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Working Paper Series Interest Rate Rules and Nominal Determinacy WP 90-01 John H. Boyd, III University of Rochester Michael Dotsey Federal Reserve Bank of Richmond This paper can be downloaded without charge from: http://www.richmondfed.org/publications/ Working Paper 90-l INTEREST RATE RULES AND NOMINAL DETERMINACY John H. Boyd, III* and Michael Dotsey University of Rochester and Federal Reserve Bank of Richmond February 1990 *John H. Boyd, III is an Assistant Professor at the University of Rochester. The authors wish to thank Marvin Goodfriend, Bennett McCallum, and Alan Stockman for their helpful comments. The views expressed in this paper are solely those of the authors and do not necessarily represent those of the University of Rochester, the Federal Reserve Bank of Richmond, or the Federal Reserve System. Abstract Monetary economists have recently begun a serious study of money supply rules that allow the Fed to adjustably peg the nominal interest rate under rational expectations. These rules vary from procedures that produce stationary nominal magnitudes to those that generate nonstationarities in nominal variables. Our paper investigates the determinacy properties of three representative interest rate rules. We use Blanchard and Kahn’s solution technique as a starting point. rectly apply, so we first modify their procedure. It doesn’t di- We then narrow the range of solutions by considering the ARMA solutions of Evans and Honkapohja and the global minimum state variable solution of McCallum. We then examine these solutions in light of the expectational stability notions emplayed by DeCanio, Bray and Evans. Two of the three classes of rules yield a unique admissible solution. The exclusion of bubbles usually rules out the general ARMA solutions present in Evans and Honkapohja and leads to unique solutions via a saddlepoint property. Nonetheless, the nonstationary money supply rules we examine do not generally yield a well determined system over all parameter values. We employ the global minimum state variable methodology Callum and Evans’ expectational stability in an effort to insure uniqueness. of Mc- Although these methods are usually in agreement, one of the nonstationary rules yields a global minimum state variable solution that is expectationally is sensitive to interest rate deviations. (non-global) unstable when the central bank Moreover, under these conditions, an alternative minimum state variable solution is expectationally the applicability of McCallum’s global procedure in thii context. stable, casting doubt on 1. Introduction Beginning with McCallum’s (1981) article, monetary economists have been able to seriously study the use of an interest rate instrument in an economic environment that incorporates rational expectations. Dotsey and King (1983,1986) and Canzoneri, Henderson and Rogoff (1983) have examined a variety of money supply specifications that allow the central bank to target or adjustably peg the nominal interest rate. Barro (1989), building on the work of Goodfriend (1987) and McCallum (1986), has used the concept of interest rate smoothing in an attempt to generate nominal variables that have statistical properties that resemble actual time series. The literature has produced a number of money supply rules that allow the Fed to adjustably peg the nominal interest rate. The rules vary from procedures that produce stationary nominal magnitudes to those that generate nonstationarities in nominal variables. Although determinacy issues are central to much of McCallum’s work, the various types of rules have not been subjected to a systematic investigation. Our paper carries out that investigation. We use Blanchard and Kahn’s (1980) procedure as a starting point for studying three representative forms of interest rate rules. Since their procedure doesn’t directly apply to our problems, the first order of business is to modify it so that it does apply. We then narrow the range of solutions using the methods of Evans and Honkapohja (1986) and of McCallum (1983). Finally, we examine these solutions in light of the expectational stability notions employed by DeCanio (1979), Bray (1982) and Evans (1985, 1986).’ By focusing on solutions that meet the nonexplosiveness criteria of Blanchard and Kahn (1980) we find that two of the three classes of rules yield a unique admissible solution. The exclusion of bubbles usually rules out the general ARMA solutions present in Evans and Honkapohja and leads to unique solutions via a saddlepoint property similar to that used in Blanchard and Kahn (1980). Nonetheless, the nonstationary money supply rules which are employed in McCallum (1986) and extended by Barro (1989) d o not generally yield a well determined system over all parameter values - there are an infinity of solutions. Focusing on solutions involving a minimal set of state variables helps somewhat, but uniqueness may still be a problem. In an effort to insure uniqueness, McCallum (1983) has proposed a subsidary principle, further refining the set of admissible solutions to those minimum state variable solutions ’ General discussions of expectational stability may be found in Blume, Bray and Easley (1982) and Frydman and Phelps (1983). 2 that hold globally - for all values of the parameters. Both McCallum and Barro appeal to the global minimum state variable methodology of McCallum (1986). However, our methodology indicates that only Barro’s model has a unique nonexplosive solution when the interest rate is pegged. This occurs because, in his model, the central bank is concerned about the variance of price level surprises. Thii concern leads to a restriction of the admissible parameter values that the coefficients on the interest rate feedback terms are allowed to have. When it comes to interest rate rules, we are not as optimistic as they are in applying McCallum’s technique. Although Evans (1986) finds that expectational stability often provides support for the subsidiary principle, that is not always the case here. In fact, under one of the nonstationary rules, we find that the global miniium solution is expectationally state variable unstable when the central bank is sensitive to interest rate deviations. Moreover, under these conditions, an alternative (non-global) minimum state variable solution is expectationally stable, casting doubt on the applicability of McCallum’s subsidiary principle in this context. The paper proceeds as follows. Section Two gives three variants of the basic model, differing only in the interest rate rule. Section Three contains our modification of Blanchard and Kahn’s eigenvalue counting rules. In Section Four, we examine the solutions to our systems and investigate their nominal determinacy properties. Section Five analyzes their expectational stability, while Section Six discusses Barro’s model. Section Seven contains concluding remarks. 