Optimal Rules for Central Bank Interest Rates Subject to Zero Lower Bound

The celebrated Taylor rule provides a simple formula that aims to capture how the central bank interest rate is adjusted as a linear function of inflation and output gap. However, the rule does not take explicitly into account the zero lower bound on the interest rate. Prior studies on interest rate selection subject to the zero lower bound have not produced derivations of explicit formulas. In this work, Taylor-like rules for central bank interest rates bounded below by zero are derived rigorously using a multi-parametric model predictive control framework. This framework is used to derive rules with or without inertia. The proposed approach is illustrated through simulations. Application of the approach to US economy data demonstrates its relevance and provides insight into the objectives underlying central bank interest rate decisions. A number of issues for future study are proposed. JEL E52 C61


1 Introduction and Motivation
The general form of the standard Taylor rule suggests that the short-term interest rate t i applied by the central bank at time t can be set according to the formula where y represents the output gap (deviation of real GDP from potential GDP as percent of potential GDP);  represents inflation rate; subscript t refers to the time the rule is applied, using information up to that time; superscript * represents the desired equilibrium value; ri  is the real interest rate; and y  ,   are coefficients associated with the output gap and inflation rate respectively. In the original publication [1] it was assumed that *0 y  , * 2%   , * 2% r  , 1 references therein]. The stated objective for inertia-based policies is interest rate smoothing, to avoid large variations in interest rates and to produce robust policy rules [3][4][5]. Additional variants of the Taylor rule containing more lagged terms of i have also appeared [6,7].
While the initial inspiration for the Taylor rule was based on fitting actual historical data, Taylor rules and some of its variants can be derived by application of optimization theory on a quadratic objective function, using a small-scale model of the economy to capture the effect of interest rate on inflation and output gap [8][9][10][11]. Such derivations have mainly focused on the effect of the specific form of the quadratic objective function on the resulting rule. This approach, however, has not been successful at producing a rigorous derivation of explicit Taylor rules when a zero lower bound (ZLB) on the interest rate is included in the optimization. Nevertheless, a number of approaches for determining an optimal interest rate subject to ZLB have been proposed, which can be broadly classified into two categories: The first category includes explicit rules that truncate to zero the interest rate TR ii  ), to ensure that a nonnegative interest rate t i is produced [12][13][14]. The rationale behind approaches in this category relies on qualitative analysis of a ZLB-constrained quadratic optimization problem or on other qualitative analysis of optimal policy effects on inflation and output gap.
The second category does not produce explicit rules; rather, it employs numerical simulation, i.e. repeated numerical solution of a ZLB-constrained optimization problem, to determine the optimal values of interest rate for inflation and output gap values in a range of interest [8,[15][16][17][18]. Most studies in this category rely on a constrained dynamic programming formulation of the underlying optimization problem, whose explicit analytical solution is hard to get.
Interesting observations were made in these studies. For example, it was observed that resulting policies may be nonlinear, (rather than piecewise linear, according to truncated Taylor rules) and more aggressive for interest rates close to ZLB (a behavior characterized as pre-emptiveness). However, a rigorous derivation of simple explicit Taylor rules subject to ZLB is, to our knowledge, not currently available.
In this paper, we rigorously derive explicit rules for interest rate subject to ZLB.
Our approach relies on a formalism known as multi-parametric (mp) programming, a technique applied by the engineering community to constrained model predictive control (MPC) [19] or constrained state estimation problems [20]. The following are the key elements of the proposed approach.
 When a ZLB is present, explicit rules can be developed that produce a value for the interest rate through application of one from a finite number of explicit formulas. These formulas entail a finite number of Taylor-like rules. To know which of these formulas will be applied at any time, one has to simply pick an entry from a look-up table, based on checking which inequality is satisfied out of a finite number of a priori developed mutually exclusive linear inequalities on the inflation and output gap.
 Various forms of Taylor-like rules result rigorously from the particular form of the quadratic objective used in MPC. For example, Taylor rules with inertia terms arise from inclusion of a quadratic penalty on the rate of change of the interest rate (rather than on the interest rate itself).
 Application of any interest rate policy, Taylor-like or not, essentially creates a closed-loop feedback controlled economy. Therefore, any policy should, at the very least, result in a stable closed loop. Additionally, it should be fairly robust, namely it should produce sensible results in the presence of discrepancies between assumed economy models and the actual economy.
In the rest of the paper we first provide some background on MPC and mpMPC, and elaborate on the small-scale economy model used. Within this setting, we derive a 5 number of Taylor-like rules, based on a number of MPC quadratic objectives, and examine their dependence on relative weights of various terms in the MPC objective.
The effect of these rules on the resulting closed-loop behavior is examined. Comparison with the standard Taylor rule and actual interest rates implemented by the Central bank is provided. Finally, future extensions are proposed.

