Design Limits in Regime&Switching cases - Quantitative and ...

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Design Limits in Regime&Switching cases - Quantitative and ...

43.5NCoptimal Cfeasible Cunfeasible C32.521.510.50­4 ­3 ­2 ­1 0 1 2 3 4Figure 2: The spectral representation of the process x t = 0:5x t 1 + " t (solid line) andfor the optimally controlled process which reduces to a white noise (dashed line). Thedotted line represents the process resulting from a feasible control. The dashed-dotline reproduces an unfeasible result.This theorem states that, in backward looking environments, even if the policymakeris able to stabilize the state variable of interest, the variance contributions atsome frequencies will necessarily exacerbate, since, the sensitivity function, S (!) 7 ;cannot be less than one at all the frequencies: in other words, it must be thatf xjC (!) > f xjNC (!) for some !:Notice that, for all the stable free dynamics, the value of the Bode’s integral isalways zero and independent of the particular model under exam. Further, it isindependent of the policy rule. Figure 2 proposes again the spectral representation ofa stable autoregressive process (the solid line) and the spectrum of the process when avariance minimized rule is adopted (the dashed line). As anticipated before, Theorem2 ensures that C’is an unfeasible control. Further, it shows that any randomly chosenfeasible control (represented, for instance, by the dotted line), even if able to lowerthe overall variance, produces some frequency trade-o¤s.7 Trough the paper, we frequently switch from the equivalent notations S e i! and S (!). Theformer is more formal, the latter is more intuitive and synthetic: in covariance stationary cases,the spectral representation and, consequently, the sensitivity function, are always real objects andfunctions of the frequency, !:7


we can state that A (!) "plays the role" of the complex and conjugate product in(10).In what follows we exploit (11) to characterize the design limits of regime switchingmodels. While we recognize that the MSAR(1) is a very simple example which hardlyallows a relevant economic application, we think it is important to consider it as astarting point because, while calculations remain tractable, it allows to derive quitegeneral considerations of regime switching cases.4 Design Limits in MSAR(1)The goal of this section is the derivation of the design limits in the case of theMSAR(1) model, de…ned in (8). When a feedback-type control is considered, (8)may be rewritten asx t = a ( t ) x t 1 & ( t ) u t + " t (12)For the sake of simplicity, we just assume & ( t ) not only state-independent but alsoequal to unity. The generic rule is on the form u t = F (L) x t 1 ; and we supposeF (L) = F so that we can …nally rewrite (12) asx t = (a ( t ) F )x t 1 + " t = (13)= a C ( t ) x t 1 + " tand we can still work in a MSAR(1) framework. Given (9), we compare:withf xjNC e i! = 2 "2f xjC e i! = 2 "2K NC 1(1 NC1 e i! )(1 NC1 e i! ) +H NC 1(1 NC2 e i! )(1 NC2 e i! )K C 1(1 C 1 e i! )(1 C 1 ei! ) +H C 1(1 C 2 e i! )(1 C 2 ei! )where C and NC correspond, respectively, to the case in which we suppose thepolicymaker intervenes through the choice of F and the case in which u t = 0. Weare interested in these two cases because, as we showed before, the Bode’s integralrepresents an aggregate measure of design limitations the policymaker must face withthe respect to the case without control.In linear time invariant (LTI) frameworks (for instance, in the analysis of model(11)), we are used to consider f yjNC (e i! ) and f yjC (e i! ) in the following forms:f yjNC (!) = f (!) j G NC (!) j 2 (14)!!andf yjC (!) = f (!) j G C (!) j 2 (15)11


