:1) P (cid:0) S K ;Yt;(cid:18) (cid:1) P (cid:0) K = 1 Y (cid:1)t;(cid:18) t+1 t+1 t+1; t+1 t+1 t+1 / j j j +P (cid:0) y S (cid:1) ;K (cid:0) Yt;(cid:18) P S (cid:1) (cid:0) K ;Yt;(cid:18) (cid:1) P K = 2 Yt;(cid:18) : t+1 Rt+1j t+1 Rt+1 t+1; t+1 Rt+1j t+1 t+1 j (cid:0) (cid:0) (cid:0) (cid:16) (cid:17) (cid:16) (cid:17) (cid:0) (cid:1) b b b (cid:150)Updating states probability (cid:15) Since I observe the data y , I observe the interest rate. If R = 1, I deduce that t+1 t+1 P K = 1 Yt+1 = 0; and P K = 2 Yt+1 = 1 t+1 t+1 j j (cid:0) (cid:1) (cid:0) (cid:1) 21

and vice versa. So I do not need to track the long history of the states because when I know the history of Yt I know for certain the history of the states. The distribution of the states at each time is degenerated. In practice, any quarterly federal funds rate that is smaller than 40bp15 is treated as a zero interest rate. 4.2 Data IusetheUnitedStatesquarterlydatafromthethirdquarterof1987(1987q3)tothethird quarter of 2010 (2010q3) for the following three series: real GDP per capita, core personal consumption expenditures (PCE) de(cid:135)ator, and nominal e⁄ective federal funds rate.16 All data are extracted from the Federal Reserve Economic Data (FRED) maintained by the Federal Reserve Bank of St. Louis. The mapping between these observable variables and the state variables in the DSGE models is (cid:1)Yobs = 100((cid:13) +Y ^ Y ^ +z^); t z;t (cid:0) z;t (cid:0) 1 t (cid:25)obs = 100((cid:25) +(cid:25)^ ); t t robs = 100(r+r^); t t where all state variables are in deviations from their ergodic steady state values (corresponding to the ergodic steady state R for the policy rate), (cid:25) ln((cid:5)) and r ln(R). (cid:17) (cid:17) I construct the real GDP per capita series by dividing the nominal GDP series by the population and the GDP de(cid:135)ator. The observable (cid:1)Yobs, the growth rate of real GDP, t corresponds to the (cid:133)rst di⁄erence in logs of this series, multiplied by 100. The log-di⁄erence of the quarterly personal consumption expenditures (PCE) core price index is the measure of in(cid:135)ation. I use the e⁄ective federal funds rate as the measure of the nominal short-term 15Admittedly, this cut-o⁄choice is arbitrary. 16I use an extended sample, starting in 1975q1, to initialize the (cid:133)lter, but the likelihood function itself is evaluated only for the period starting in 1987q3, conditional on the previous sample. 22

rates. 4.3 Prior Choice Tables1and2(columnstwotofour)summarizethepriordistributionsof eachparameter in the regime-switching DSGE model. I (cid:133)x the coe¢ cient of relative risk aversion (cid:27) at 2. I use Gamma distributions for the prior distributions of the parameters that economic theory suggests must be positive. For those parameters that are de(cid:133)ned over the interval [0;1], I use the Beta distribution. For the standard deviation of the structural shocks, I use the Inverse-Gamma distribution. Theergodicmeanforin(cid:135)ationiscenteredat2%,consistentwiththeFederalOpenMarket Committee(cid:146)s long-run goal. The steady state annualized growth rate of output is centred at 2.5%. The prior distribution of the discount factor implies the mean of the annualized real interest rate is 2%. IfollowDelNegroandSchorfheide(2008)tochoosethepriorsforthestandardparameters in the DSGE models. Table 1 contains two nonstandard parameters (P and P ) speci(cid:133)c 11 22 to this regime-switching model, which controls the Markov switching probability of staying in the normal regime at time t+1 when it is in the normal regime at time t and the Markov switching probability of staying in the ZIRP regime at time t + 1 when it is in the ZIRP regime at time t. P is centered at 0:986, which implies an expected duration of staying in 11 the normal regime is 17:86 years. P is centered at 0:825 at prior, which implies an expected 22 duration of staying in the ZIRP regime is 5:7 quarters, consistent with what is observed in the data by 2010Q3. The prior for the price rigidity parameter, (cid:16) ; is centred at 0:5 with a standard deviation p of 0:1, as in Smets and Wouters (2007). The interest rate smoothing parameter, (cid:26) ; is r centered at 0:7. The interest rate feedback to output growth, (cid:30) ; is centred at 0:4, and the y feedback to in(cid:135)ation, (cid:30) ; is centred at 1:5 at priors. (cid:25) 23