Preliminaries: Model Predictive Control (MPC) and
Taylor rules MPC is a class of model-based feedback control algorithms for systems with constraints [21,22]. MPC finds the value of the manipulated input (interest rate in our case) of a controlled process at each point in time by setting up and solving a constrained optimization problem at that time. The optimization involves an objective function (usually quadratic) over a future horizon. The objective contains terms involving future predictions of the controlled variables (output gap and inflation in our case) as well as penalty terms on manipulated inputs within the horizon. Future output predictions are established in terms of a model and existing measurements.
As will be made clear below, MPC (also known as "open-loop optimal feedback") differs from stochastic dynamic programming (also known as "closed-loop optimal feedback") in that MPC does not explicitly account for information that is now expected to be available in the future, thus avoiding the computational complexity of the nested optimization (curse of dimensionality from Bellman's principle of optimality) which burdens stochastic dynamic programming.
Next, we first provide a description of the model we use, and subsequently explain its use in formulating the MPC optimization.

Economy model structure
A semi-empirical linear model around a baseline can describe the evolution of the economy as (2) and (3), can be written as Using the above model, the optimal k -step-ahead prediction for the state x with initial condition stands for the expected value of x at time tk  using all information available at time t . The above prediction will be used in the formulation of the MPC objective below.
It should be noted that the idea here is not to fully explain the complex dynamics of the economy with such a simple linear model. Rather, the intended use of the above model is to help understand how optimal monetary policies are affected by various objective functions and by a ZLB on the interest rate when constrained MPC is used to derive such policies. The dimension of the state vector x is also limited to two, so that the solution of the constrained MPC optimization problem can be easily understood graphically in 2-D and 3-D plots using the mpMPC approach. 8

Economy model calibration
The economy model expressed by eqns. (2) and (3) The eigenvalues of A are 0.58 and 1.05, suggesting that the economy model for the US economy is mildly unstable. Consequently, whatever control policy ones chooses to control the US economy, such a policy must be, at the very least, a stabilizing policy. We develop such a policy below via MPC.

Formulation of MPC optimization
The central bank's generalized loss function projected to infinity at time t is generally of subject to the model constraints the unstable mode stabilization constraints 12 uu , ,..., the input move restriction constraints and the inequality constraints

Taylor rules from MPC without zero lower bound
In the absence of ZLB, eqn. (16), and without penalty on the change of interest rate where ( at time t , which is clearly a Taylor-like rule, as in eqn. (1). It is also clear that functions of the economic model matrices A , B , and of the weights R ,  , given N , m and  .

Choice of prediction horizon length,
N For an unstable system such as the one described by eqns. (2) and (3), the horizon length, N , should be made long enough to ensure that the MPC optimization problem is feasible and ensure closed-loop stability. Systematic methods can be used for selecting N [26][27][28].
In all subsequent developments we will consider 80 N  .  Table 2 substantiates this claim by example, showing that the closed-loop poles remain almost unchanged after increasing the value of m beyond 4. The associated Table 3 shows the resulting coefficient for the Taylor-like solution provided by MPC.
Second, it has been rigorously shown that keeping m small improves the robustness of the closed loop, namely it helps maintain closed-loop stability in the presence of discrepancies between the model used by MPC and the actual system under control [29][30][31].
In all subsequent developments we will consider 4 m  .

Choice of discount factor, 
Following the literature [15,16] we use a value of the discount factor 0.99

 
, except in situations where we explicitly specify a different value. We will comment below on how different values of  affect the resulting Taylor rules and closed-loop stability and performance.

Effects of MPC objective function weights on resulting Taylor rules
For the choice of 80 N  , 4 m  , and 0.99 , discussed in the preceding sections, we now proceed to examine the effect of R and  on the resulting Taylor rules, via eqn.
14 (24). Following the calculations in Appendix A, the matrices H and F in eqn. (22) are calculated as functions of R and  , and coefficients of the output gap and inflation in the Taylor rule or eqn. (1) are expressed analytically in terms of R and  , as respectively, where the values of the corresponding parameters are shown in Table 4. In general, the numerator and denominator for y  and   are polynomial functions of The following general observations can be made on Figure 3 and Figure 4:  (25) and (26) can be derived in the same way. As shown in Figure 5 and Figure 6, the original . This issue will be explored elsewhere.