where f (!) = 2 and j G NC (!) j 2 and j G C (!) j 2 represent the complex and conjugateproducts of the transfer functions of the, respectively, unconstrained and constrainedsystems. As described in (5), in linear frameworks, the sensitivity functionis usually derived by the knowledge of the transfer functions of the controlled anduncontrolled models. In our case, even if we deal with nonlinear objects, we canstill de…ne the complex and conjugate product of the sensitivity function via theknowledge of the spectra, as described in (6), wheref xjC (!)f xjNC (!)=K C 1(1 C 1 e i! )(1 C 1 ei! ) +H C 1(1 C 2 e i! )(1 C 2 ei! )!K NC 1(1 NC1 e i! )(1 NC1 e i! ) +H NC 1(1 NC2 e i! )(1 NC2 e i! )The Bode’s integral can now be computed:RKB = log j S (!) j 2 d!:! 1Taking the log of j S (!) j 2 , we end up with an expression on the formwherelog j S (!) j 2 = log X C (!) + log Y NC (!) log Y C (!) log X NC (!)X C (!) = K C 1 C 2 e i! 1 C 2 e i! +H C 1 C 1 e i! 1 C 1 e i!Y NC (!) = 1 NC1 e i! 1 NC1 e i! 1 NC2 e i! 1 NC2 e i!Y C (!) = 1 C 1 e i! 1 C 1 e i! 1 C 2 e i! 1 C 2 e i!X NC (!) = K NC 1 NC2 e i! 1 NC2 e i! +H NC 1 NC1 e i! 1 NC1 e i! :The Bode’s constraint can therefore be rewritten asRRRKB = log X C (!) d! + log Y NC (!) d! log Y C (!) d!Rlog X NC (!) d!:The terms Y NC (!) and Y C (!) appear familiar because they characterize the transferfunctions of LTI models: By simple algebra we can writelogY NC (!) = log j 1 NC1 e i! j 2 j 1 NC2 e i! j 2 = log j e i! NC1 j 2 j e i! NC2 j 212


which allows to immediately apply the Wu and Jonkheere lemma (see Brock et al.(2006) for details).Lemma 4Rlog j e i!r j 2 d! = 0 if j r j< 1; 2 log j r j 2 otherwise.Now we can writeRlog Y NC (!) d! = 4 P v ilog j NCv ij; i 2 v i if j NCi j> 1; 8i = 1; 2:Similar observations apply to Y C (!), so thatRlog Y NC (!) d!Rlog Y C (!) d! = P4v ilog j NCv ijPr ilog j C r ij ; (16)i 2 fv i g if j NCi j> 1; i 2 fr i g if j C i j> 1; 8i = 1; 2:From (20) one is tempted to conclude that one part of the design limits formula resemblesthe linear framework’s result: the Bode’s integral is zero in case of a (global)stable free dynamics. Notice, however, that, contrary to Brock et al.(2006), we usedthe notion of the spectral representation to get the complex and conjugate productof the sensitivity function and the condition of stationarity is necessary to derive thespectral representation. Therefore, the condition of global stationarity must hold inboth the uncontrolled and controlled model. It follows that, as in linear frameworks,the Bode’s integral is zero in case of a stable free dynamics, while we cannot stateanything regarding unstable uncontrolled models. This observation does not necessarilyrule out interesting comparisons with positive Bode values in the underlyingmodels: instabilities in each AR(1) model are neither a necessary nor a su¢ cientcondition for global instability (Costa et al. (2005)).We are now left with the terms X C (!) and X NC (!) : Let’s start from the former.which is on the formso thatX C (!) = K C 1 C 2 e i! 1 C 2 e i! +H C 1 C 1 e i! 1 C 1 e i!RX C (!) = A C + B C cos !log X C (!) d! =Rlog A C + B C cos ! d! (17)13


whereA C = K C (1 + C22 ) + H C 1 + C21B C = 2 K C C 2 + H C C 1 : (18)Similarly,RRlog X NC (!) d! = log A NC + B NC cos ! d! (19)From Gradshteyn and Ryzhik (1980) 11Rlog (a + b cos !) d! = 2 log a + p a 2 b 22(20)so thatRlog X C (!) d!Rlog A C + B C cos ! Rlog X NC (!) d! =Rlog A NC + B NC cos ! d! =A C + p A= 2 logC2 B C2A NC + p 12A NC2 B NC2The above results can be collected and summarized in the following Bode’s integraltheorem for regime-switching cases.Theorem 5 Given model (15), the value of the Bode’integral corresponds toKB = 2 log AC + p A C2 B C2 PA NC + p A NC2 B + 4 log j NCC2 v ijv iwith i 2 fv i g if j NCi j> 1; i 2 fr i g if j C i j> 1; 8i = 1; 2where A NC ; A C ; B NC and B NC are de…ned as in (22).Pr ilog j C r ij (21)Theorem 5 de…nes design limits in case of a MSAR(1) model. Its generic analyticformula depends on the policy rule; a (i) and p ij with i; j = 1; 2: This leads us to noticetwo important di¤erences with the respect to the LTI cases. First, the Bode’s integralis model speci…c, and therefore subject to model misspeci…cations, even in backwardlooking cases. Second, the presence of the terms A C and B C suggests that the value ofthe Bode’s integral can be a¤ected by the policy rule, consequently, one can associateto it the role of an endogenous constraint, similar to an externality e¤ect, rather than11 See Gradshteyn and Ryshink (1880), page 527.12 Notice that the explicit solution of the integral exists only for A jBj and A c jB c j : Thisrestriction, however, is not binding in any of the simulations presented next.14