2. The Model The basic economic structure is similar to McCallum (1981, 1986), and Goodfriend (1987). The real side of the economy is depicted by a Fisher relationship between the nominal rate of interest, the stochastic real rate of interest, and the expected rate of inflation. Thus &= =+&I%+1 -Pt + rt (1) where & is the nominal rate of interest, a + tt is the real rate of interest, a serially independent stochastic process with mean a, pt is the logarithm of the current price level, and Etpt+r is the expectation of the log of next period’s price level, conditional on current information. The current information set includes observations on all current and past values of the endogenous variables and the stochastic disturbances. The demand for money is given by 3 where mt is the logarithm of nominal money balances and vt is a mean zero white noise disturbance term. The initial money supply m. is given. The disturbance term includes the effects of changes in income on the demand for money as well as shifts in taste and transactions technology. The model is closed by specifying the money supply process. Here we choose to examine three different candidates. mi=b+X(&-R*). rni = rnt-l + A(& - A!‘), (34 m0 given m,d = w-1 + A(& - G-I&), (3b) m0 given. (34 Equations (3a) and (3b) are in the spirit of McCallum (1986)’ while (3~) is somewhat representative of the specification used in Goodfriend (1987)) Dotsey and King (1983,1986), and Canzoneri, Henderson, and Rogoff (1983). The parameter X determines how much the monetary authority reacts to changes in the interest rate. In all three specifications we approximate the behavior of a peg by letting X + 00. Equation (3a) implies a stationary money supply, while (3b) and (3~) yield nonstationary movements in money. As will be shown below, equation (3b) produces determinacy problems not found in the other two systems. 3. Root Counting Rules In order to analyze the determinancy properties of our interest rate rules we need a methodology. The methodology that we find most appealing is that of Blanchard and Kahn (1980)) since it rules out explosive solutions and is carefully based on a general theory of uniqueness for a system of linear difference equations. conditional expections of future x do not explode. They require that the Specificially, &zt+i is polynomially bounded in the sense that for all t there exist random variables Z, and integers w with IE,z,*I 5 (1 + i)“‘Zt for all i. The nonexplosiveness criteria seems sensible to us since it mimics the role of transversality conditions in well specified optimization models. Also, as an empirical matter, analyzing explosive bubbles does not seem particularly relevant. To facilitate comparison with Blanchard and Kahn (1980)’ we refer to variables with &6+1 = zt+r as predetermined and those with E,z,+r # zt+l as non-predetermined. course, Etzt = xt for all x. Of Blanchard and Kahn assume that the initial values of the predetermined variables are given, while the initial values of non-predetermined variables are not given. We break camp with them here. Our analysis in this section shows that it is not the stochastic properties of the variables which matter, but rather the presence or absence of initial conditions. 4 Blanchard and Kahn’s formulation is not immediately applicable to our set of problems since systems involving (3b) or (3~) fall under their example C due to the presence of terms containing E tP t+s. As they point out, problems in thii category can not be addressed by the counting rules in their theorem. We are, therefore, required to modify their methodology. Our method will prove useful for a wide class of models that include past expectations of current and future variables and is of interest in and of itself. Although our method is applicable to a much wider variety of systems, the theorem we present is optimized for the systems set forth in section 2. In addition to meriting investigation in their own right, the interest rate rules we analyze in sections 4 and 5 serve double duty as interesting examples of our methods. Our starting point is the solution procedure of Blanchard and Kahn (1980). The matrix multiplying the current variables is put into Jordan form J, with the eigenvalues listed in order of increasing modulus. The matrix that performs the diagonaliiation is denoted S. Blanchard and Kahn’s procedure is to partition J into blocks corresponding to the ni, eigenvalues inside or on the unit circle, and the noa eigenvalues outside the unit circle. The vector of variables gives the nwe p redetermined variables first, followed by the nnonnon-predetermined variables. Partition S accordingly. It is crucial that the nd x nm matrix Ss2 be of full rank. Blanchard and Kahn then restrict their attention to the polynomially bounded solutions of the system. They find that there is a unique solution when nM = nnon, there is an inEnity of solutions when nd < nnon, and, for almost all initial conditions, there is no solution when nd > nm. The importance of their rank condition cannot be underestimated. When the rank condition fails, the ill-behaved parts of the solution cannot be ruled out through polynomial boundedness. The root counting procedure fails. The following simple example illustrates this. Consider the system zt+r = 2xt and