OPTIMALITY AND TAYLOR RULES.

Blake, Andrew P.

Andrew P. Blake [*]

This paper discusses the role of forecasts in the control of inflation. Much has been made of variations on the so-called Taylor rule for inflation control. Forward-looking Taylor rules are reconciled with optimal control using a class of rules described as error-correcting Taylor rules.

Introduction

This paper is about monetary policy rules, and in particular interest rate rules of a generic sort which have become known as Taylor rules (Taylor, 1993). The Taylor rule (in its original form and many later variations) has provided the valuable service of focusing attention on the appropriate specification of interest rate rules that can adequately control inflation.

It is useful to begin by expositing the general form of rule used in the literature. In Taylor (1993) the following policy rule is analysed:

[i.sub.t] = r + [[pi].sub.t] + [[kappa].sub.y][y.sub.t] + [[kappa].sub.[pi]]([[pi].sub.t] - [[pi].sup.*]) (1)

where [i.sub.t] is the nominal interest rate at time t, [[pi].sub.t] the inflation rate, [y.sub.t] the output gap, r the equilibrium real interest rate and [[pi].sup.*] the target inflation rate. The values chosen for [[kappa].sub.y] and [[kappa].sub.[pi]] were both 0.5 and both of r and [[pi].sup.*] set at 2 per cent per annum, reflecting the United States' historical experience. For the United Kingdom the equilibrium real interest rate is closer to 3 per cent, and the avowed inflation target 2.5 per cent.

As an empirical regularity for the United States, (1) has much to recommend it. [1] A particular advantage is that it looks very like an interest rate reaction function that could feasibly be used in monetary policymaking. This indicates that whatever the monetary authorities are reacting to in the short term can be translated very straightforwardly to a simple rule. Even so, the rule is open to interpretation: is the coefficient on the output gap a response to anticipated inflation or does it reflect a concern for output stabilisation? More generally, does a feedback on any variable indicate a concern for that variable or its use as an indicator? The foreign exchange rate is an example of such a variable.

It should be emphasised that Taylor was by no means the first exponent of this approach. The use of simple feedback rules for interest rates was suggested in Phillips (1954) and Phillips (1957), and was as widespread in the 1980s as the use of optimal control was in the 1970s, although the targeting regimes were many and varied. The effectiveness of a number of alternative policy regimes was studied by Weale, Blake, Christodoulakis, Meade, and Vines (1989), but direct inflation targeting as the main policy option was relatively under-investigated before the mid-1990s. More recently, the specific open economy inflation targeting problem has been investigated by inter alia Blake and Westaway (1996), Svensson (2000) and Ball (1999). This article seeks to bring together a number of strands of the literature, by providing a unified framework for policy rules and their implications.

To facilitate this in what follows, the rules will be reparameterised to a common representation. Notice that even (1) has an alternative representation, and can be written:

[i.sub.t] = (r + [[pi].sup.*]) + [[kappa].sub.y][y.sub.t] + (1+[[kappa].sub.[pi]])([[pi].sub.t] - [[pi].sup.*]) (2)

by adding and subtracting [[pi].sup.*] to the right-hand side. Here the nominal interest rate is set equal to its equilibrium (r + [[pi].sup.*]), with a coefficient greater than unity on inflation deviations from target. The constant in the rule, (r + [[pi].sup.*] reflects the policy regime in so far as it includes the inflation target. In much of the literature both and [[pi].sup.*] are set to zero.