the exogenous constraint, typical of linear frameworks. This certainly constitutes anadditional motivation to consider the analysis of design limits important in any policyevaluation exercise.In the next section, we present some simulations in order to understand the behaviorof the Bode’s integral. Unless otherwise stated, we consider the varianceminimizing rule as the candidate one: We think it is useful here to make a digressionon the characteristics of the variance of a regime-switching model and on theability of any rule of the type described in (15) to control it. The total variabilitycan be decomposed in two parts: the variability-within the underlying models andthe variability-between the models, linked, respectively, to the shock process " t andthe Markov Chain, t . The variability-within depends, of course, on the autoregressivestructure of the underlying models, a (1) and a (2). The variability between themodels is related to the measure or amount of the switching, once it occurs. Inother words, the closer the models in terms of variability (in our simple case, thecloser ja (1)j and ja (2)j), the lower is the amount of the transition variability. Thereis, however, a second component of the transition variability, which is given by theprobability of the switching. The higher p 12 and p 21 (or correspondingly, the lower p 11and p 22 ), the more frequent is the switching: indeed, the unconditional probability ofswitching is given by 1 p 12 + 2 p 21 : Notice that, by (15), the policy rule cannot a¤ect(reduce) the transition probabilities and therefore, the frequency of the switching.Those observations will turn out to be important for what follows.5 The dynamics of the Bode’s integralGiven the complexity of analytical formulae, we present some simulations related tomodel (15), with F corresponding at the variance minimizing rule and 2 " = 1 . InSection 5.1.1 we consider the case in which the transition probabilities p 11 and p 22are both set equal to 0:5 while we consider di¤erent combinations of a (1) and a (2) :We call it the case of symmetric transition probabilities. In Section 5.1.2 we proposethe same analysis in the case in which the transition probabilities are both greaterand less than 0:5; so to investigate the e¤ects of di¤erent transition probabilities onthe Bode’s integral. Finally, we study the e¤ects of di¤erent simple policy rules onthe measure of design limits.5.1 The Bode’s integral across the models5.1.1 The case of symmetric transition probabilitiesIn this section we suppose symmetric transition probabilities ( p 11 = p 22 = p 21 =p 12 = 0:5). The following …gures represent three simulated cases. For each case weuse three graphs: the …rst one refers to the global MSAR(1) model, while the othertwo refer to the respective underlying models. In all of them the solid line represent15


the spectral density of the free dynamics of the model while the dashed line illustratesthe spectral density of the constrained processes when the variance minimizing ruleis applied.Figure (3) shows that the control is able to eliminate all the temporal dependences.In this case the two underlying models are close to each other and the variation due tothe switching appears to be negligible: the total variability of the controlled processis only slightly above 1 13 .Figure 3: Spectrum ofMSAR(1), a (1) = 0:5;a (2) = 0:2:Figure 4: Spectrum of theunderlying model 1:AR(1) with a = 0:5:Figure 5: Spectrum of theunderlying model 2:AR(1) with a = 0:2:The Bode’s integral is negative, even if the controlled spectrum presents some frequencytrade-o¤s (stabilization is improved at the low frequencies but exacerbated atthe high ones). A negative Bode’s integral implies that, even though some frequencytrade-o¤s result due to policy intervention, overall, the frequency-speci…c variabilitycontributions are reduced in comparison with the free dynamics case, in the sensethat, while we are able to reduce the overall variability, the frequency-speci…c tradeo¤sthat result appear diminished.In the underlying models we observe very di¤erent frequency-speci…c dynamics.Even if, in our set up, the policymaker is supposed to care only about global dynamics,he may want to take into account the spectra of the underlying models, both of whichviable, given the assumption of ergodicity of the Markov Chain, in particular, if heassociates di¤erent losses to di¤erent frequency ranges.13 In the limiting case, in which the models are identical (so that there is no variation due to theswitching), the controlled process would result a white noise with unit variance, which constitutes,therefore, the lower bound for the variance of the controlled regime switching model.16


