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 l**in**e) **and**for the optimally controlled process which reduces to a white noise (dashed l**in**e). Thedotted l**in**e represents the process result**in**g from a feasible control. The dashed-dotl**in**e reproduces an unfeasible result.This theorem states that, **in** backward look**in**g environments, even if the policymakeris able to stabilize the state variable of **in**terest, the variance contributions atsome frequencies will necessarily exacerbate, s**in**ce, 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 **in**tegral isalways zero **and** **in**dependent of the particular model under exam. Further, it is**in**dependent of the policy rule. Figure 2 proposes aga**in** the spectral representation ofa stable autoregressive process (the solid l**in**e) **and** the spectrum of the process when avariance m**in**imized rule is adopted (the dashed l**in**e). As anticipated before, Theorem2 ensures that C’is an unfeasible control. Further, it shows that any r**and**omly chosenfeasible control (represented, for **in**stance, by the dotted l**in**e), 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 **in**tuitive **and** synthetic: **in** covariance stationary **cases**,the spectral representation **and**, consequently, the sensitivity function, are always real objects **and**functions 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 switch**in**gmodels. While we recognize that the MSAR(1) is a very simple example which hardlyallows a relevant economic application, we th**in**k it is important to consider it as astart**in**g po**in**t because, while calculations rema**in** tractable, it allows to derive quitegeneral considerations of regime switch**in**g **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-**in**dependent 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 + " t**and** 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 **in**tervenes through the choice of F **and** the case **in** which u t = 0. Weare **in**terested **in** these two **cases** because, as we showed before, the Bode’s **in**tegralrepresents an aggregate measure of design limitations the policymaker must face withthe respect to the case without control.In l**in**ear time **in**variant (LTI) frameworks (for **in**stance, **in** the analysis of model(11)), we are used to consider f yjNC (e i! ) **and** f yjC (e i! ) **in** the follow**in**g forms:f yjNC (!) = f (!) j G NC (!) j 2 (14)!!**and**f 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, unconstra**in**ed **and** constra**in**edsystems. As described **in** (5), **in** l**in**ear frameworks, the sensitivity functionis usually derived by the knowledge of the transfer functions of the controlled **and**uncontrolled models. In our case, even if we deal with nonl**in**ear 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 **in**tegral can now be computed:RKB = log j S (!) j 2 d!:! 1Tak**in**g 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 constra**in**t 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 l**in**ear framework’s result: the Bode’s **in**tegral 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 **in**both the uncontrolled **and** controlled model. It follows that, as **in** l**in**ear frameworks,the Bode’s **in**tegral is zero **in** case of a stable free dynamics, while we cannot stateanyth**in**g regard**in**g unstable uncontrolled models. This observation does not necessarilyrule out **in**terest**in**g comparisons with positive Bode values **in** the underly**in**gmodels: **in**stabilities **in** each AR(1) model are neither a necessary nor a su¢ cientcondition for global **in**stability (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 follow**in**g Bode’s **in**tegraltheorem for regime-switch**in**g **cases**.Theorem 5 Given model (15), the value of the Bode’**in**tegral 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 **in**tegralis model speci…c, **and** therefore subject to model misspeci…cations, even **in** backwardlook**in**g **cases**. Second, the presence of the terms A C **and** B C suggests that the value ofthe Bode’s **in**tegral can be a¤ected by the policy rule, consequently, one can associateto it the role of an endogenous constra**in**t, similar to an externality e¤ect, rather than11 See Gradshteyn **and** Rysh**in**k (1880), page 527.12 Notice that the explicit solution of the **in**tegral exists only for A jBj **and** A c jB c j : Thisrestriction, however, is not b**in**d**in**g **in** any of the simulations presented next.14

the exogenous constra**in**t, typical of l**in**ear frameworks. This certa**in**ly 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 underst**and** the behaviorof the Bode’s **in**tegral. Unless otherwise stated, we consider the variancem**in**imiz**in**g rule as the c**and**idate one: We th**in**k it is useful here to make a digressionon the characteristics of the variance of a regime-switch**in**g 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-with**in** the underly**in**g models **and**the variability-between the models, l**in**ked, respectively, to the shock process " t **and**the Markov Cha**in**, t . The variability-with**in** depends, of course, on the autoregressivestructure of the underly**in**g models, a (1) **and** a (2). The variability between themodels is related to the measure or amount of the switch**in**g, 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 switch**in**g. The higher p 12 **and** p 21 (or correspond**in**gly, the lower p 11**and** p 22 ), the more frequent is the switch**in**g: **in**deed, the unconditional probability ofswitch**in**g 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 switch**in**g.Those observations will turn out to be important for what follows.5 The dynamics of the Bode’s **in**tegralGiven the complexity of analytical formulae, we present some simulations related tomodel (15), with F correspond**in**g at the variance m**in**imiz**in**g 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 comb**in**ations 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 greater**and** less than 0:5; so to **in**vestigate the e¤ects of di¤erent transition probabilities onthe Bode’s **in**tegral. F**in**ally, we study the e¤ects of di¤erent simple policy rules onthe measure of design limits.5.1 The Bode’s **in**tegral 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 follow**in**g …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 underly**in**g models. In all of them the solid l**in**e represent15