All of the structural shocks follow AR(1) processes. Their autocorrelation coe¢ cients are centred at 0:75, with the exception of productivity shocks whose autocorrelation coe¢ cient is centered at 0:4; because this process characterizes the transitory shock to the growth rate of the technology process. 4.4 Parameter Posterior Distribution In order to obtain the posterior distribution of the parameters, I (cid:133)rst obtain the posterior mode by maximizing the likelihood function. The last column of tables 1 and 2 report the posteriormodeofeachparameter. IthenusetherandomwalkMetropolisHastingsalgorithm to draw from the posterior distributions. I store those parameter draws and use them for simulation exercises discussed later. The Markov switching probabilities are well identi(cid:133)ed because, although the priors are concentrated at their mean, the posterior modes of the transition probabilities are very distinguishable from the prior means. The posterior distributions indicate that the expected duration of staying in the normal regime is 25:45 quarters, and the expected duration of staying in the ZIRP regime is 4:44 quarters. One may argue that data seem to suggest that we have been in the ZIRP regime for at least 20 quarters (from 2009Q1 to 2013Q4). There are two reasons why the estimated duration is substantially shorter than what has happened in actuality. First, the data in my estimation stops at the third quarter of 2010, by which there were only 6 quarters of zero interest rate policy. Second, I treat the 8 quarters from 2002Q4 to 2004Q3 as a ZIRP regime (quarterly FFR is less than 40 basis points) so that we have observations of exiting the ZIRP regime. The time of staying in the ZIRP regime is also short here. 24

4.5 The E¢ cacy of the ZIRP in DSGE models Having estimated the DSGE models, I abstract the (cid:133)ltered states, and, starting from 2010Q4, I simulate the U.S. economy forward for 20 quarters under two scenarios. Under the (cid:133)rst scenario, there is no intervention from the central bank, and all of the structural shocks are zero. So, output should gradually go back to its long-term trend, and in(cid:135)ation andinterestratesshouldgraduallygobacktotheirsteadystates. Underthesecondscenario, the economy is under the intervention of ZIRP by the central bank. One complication of the simulation in the regime-switching DSGE model is that agents have uncertainty over the future states. There are 2t possible states at time t. To maintain tractability, I combine the states with similar history and only keep track of 16 states at each period.17 The predicted path of the macro variables is thus the probability weighted average of those 16 states. When I simulate the U.S. economy under the ZIRP for an extended period, I consider keeping interest rates at zero for four, (cid:133)ve, and six quarters at the regime-switching model and keeping interest rates at the 2010Q3 level for four, (cid:133)ve, and six quarters in the model wheretheZIRPisimplementedbythePFRE.Intheregime-switchingmodel, ateachperiod, agents ex ante always attach a non-zero probability of exiting the ZIRP regime in the next period, and the ZIRP regime is realized for four, (cid:133)ve, or six quarters ex post. In the PFRE model, agents know that the ZIRP will be kept for four, (cid:133)ve, or six quarters. I choose four, (cid:133)ve, and six quarters because participants of the Blue Chip Survey of Professional Forecasters (Blue Chip Survey) expected the ZIRP to last four or (cid:133)ve quarters at the end of 2010 when LSAP II was implemented. However, since the 2011 October FOMC meeting, the Blue Chip survey suggests that the expected number of quarters until the lift-up of ZIRP is seven or more (top-coded). Figure 1 and (cid:133)gure 2 show the predicted paths of the output growth rate, in(cid:135)ation, 17See Schorfheide (2005) for how this tractability can be achieved. 25