Etpt+r = pt/2 where zt is predetermined and pt is non-predetermined. Since nd = nnon= 1, root counting would yield a unique polynomially bounded solution for all initial conditions. However, for zs # 0, there is no polynomially bounded solution since zt = 2%s. When z,-, = 0, there are many polynomially bounded solutions. To see the indeterminacy, let wt be an arbitrary martingale, so Ecwt+1 = wt. Consider pt = c~(1/2)~wtand Z~ = 0 for CYarbitrary. This is a solution she &pt+l = pt/2, and is clearly polynomially bounded.2 Our problems appear to be only slightly different from Blanchard and Kahn%. Nonetheless, their results do not directly apply to our problems due to the presence of future 2 Martingale solutions are considered in Pesaran (1981) and Gourieroux, Monfort (1982). See Pesaran’s (1987) book for a complete exposition. Laffont and 5 expectations. Fortunately, similar results do apply. ’ Let Pt be the m-vector m+-vector, of variables, n, an m-vector of disturbance terms, F an Q an m* x 2m matrix and let A, 23, C and D be m x m matrices. The equations we are interested in will have the form P:+I = Apt + BW’t+l + CM’t+2 s.t. + %+l, + D&+&+2 Q [ PO Eopl I = F (4) where Xc = EoPl. All three of our cases fit into this format, as does Blanchard and Kahn’s second example C, Pt+l = CU(E~P,+~ - E,P,+,) + ct. The initial condition on mt can be written in terms of PO and EoPl by using (1) and (2). This yields the initial condition (1 + c)po - cEopl = m + ca + cro - vo. Start by applying Et to (4). Thii yields E*fi+l = APt + BE&+1 + (C + D)Et(Et+lPt+2) + Etilt+l. (5) Provided C + D is invertible, the substitution Xt = EtPt+l puts this into the Blanchard and Kahn format E*[$]=[ -(C +‘D)-‘A (C + D&(1 - B) ] [z] + [dt,,]’ To study uniqueness, we consider the case with fit = 0 and F = 0. Equation (4) becomes [A-D] [2l]=[AJl [z]+!WlE: [zl] while equation (5) becomes -(C +‘D)-‘A (C + D+(I -B) pt X, I[ I. (7) If there are two solutions to (4)’ their difference must solve (6). We cannot blindly apply the Blanchard and Kahn results to equation (7). Some of its solutions may not satisfy (6). Our method is to first solve (7), and then substitute into (6) to see if it imposes any additional restrictions. Let K= -(C +‘D)-‘A (C + D)“(I - B) 1. Let J be its Jordan form with eigenvalues arranged in order of increasing modulus and let S satisfy SJ = KS. Partition both J and S into four blocks each so that Jll and Srr are m,, x ~YQ,and 522 and Ss2 are rnd x mat. ’ Our technique also applies to the recalcitrant example C of Blanchard and Kahn. 6 THEOREM. Suppose that K is invertible and QS has full rank. Lf m,, > m’, there are infinitelymany polynomially bounded solutions, if mi, < m’ most initialconditions do not admit any polynomially bounded solutions, while if q, = m* < m with Sll - D& invertible, all initial conditions yield a uniquepolynomially bounded solution. First consider the case m,, c m*. Take the expectation PROOF. equation (4) and multiply by S-l. EoS-’ Setting & = S-r [1 it with initial condition QSRc = F. polynomially bounded and deterministic. [ 1 Eopt Eopt+l at time zero in , thii yields &+r = J& + If there is a solution to (4)’ & will be By subtracting a particular solution, we may assume fit = 0 without loss of generality. Then & = J*&, and polynomial boundedness requires that the last m d entries in & be 0. The inital condition QS& = F then gives us m* equations in min unknowns. Since QS has full rank, it will usually have no solutions. It follows that (4) will usually have no solutions. Now suppose mi, > m*. It is tedious but straightforward to use Blanchard and Kahn’s procedure to show that at least one solution to (4) exists. Again consider the homogeneous equation obtained by taking expectations at time zero and multiplying by S-l. polynomially bounded solution to thii with initial condition QS& Any = 0 can be added to a solution to (4) to obtain another solution. As long as the last moutentries in & are 0, & will be polynomially bounded. Since there are m* < mi, equations in mi,, unknowns, there are many such Rc, and the solution is not unique. Finally, suppose mi,, = m* 5 m. Again, there is at least one solution. We restrict our attention to the associated homogeneous stochastic difference equation (6). Pesaran (1987) shows that the general solution to (6) is = SJ*Mt = K’SMt where M* is an arbitrary martingale. It is easily verified that [I, -D]K Substituting in (6) we obtain [I, -D]S J*+l(Mt-Mt+l) = [A, B] + [0, C]K. = 0. Again, the last rnoG entries of M* must be zero in view of polynomial boundedness. Since rni, 6 m, we have [I, -D]S J*+l(W-Mt+~) = (%I - DS21)~;I-1(M~,t+~ --Ml,:) = 0. Then the invertibility of Srr - D&l and Jll implies Mrt =Mr,*+r. Finally, plugging in the initial conditions, we see that M* = 0, and the solution is unique. QED One interesting aspect of this type of equation is that the distinction between predetermined and non-predetermined variables is unimportant in the solution procedure. We can only discover which variables are which type by solving the equations. What was important was the presence or absence of intitial conditions. The predetermined/non-predetermined 7 distinction is important to Blanchard and Kahn only through the initial conditions. If there are predetermined variable without initial conditions, trying to use the distinction causes problems. A one-sector deterministic optimal growth problem illustrates thii problem. Lmearizing the Euler equations, we obtain a second-order difference equation in the capital stock kt. Introducing zt = Ict+r as a variable gives a first-order system of the type considered by Blanchard and Kahn. Th e saddlepoint property will typically hold, and there will be one root inside the unit circle and one root outside it. However, since everything is deterministic, all variables are predetermined and root-counting implies that there are (usually) no polynomially bounded solutions since there are two predetermined variables, kt and zt. Yet it is well-known that the linearized system will have a unique polynomially bounded solution. The problem is that the Blanchard and Kahn framework requires us to impose an initial condition on the future capital stock x t. Of course, only one such initial condition is consistent with optimality. Any other initial condition will not yield a solution. Our framework handles the optimal growth problem correctly by only specifying one initial condition, for current capital. Our root counting implies that there is a unique solution the correct result. 