the spectral density of the free dynamics of the model while the dashed l**in**e illustratesthe spectral density of the constra**in**ed processes when the variance m**in**imiz**in**g ruleis applied.Figure (3) shows that the control is able to elim**in**ate all the temporal dependences.In this case the two underly**in**g models are close to each other **and** the variation due tothe switch**in**g 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 theunderly**in**g model 1:AR(1) with a = 0:5:Figure 5: Spectrum of theunderly**in**g model 2:AR(1) with a = 0:2:The Bode’s **in**tegral 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 **in**tegral implies that, even though some frequencytrade-o¤s result due to policy **in**tervention, 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 dim**in**ished.In the underly**in**g 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 **in**to account the spectra of the underly**in**g models, both of whichviable, given the assumption of ergodicity of the Markov Cha**in**, **in** particular, if heassociates di¤erent losses to di¤erent frequency ranges.13 In the limit**in**g case, **in** which the models are identical (so that there is no variation due to theswitch**in**g), the controlled process would result a white noise with unit variance, which constitutes,therefore, the lower bound for the variance of the controlled regime switch**in**g model.16

Figure 6: Spectrum ofMSAR(1), a (1) = 0:8;a (2) = 0:2:Figure 7: Spectrum of theunderly**in**g model 1:AR(1) with a = 0:8:Figure 8: Spectrum of theunderly**in**g 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 l**in**e **in** …gure 6, is 1.333. The Bode’s **in**tegral is,aga**in**, negative **and** lower than the Bode’s **in**tegral of the …rst case. Figure 6 showshow the control exacerbates the variability at the low frequencies. This is due, **in**particular 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 theunderly**in**g model 1: AR(1)with a = 0:5:Figure 11: Spectrum of theunderly**in**g 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 underly**in**g modelshave opposite coe¢ cients **and** the transition probabilities are ‡at. Therefore, thebest response **in** terms of the m**in**imization of the variance is not to **in**tervene **and**the Bode’s **in**tegral is zero by de…nition. The variation between the two underly**in**g17

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’**in**tegral 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’**in**tegral 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 **in**vestigate the behavior of the Bode’s **in**tegral value across di¤erentmodels: we keep …xed model 1’s autoregressive coe¢ cient, a (1) ; **and** consider thedynamics of the Bode’s **in**tegral constra**in**t as the second model’s coe¢ cient (a (2))changes.Tables 1 **and** 2 show the values of the Bode’s **in**tegral, 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 **in**tegral is always negativeor equal to zero. This is, at …rst glance, counter**in**tuitive: the assumption ofthe switch**in**g regimes **in**troduces an additional source of uncerta**in**ty **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 **in**tegral are symmetric around a (2) = 0.The respective complete dynamics of the Bode’s **in**tegral is depicted **in** …gures 12**and** 13. Let’s consider the latter. From very negative values, mov**in**g from a (2) =0:9 towards 0 the Bode’s **in**tegral **in**creases very fast till it reaches 0 when a (2) =0:5: This is the case we discussed **in** …gure 9: the policymaker does not **in**tervene.Cont**in**u**in**g mov**in**g rightward, the Bode’s **in**tegral slightly dim**in**ishes till a (2) = 0;after which it **in**creases aga**in** until it reaches 0 at a (2) = 0:5: Here the two modelsco**in**cide, no model uncerta**in**ty is present so that Bode is zero from Theorem 2.In other words, as shown **in** the …gures 12 **and** 13, the Bode’s **in**tegral 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 case**in** which there is no switch**in**g **and** the two underly**in**g models are identical.18