and the federal fund rate under the ZIRP generated by the regime-switching model and the PFRE model. The red lines in those two (cid:133)gures are the predicted paths of those macro variables without the ZIRP. The blue, green, and magenta lines are the predicted paths with the ZIRP for four, (cid:133)ve, and six quarters, respectively. The black dots are the actual observations. Allvariablesarepercentagemeasuredquarterlyexcepttheoutputgrowthrate, which is annualized. Zero interest rate policy is e⁄ective in boosting output and in(cid:135)ation. Bothofthemodelsconsideredsuggestsubstantiale⁄ectsoftheZIRP.Thee⁄ectsarestronger if the ZIRP is kept for a longer period.18 Also the e⁄ects are not monotonic: Although the overall e⁄ects are positive, the stimulus to output growth rate is most signi(cid:133)cant in the near term, and the positive e⁄ects gradually revert back.19 In early periods for in(cid:135)ation, the e⁄ects are negative, however, the e⁄ects of ZIRP on in(cid:135)ation quickly turn positive and also are long-lasting. Notice that due to the lack of macro shocks, none of the predicted paths should be expected to match the actual data, but the predicted paths generated by the regime-switching model are still close to the actual realizations, which is a desirable feature of the model because it implies that we only need small structural macro shocks to match the observed data. Figure 3 summarizes the e⁄ects of the ZIRP in the regime-switching DSGE models. At eachtimeofthesimulatedpath, Itakethedi⁄erenceofthemacrovariableswithandwithout the ZIRP intervention, and sum up over 20 quarters. This (cid:133)gure plots the total e⁄ects. The squares stand for the mean e⁄ects and the circles, which are the 90 percent credible sets, re(cid:135)ect the uncertainty of posterior parameter draws. Blue, green, and magenta represent the macro e⁄ects of the policy of keeping interest rates at zero for four, (cid:133)ve, and six quarters, respectively. The regime-switching DSGE model suggests that keeping interest rates at zero 18Notice that in the regime-switching model, the future paths of macro variables are expected, while in the perfect foresight model, there is no uncertainty. 19See (cid:133)gure 3. 26

for four, (cid:133)ve, and six quarters on average20 increases the output growth rate by 0:064%, 0:09%, and 0:12%; respectively21 and in(cid:135)ation by 0:544%, 0:72%, and 0:90%; respectively22 over the course of 20 quarters cumulatively. The e⁄ects implied by the upper bound of the 90% credible sets can be very signi(cid:133)cant. Figure 2 shows that the non-regime-switching model where the ZIRP is implemented by the PFRE suggests much stronger stimulus on output growth rate at peak and on in(cid:135)ation: On average, keeping ZIRP for four, (cid:133)ve, and six quarters increases output growth rate by 2:62%, 5:00%, and 9:41% at peak, respectively, though the total cumulative e⁄ect gradually reverts back, reaching almost zero after 20 quarters.23 The e⁄ect of the ZIRP on in(cid:135)ation implied by the PFRE model is always large and long-lasting. On average, the cumulative e⁄ect on in(cid:135)ation of keeping ZIRP for four, (cid:133)ve, and six quarters24 is 8:53%, 17:92%, and 36:29%; respectively. As mentioned earlier, those two models are fundamentally di⁄erent in how agents formulate their expectations about future monetary policy. The central bank(cid:146)s "extended period" language is treated as completely credible by the agents in the PFRE model, whiletheregime-switchingmodelabstractsfromthecentralbank(cid:146)sforwardguidance. Notice that the PFRE model generates a spurious rise in in(cid:135)ation. The predicted path generated by the PFRE is explosive while the predicted path generated by the regimeswitching model is closer to the actual path; therefore, I argue that the regime-switching DSGE model is better suited to analyzing the e⁄ects of ZIRP. 4.6 Robustness 20Average over 500 simulations. 21Peak at 0:35%, 0:44%, and 0:54%; respectively. 22Peak at 0:544%, 0:72%, and 0:90%; respectively. 23However, the stimulus on output level is positive and long-lasting. On average, keeping ZIRP for four quarters increases output level by 25:01% over the course of 20 quarters cumulatively. 24In the regime-switching model, zero interest rates are realized ex post. Ex ante, agents always expect to exit zero-interest-rate regime with some non-zero probability. 27

