Implied Historical Federal Reserve Bank Behavior Under Uncertainty

Transcription

1 Proceeding of the 2009 IEEE International Conference on Sytem, Man, and Cybernetic San Antonio, TX, USA - October 2009 Implied Hitorical Federal Reerve Bank Behavior Under Uncertainty Muhittin Yilmaz, Selahattin Ozcelik and Nuri Yilmazer Frank H. Dotterweich College of Engineering Department of Electrical Engineering and Computer Science Texa A&M Univerity-Kingville, Kingville, Texa muhittin.yilmaz@tamuk.edu and kfny000@tamuk.edu Frank H. Dotterweich College of Engineering Department of Mechanical and Indutrial Engineering Texa A&M Univerity-Kingville, Kingville, Texa kfo000@tamuk.edu Abtract Thi paper preent the implied United State Federal Reerve Bank (FED) policy behavior under multiplicative model and hock uncertainty, defined through performance objective, cae by uing hitorical data. Robut ytem theory framework are ued to empirically tudy the characteritic of the FED hort-term interet rate-inflation dynamic under different circumtance by uing a ingle-input ingle-output model. The main reult of thi tudy demontrate that the hitorical FED action wa conervative under model and hock uncertainty. Index Term Inflation dynamic, uncertainty, US FED I. INTRODUCTION Economic dynamic and optimization have been a central concern for the economic policy maker, epecially for central banker and government, to tabilize the overall behavior of the economy under uncertainty. It i pointed out that although policy maker acknowledge model uncertainty, the type of uncertainty can not be characterized by ubjective or objective probability ditribution, which the policy maker have limited information [1], [2]. Moreover, the term rik i ued to refer to the outcome for known probabilitie and the term uncertainty i ued for the outcome for unknown probabilitie [3]. Furthermore, it ha been propoed that the policy maker hould act conervatively, i.e., make maller change than the one uggeted by the noal model, during the economic tabilization problem involving uncertainty [4]. Mainly, there are two major framework to addre model uncertainty in economic policy analyi; Bayeian analyi and robut ytem framework. Bayeian analyi ha widely been ued to optimize the monetary policie. Thi approach ue an aigned a-priori probability ditribution for each model parameter and optimize the monetary policy by imizing the expected lo againt thee probability ditribution. However, the prior probability ditribution aignment to each model parameter i difficult and become intractable for ytem with large number of parameter. Moreover, the parametric nature of thi approach i prone to more difficult cae under different regime of the economic dynamic. Advanced etup uch a varying parameter and learning to update poterior attempt to mitigate the hortcog of thi approach [5], [6]. The Bayeian analyi for macroeconomic tabilization ha hitorically hown that the monetary policy hould be le aggreive when model uncertainty exit. Robut ytem framework preent a ytematic approach to handle uncertainty in a wort-cae cenario, offering an alternative to Bayeian analyi. Economic policy analyi ha been concurrently developed with available mathematical reource. When fixed-model optimal control framework proved to be inefficient to explain real-life paradigm due to it deficiencie related to model variation and uncertainty, robut ytem framework including model uncertainty and performance objective appeared to be the olution tarting in the early 80, with Zame [7], [8], [9]. However, economic dynamic analyi by uing robut ytem framework have recently become the focu of attention. The robutne of the monetary deciion dynamic under model uncertainty ha been tudied in different form. Since the fixed model did not account for the uncertainty, there wa a practice of averaging the policy rule for a number of plauible mipecified model [10]. Although thi tudy did not include formal robut ytem technique, it clearly emphaized the concern for robutne under model uncertainty. Robut ytem framework ha been utilized by a number of tudie [11], [12], [13], [14], [15]. One of the tudie [14] ued the the tructured model uncertainty in robut ytem H and l 1 framework but ued the explicit Taylor rule 1 a the feedback rule to analyze the characteritic of the correponding policy rule..their reult howed that the robut ytem optimization yield more aggreive policy rule 1 The generalized Taylor rule i an etimation of the policy rule over year and i given a i t = g π π t + g yy t, where i t i the noal interet rate, π t i the average of lat four quarter inflation rate, y t i the output gap, and contant g π =1.5 and g y = /09/$ IEEE 4588