Figure 6: Spectrum ofMSAR(1), a (1) = 0:8;a (2) = 0:2:Figure 7: Spectrum of theunderlying model 1:AR(1) with a = 0:8:Figure 8: Spectrum of theunderlying model 2:AR(1) with a = 0:2:Figures 6, 7 and 8 refer to the second simulation. Model 1 is now quite persistent,while the second one is stabler. In this case the variation between the models is higherwith the respect to the previous case and the variance of the controlled process, whosespectrum is depicted by the dashed line in …gure 6, is 1.333. The Bode’s integral is,again, negative and lower than the Bode’s integral of the …rst case. Figure 6 showshow the control exacerbates the variability at the low frequencies. This is due, inparticular to the bad performance in correspondence of the low frequencies in themodel 2, as shown by …gure 8. However, high and medium frequencies componentsresult much stabler than the free dynamics case.Figure 9: Spectrum ofMSAR(1), a (1) = 0:5;a (2) = 0:5:Figure 10: Spectrum of theunderlying model 1: AR(1)with a = 0:5:Figure 11: Spectrum of theunderlying model 2: AR(1)with a = 0:5:The last case, to which …gures 9, 10 and 11 refer, deserves particular attention.The MSAR(1) behaves already as a white noise because the two underlying modelshave opposite coe¢ cients and the transition probabilities are ‡at. Therefore, thebest response in terms of the minimization of the variance is not to intervene andthe Bode’s integral is zero by de…nition. The variation between the two underlying17


a (2) -0.9 -0.8 -0.6 -0.4 0 0.4 0.6 0.8 0.9KB -0.041 0 -0.061 -0.149 -0.232 -0.149 -0.061 0 -0.041Table 1: The Bodes’integral with a(1)=0.8 and ‡at transition probabilitiesa (2) -0.9 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 0.9KB -0.256 -0.064 0 -0.016 -0.03 -0.016 0 -0.064 -0.256Table 2: The Bodes’integral with a(1)=0.5 and ‡at transition probabilitiesmodels is not reduced at all, so that the variance of the controlled global processreaches quite high values compared to the …rst case.Next we investigate the behavior of the Bode’s integral value across di¤erentmodels: we keep …xed model 1’s autoregressive coe¢ cient, a (1) ; and consider thedynamics of the Bode’s integral constraint as the second model’s coe¢ cient (a (2))changes.Tables 1 and 2 show the values of the Bode’s integral, given p 11 = p 22 = 0:5; and,respectively, a (1) = 0:8 and a (1) = 0:5.Several considerations can be advanced. First, the Bode’s integral is always negativeor equal to zero. This is, at …rst glance, counterintuitive: the assumption ofthe switching regimes introduces an additional source of uncertainty and it is reasonableto expect that this may cause an exacerbation to design limits. Second, in bothtables, given a (1) ; the values of the Bode’s integral are symmetric around a (2) = 0.The respective complete dynamics of the Bode’s integral is depicted in …gures 12and 13. Let’s consider the latter. From very negative values, moving from a (2) =0:9 towards 0 the Bode’s integral increases very fast till it reaches 0 when a (2) =0:5: This is the case we discussed in …gure 9: the policymaker does not intervene.Continuing moving rightward, the Bode’s integral slightly diminishes till a (2) = 0;after which it increases again until it reaches 0 at a (2) = 0:5: Here the two modelscoincide, no model uncertainty is present so that Bode is zero from Theorem 2.In other words, as shown in the …gures 12 and 13, the Bode’s integral dynamicswith the respect to a (2) presents two peaks: one corresponds to the case in whichthe MSAR(1) is already perceived as a white noise, the other one represents the casein which there is no switching and the two underlying models are identical.18