The advantage of estimating the model using data up to 2010Q3 is that we can compare the predictive distributions of the macro variables derived from the model with the actual observables because we have some realizations of the macro variables of interest. Obviously the Markov transition probability estimated depends on the number of quarters of zero interest rates. Also the expected quarters until the lift-o⁄of ZIRP have changed. A natural question to ask is do the implications of ZIRP change if we estimate the model up to the current data and calibrate the quarters of ZIRP according to current market expectations? I estimate the same model using the data up to the second quarter of 2013, and then simulate the economy forward with and without the ZIRP. ZIRPis realized for seven or eight quarters becausesincethe2011OctoberFOMCmeeting, themedianresponseoftheBlueChipsurvey suggests that the expected number of quarters until the lift-o⁄of ZIRP is seven or more. Figure 4 shows the predicted paths of the output growth rate, in(cid:135)ation, and the federal fund rates with and without the ZIRP generated by the regime-switching model. The red lines are the predicted paths of those macro variables without the ZIRP. The blue and green lines are the predicted paths with the ZIRP for seven and eight quarters, respectively. On average, keeping ZIRP for seven and eight quarters increases output growth rate by 0:105% and 0:137%; respectively, and in(cid:135)ation by 0:666% and 0:794%; respectively, over 20 quarters cumulatively. So the regime-switching model suggests that ZIRP is e⁄ective in boosting economic activity. 5 Conclusions Given the novelty of unconventional monetary policies, it is critical for economists to construct models capable of assessing their e⁄ectiveness and guiding policy. This paper develops a new approach to modeling the ZIRP that not only (cid:133)ts the macro data featuring a persistent period of extremely low interest rates, and generates a predicted path closer to 28

the actual path, but also provides a plausible mechanism for modeling the exit of the zero interest rate policy. I (cid:133)nd that the Federal Reserve(cid:146)s decision to maintain a zero interest rate for a lengthy period is likely to be e⁄ective in boosting economic activity. References Adam, K., and Billi, R. (2007). (cid:145)Discretionary Monetary Policy at the Zero Lower Bound on Nominal Interest Rates(cid:146), Journal of Monetary Economics, 54, 728-752. An, S. and Schorfheide, F. (2007a.) (cid:145)Bayesian analysis of DSGE models(cid:146), Econometric Reviews, vol. 26(2-4), pp. 113-72. Aruoba, B. and Schorfheide, F. (2012) (cid:145)Macroeconomic Dynamics Near the ZLB: A Tale of Two Equilibria(cid:146), working paper. AndrØs, J., L(cid:243)pez-Salido, J. and Nelson, E. (2004). (cid:145)Tobin(cid:146)s imperfect asset substitution in optimizing general equilibrium(cid:146), Journal of Money, Credit and Banking, vol. 36(4), pp. 665-90. Baumeister, C. and Benati, L. (2010). (cid:145)Unconventional monetary policy and the Great Recession(cid:146), ECB Working Paper 1258. Bernanke, B. and Reinhart, V. (2004). (cid:145)Conducting Monetary Policy at Very Low Short- Term Interest Rates(cid:146), American Economic Review, vol. 94(2), pp. 85-90. Blake, A. (2012). (cid:145)Fixed Interest Rates over Finite Horizons(cid:146), Bank of England Working Paper 454. Braun, R. A., and K(cid:246)rber, L. M. (2011) (cid:145)New Keynesian Dynamics in a Low Interest Rate Environment(cid:146), Journal of Economic Dynamics and Control, 35(12), pp. 2213-2227. Carlstrom, C., Fuerst, T., and Paustian, M. (2012) (cid:145)In(cid:135)ation and Output in New Keynesian Models with a Transient Interest Rate Peg(cid:146), Federal Reserve Bank of Cleveland Working Paper 12-34. 29

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Farmer, R., Waggoner, D. and Zha, T. (2011). (cid:145)Minimal state variable solutions to Markovswitching rational expectations models(cid:146), Journal of Economic Dynamics and Control, Elsevier, vol. 35(12), pp. 2150-2166. FernÆndez-Villaverde, J., Gordon, G., Guerr(cid:243)n-Quintana, P. and Rubio-Ram(cid:237)rez, J. (2012). (cid:145)Nonlinear Adventures at the Zero Lower Bound(cid:146), NBER working paper 18058. Foerster, A., Rubio-Ram(cid:237)rez, J., Waggoner, D. and Zha, T. (2012). (cid:145)Perturbation Methods for Markov-Switching Models(cid:146), working paper. Hamilton, J. D. (1994). Times Series Analysis. Princeton University Press, Princeton, NJ. Justiniano, A., Primiceri, G. and Tambalotti, A. (2010). (cid:145)Investment shocks and business cycles(cid:146), Journal of Monetary Economics, vol. 57(2), pp. 132-45. Justiniano, A., Primiceri, G. and Tambalotti, A. (2011). (cid:145)Is there a trade-o⁄between in(cid:135)ation and output stabilization?(cid:146), Working Paper, Federal Reserve Bank of Chicago. Kim, C. and Nelson, C. (1999). (cid:145)State-Space Models with Regime Switching: Classical and Gibbs-Sampling Approaches with Applications(cid:146), The MIT Press (May 7, 1999). Judd, K., Maliar, L. and Maliar, S. (2011). (cid:145)A Cluster-Grid Projection Method: Solving Problems with High Dimensionality(cid:146), Working Paper, Stanford University, 15965. Liu, Z., Waggoner, D. and Zha, T. (2011). (cid:145)Sources of macroeconomic (cid:135)uctuations: A regime-switching DSGE approach(cid:146), Quantitative Economics, Econometric Society, vol. 2(2), pp. 251-301, 07. Reifschneider, D. and Williams, J. (2000). (cid:145)Three lessons for monetary policy in a lowin(cid:135)ation era(cid:146), Conference Series ; [Proceedings], Federal Reserve Bank of Boston, pp. 936-978. Schorfheide, F. (2005). (cid:145)Learning and Monetary Policy Shifts,(cid:146)Review of Economic Dynamics, 8(2), 2005, pp. 392-419. Smets, F. and Wouters, R. (2003). (cid:145)An estimated dynamic stochastic general equilibrium model of the Euro Area(cid:146), Journal of the European Economic Association, vol. 1(5), pp. 1123-75. 31