2 than Linear Quadratic Gauian framework and that the variou uncertainty decription ignificantly altered the correponding robut policy rule. The robut ytem optimization, identification, and model (in)validation framework can identify and optimize different economic problem involving model uncertainty [16], can quantify the uncertainty in a ytematic way, and can tet the model variation with repect to the uncertainty bound for the future output. Thi paper extend the robut ytem model identification and (in)validation concept [17] to analyze the implied and hitorical FED behavior under model and hock uncertainty. Section 2 cover the robut ytem framework, Section 3 follow with the macroeconomic tabilization problem decription and analyi in detail and Section 4 conclude. II. ROBUST SYSTEMS FRAMEWORK Robut ytem framework deal with uncertainty in a typical linear fractional tranformation (LFT) framework [18], combining the ytem noal model, a bounded uncertainty, and a controller. Noal performance implie the achieved performance objective for the noal plant only and it can be checked by the norm of the appropriate performance objective function. Robut tability enure the ytem tability for all model of the plant family, a linear time-invariant model with a bounded uncertainty. The mall-gain theorem tet the robut tability by perforg an appropriate tranfer function norm tet. Robut performance mean that the LFT ytem i both table and achieve the deired objective for all poible plant model imultaneouly. Conveniently, the robut performance of a ytem can be detered by uing the tructured ingular value tet for the tructured uncertainty cae. Aume that a ytem i decribed by w e = G 11 G 12 G 13 G 21 G 22 G 23. z d (1) y G 31 G 32 G 33 u }{{} G z = Δ.w (2) u = K.y (3) where the tranfer function matrix G denote the augmented plant model including the performance and uncertainty weighting function, Δ denote the norm-bounded uncertainty uch that Δ B Δ, where. B Δ = {Δ : Δ 1}, and K denote the controller. While z, w, u and y ignal are elf-explanatory, d and e ignal repreent the input and output ignal, repectively, to meaure the performance pecification. Then, the LFT framework, hown in Fig. 1, controller loop can be [ cloed to] achieve the lower LFT repreentation G11 G F l (G, K) = 12 +[G G 21 G 13 G 23 ] K(I G 33 K) 1 [G 31 G 32 ] [ ] 22 [ w z uch that = F e l (G, K).. d] Fig. 1. z d u Δ G K w e y The Robut Sytem Framework. Robut performance problem 2 are typically handled by uing one block for the model uncertaintie (Δ U ) and another (fictitiou) block (Δ P ) for the performance objective, [ yielding ] ΔU 0 the augmented uncertainty tructure (Δ) aδ=, 0 Δ P where Δ U B Δ, Δ P C md me, where m d and m e denote the length of the d and e variable, the performance objective variable, a hown in Fig. 1. The robut performance problem with tructured uncertainty i olved by uing the μ-ynthei framework [18]. Thi framework define a tructured ingular value, μ(.), μ Δ (F l (G, K)) =. 1 {σ(δ) : Δ B Δ, det(i F l (G, K)Δ) = 0} (4) and tate that the ytem i table and ha σ(t de ) < 1, where σ(.) denote the maximum ingular value of the aociated tranfer function matrix, if and only if F l (G, K) i table and max μ Δ (F l (G, K)(jω)) < 1 ω Therefore, it olve K tabilizing max ω μ Δ (F l (G, K)(jω)) (5) Since the exact calculation of μ Δ (.) i not known yet, it can be approximated by K tabilizing max ω μ Δ(F l(g,k)(jω)) {}}{ D D σ(df l(g, K)(jω)D 1 ) (6) after a proper D-caling 3 and thi approximation i performed by D-K iteration 4 : K tabilizing DF l(g, K)D 1 D D (7) 2 Economic ub-dynamic may involve untructured or tructured uncertainty. If there i no relationhip aumed among the uncertainty variable, i.e., Δ contain all nonzero element, it yield a well-known H problem [19], [20], eaier to olve but yield more conervative reult. On the other hand, if the uncertainty block i aumed to have ome tructure, i.e., Δ ha ome internal tructure, it yield a μ-ynthei problem [18], harder to olve but yield better reult. 3 The element of D-caling matrix can take nonzero complex value without changing the upper bound and the D matrix can be any real, rational, table and imum-phae tranfer function. 4 The drawback of D-K iteration are that the μ Δ (.) i approximated by it upper bound and that the iteration may get tuck at a local imum. 4589