Figure 12: The dynamics of the Bode’s integralconstraint with a (1) = 0:8; p 11 = 0:5 and p 22 = 0:5:Figure 13: The dynamics of the Bode’s integralconstraint with a (1) = 0:5; p 11 = 0:5 and p 22 = 0:5:The dynamics of the Bode’s integral may have a meaningful interpretation. Wealready know that the lower is the value of the Bode’s integral, the better is thecontrol in the sense of producing limited frequency speci…c trade-o¤s. Our job isto understand how the introduction of model uncertainty, in the form of switchingregimes, may reduce design limits when p 11 = p 22 = 0:5. The key point to understand19


the above dynamics is to recall the distinction of the two types of variability the policymaker is subject to: the variability-between and the variability-within the models.The more the models are diverse, in terms of variability, the more the varianceminimizing rule is e¤ective in reducing the variability between the models and thereforethe overall uncertainty reduces in the sense that the component of the totalvariability due to the switching is smaller. Using the information-theoretic interpretationprovided in time domain, we can say that reducing the variability between themodels corresponds to modifying the Markov Switching model in such a way thatit embeds a less complex message, or, equivalently, that its forecastability has beenimproved.When the two underlying models are close to each other, the control of the policymakercannot be so e¤ective in the reduction of the transition probability which isalready limited. We can now reinterpret the two points in which Bode is zero: in thecase of no switching, there is no transition probability to reduce, in the case in whicha (1) = a (2) the models behave in the opposite way, the transition probability isvery high but it cannot be reduced because the policymaker doesn’t intervene. Thefact that the Bode’s integral can take negative values does not imply that the introductionof model uncertainty has improved the performance. The Bode’integral isa measure of relative performance (with the respect to the unconstrained case) andnegative values represent those cases in which the policymaker is able to reduce thefrequency-speci…c trade-o¤s, even if such trade-o¤s are still present.Figure 14: The dynamics of the Bode’sintegral constraint with a (1) = 0; p 11 = 0:5and p 22 = 0:5:Figure 14 concludes this section presenting the limiting case in which a (1) = 0 sothat, when also a (2) = 0; the combination of models for which there is no switching20


coincides with the one for which the policymaker is indi¤erent between intervening ornot into the model. This case will result important as a comparison for the followinganalysis.5.1.2 The Bode’s integral across the probabilities of switchingIn this section the restriction of symmetric transition probabilities is relaxed. In particularwe consider two cases. In the …rst one, we diminish the overall unconditionalprobability of switching by setting p 11 = p 22 > 0:5 and equal to 0:8: In the secondone the unconditional probability of switching is set to 0:7:Figure 15: The dynamics of the Bode’s integralconstraint with a (1) = 0:8; p 11 = 0:8 andp 22 = 0:8:21


Figure 16: The dynamics of the Bode’s integralconstraint with a (1) = 0:5; p 11 = 0:8 andp 22 = 0:8:Figure 17: The dynamics of the Bode’s integralconstraint with a (1) = 0; p 11 = 0:8 andp 22 = 0:8:The comparison of …gures 15, 16 and 17 with the relative cases analyzed in theprevious section, shows that the qualitatively dynamics inside each graph does notvary, but, quantitatively, the values of the Bode’s integrals are always reduced whenthe unconditional probability of switching is lower (except for the cases in which theBode’s integral is zero). In order to draw more general considerations, we propose22


next …gures 18, 19 and 20 which depict our candidate three cases in presence of highunconditional probability of switching, equal to 0:7 (p 11 = p 22 = 0:3).Figure 18: The dynamics of the Bode’s integralconstraint with a (1) = 0:8; p 11 = 0:3 andp 22 = 0:3:Figure 19: The dynamics of the Bodes’integralconstraint with a (1) = 0:5; p 11 = 0:3 andp 22 = 0:3:23