4. The Solutions Now that we are armed with an appropriately modified Blanchard-Kahn result, we are ready to investigate the relations between the three types of solutions to our systems - the Blanchard-Kahn type, the Evans-Honkapohja ARMA solutions and McCallum’s minimum state variable solutions. (a) The stationary money supply rule Equations (l), (2), and (3a) yield a reduced form expression for prices of l [b-XR*+(X+c)Etpt+l+(X+c)(a++ut]. I%= 1+x+c This expression is easily put in the framework above, yielding Em+1 = &(1+ (8) x + C>Pt - WI where wt = b - AR’ + (A + c)(a + rt) - ut. The root of thii equation is (1 + X + c)/(A + c). For large X, this root is outside the unit circle. Since there is precisely one non-predetermined endogenous variable at time t, there is a unique nonexplosive solution. It is pt = (A + c)a + b - AR* + (Xl++c~+;vf. 8 Thii is also the global minimum state variable solution derived in McCallum (1983,1986). When R* = a, this solution will approach a limit as X + 00. This limit apprmcimates the solution for large X. It is not to be confused with the solution to the lit (8), which is without obvious economic meaning. directly incorporate of equation The money supply rule (3a) does not a trend rate of money growth. It makes intuitive sense that the nominal interest rate target is the expected real rate of interest. In this case the nominal interest rate is & = R* + (rt +ut)/(l +X +c) and will fluctuate randomly around its target. Even though b is not its target, the money stock m = b + X(rt + ut)/(l + X + c) likewise fluctuates randomly about b. The limiting value of the price level as X + 00, pt(oo), is well-defined for R’ = a and equal to Ptb) = ac -I- b+ rt. As in other interest rate pegging literature, the price level is unaffected by the money demand disturbance. The limiting behavior of the price level is also seen to be equal to the price level of a money supply rule in which the central bank buys and sells bonds at a nominal interest rate that is expected to produce a money supply of b. That is, if the monetary authority chooses to buy and sell bonds at a nominal interest given by Rt = +t-lpt - b), the price level pt (00) and interest rate a = R’ emerge as the unique polynomially bounded solution when c > 0. The interpretation of the limiting value of (9) as a peg is thus well motivated. While (9) represents the unique nonexplosive solution to (8), there are infinitely many explosive solutions. The class of these with finite degree ARMA representations can be found by employing the method used in Evans and Honkapohja (1986). First eliminate the constant term with the transformation qt = pt - (UC+ b). The finite degree ARMA solutions are then found by substituting qt = CfE,~qt,i + Cf=,(biut-~+ cirt+). The general ARMA solution is given by Pt = &(I+ A + +t-1 - (b - AR’) + w-l] - (a + w-l) + bout + car: where &, and COare arbitrary. The nonexplosive solution (9) solves this for bo = -l/(1 X + c) and co = (X + c)/(l 00) + + X + c). In o th er cases, it is easy to see that the difference equation for pt is explosive for positive X. Arbitarily limiting the solution to bounded price levels, one could solve (8) forward as in Sargent (1979) and end up with equation (9). 9 Alternatively, restricting the solutions to be expectationally stable in the sense of EWIU (1985) rules out solutions other than (9). (b) Nonstationary money supply rules We next analyze rule (3b). The reduced form equation for prices derived from (1)) (2)) and (3b) is pt= ’ 1+x+c [Ata - R*) + (I+ c)pt-1 - c&-a - crt-1+ ut-1+ (X + c)rt - + (A + c)Etpt+l ut]. 01) In this case, it’s simpler to walk through the steps of the theorem rather than invoke it directly. Update, take expectations conditional on date t information, substitute zt = Etpt+l, and rearrange to obtain the matrix form [~~]=i%[ -(IO+ c) 1+ x+c x + 2c ] [:I +& [A(R’-.P+.rt-Vr] This new system is in our framework and has eigenvalues p= 02) 1 and Y = (1 + c) / (X + c). Since at most one eigenvalue will be outside the unit circle, there is a temptation to use Blanchard and Kahn’s proposition 3 to conclude there are an infinity of nonexplosive solution. This would be an error. The transition from (11) to (12) has introduced some extraneous solutions which solve (12) but not (11). Let pt be any solution to (12) which also solves (11). As before, any other solution to (12) can be obtained by adding #rnlt+vtrn2t where mlt and rnzt are martingales and p and v are the eigenvalues. Consider first the case when 1~1> 1. The polynomial boundedness requirement implies m2t = 0. Now substitute in (11), and use the fact that pt solves (11) to obtain ml,t+r = mlt. It follows that (11) has a unique solution, provided we know the initial price level. Provided we know the initial price level, the extra constraint imposed by (11) has the same effect on determinacy as reducing the number of non-predetermined variables by one. Roughly speaking, (11) turns pt into a predetermined variable. Now consider the case where 1~1< 1. Again substituting in (11), and using the fact that Y = (1 + c)/(X + c), we obtain Xvt+1m2,t+l+ml,t+l = Xvt+1m2t+mlt. Even given an initial price level, there are an infinity of solutions to (11). In particular, taking the lit as A 3 00 will not produce a unique solution. The solution to an interest rate peg with this underlying money supply rule will suffer from nominal indeterminacy. The myriad of nonexplosive finite-degree ARMA solutions can be found by extending the methodology of Evans and Honkapohja (1986) so that it encompasses equation (11). 