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Table 1: Parameter Prior and Posterior Distribution: Structural Parameters. Prior Posterior Dist Mean Std Mode 400(cid:13) G 2.5 0.5 2.2951 400(cid:25) G 2.0 0.5 1.8648 400((cid:12) 1 1) G 1.0 0.25 0.9141 (cid:0) (cid:0) (cid:23) G 2.0 0.5 1.8318 (cid:16) B 0.5 0.1 0.8962 p (cid:26) B 0.7 0.1 0.7368 r (cid:30) G 1.5 0.25 1.8036 (cid:25) (cid:30) G 0.4 0.2 0.3878 y P B 0.986 0.05 0.9607 11 P B 0.825 0.10 0.7749 22 33

Table 2: Parameter Prior and Posterior Distribution: Shock Process Parameters. Prior Posterior Dist Mean Std Mode (cid:26) B 0.4 0.2 0.4471 z (cid:26) B 0.75 0.1 0.8845 b (cid:26) B 0.75 0.1 0.7338 g 100(cid:27) IG1 0.5 4 0.6779 z 100(cid:27) IG1 0.5 4 0.9235 b 100(cid:27) IG1 0.5 4 0.2551 m 100(cid:27) IG1 0.5 4 0.8033 g 34

6 Figures Fig. 1: Simulation of the U.S. economy forward from 2010Q4 under the ZIRP intervention in the regime switching DSGE model. The red lines show the mean of predicted paths of the macro variables without shocks and under no intervention. The blue, green, and magenta lines are the predicted paths with the ZIRP for four, (cid:133)ve, and six quarters, respectively.The black dots are actual observations. 35

Fig. 2: Simulation of the U.S. economy forward from 2010Q4 under the ZIRP intervention implemented by the PFRE. The red lines show the mean of predicted paths of the macro variables without shocks and under no intervention generated by the DSGE models. The blue, green, and magenta lines are the predicted paths with the ZIRP for four, (cid:133)ve, and six quarters, respectively. 36

Fig. 3: Summaryofmacroe⁄ectsofZIRPintheregime-switchingDSGEmodel. The squares stand for mean e⁄ects and the circles, which are the 90 credible sets, re(cid:135)ect the uncertainty of posterior parameter draws. Blue, green, and magenta represent the macro e⁄ects of the policy of keeping interest rates at zero for four, (cid:133)ve, and six quarters, repectively. The Macro Effects of Zero Interest Rate Policy 2.5 2 1.5 1 0.5 0 Mean Effects of Keeping ZIRP for 4 Quarters 90% Credible Sets of Effects of Keeping ZIRP for 4 Quarters Mean Effects of Keeping ZIRP for 5 Quarters 90% Credible Sets of Effects of Keeping ZIRP for 5 Quarters Mean Effects of Keeping ZIRP for 6 Quarters -0.5 90% Credible Sets of Effects of Keeping ZIRP for 6 Quarters -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Cumulative Change to Annualized Real Output Growth Rate over 20 Quarters 37