3 III. THE MACROECONOMIC STABILIZATION PROBLEM The United State FED ha been mandated by law to conduct the monetary policy to maintain low inflation level while enuring the growth of the economy by utilizing it variou policy control input, the hort-term interet-rate being the mot effective one. Although there are many economic variable to optimize, the main objective of the FED are the inflation and growth parameter. Thi tudy aume that the overall economic dynamic between the FED hort-term interet rate and the annual inflation can be repreented by a ingle-input ingle-output (SISO) and a deteritic model and compare the hitorical and implied FED behavior during the inflation tabilization problem under model uncertainty. Furthermore, the propoed framework alo treat all other economic variable a internal dynamic parameter whoe effect are reflected in thi SISO model. The inflation tabilization problem utilize the noal ytem model [17] tranfer function, which wa already analyzed and validated for the repective hitorical data: Y [z] U[z] = 0.522z z z z z z z z (8) where Y [z] i the annual inflation output variable and U[z] i the FED hort-term interet rate change. Since the -norm i tranferable in continuou and dicrete-time, the bilinear tranformation, z = e T 1+ T 2 1 T 2 or 2(z 1) T (z+1), i ued to obtain G N (), the continuou-time equivalent ytem model of G[z], a G N () = (9) and it can eaily be treated a G() =G N ()(1+Δ()), where Δ() B Δ to utilize the multiplicative uncertainty cae. It hould be noted that the comparion imulation are done in the dicrete-time after the continuou-time optimization problem i olved. Once the underlying economic model ha been detered, the correponding noal performance, robut tability and robut performance iue can be preented by uing the robut ytem theory a defined in Sec. II to analyze the implied nature of the FED policy. A number of different uncertainty relationhip and propertie, different weighting function, and variou exogenou input can be utilized to interpret the FED policy choice. For example, the diturbance rejection problem block diagram including the noal plant (G N ()), a controller (K()), a multiplicative uncertainty term (Δ G ), an uncertainty weighting function (W G ()), a hock (diturbance) weighting function (W d ()) and an error weighting function, (W e ()) i hown in Fig. 2. The diturbance or hock, d(), can be anything to affect the inflation rate, uch a a udden jump in oil price or natural diater in large area. The robut performance objective hould be detered to correctly place the appropriate weighting function. In general, many performance objective are defined by uing weighted d() W G () z ΔG w W d () GN () + + W e ()e() u() Fig. 2. K() y() The control ytem interconnection tructure. tranfer function which, in turn, are incorporated to the controller deign. Weighting function can be een a frequencydependent caling of the correponding tranfer function or ignal. However, there i no exact etablihed relationhip other than ome guideline between the imum teady-tate error and the weighting function value, the exact weighting function election for different performance objective i till an open reearch area. Therefore, the weighting function election proce hould be iterated to reflect the real-life ituation. Since the FED project a monetary policy aig low inflation level and utainable Gro Dometic Product (GDP) growth rate, the correponding inflation targeting problem are uppoed to optimize the inflation dynamic under poible diturbance while maximizing the GDP output. However, the current model focue olely on the inflation dynamic, i.e., the dynamic between the FED hort-term interet rate and the annual inflation, the imum tracking error due to the diturbance input (exogenou hock) at the inflation output i taken one of the performance objective and the error weighting function i choen to be 1 +1 W e () = (10) a low-pa filter, to emphaize the teady-tate error, i.e., low frequency behavior, which i aumed to be the main concern and i placed on the path of the weighted error output, e(). The diturbance weighting function W d () can be placed on the diturbance input path to define the nature of diturbance. For example, if the noie type diturbance, i.e., highly volatile change, i the main concern, a high-pa filter tranfer function hould be ued. On the other hand, if utained low frequency hock, i.e., teady change, are of the main concern, the W d () hould be adjuted properly to reflect the diturbance pectrum. Once the diturbance weighting function i detered, the diturbance bound d() 2 1 can typically be taken. For implicity and brevity of the FED policy analyi, W d () =1i taken for thi ection, i.e., all poible diturbance variation are penalized equally. The uncertainty weighting function i detered according to the model uncertainty characteritic uch that low and high frequency model uncertainty component can be reflected. Since the low frequency behavior i much better undertood than the high frequency behavior, thi example aume that the model uncertainty at low frequencie i 10 time maller with repect to the one at high frequencie, i.e., the uncertainty 4590