Figure 12: The dynamics of the Bode’s **in**tegralconstra**in**t with a (1) = 0:8; p 11 = 0:5 **and** p 22 = 0:5:Figure 13: The dynamics of the Bode’s **in**tegralconstra**in**t with a (1) = 0:5; p 11 = 0:5 **and** p 22 = 0:5:The dynamics of the Bode’s **in**tegral may have a mean**in**gful **in**terpretation. Wealready know that the lower is the value of the Bode’s **in**tegral, the better is thecontrol **in** the sense of produc**in**g limited frequency speci…c trade-o¤s. Our job isto underst**and** how the **in**troduction of model uncerta**in**ty, **in** the form of switch**in**gregimes, may reduce design limits when p 11 = p 22 = 0:5. The key po**in**t to underst**and**19

the above dynamics is to recall the dist**in**ction of the two types of variability the policymaker is subject to: the variability-between **and** the variability-with**in** the models.The more the models are diverse, **in** terms of variability, the more the variancem**in**imiz**in**g rule is e¤ective **in** reduc**in**g the variability between the models **and** thereforethe overall uncerta**in**ty reduces **in** the sense that the component of the totalvariability due to the switch**in**g is smaller. Us**in**g the **in**formation-theoretic **in**terpretationprovided **in** time doma**in**, we can say that reduc**in**g the variability between themodels corresponds to modify**in**g the Markov Switch**in**g model **in** such a way thatit embeds a less complex message, or, equivalently, that its forecastability has beenimproved.When the two underly**in**g 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 re**in**terpret the two po**in**ts **in** which Bode is zero: **in** thecase of no switch**in**g, 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 **in**tervene. Thefact that the Bode’s **in**tegral can take negative values does not imply that the **in**troductionof model uncerta**in**ty has improved the performance. The Bode’**in**tegral isa measure of relative performance (with the respect to the unconstra**in**ed case) **and**negative 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’s**in**tegral constra**in**t with a (1) = 0; p 11 = 0:5**and** p 22 = 0:5:Figure 14 concludes this section present**in**g the limit**in**g case **in** which a (1) = 0 sothat, when also a (2) = 0; the comb**in**ation of models for which there is no switch**in**g20

co**in**cides with the one for which the policymaker is **in**di¤erent between **in**terven**in**g ornot **in**to the model. This case will result important as a comparison for the follow**in**ganalysis.5.1.2 The Bode’s **in**tegral across the probabilities of switch**in**gIn this section the restriction of symmetric transition probabilities is relaxed. In particularwe consider two **cases**. In the …rst one, we dim**in**ish the overall unconditionalprobability of switch**in**g by sett**in**g p 11 = p 22 > 0:5 **and** equal to 0:8: In the secondone the unconditional probability of switch**in**g is set to 0:7:Figure 15: The dynamics of the Bode’s **in**tegralconstra**in**t with a (1) = 0:8; p 11 = 0:8 **and**p 22 = 0:8:21

Figure 16: The dynamics of the Bode’s **in**tegralconstra**in**t with a (1) = 0:5; p 11 = 0:8 **and**p 22 = 0:8:Figure 17: The dynamics of the Bode’s **in**tegralconstra**in**t with a (1) = 0; p 11 = 0:8 **and**p 22 = 0:8:The comparison of …gures 15, 16 **and** 17 with the relative **cases** analyzed **in** theprevious section, shows that the qualitatively dynamics **in**side each graph does notvary, but, quantitatively, the values of the Bode’s **in**tegrals are always reduced whenthe unconditional probability of switch**in**g is lower (except for the **cases** **in** which theBode’s **in**tegral is zero). In order to draw more general considerations, we propose22

next …gures 18, 19 **and** 20 which depict our c**and**idate three **cases** **in** presence of highunconditional probability of switch**in**g, equal to 0:7 (p 11 = p 22 = 0:3).Figure 18: The dynamics of the Bode’s **in**tegralconstra**in**t with a (1) = 0:8; p 11 = 0:3 **and**p 22 = 0:3:Figure 19: The dynamics of the Bodes’**in**tegralconstra**in**t with a (1) = 0:5; p 11 = 0:3 **and**p 22 = 0:3:23