Taylor rules and resulting closed-loop stability
For any rule proposed, it is important to determine, at the very least, whether such a rule results in a stable closed loop. Combination of the Taylor rule in eqn. (1) with the simple economy model, eqn. (4), yields (Appendix B) the closed loop structure It can be shown (Appendix B) that both eigenvalues of CL A are inside the unit disk, i.e.
the closed-loop system is stable, if and only if 2.1 0.12 8.5 0.06 as illustrated in Figure 7. This is in agreement with the well established Taylor principle that the central bank should raise its interest rate more than one-for-one with increase in inflation [34,35]. Figure 4 shows that this requirement is satisfied for all combinations of the MPC weighting parameters R and  . In fact, Figure 8 illustrates that the stability conditions, eqns. (29) and (30) , as illustrated in Figure 9, which shows that as the value of  is reduced, the value of R should not be too small, to avoid closed-loop instability.
It is interesting to note that as R , namely high values of interest rate are heavily penalized, the closed loop remains stable, due to the stabilizing equality constraint, eqn. (14). For R , eqns. (25) and (26)  Following the preceding observations, it should be noted that the widespread practice of using a discount factor  may be more problematic than realized, in the sense that it may not result in robustly stabilizing strategies. This situation, namely the need to shape weights of the terms in the MPC objective in an increasing rather than decreasing fashion in order to ensure robustness, has been rigorously analyzed in the past [30,31] and should be explored further.

Taylor rules from MPC with zero lower bound
When the interest rate must satisfy a ZLB constraint, the optimization problem to be solved by MPC entails the objective in eqn. (11), the equality constraints in eqns. (12)- (15), and the inequality constraint in eqn. (16). It can be shown (see Appendix C) that for 0 S  , the entire optimization problem can be cast in the form subject to Eqns. (31) and (32) suggest that the optimization problems solved by MPC at successive points in time differ only by the right-hand side of eqn. (32), which is affine in the state t x . No single formula exists for the explicit solution of all of these problems.
However, the optimal solution can be expressed explicitly at each point as and Therefore, determining the active and inactive constraints in eqn. (32), and consequently which is a Taylor-like rule, or as namely at the ZLB value.
To our knowledge, the above development is the first rigorous derivation of an explicit Taylor-like rule that satisfies the ZLB without resorting to either ad hoc clipping of the interest rate value produced by a Taylor rule [12][13][14] or numerical simulation [8,[15][16][17][18]].

Taylor rules with inertia from MPC
A simple form of a Taylor-like rule with an inertia term is where are functions of A , B , S , and  ; and the vector x is defined as In the absence of a ZLB, the minimum in the optimization problem in eqn.
A parametric analysis similar to that in section 3.
respectively, where the values of the corresponding parameters are shown in Table 5.

Inertia-based rules and resulting closed-loop stability
For Taylor rules with inertia as in eqn. (38) the corresponding closed-loop is as shown in Figure 13. As in section 3.1.4, it is also found that all combinations of S and  result in stabilizing monetary policies. Eqn. (47) is the counterpart of eqn. (29) and has been derived before in a different setting, using a rational expectations approach [36].
It is again interesting to note that as S , namely as aggressive changes in the value of interest rate are heavily penalized, the closed loop remains stable, due to the stabilizing equality constraint, eqn. (14).

Taylor rules with inertia from MPC with zero lower bound
Following the same approach as in section 3.2, the optimization problem with eqn. (11) with 0 R  , 0 S  , subject to the equality constraints in eqns. (12)- (15), and the inequality constraint in eqn. (16) can be cast in the form subject to

Numerical Simulations
The objective of this section is to illustrate the interest rate rules resulting from application of the methodology we outlined in the previous section. Emphasis is placed on directly including the ZLB constraint in the development of explicit rules.

Taylor rules form MPC with ZLB
The optimization problem defined by eqn. (31) with inequality constraints given by eqn.  Table 6 through Table 11. The corresponding linear polytopes are illustrated in Figure 14 through Figure 19. Of these tables, Table 7, corresponding to Figure  Further comparison of these figures reveals that the optimal rules follow an asymmetric pattern for small values of R ( Figure 14, Figure 16, Figure 18), as has also been observed in a number of numerical studies with 0 R  [8,13,17,37]. However, this asymmetry practically disappears (i.e. it would be observable only for unrealistically large output gaps) for large values of R ( Figure 17, Figure 19), namely for very sluggish policies.  The infeasibility polytope remains the same. From Table 12 and Figure 20 it can be 26 concluded that in polytopes of low inflation and negative output gap, if the lagged interest rate 1 t i  is high (polytopes 4 and 6), the optimal rule becomes less aggressive than the rule in the unconstrained case. However, for low 1 t i  , the optimal rule is just a truncation to zero of the unconstrained case, eqn. (41). Also, in polytope 5, characterized by low inflation, high output gap, and high 1 t i  , the optimal rule is more aggressive than the rule in the unconstrained case, eqn. (41). Therefore, an important conclusion is that for rules with inertia ( 0 S  ), the optimal policy becomes asymmetrical with respect to both lagged interest rate and output gap for low inflation economic conditions.