Fig. 4: Simulation of the U.S. economy forward from 2013Q3 under the ZIRP intervention Fig. 4: Simulation of the U.S. economy forward from 2013Q3 under the ZIRP intervention in the regime switching DSGE model. The red lines show the mean of predicted paths of the in the regime switching DSGE model. The red lines show the mean of predicted paths of the macro variables without shocks and under no intervention. The blue and green lines are the macro variables without shocks and under no intervention. The blue, green, and magenta predicted paths with the ZIRP for seven and eight quarters, respectively. lines are the predicted paths with the ZIRP for four, five, and six quarters, respectively. 38 38

7 Appendix: Proof that Schorfheide (2005) and Liu et al. (2011) give rise to the same solution This section assumes that the only regime-switching parameter is the target steady state interest rate Schorfheide (2005): ^ ^ ^ R = (cid:26) R +(1 (cid:26) )’ (cid:25)^ +(1 (cid:26) )’ y^ +" +(1 (cid:26) )(1 ’ )R t R t (cid:0) 1 (cid:0) R (cid:25) t (cid:0) R y t R;t (cid:0) R (cid:0) (cid:25) t(cid:3) ^ = (cid:26) R +(1 (cid:26) )’ (cid:25)^ +(1 (cid:26) )’ y^ +" ; R t (cid:0) 1 (cid:0) R (cid:25) t (cid:0) R y t (cid:3)R;t where R R 1 2 " = " +(1 (cid:26) )(1 ’ ) log ; log ^e : (cid:3)R;t R;t (cid:0) R (cid:0) (cid:25) R R s;t (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) Solution by gensys can be written as below, where I assume the (cid:133)rst shock is " : (cid:3)R;t 1 y = (cid:2) y +(cid:2) z +(cid:2) (cid:2)s(cid:2) E z t 1 t 1 0 t y f z t t+s (cid:0) s=1 X 1 = (cid:2) y +(cid:2) z +(1 (cid:26) )(1 ’ )(cid:2) 1 (cid:2)s(cid:2) 0 log R 1 ;log R 2 Ps: 1 t (cid:0) 1 0 t (cid:0) R (cid:0) (cid:25) y f z2 3 R R s=1 0 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) X 6 7 4 5 So the constant is R R 1 2 (cid:2) (K ) = (1 (cid:26) )(1 ’ )(cid:2) log ;log e c t (cid:0) R (cid:0) (cid:25) 0 (cid:1) 1 (cid:1) R R s;t (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) 1 = (1 (cid:26) )(1 ’ )(cid:2) 1 (cid:2)s(cid:2) 0 log R 1 ;log R 2 Pse : (cid:0) R (cid:0) (cid:25) y f z2 3 R R s;t s=1 0 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) X 6 7 4 5 Now I will prove that Liu, Waggoner, and Zha (2011) give rise to the same solution. Assuming the (cid:133)rst row of the equilibrium conditions is for the federal funds rate: 39

(1 (cid:26) )(1 ’ ) log R1 ;log R2 (cid:0) (cid:0) R (cid:0) (cid:25) R R 2 (cid:0) 0 ;0 0 (cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) 1 3 y t (cid:0) 1 0 y t 1 . = (cid:0) + 6 B . . C 7 " ^e s;t # " 0 P #" ^e s;t 1 # 6 B C 7 (cid:0) 6 @0:::; I A 7 6 2 7 4 5 (cid:9) 0 z (cid:5)(cid:17) t + t : " 0 I 2 #" (cid:23) t # " 0 # Q 0 Perform QZ decomposition on (cid:0) and (cid:0) and then premultiply both sides by : 0 1 " 0 I 2 # (1 (cid:26) )(1 ’ ) log R1 ;log R2 (cid:0) (cid:0) R (cid:0) (cid:25) R R Q nxn 0 2 Q 0 (cid:3)Z 0 ;0 0 (cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) 1 3 y t . " 0 I 2 # 6 B . . C 7 " ^e s;t # 6 B Cnx2 7 6 @[0] :::; I A 7 6 2xn 2 7 4 5 Q 0 Q(cid:10)Z 0 y Q 0 (cid:9) 0 z (cid:5)(cid:17) = nxn 0 0 t (cid:0) 1 + nxn t + t ; " 0 I 2 #" 0 P #" ^e s;t 1 # " 0 I 2 # " 0 I 2 #" (cid:23) t # " 0 #! (cid:0) and thus: (1 (cid:26) )(1 ’ ) log R1 ; log R2 (cid:0) (cid:0) R (cid:0) (cid:25) R R 2 (cid:3)Z 0 Q0 0 (cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) 1 3 y t . (7.1) 6 B . . C 7 " ^e s;t # 6 B C 7 6 @0 I A 7 6 2 7 4 5 (cid:10)Z 0 y Q(cid:9) 0 z Q(cid:5)(cid:17) = 0 t (cid:0) 1 + t + t : " 0 P #" ^e s;t 1 # " 0 I 2 #" (cid:23) t # " 0 # (cid:0) Let w = Z y ; and w = Z y : 7.1 becomes: t 0 t t 1 0 t 1 (cid:0) (cid:0) (1 (cid:26) )(1 ’ ) log R1 ; log R2 (cid:0) (cid:0) R (cid:0) (cid:25) R R (cid:3)w t +Q0 0 (cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) 1^e s;t . . B . C B C = (cid:10)w +Q@(cid:9)z +Q(cid:5)(cid:17) ; A t 1 t t (cid:0) 40