Figure 20: The dynamics of the Bode’s **in**tegralconstra**in**t with a (1) = 0; p 11 = 0:3 **and** p 22 = 0:3:When the probability of switch**in**g is **in**creased, the Bode’s **in**tegral takes positivevalues. In other words, the presence of an highly uncerta**in** state of the economy,**in** which there occurs frequent switch**in**g between one model **and** the other, impedesthe action of the policymaker to be e¤ective **in** reduc**in**g (or, at least, non exacerbat**in**g)the aggregate measure of design limits, characterized by the value of the Bode’s**in**tegral. This consideration appears very **in**terest**in**g, because it contributes **in** theunderst**and****in**g the l**in**ks between the Bode’s **in**tegral **and** the uncerta**in**ty faced bythe policymaker, given, **in** our case, by the high probability of switch**in**g. As statedbefore, **in** l**in**ear frameworks, it has been proved that the Bode’s **in**tegral associated toa certa**in** model corresponds to the di¤erence of the **in**formation-theoretic (or Shannon)entropy of the controlled **and** uncontrolled model. It is possible to prove thatthe **in**formation-theoretic **in**terpretation of the Bode’s **in**tegral still holds **in** regimeswitch**in**g**cases** (Pataracchia 2008b). In other words, high probability of switch**in**gcorresponds to high level of entropy of the regime switch**in**g model as if the action ofthe policymaker is not only not able to reduce the uncerta**in**ty associated to it, butit further exacerbates it.The dynamics of the Bode’s **in**tegral is opposite with the respect to the low switch**in**gcase. When models are very diverse **in** terms of variability, reduc**in**g the variancebetween them does not necessarily reduce uncerta**in**ty. This may be due to the factthat when the noise associated to the frequent switch**in**g is very high, reduc**in**g variabilitybetween the models may further worsen estimation **and** forecastability, **in**creas**in**g,therefore, the entropy associated to the model.24

5.2 The Bode’s **in**tegral across the policy rulesIn the previous sections we stressed how the dynamics of the design limits **in** MarkovSwitch**in**g contexts depends on the models’parameters (both **in** terms of autoregressivestructure **and** transition probabilities).A second important feature of design limits aris**in**g **in** switch**in**g 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 constra**in**t he issubject to. Figure 21 shows the dynamics of the Bode’s **in**tegral 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 m**in**imiz**in**g rule. In absence of switch**in**g (a (1) = a (2) = 0:8), the dynamicsof the Bode’s **in**tegral converges to zero (we are back to the l**in**ear framework). Aswe move leftward, the variability between the models becomes important **and** thedynamics of design limits varies substantially accord**in**g to the policy rule. Noticethat the variance m**in**imiz**in**g rule (the solid l**in**e case) always dom**in**ates 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 **in**tegralconstra**in**t 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 m**in**imiz**in**g rule alwaysproduces the smallest amount of design limits does not always hold, as …gure 22shows. It depicts the measure of the Bode’s constra**in**t with the same set of rulesconsidered before when both models are quite unstable.25

Figure 22: The dynamics of the Bode constra**in**t fordi¤erent policies.[a (1) = 0:8; p 11 = 0:3 **and** p 22 = 0:2]When the probability of switch**in**g is high, the variance m**in**imiz**in**g rule is the one,among those proposed, which determ**in**es the highest level of fundamental limitationsof the design. As we already discussed, higher design limits imply higher exacerbationof the **in**formation theoretic entropy of the model under control, worse forecastability**and** the exacerbation of the volatility at some frequencies, which is probably the more**in**tuitive cost that economists can associate to design limits.To conclude, we state that propos**in**g the m**in**imization of design limits throughthe m**in**imum Bode’s **in**tegral as an object is certa**in**ly a too strong argument. **Design**limits may, nonetheless, constitute an important externality e¤ect to take **in**toconsideration along with conventional monetary policy exercises, **in** particular, as weshowed, **in** presence of an highly uncerta**in** state of the economy represented by anhigh unconditional probability of switch**in**g among potential states.6 ConclusionIn this paper we extend the theory of design limits **in** regime-switch**in**g contexts,deriv**in**g the analogue of the Bode’s **in**tegral value, recently **in**troduced **in** macroeconomicstudies by Brock **and** Durlauf (2004). The Bode’s **in**tegral value quanti…es theamount of the frequency trade-o¤s (design limits) the controller (typically, a centralbank) has to face while stabiliz**in**g the economic system under exam. Positive valuesof the Bode’s **in**tegral 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 str**in**gent limits.The ma**in** message we want to convey is that the analysis of the design limits **in**regime switch**in**g **cases** can be viewed as a general framework **in** which, the l**in**ear caseconstitutes just a special case where the potential models co**in**cide.While we described the general procedure, we show explicitly the computation forthe simple case of a Markov Switch**in**g model with two possible AR(1) states wherethe policymaker has to choose a stabiliz**in**g, model-**in**variant rule before know**in**g therealization of the Markov Cha**in**.Two ma**in** features, peculiar to regime-switch**in**g **cases** are revealed. First, thebehavior of the Bode’s **in**tegral strongly depends on the particular model considered.Second, di¤erent policy rules shape the spectral density of the process under exam **in**di¤erent ways, a¤ect**in**g the measure of design limits. Therefore, contrary to l**in**earframeworks, design limits are not **in**dependent of the control of the policymaker.Given the optimal variance-m**in**imiz**in**g rule, the dynamics **and** the sign of theBode’s **in**tegral is strictly related to the probability of switch**in**g. When the transitionprobabilities are such that, once the economy …nds itself **in** one model is it very likelythat it is go**in**g to stay rather than switch to the other one, the variance-m**in**imiz**in**gpolicy is associated to less str**in**gent frequency trade-o¤s **and** the value of the Bode’s**in**tegral is typically negative. The key po**in**t 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 underly**in**g 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 **in**tegral value.Di¤erent is the conclusion **in** presence of an high unconditional probability ofswitch**in**g between the models. In those **cases**, the stabilization has a price **in** termsof frequency speci…c performance. The values of the Bode’s **in**tegral are typicallypositive.Model uncerta**in**ty **in** the form of Markov Switch**in**g regimes has, therefore, twoma**in** e¤ects **in** terms of design limits. The **in**troduction 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 **and**the Bode’s **in**tegral is typically negative. The variance m**in**imiz**in**g rule, however,cannot a¤ect or reduce the variability due to the switch**in**g because, by construction,it cannot modify the transition probabilities. It follows that **in** the **cases** **in** which theprobability of switch**in**g 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 **in**tegral value **in** regime switch**in**gmodels 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 **in**tegral27