Figure 20: The dynamics of the Bode’s integralconstraint with a (1) = 0; p 11 = 0:3 and p 22 = 0:3:When the probability of switching is increased, the Bode’s integral takes positivevalues. In other words, the presence of an highly uncertain state of the economy,in which there occurs frequent switching between one model and the other, impedesthe action of the policymaker to be e¤ective in reducing (or, at least, non exacerbating)the aggregate measure of design limits, characterized by the value of the Bode’sintegral. This consideration appears very interesting, because it contributes in theunderstanding the links between the Bode’s integral and the uncertainty faced bythe policymaker, given, in our case, by the high probability of switching. As statedbefore, in linear frameworks, it has been proved that the Bode’s integral associated toa certain model corresponds to the di¤erence of the information-theoretic (or Shannon)entropy of the controlled and uncontrolled model. It is possible to prove thatthe information-theoretic interpretation of the Bode’s integral still holds in regimeswitchingcases (Pataracchia 2008b). In other words, high probability of switchingcorresponds to high level of entropy of the regime switching model as if the action ofthe policymaker is not only not able to reduce the uncertainty associated to it, butit further exacerbates it.The dynamics of the Bode’s integral is opposite with the respect to the low switchingcase. When models are very diverse in terms of variability, reducing the variancebetween them does not necessarily reduce uncertainty. This may be due to the factthat when the noise associated to the frequent switching is very high, reducing variabilitybetween the models may further worsen estimation and forecastability, increasing,therefore, the entropy associated to the model.24


5.2 The Bode’s integral across the policy rulesIn the previous sections we stressed how the dynamics of the design limits in MarkovSwitching contexts depends on the models’parameters (both in terms of autoregressivestructure and transition probabilities).A second important feature of design limits arising in switching regimes is thedependence on the particular policy rule: through the policy choice the policymakernot only shapes the spectral characteristics of the model, but he does it in a waywhich a¤ects also the measure of frequency trade-o¤s, therefore the constraint he issubject to. Figure 21 shows the dynamics of the Bode’s integral across several modelswhen di¤erent policies of the type described in (15) are considered. We supposea (1) = 0:8; p 11 = 0:2 and p 22 = 0:8 and we compare di¤erent policies with thevariance minimizing rule. In absence of switching (a (1) = a (2) = 0:8), the dynamicsof the Bode’s integral converges to zero (we are back to the linear framework). Aswe move leftward, the variability between the models becomes important and thedynamics of design limits varies substantially according to the policy rule. Noticethat the variance minimizing rule (the solid line case) always dominates all the otherproposed rules while the farther is the policy rule from the optimal one, the worstthe performance in terms of design limits.Figure 21: The dynamics of the Bode’s integralconstraint for di¤erent policies.[a (1) = 0:8; p 11 = 0:2; p 22 = 0:8] :In general cases, however, the statement that the variance minimizing rule alwaysproduces the smallest amount of design limits does not always hold, as …gure 22shows. It depicts the measure of the Bode’s constraint with the same set of rulesconsidered before when both models are quite unstable.25


Figure 22: The dynamics of the Bode constraint fordi¤erent policies.[a (1) = 0:8; p 11 = 0:3 and p 22 = 0:2]When the probability of switching is high, the variance minimizing rule is the one,among those proposed, which determines the highest level of fundamental limitationsof the design. As we already discussed, higher design limits imply higher exacerbationof the information theoretic entropy of the model under control, worse forecastabilityand the exacerbation of the volatility at some frequencies, which is probably the moreintuitive cost that economists can associate to design limits.To conclude, we state that proposing the minimization of design limits throughthe minimum Bode’s integral as an object is certainly a too strong argument. Designlimits may, nonetheless, constitute an important externality e¤ect to take intoconsideration along with conventional monetary policy exercises, in particular, as weshowed, in presence of an highly uncertain state of the economy represented by anhigh unconditional probability of switching among potential states.6 ConclusionIn this paper we extend the theory of design limits in regime-switching contexts,deriving the analogue of the Bode’s integral value, recently introduced in macroeconomicstudies by Brock and Durlauf (2004). The Bode’s integral value quanti…es theamount of the frequency trade-o¤s (design limits) the controller (typically, a centralbank) has to face while stabilizing the economic system under exam. Positive valuesof the Bode’s integral imply necessary exacerbations of the contribution of some fre-26