and thus: (1 (cid:26) )(1 ’ ) log R1 ; log R2 (cid:3) (cid:3) w (t) (cid:0) R (cid:0) (cid:25) R R 11 12 1 Q00 (cid:2) 0 (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) 1^e s;t +(cid:9)z t +(cid:5)((cid:17)7 t .12) " 0 (cid:3) 22 #" w 2 (t) # (cid:0) . . BB . C C BB C C @@ A A xt (cid:10) (cid:10) w (t 1) 11 12 1 | {z } = (cid:0) : " 0 (cid:10) 22 #" w 2 (t 1) # (cid:0) Let M = (cid:10) 1(cid:3) and solve forward: (cid:0)22 22 1 w (t) = E Ms 1(cid:10) 1x (t+s) 2 (cid:0) t (cid:0) (cid:0)22 2 " # s=1 X 1 = Ms 1(cid:10) 1x (t+s) : (cid:0) (cid:0) (cid:0)22 2 " # s=1 X Replace x with their de(cid:133)nition and use the fact E (cid:17) = 0 : t t t+s (1 (cid:26) )(1 ’ ) log R1 ; log R2 (cid:0) R (cid:0) (cid:25) R R 1 = (cid:0) E t 2 Ms (cid:0) 1(cid:10) (cid:0)22 1Q 2 (cid:1) 0(cid:9)Z t+s +0 (cid:2) 0 . (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) 1^e s;t+s 13 6X s=1 B B . . C C7 6 B B C C7 4 @ @ A A5 (1 (cid:26) )(1 ’ ) log R1 ; log R2 (cid:0) R (cid:0) (cid:25) R R 1 = (cid:0) 2 Ms (cid:0) 1(cid:10) (cid:0)22 1Q 2 (cid:1) 0(cid:9)Z t+s +0 (cid:2) 0 . (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) 1^e s;t+s +(cid:5)(cid:17) t+s 13; 6X s=1 B B . . C C7 6 B B C C7 4 @ @ A A5 and thus: 41

1 Q (cid:5)(cid:17) = (cid:10) Ms 1(cid:10) 1Q ((cid:9)(E z E z )) 2 (cid:1) t+1 22 (cid:0) (cid:0)22 2 (cid:1) t+1 t+s (cid:0) t t+s s=1 X (1 (cid:26) )(1 ’ ) log R1 ;log R2 (cid:0) R (cid:0) (cid:25) R R 1 + (cid:10) 22 Ms (cid:0) 1(cid:10) (cid:0)22 1Q 2 (cid:1) 00 (cid:2) 0 . (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) 1(E t+1 ^e s;t+s (cid:0) E t ^e s;t+s )1: X s=1 BB . . C C BB C C @@ A A If the solution is unique: Q (cid:5) = (cid:8)Q (cid:5): 1 2 (cid:1) (cid:1) Premultiplying 7.2 by [I (cid:8)] : (cid:0) (cid:3) (cid:3) (cid:8)(cid:3) w (t) 11 12 22 1 (cid:0) " 0 I #" w 2 (t) # (1 (cid:26) )(1 ’ ) log R1 ; log R2 Q (cid:8)Q (cid:0) R (cid:0) (cid:25) R R 1 (cid:1) (cid:0) 2 (cid:1) 00 (cid:2) 0 (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) 1^e s;t 1 (cid:0) " 0 # . . BB . C C BB C C @@ A A (cid:10) (cid:10) (cid:8)(cid:10) w (t 1) Q (cid:8)Q 11 12 22 1 1 2 = (cid:0) (cid:0) + (cid:1) (cid:0) (cid:1) (cid:9)z t " 0 0 #" w 2 (t 1) # " 0 # (cid:0) 0 : (cid:0) " E t 1 s=1 Ms (cid:0) 1(cid:10) (cid:0)22 1x 2 (t+s) # (cid:2)P (cid:3) Finally, 42