Remarks on rules from MPC
The following can be observed in the results of sections 4.1 and 4.2.
 Polytope 1, where no constraint is active, grows in size with increasing R or S .
 The policy becomes sluggish and the size of polytopes 2, 4 and higher decreases as R or S increase.
 For any MPC formulation, situations may arise in which either a negative interest rate would be optimal (when the ZLB is not explicitly included in the optimization) or a stabilizing interest rate at or above the ZLB is not feasible (when the ZLB is explicitly included in the optimization). It can be shown  ) that clipping to zero is optimal interest rate for nearly all economic points while in Figure 15, Figure 17 and Figure 19 ( 0.55 R  ) more of the economic data indicate non-zero interest rate due to the policy rule being sluggish.

Illustration of proposed approach
The first set of simulations serves to simply illustrate the effects of ZLB on the closedloop system. Simulations are shown using the rules presented in Table 6 through Table   11, as well as the rules with inertia shown in Table 12 along with five additional rules with similar structure but different MPC weights R and S (not shown in Table 12 for brevity). For this set of simulations the economy is considered to be at 3.7 y  and 1.9

 
in year 1, corresponding to 2009Q1. The results are summarized in Figure 21 and Figure 22. The resulting sums of squared errors (discrepancies between actual and desired values) are summarized in Table 13 and Table 14.  Figure 24, confirm the preceding assertions for both cases. It is also interesting to note that even though the interest rate in the first case is stabilizing, recovery of the economy is very slow due to the effect of ZLB (inflation stabilization, in particular, takes many years).

Comparison with historical data
We use real-time data available to the central bank at the time of making a decision on the interest rate, for the period 1987Q4:2008Q4. For output gap we use Greenbook data over the period 1987Q4:2005Q4; for the remaining period we consider CBO data [39].
The real-time inflation data is also taken from the same publication.
We focus on the interest rate rule with inertia, eqn. (38), with * 1.  Table 15. Figure 25 compares the interest rate resulting from fitting eqn.  Figure 27.
It is also interesting to examine whether additional insight may be gained by fitting data over short periods for which large residuals result from fitting the entire data set. One such period with large residuals is 2000Q1:2004Q4. nonetheless places all emphasis on output gap (growth). The actual policy implemented over that period and its role on stimulating over-expansion of the economy has been the subject of intense discussion [40].

Conclusions and Future Work
The main issue addressed in this work is the effect of zero lower bound on the optimal interest rate determined by a central bank. We address this issue in a multi-parametric model predictive control framework, which allows the derivation of explicit feedback rules even when inequality constraints are present. Application of this framework to a simple model of the US economy produced a number of Taylor suggest that a small number of simple Taylor-like rules can be applied at each time, depending on the state of the economy. However, it was also shown that simply setting to zero negative interest rates produced by unconstrained Taylor rules is optimal in situations of negative output gap, as happened recently. Furthermore, it was observed, as has been noted elsewhere, that rules with inertia appear to better capture past decisions by the central bank. Such rules have been systematically derived here by considering penalties on the rate of interest rate change in the MPC objective function.
A number of issues touched in this work warrant further investigation, such as the following:  The inverse problem: Given a suggested Taylor-like rule, what objective function, as in eqn. (11), is minimized? A promising approach is suggested in section 3.1.5.
 Robust stability and performance: There is a vast body of work in the automatic control community addressing the robustness issue, namely how a controller performs when the model assumed in controller design has quantifiable uncertainty.
 Modeling and selection of controlled variables: Should the pair output gap and inflation be the main focus or could variables such unemployment [5] be central in controlling an economy?
 Policy adaptation: The main attractiveness of a fixed rule is its simplicity and predictability [38]. However, such a rule may become sub-optimal over time, as the economy or disturbance models change [10]. Can a fixed rule be replaced by a fixed rule adaptation policy that maintains robustness?
We hope to address the above issues in forthcoming publications.

Acknowledgement
The authors would like to acknowledge financial support from Department of Chemical and Biomolecular Engineering at the University of Houston. Helpful comments by Prof.
David Papell (Department of Economics, University of Houston) are also gratefully acknowledged.

Appendix E. Closed-loop stability for inertial Taylor-like rule
The interest rate rule is The characteristic equation for the matrix CL A is given by Closed-loop stability is guaranteed (by the Jury-Routh-Hurwitz stability criterion) if and only if