10 First remove the constant term with the transformation qt = pt - X(R’ - @/(A - 1) (the time dependence arises from the unit root). Then set qt = Cfzlqqt-i + C~=,(biut-i+ cirt-i)a It is easily verified that there are at most two non-zero lags. There are three solutions, dependingonwhetheral = (l+X+2c)/(A+c) oral solves (X+c)af-(l+X+b)u~+l+c = 0. The first is the most general form and is given by Qt = &[(I + X + 2C)qt-1- (I+ c)qt-2 + crt-2 - ut-f + ut-1 - (A + c)r+l] + h&h - cut-l/(X + c)] + c0[rt - cc-l/(X + c)] (13) where b and co are arbitrary. The roots of this difference equation are 1 and (1+c)/(X+c). When I1+ cl < I;\+ cl, this solution is nonexplosive. The other two solutions are actually specializations of (13).The second solution has = 1,and is the solution picked by McCallum. Thii solution obeys a1 Qt = !?t-1 + Thiisolutionoccurswhenbo &--[(A - l)vt + (A + c)rt + vt-1 - crt-l]. (14 = (A-l)/(l+c) andco = (X+c)/(l+c). Choosing thesevalues implies that (13) can be obtained from (14)by multiplying (14)by I - (1+ c)L/(X+ c), where I and L are the identity and lag operators, respectively.’ The third solution has al = (1+ c)/(X+ c)and obeys Qt= 1+c x+t-1+ L[ut-l x+c - crt--l]+ rt. When II+ cl< IX+cI, th is is also a minimum state variable solution. However, since it fails to be admissible for some parameter values (X < 1),it is not the global minimum state variable solution of McCallum. Similarly, choosing be = 0 and CO= 1 implies that (13)can be obtained from (15) by multiplying it by (I - L). Note that adding a constant term to (15) also gives an equation that solves (l3), since the constant disappears upon application of I - L. Further note that this would fail for (15’) since the constant term in (13’) is zero. Transforming back to pt yields, respectively Pt = $-[X(R’ - a)+ (1+ X + 2c)pt-1 - (1+ c)pt-2 + a-t-2 - ut-2 + q-1 - (A + c)rt-I] ’ To further see that McCallum’s solution is not the unique nonexplosive solution when X < 1, one can add the linear combination of martingales #mu + ytrnzt to equation (14) where mrt = Cf,lari and m2t = Cf,rflvj with ar and P arbitrary. Since 1~1, 1~15 1, this is also a nonexplosive solution. Also, a little algebra confirms that the solution with arbitrary martingales can be transformed into equation (13). 11 + bo[vt- cut-l/(X+ c)] + c0[rt - m-l/(X + c)]. Pt =- *- $--[(A a) + Pt-1+ AR pt = $---[&-(I (13’) - 1)ut + (A + c)rt + ut-1 - crt-11. (14’) + c)(R* - a) + (1 + c)pt-1 + ut-1 - crt-11+ rt. (15’) Perhaps a more intuitive way of seeing the underlying indeterminacy is to iterate (12 forward n periods and to examine the system. Let nt = [ crt _ ut ’ ]D”dz=[A(Rp-.)l for notational simplicity. Then (12) becomes Et [ -] = S-lJ”S [E] + $-[s-@)sz + s-‘J”-‘sflt] (16) where J is the diagonal matrix composed of the eigenvalues of (12) and S-l represents the matrix of corresponding eigenvectors. They are given by s-1 =[ Xfc l’tc 1 1 1. Examining the tist row of (16) yields GPt+n = (A + C)(a= - l)&pt+1+ [(I + c) - (A + +qPt +A[=-n](R*-a)+(#-‘-l)(crt-ut) (17) where (Y = (1 + c)/(X + c) and we have substituted zt = Etpt+l. Since c > 0, we observe that larj> 1 for -1 - 2c < X < 1. Dividing (17) by cr” and letting n + 00, we obtain Etn+l =pt+&R*-u)+‘;;cVf’ which is the global minimum state variable solution in equation (14’). The other case of interest is when X < -1 - 2c or A > 1, which implies Ial < 1. Requiring R* = a and letting n + co, equation (17) becomes Em+1 = +-[(I + c)pt - (crt - ut) + (A - l)Etz-+)]. Recalling that R* = a, we see that this agrees with (15’) when &p(co) = 0. However, there exists an infinity of solutions for Etpt+l, one for each arbitrary choice of Etp(co). Interestingly, if one constrains oneself to the choice Etp(oo) = Etp:+l, the global miniium state variable solution (14’) is obtained. this solution. Essentially, McCallum’s (1983) procedure picks 12 The global minimum state variable procedure for the above model works in the following manner. The procedure works by perturbing the model and examining a more general specification of the money supply rule rnf = hmt-l + X(& - R’). The procedure then picks the solution associated with the eigenvalue that does not go to zero when h is zero. When h = 0, we know the nonexplosive solution is unique. The procedure indicates that the solution associated with the positive root of the difference equation for the price level is the correct one and implies that the coefficient on pt-1 in the solution for pt is one. As mentioned, this means that Etpt+i = Etpt for all j and hence J-G(=) = Etn+r. In many instances the global miniium state variable methodology has intuitive appeal. A casual look out the window doesn’t seem to indicate that indeterminacy is an important aspect of the economic environment. Adopting a solution that is robust for various values of preference or technology parameters seems sensible since unique solutions often characterize well-defined optimization problems. In the case of interest rate (money supply) rules the parameters in question, h and X, are just arbitrarily chosen by the modeler or policymaker. In a standard optimizing model with money in the utility function choosing h = 1 does not violate any transversality conditions, yet leads to a nominal indeterminacy. An alternative interpretation to McCallum (1986) is that a money supply rule given by (3b) is not well specified, rather than hi interpretation that hi particular solution should be chosen. Inferences drawn from the solution when h = 0 may not be relevant for a solution when h = 1 as economic systems that are stationary are quite different from those that are not. (c) An alternative nonstationary money supply rule If instead of (3b) we close our economic system with equation (3c), the results are strikingly different. The reduced form becomes pt = [(I + c)pt-I + (A + c)&pt+l - XEt-lpt+l - q-1 + (A - c)Et-IP~ + ut-1 + (A + c)rt - ut]/(l+ Updating puts this in our form with A = (1 + c)/(l C= -X/(l+X+c) and D = (X+c)/(l+X+c). [2$]=:[ x + c)* + x + c), B = (A - c)/(l (18) + A + c), Taking expectations and using zt s Etpt+l -(1"+c) 1;2c ] [ft]+ [rt_o,/c]. 