4 weighting function for the H and μ-ynthei problem are hown in Fig. 5. A een from Fig. 5, the controller deign for the ame ytem with different uncertainty tructure yielded a lower μ bound than H bound. Thi i an expected reult ince the μ-ynthei framework exploit the uncertainty tructure and alway find a lower or equal value of H bound. For thi imulation, the wort-cae μ bound i equal to Since μ = < 1, it implie that the ytem atifie the robut performance and that there i till ome room for performance improvement. Fig. 3. The uncertainty and performance weighting function. weighting function reflect the 10% uncertainty at the low frequencie and 100% uncertainty for the high frequencie a given in (11). The error and uncertainty weighting function are alo hown in Fig. 3. W G () = (11) +1 Once the weighting function are properly defined, the control ytem in Fig. 2 can be converted into an LFT framework a hown in Fig. 4. The macroeconomic tabilization problem in robut ytem frame- Fig. 4. work. Δ pδg w() z() d() e() P () u() y() K() The augmented tranfer function matrix (P ()) in Fig. 4 can be eaily found by finding the tranfer function between each input-output pair; P () = 0 0 G N ()W G () W e () W d ()W e () G N ()W e () 1 W d () G N () (12) The LFT tructure in Fig. 4 can be eaily olved by uing the Matlab robut control toolbox [21], deigning H and μ controller, for the untructured and tructured uncertainty cae, repectively. The correponding bound of elected Fig. 5. The H and μ bound. The weighting function pecify the a-priori known nature of each variable and do not exit in the actual cloed-loop ytem. Therefore, when the overall cloed-loop ytem i imulated for reference or diturbance input, all weighting function are removed a hown in Fig. 6. R[z] + E[z] K[z] Fig. 6. G[z] U[z] GN [z] Δ G D[z] + + Y [z] The macroeconomic tabilization cloed loop ytem. The cloed loop ytem i propoed a an inflation tracking ytem to analyze the hitorical and implied FED behavior and i imulated aug that the hitorical inflation i the reference input, R[z], and the etimate of the inflation i the output variable, Y [z], and the FED fund rate change i the control action, U[z]. The inflation tracking problem performance can be tudied by uing the input-output inflation dynamic, Y [z] R[z], the top plot in Fig. 7. The inflation-fed fund rate 4591

5 dynamic can be tudied by uing the deired inflation input, i.e., the hitorical inflation rate, and the correponding FED U[z] control action, R[z], the bottom plot in Fig. 7. A een from Fig. 7, the input-output inflation repone ha ome tracking error due to looely choen weighting function, epecially the error weighting function. When the correponding implied and hitorical FED control action are compared, the reult ugget that the implied FED action i more aggreive. It hould be noted that thee imulation are cloely dependent on choen weighting function. bank policy. Thu, it can be concluded that, under the cae of model uncertainty and choen weighting function, the FED policy tend to be more cautiou and le reactionary, a mot economit believe, than the policie uggeted by a noal model. Fig. 8. The H and μ bound for the (implied) optimal FED behavior. Fig. 7. The cloed loop ytem imulation. The ame control ytem can be ued to analyze the implied FED fund rate behavior againt model uncertainty. Since the ytem reference ignal i hitorical inflation rate, the ame inflation data i expected to appear at the ytem output if the FED fund rate policy wa optimal with repect to the uncertainty and performance objective choen for the underlying problem. Thi cae can be analyzed by adjuting the error weighting function, W e (), o that perfect inflation tracking, i.e., zero teady-tate error, occur. Then, the implied FED fund rate change veru hitorical FED fund rate change can be compared. Since the tracking error can be adjuted by changing the error weighting function, the new weighting function i adjuted to 1 +1 W e () = (13) The correponding wort-cae μ bound i found to be , hown in Fig. 8, and the correponding imulation for the inflation tracking and FED fund rate change are given in Fig. 9. A een from Fig. 9, the inflation tracking problem achieved it goal, i.e., the error between the deired and hitorical inflation i almot negligible. The correponding actual hitorical FED fund rate change are much more moother and maller in magnitude than the optimally implied FED fund rate change, clearly implying that the very conervative central IV. CONCLUSION The main reult of thi paper are that the formal robut controller framework offer another avenue to invetigate the uncertainty iue in macroeconomic tabilization problem, in particular, inflation tracking problem, and another upporting evidence of the FED behavior under model uncertainty when it action are limited to inflation output only, which i more conervative. The current analyi can be enhanced by utilizing a multiinput multi-output model to compare the FED behavior in different framework uch a the robut ytem framework, Bayeian analyi and rule-baed policy reult and in different tability and performance objective norm, i.e., a multiobjective problem. Although the uncertainty i modeled a a model uncertainty, the individual model parameter can be aumed varying from the noal value by ome tandard deviation range uch a 1-tandard deviation or 2-tandard deviation. Thee deviation can be treated a parameter uncertaintie and the correponding optimization problem can be preented in tatitical accuracy range, which i currently preferred by the policy maker. REFERENCES [1] M. Friedman, The effect of a Full-Employment Policy on Economic Stability: A formal Analyi, In Friedman, Milton(ed.), Eay in Poitive Economic, Univerity of Chicago Pre, Chicago, IL, [2] M. Friedman, A Program for Monetary Stability, Fordham Univerity Pre, [3] F.H. Knight, Rik, Uncertainty and Profit, Houghton Mifflin Company,

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