**and** a possible further extension **in** which the m**in**imization 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 derived**and** 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 m**in**imization of the Bode’s**in**tegral values as a pure object. However, we regard the communication of the frequencyspeci…c e¤ects of any policy rule, **in**clud**in**g 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 uncerta**in**ty, which can berepresented by frequent switch**in**g among potential states.References[1] Bode, H.W. (1945), "Network Analysis **and** Feedback Ampli…er **Design**", VanNostr**and**, Pr**in**ceton, NJ.[2] Br**and**t, 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** **Limits**to Optimal Policy", The Manchester School, Vol. 72, Supplement 1, 1-18.[4] Brock, W. A. **and** Durlauf S. N., (2005), "Local Robustness analysis: Theory**and** Application", Journal of Economics Dynamics & Control 29, 2067-2092.[5] Brock, W. A., Durlauf S. N. **and** Rond**in**a G., (2006), "**Design** **Limits** **and** DynamicPolicy Analysis", mimeo.[6] Costa, O.L.V., Fragoso, M.D., Marques, R.P. (2005), "Discrete Time MarkovJump L**in**ear Systems", Probability **and** Its Application, Spr**in**ger Series.[7] Farmer R., Waggoner D. **and** Zha T., (2006), "M**in**imal State Variable Solutionsto Markov Switch**in**g Rational Expectation Models", mimeo.[8] Francq, C. **and** Zakoian J.M, (2001),"Stationarity of Multivariate MarkovSwitch**in**g ARMA Models", Journal of Econometrics, 102, 339-364.GradshteynI.S. **and** Ryzhik, I.M., (1980), "Table of Integrals, Series, **and** Products-Corrected**and** 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** theBus**in**ess Cycle", Econometrica, 57, 350-384.28

[11] Hansen L. P. **and** Sargent T. J., (2008), "Robustness", Pr**in**ceton UniversityPress.[12] Otrok C. (2001), "Spectral Welfare Cost Function", International Economic Review42, 2, 345-367.[13] Otrok C., Ravikumar B. **and** Whiteman C., (2002), "Habit formation: a Resolutionof Equity Premium Puzzle", Journal of Monetary Economics, 49, 1261-1288.[14] Pataracchia B., (2008a), "The Spectral Representation of Markov Switch**in**gARMA Models", Work**in**g Paper N. 258, Marzo 2008, Quaderni del Dipartimentodi Economia Politica, Universita’degli Studi di Siena.[15] Pataracchia B., (2008b), "Risk Aversion across Frequencies", unpublished.[16] Priestley M. (1981), "Spectral Analysis **and** Time Series". Academic Press.Orl**and**o, Florida.[17] Sargent T. J., (1987), "Macroeconomic Theory". Academic Press. London, UK.[18] Wu B. **and** Jonckheere E. (1992), "A simpli…ed Approach to Bode’s Theoremfor Cont**in**uous **and** Discrete Time Systems", IEEE Transactions on AutomaticControl, Vol. 37 No. 100, pp. 1797-1802.[19] Zang G. **and** Iglesias P.A., (2003), "Nonl**in**ear extensions of Bode’s **in**tegral basedon an **in**formation theoretic **in**terpretation", System **and** Control Letters, Vol. 50(1), pages 11-19.29