(1 (cid:26) )(1 ’ ) log R1 ; log R2 (cid:3) 1 (cid:3) 1((cid:3) (cid:8)(cid:3) ) Q (cid:8)Q (cid:0) (cid:0) R (cid:0) (cid:25) R R y t +Z (cid:0)11 (cid:0)11 12 (cid:0) 22 1 (cid:1) (cid:0) 2 (cid:1) 00 0 (cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) 1^e s;t 1 " 0 I #" 0 # . . BB . C C BB C C (cid:3) 1 (cid:3) 1((cid:3) (cid:8)(cid:3) ) (cid:10) @@ (cid:10) (cid:8)(cid:10) A A = Z (cid:0)11 (cid:0)11 12 (cid:0) 22 11 12 (cid:0) 22 Z y 0 t 1 " 0 I #" 0 0 # (cid:0) (cid:3) 1 (cid:3) 1((cid:3) (cid:8)(cid:3) ) Q (cid:8)Q +Z (cid:0)11 (cid:0)11 12 (cid:0) 22 1 (cid:1) (cid:0) 2 (cid:1) (cid:9)z t " 0 I #" 0 # (cid:3) 1 (cid:3) 1((cid:3) (cid:8)(cid:3) ) 0 Z (cid:0)11 (cid:0)11 12 (cid:0) 22 (cid:0) " 0 I #" E t 1 s=1 Ms (cid:0) 1(cid:10) (cid:0)22 1Q 2 (cid:9)z t+s # (cid:1) Z (cid:3) (cid:0)11 1 (cid:3) (cid:0)11 1((cid:3) 12 (cid:2) (cid:0) P(cid:8)(cid:3) 22 ) (cid:3) (cid:0) " 0 I # (cid:1) 0 (1 (cid:26) )(1 ’ ) log R1 ;log R2 2 (cid:0) R (cid:0) (cid:25) R R 3 : 6 E t 2 1 s=1 Ms (cid:0) 1(cid:10) (cid:0)22 1Q 2 0 (cid:2) 0 (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) 1^e s;t+s 3 7 6 (cid:1) . . 7 6 6P B . C 7 7 6 6 B C 7 7 4 4 @ A 5 5 By simplifying notation, I can rewrite the above equation as: (1 (cid:26) )(1 ’ ) log R1 ;log R2 (cid:0) R (cid:0) (cid:25) R R y t = (cid:2) 1 y t 1 +(cid:2) 0 0(cid:9)z t +0 (cid:2) 0 (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) 1^e s;t 1 (cid:0) . . B B . C C B B C C @ @ A A (1 (cid:26) )(1 ’ ) log R1 ;log R2 (cid:0) R (cid:0) (cid:25) R R 1 +(cid:2) y (cid:2)s f(cid:0) 1(cid:2) z E t 0(cid:9)z t+s +0 (cid:2) 0 (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) 1^e s;t+s 1; . X s=1 B B . . C C B B C C @ @ A A where (cid:3) 1 (cid:3) 1((cid:3) (cid:8)(cid:3) ) (cid:10) (cid:10) (cid:8)(cid:10) (cid:2) = Z (cid:0)11 (cid:0)11 12 (cid:0) 22 11 12 (cid:0) 22 Z ; 1 0 " 0 I #" 0 0 # 43

(cid:3) 1 (cid:3) 1((cid:3) (cid:8)(cid:3) ) Q (cid:8)Q (cid:2) 0 = Z (cid:0)11 (cid:0)11 12 (cid:0) 22 1 (cid:1) (cid:0) 2 (cid:1) ; " 0 I #" 0 # (cid:3) 1 (cid:3) 1((cid:3) (cid:8)(cid:3) ) (cid:2) = Z (cid:0)11 (cid:0)11 12 (cid:0) 22 ; y (cid:0) " 0 I # (cid:2) = M; f and (cid:2) = (cid:10) 1Q : z (cid:0)22 2 (cid:1) This expression is exactly the same as treating ^e as a shock as in Schorfheide (2005). s;t+s 44