13 The eigenvectors for this system are ~1= 1 and Y = (1 + c)/c. As both are non-zero, the matrix K is non-singular. It is easily verified that transforms K into Jordan form. Since c > 0, we have one eigenvector outside the unit circle. Since S - DS = l/(1+ X + c)# 0 and the first column of QS is 1 + c - c = 1,the hypotheses of the theorem hold. Since we have one initial condition and one root outside the unit circle, there is a unique polynomially bounded solution from a given initial money stock. (This could also be easily verified by looking for additional solutions of the form pt + ptmlt + vtmzt where pt is a solution to (18). The polynomial boundedness implies m2t = 0. Substituting in (18)shows that ml,t+l =mLt.] An extension of the Evans and Honkapohja (1986)technique yields the following second-order solution pt = [(1+ 2c)pt-1 - (1+ c)pt-2 - ut-2]/c + ut4/(X + c) + rt-2 - rt-1 + b0[vt - (A + Xc + c2)vt-l/c(X + c)] + c0[rt - (A + Xc + c’)rt-l/c(X + c)] (19) where b. and c,-,are arbitrary. As in the previous section, there are two first-order equations solving (18).The 6rst is the global minimum state variable solution of McCallum, Pt = Pt-1+ $--[(A - 1)~~+ (A + +t + vt-I - crt-11 (20) and the second is the explosive solution 1+c Pt = cpt-1 ut-1 - vt + c - rt. (21) As before, (19) may be obtained from (20) by setting b = (A - l)/(l+ c) and CO = c 1 an d multiplying (20) by I - (1 + c)L/ c, while (19) is obtained from (21) 0 +M1+ by setting &, = -1 and co = 0 and multiplying by I - L. However, unlike the previous section, (21)always yields an explosive solution. Although (20) is a well-defined nonexplosive solution for finite but arbitrarily large X, the variance of the price level goes to infinity as X --) 00. As McCallum (1986) notes, this result is model specific. An augmented model that includes an aggregate supply term containing pt - Et-Ipt would yield a price level with finite variance no matter how large X became. As before, letting X + 00 can be interpreted as having the central bank buy and 14 sell bonds at a nominal interest rate that will yield a money supply equal to last period’s money supply. When the aggregate supply term is included, buying and selling bonds at & = $Lpt C - m-l] yields an equation similar to (20)as the liiiting (d) A comparison solution of this augmented model. of all three rules From an intuitive standpoint our results are a little puzzling. Responding to interest rate deviations from a preset value is a well-defined procedure when the underlying money supply rule is stationary, yet yields indeterminacy when the money supply is nonstationary. This indeterminacy occurs even when R’ = a the expected real interest rate. Yet a nonstationary money supply rule that responds to deviations of the nominal interest rates from last period’s expectation of the current nominal interest rate yields unique nonexplosive solutions with the property that Et-l& = a in equilibrium. With respect to the latter puzzle our conjecture is that specifying the nominal interest rate target at Et-l& places restrictions on beliefs that are not present with money supply rule (3b). For example today’s expectations of future money supplies, mt+l, are equal to m with money supply rule (3~). With specification (3b), Etmt+n = mt + XC~=~(Et&+j - a). Deviations of Et&+j from a cumulate in expectations of future money. With X > 1 thiileads to an entire family of pricelevel pathsthatareconsistent withvarious departure ofE&+i from a. Essentially, money supply rule (3b) d oes not pin down the future and hence does not uniquely define current nominal quantities when X > 1. This inability to uniquely define expectations of future money is potentially a problem for money supply rule (3a). However, deviations of Et&+j from R’ = a, do not cumulate. That is, it is impossible for E tm t+j to stray too far from the value b. Apparently this is sufficient to guarantee uniqueness and that Et-l& = a (when R* = u). An alternative way of examining the difference between rules (3b)and (3~)would be to posit a hybrid money supply rule mi = m-1 + @t - Et-&) + b(&-& - a), mo given. With this rule the eigenvalues are (1 + c)/(X, + c) and 1 implying that -1 - 2c < X2 < 1 is needed for uniqueness of the nonexplosive solution. Considering only positive values of X2, deviations of Et& +n from a can potentially accumulate in the expectations of mt+,. However with X2 < 1 these deviations are insufficient to generate nonuniqueness. with X2 < 1,Et&+n = a. Hence 15 From our discussion one might also draw the conclusion that observing nonstationary money is inconsistent with the central bank responding to nominal interest rate deviations around some arbitrary value. This is not the case, since one could append a nonstationary money control error, xt = xt-1 + et, to equation (3a) without influencing the discussion concerning uniqueness. Interestingly, the rule (3a) with a random control error is identical to rn5= w-1 + A(& - R*) - A(&-1 - R’) + et, mo given. The eigenvalues when thii rule is used are (1 + X + c)/(X + c) and 1, which implies a unique nonexplosive solution. 5. Barro’s Model In a recent article, Barro (1989) builds on the work of Goodfriend (1987) and McCallum (1986) in an attempt to generate a money supply rule that produces time series properties of nominal variables that are consistent with actual observations. He also employs McCallum’s (1983) procedure to isolate a unique global minimum state variable solution. We have seen that this procedure can not always be relied upon to guarantee a unique nonexplosive solution and, as we will show in the next section, does not always produce expectationally stable solutions. It is, therefore, worth investigating Barro’s extension with our methodology. For our purposes, a simplified version of Barro’s money supply rule can be written as m,b= mt-l + X1(& - R’) - As(&-1 - R*), mo given where we have altered the model by holding the interest rate target constant. The eigenvalues for the system using this money supply rule and (1) and (2) are (1 +X3 +c)/(& +c) and 1. Clearly not all values of AS and X1 will produce a unique nonexplosive solution. In Barro’s model, however, the central bank is concerned about two objectives. One is to smooth nominal interest rates by minimizing E(& - R*)2. The other is to miniize the variance of unexpected price level movements, i.e., E(pt - Et-1pt)2. The first concern implies the As should be set arbitrarily large, while the second implies that Xr = (1 + Xa)B - c(1 - 8) with 8 = ~:/(a: of Xl/h + a:). As X3 gets arbitrarily large, the limiting value = 8. The restriction placed on parameter values by a concern for price level surprises guarantees uniqueness for thii model. Hence, in searching for money supply rules that yield nominal determinacy, one can employ reasonable loss functions that generate the appropriate parameter restrictions. 