quency ranges to the total ‡uctuations of the model. Negative values do not implythe absence of frequency trade-o¤s, but denote less stringent limits.The main message we want to convey is that the analysis of the design limits inregime switching cases can be viewed as a general framework in which, the linear caseconstitutes just a special case where the potential models coincide.While we described the general procedure, we show explicitly the computation forthe simple case of a Markov Switching model with two possible AR(1) states wherethe policymaker has to choose a stabilizing, model-invariant rule before knowing therealization of the Markov Chain.Two main features, peculiar to regime-switching cases are revealed. First, thebehavior of the Bode’s integral strongly depends on the particular model considered.Second, di¤erent policy rules shape the spectral density of the process under exam indi¤erent ways, a¤ecting the measure of design limits. Therefore, contrary to linearframeworks, design limits are not independent of the control of the policymaker.Given the optimal variance-minimizing rule, the dynamics and the sign of theBode’s integral is strictly related to the probability of switching. When the transitionprobabilities are such that, once the economy …nds itself in one model is it very likelythat it is going to stay rather than switch to the other one, the variance-minimizingpolicy is associated to less stringent frequency trade-o¤s and the value of the Bode’sintegral is typically negative. The key point is that there is an additional source ofvariability that the control may reduce: the variation-between the models. By thatwe refer to the variation due to the diverse characteristics of the underlying models.The more similar are those, the lower the variation-between. The more the control isable to reduce it, the lower is the Bode’s integral value.Di¤erent is the conclusion in presence of an high unconditional probability ofswitching between the models. In those cases, the stabilization has a price in termsof frequency speci…c performance. The values of the Bode’s integral are typicallypositive.Model uncertainty in the form of Markov Switching regimes has, therefore, twomain e¤ects in terms of design limits. The introduction of an additional potentialmodel enlarges the sources of variabilities that may be controlled so that the optimalrule is able to reduce the additional source of variability between the models andthe Bode’s integral is typically negative. The variance minimizing rule, however,cannot a¤ect or reduce the variability due to the switching because, by construction,it cannot modify the transition probabilities. It follows that in the cases in which theprobability of switching is high, there is an important source of variation that cannotbe controlled. In this case, the stabilization has a price in terms of frequency-speci…cperformance.The second peculiar characteristics of the Bode’s integral value in regime switchingmodels is represented by its dependence on the policy rule. In this sense, thefrequency-speci…c performance plays the role of an externality of the policymaker’saction. This observation can open the way to a reconsideration of the Bode’s integral27


and a possible further extension in which the minimization of the frequency tradeo¤smay be associated a relative weight in the target vector of a monetary economicanalysis. The examples shown make clear that general conclusions cannot be derivedand that each particular case should be evaluated in order to derive policy relevantconsiderations.In any case, we do not have the conceit to propose the minimization of the Bode’sintegral values as a pure object. However, we regard the communication of the frequencyspeci…c e¤ects of any policy rule, including the knowledge of design limits,as an important practice which should contribute to a more complete monetary policyevaluation analysis, in particular in presence of high uncertainty, which can berepresented by frequent switching among potential states.References[1] Bode, H.W. (1945), "Network Analysis and Feedback Ampli…er Design", VanNostrand, Princeton, NJ.[2] Brandt, A. (1986). "The stochastic Equation Y n+1 = A n Y n + B n with stationarycoe¢ cients". Advances in Applied Probability 18, 221-254.[3] Brock, W. A. and Durlauf S. N., (2004), "Elements of a Theory of Design Limitsto Optimal Policy", The Manchester School, Vol. 72, Supplement 1, 1-18.[4] Brock, W. A. and Durlauf S. N., (2005), "Local Robustness analysis: Theoryand Application", Journal of Economics Dynamics & Control 29, 2067-2092.[5] Brock, W. A., Durlauf S. N. and Rondina G., (2006), "Design Limits and DynamicPolicy Analysis", mimeo.[6] Costa, O.L.V., Fragoso, M.D., Marques, R.P. (2005), "Discrete Time MarkovJump Linear Systems", Probability and Its Application, Springer Series.[7] Farmer R., Waggoner D. and Zha T., (2006), "Minimal State Variable Solutionsto Markov Switching Rational Expectation Models", mimeo.[8] Francq, C. and Zakoian J.M, (2001),"Stationarity of Multivariate MarkovSwitching ARMA Models", Journal of Econometrics, 102, 339-364.GradshteynI.S. and Ryzhik, I.M., (1980), "Table of Integrals, Series, and Products-Correctedand Enlarged", Academic Press.[9] Gradshteyn I. and Ryzhik I., (1980), "Tables of Integrals, Series, and Products",Academic Press, London.[10] Hamilton J. D., (1989), "A new Approach to the Economic Time Series and theBusiness Cycle", Econometrica, 57, 350-384.28


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