16 6. Expectational Stability Evans (1985) proposes an alternative to McCallum’s particular solution from a class of solutions. (1983) procedure for choosing a He requires that any solution be expecta- tionally stable. Take a small deviation from a rational expectations solution. Use this to solve your system, and then update expectations. This process must converge to the same rational expectations solution. The time t expectation of pt+l at the N-th stage will be denoted Ef’pt+l. We will restrict our attention to linear expectations functions having the form E:pt+l = aN + PNPt -t vNet + 6Nut where et = [(A + c)rt - vt]/(l + X + c) and ut = (ut - crt)/(l (22) + X + c). Updating, taking expectations at time t, and substituting in (22), we obtain (a) Trend stationary money supply Equation (8) with R’ = a can be written as Pt = AI + Al&t+1 where & = (b+cu)/(l+X+c) is expectationally the qC?CtatiOIlS and Al = (X+c)/(l+X+c). + et (24) We can easily check to see if (10) stable as X + 00. Since ut does not appear in (24), we may omit it from fUIlCtiOIlwithout 1oSSOf generality, leaving Eypt+l = (YN+ @Nptf yNet. Update (24), take expectations at time t and use (23) with 6~ = 0 to get Efv+lpt+l. This yields the recursive relationships aN+l = & pN+l = Al& -I- Ar(l + /?N)QN (25) 7N+l = &PN~N- Thii system has two types of steady state solution. The frrst is a = b + cu, p = 0 and 7 = 0. Obviously, this is implied by equation (10). The second has a = -&/Al, /3 = l/Al and 7 arbitrary. Linearizing (25), we see that the three roots for the first steady state are Al, 0 and 0. Thus for IAll < 1, aN, /?N and 7~ converge to their steady state values as N -+ 00. Since IAll < 1 for arbitrarily large A, the interest rate rule (3a) is expectationally stable. Similarly, linearizing about the second steady state yields roots 1 + Al, 2 and 1. This second steady state is unstable since one of the roots is larger than 1. 17 (b) Non-stationary money supply We next check to see if the minimal state variable solution of (11)is expectationally stable. When R* = a, (11)has the form Pt = &%lpt + &Etpt+l + &a-l + w-l + et (26) where B1 = -c/(1+X + c),B2 = (A+ c)/(l + X + c),BS = (1+ c)/(l+ X + c),and et ad ut are as before. Updating (26), taking the expectation at time t and using (22) and (23) yields the recursive relationships aN+l = [Bl + B2(1+ pN+l= pN)]aN &~N+Bz&+& 7N+l = (Bl + &PN)~N 6 N+l = (Bl + B~BN)~N + 1. This iterative system has two types of stationary solutions: P = I,7 The first has Q = 0, = 0 and 6 = (1+ X + c)/(l+ c) (giving (14’)); the second has CYarbitrary, @ = (1 + c)/(X + c), 7 = 0 and 6 = (1 + X + c)/(X + c) (giving (15’)). Linearizing around the first steady state we obtain eigenvalues B1 + 2B2 (twice) and B1 + B2 (twice). These eigenvalues are (2X + c)/(l+ X + c ) and X/(1+ A + c). When -(1 + c)/2< X < 1,this solution is expectationally stable while for X > 1 or X < -(1 + c)/2, it is expectationally unstable. The second type of solution is not unique, but rather a continuum. Because of t&ii, we only ask for stability in the sense that small deviations from a member of the solution set lead to convergence to another member of that set. These solutions have eigenvalues 1, (2 + c)/(l + A + c) and l/(1 + X + c) (t wice). This system is expectationally unstable for -(2 + c) < X < 1. When X < -(2 + c) or X > 1,three of the eigenvalues have modulus less than 1. Only stability of aN is still in doubt because of the unit eigenvalue.5 However, given initial CQwe find that N-l aN = m n [BI + &(I + Pi)]. i=O Thus lo&N/~] = Nflo$% + Bz(l +bi)]. i=O ’ It pays to be careful here. Evans (1985) claims stability with a unit root in hi Proposition 2, part II. However, actual calculation of the solution reveals that the equilibrium set is not stable. Rather, his system converges to the other steady stateifa isbelowits steady state value, while his system blows up if a is above its steady state value. 18 We now apply the ratio test to the sum. Since fii+r = B1/3i+ B# is (log[Bl + I&(1+ B&i + B& + Bs)])/(log[Bl+ Bt(l +&)I). + Bs, the relevant ratio Using L’H6pital’s rule and the facts that pi + p = (1 + c)/(X + c) and fl = B1/3 + B2P2 + Bs allows us to obtain the limit Bl+ 2B2p = (2 + c)/(l+ X + c). Since thii has modulus less than one when X > 1, the series, and hence the product, converges. It follows that the solution set is stable. In particular, (15’) gives the expectationally stable solution for large X, while the miniium state variable solution (14’) is expectationally unstable. (c) Alternative non-stationary money supply Using (3~) implies a different outcome. Equation (18)can be written as Pt = GE,-,p, + C2E t-1 pt+l + C&n+1 cl = (A - c)/(l + X + c), Cz = -X/(1 (1 + c)/(l + Cot-1 + w-l + X + c), C, = (X + c)/(l + et, + X + c) and C4 = + X + c) with ut-1 and et as before. The presence of the constant term creates additional difficulties since this equation has a unit root. To handle it, we include a trend term in the expectation function. Thus EFp t+l = ~N+PNpt+rNet+SNut+tlNt. Proceeding as before, we obtain the following recursive relationships: aN+l = [cl P N+l = cl@, + + 7N+l = [Cl + 6N+1 = tl~+l= (c2 + (c2 &)(I + + fh)]aN c2)& (c2 + cS)/&V]7IV cS)@N]bN + + + VN c, [cl + (c2 + 1 [Cl + (C2 + Cs)(l + PN)]~N. Again, there are two steady state solutions corresponding to (20) and (21). The first has a! = 0, p = 1, 7 = 0, 6 = l/C 4 and q = 0 while the second has a arbitrary, p = (1 + c)/c, 7 = 0, 6 = (1 + X + c)/c and q = 0. Linearizing around the minimum state variable solutions in (20), the conditions for expectational stability are ]Cr + Z(C2 + C3) 1= 1(X + c)/(l + X + c)I < 1 and ]Ci + C, + C3] = IX/(1 + X + c)] < 1. These conditions are satisfied for positive values of X. The solution corresponding to (21) requires that ](X + c)/(l +X + c)] < 1 and I(2 + X + c)/(l+ X + c)] < 1. These cannot be simultaneously satisfied. 7. Summary and Conclusions This paper provides a detailed examination of the nominal determinacy properties of various interest rate rules. The class of rules examined is broad enough to essentially span 19 most of the literature on this subject. Our analysis indicates that care should be taken when specifying policies in which the central bank responds to nominal interest rates. This is especially true when the underlying money supply rule displays nonstationarity, since determinacy issues are sensitive to specification of the interest rate feedback term. In thii case the preferred model includes responses to deviations from last periods expectation of the current nominal interest rate rather than responses to some arbitrary target. If for other reasons, as in Barre (1989), it is desirable to explore how monetary responses to an exogenous interest rate target affect economic outcomes then the investigator should be careful to restrict the admissable parameter values that are assigned to feedback coefficients. In examining this class of policies we have drawn from a wide range of literature that deals with the solutions to rational expectations models. This literature ranges from the undetermined coefficient approach of Lucas to the martingale method of Pesaran (1987) and the general ARMA solutions of Evans and Honkapohja (1986). We have also looked at the expectational stability properties of the models and find that there is a one to one correspondence between unique nonexplosive solutions and expectationally stable solu- tions. When uniqueness is a problem we find that under a peg the general class of ARMA solutions are expectationally stable, but that McCallum’s solution is not. To rigorously examine the question of uniqueness we have extended the counting rule methodology of Blanchard and Kahn (1980) to models that include past expectations of current and future variables. Since rational expectations models with this attribute are fairly common, our methods used to establish our theorem should be useful in a variety of other contexts. Appendix: Blanchard and Kahn Revisited Although Blanchard and Kahn’s result is not implied by our result, or vice-versa, our method does apply to their system. A close reading of the Blanchard and Kahn paper reveals that the predetermined/non-predetermined distinction is only required when the number of predetermined variables is equal to the number of roots inside the unit circle. However, our method shows that predeterminedness is not required here either. Suppose there are wz’variables Xt with initial conditions and m variables without initial conditions Pt. These correspond to the predetermined and non-predetermined variables, respectively. The system considered by Blanchard and Kahn is 20 For uniqueness, we may consider the homogeneous case (f2t = 0) without loss of generality. $i to obtain EtQt+1 = At& The Jordan form of A is .7 = S’lAS [ I and the general solution to the homogeneous equation is Qt = SJtMt where Mt is an Apply Et and set Qt = arbitrary martingale. If md = m, the last m entries of Mt are zero. Substituting back in the original equation, we obtain SrrJ’+lMr,t+r = [ASJ'Mlt]l= SrrP’Mrt. invertible, this implies Mr,t+r =Mrt. the m’-vector Mlo. When Srr is The solution is deterministic and is determined by The m’ initial conditions imply Mlt =Mrc = 0. It follows that the inhomogeneous system has a unique solution. Our method also applies to their first example C. It is Yt = uEt-lYt - &. For simplicity, suppose the forcing process 2, obeys Et-l& = 0. There are no initial conditions. Applying Et-1 to the equation yields 0 = (1 - a)Et-lYt. Unless a = 1, Et-lYt = 0. Substituting back in the homogeneous equation yields Yt = -& as the unique solution. The second example C, P t+r = a(E&+2 - E&+1) + et doesn’t quite fit the statement of our theorem since zero is a root of K. Nonetheless, the same techniques apply and yield a unique solution provided Pc is given. References Robert J. Barre (1989), Interest Rate Targeting, J. Mon. Econ. 23, 3-30. Olivier J. Blanchard and Charles M. Kahn, The solution of linear difference models under rational expectations, Econometrica 48 (1980), 1305-1311. Lawrence E. Blume, David Easley and Margaret Bray, Introduction to the stability of rational expectations equilibrium, .I. Econ. Theory 26 (1982), 313-317. Margaret Bray, Learning, estimation, and the stability of rational expectations, Theory 26 (1982), 318-339. J. Econ. 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Phelps, “Individual expectations and aggregate outcomes: An introduction to the problem,” in IndividualForecastingand Aggregate Outcomes (R. Frydman and E. S. Phelps, eds.), Cambridge University Press, Cambridge, 1983. Marvin Goodfriend, Interest rate smoothing and price level trend stationarity, J. Mon. Econ. 19 (1987), 335-348. C. Gourieroux, Jean-Jaques Laffont and A. Monfort, Rational expectations in dynamic linear models: Analysis of the solutions, Econometrica50 (1982), 409-425. Bennett T. McCallum, Price level determinacy with an interest rate policy rule and rational expectations, J. Mon. Econ. 8 (1981), 319-329. Bennett T. McCallum, On non-uniqueness in rational expectations models: An attempt at perspective, J. Mon. Econ. 11 (1983), 134168. Bennett T. McCallum, Some issues concerning interest rate pegging, price level determinacy, and the real bills doctrine, J. Mon. Econ. 17 (1986), 135-160. lK~~g%saran, Identification of rational expectations models, J. Econometrics 16 (1981), . M. H. Pesaran (1987), The Limits to Rational Expectations, Basil Blackwell, New York Thomas Sargent (1979), Macroeconomic Theory, Academic Press, New York.
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