Of course such a rule doesn't fit always and everywhere. It may be rescued as an empirical relation by suitable generalisation. An important one is that it has been argued that policymakers target the forecast rather than the current inflation rate. Thus forward-looking Taylor rules have been investigated, notably by Clarida, Gali and Gertler (1998) and Clarida, Gali and Gertler (2000). One possible parameterisation is:

[i.sub.t] = (r + [[pi].sup.*]) + [[kappa].sub.y][y.sub.t] + [[kappa].sub.[pi]] ([E.sub.t][[pi].sub.t+k]-[[pi].sup.*]). (3)

A different version (where [[kappa].sub.y] is set to zero for the most part) was investigated by Batini and Haldane (1999) (and later by Batini and Nelson, 2000). These authors arrived at a number of interesting conclusions; Clarida, Gall and Gertler (1998) argue that the empirical fit is improved using this specification, Batini and Haldane (1999) that this form of rule has superior, near optimal, operational characteristics relative to (1).

Both of these observations have important implications. The results in Clarida, Gall and Gertler (1998) support the view that the simple Taylor rule is insufficient to describe properly central bank behaviour. The forecast element indicates that quite complex rules are being used to set policy. A rule of this sort needs the policymaker to forecast conditional on its own implemented policy. [2] Further, the question of optimality is an important one. For a (log-)linear closed economy Ball (1997) pointed out that symmetric coefficients are unlikely to be optimal. His argument is that optimal (in his terminology efficient) policy is a feedback on the state with coefficients that reflect the objective function. This follows as there always exists a closed-form solution to any model (rational expectations or not) which reflects the structural form and the policy regime. This is an immediate consequence of any optimal policy being a feedback on the state. Establishing what constitutes the state is therefore crucial to assessing whether forecast based rules can be close to optimal. In what follows the forecast is expressed as a closed form function of the state as well as the rational expectation and the policy feedback.

A further generalisation is to include policy inertia. Interest rate persistence is often modelled by a relationship such as (see, e.g. Woodford, 1999):

[i.sub.t] = [[gamma]i.sub.t-1] + (1 - [gamma])[(r + [[pi].sup.*]) + [[kappa].sub.y][y.sub.t] + (1 + [[kappa].sub.[pi]]([[pi].sub.t+k] - [[pi].sup.*] (4)

where k [greater than or equal to] 0 allows the forecast targeting approach to be nested. Such policy inertia (or smoothing) is often included to improve the empirical fit. With forward targeting this then gives five parameters to be chosen rather than the original three. Smoothing can easily be empirically justified. Policymakers often appear to display a marked reluctance to make large or sudden movements in interest rates. Equally, it could be argued that such movements are either a proper reaction to sufficiently important news or reflect the fact that a policymaker keen to establish (or maintain) credibility would be happy to make large movements. Of course, the case can also be made for the converse!

Taylor-type policy rules, and forecast based variants, have therefore become a serious policy proposal. Forecasts have taken a central role in describing the policy process, particularly monetary policy in the United Kingdom (Britton, Fisher and Whitley, 1998). It is therefore important to assess available policy proposals in a common framework. This includes optimal policies as well as more simple Taylor-type rules. We do that here, using a simple numerical model. We also assess the role of forecasts conditional on endogenous or exogenous policy, and how these could be used in policymaking.

To do this, a common framework is developed for the analysis of the policy rules described. This is a version of a vector error correction model (VECM), cast in statespace. Such a VECM representation gives rise to a very simple interpretation of the reduced form model and is used to illustrate each of the feedback regimes. The model used is an ad hoc open economy supply side model, based on two period overlapping wage contracts as posited by Fuhrer and Moore (1995). Previous work based on this explicit framework includes Fuhrer and Moore (1995), Blake and Westaway (1996), Fuhrer (1997), and Batini and Haldane (1999). Other work has used a reduced form supply side which can be interpreted as a version of the Fuhrer-Moore supply side including Bean (1998) and Clark, Goodhart and Huang (1999). The interest in such a supply side stems from the results postulated by Ball (1994) of costless (even beneficial) disinflations from Keynesian dynamics. Optimal control solutions for the model are derived and re-interprete d in the VECM framework and inflation targeting regimes are shown to have similar properties.

The remainder of the article is set out as follows. In the following section a representative open economy model is briefly described. The forward-looking Taylor rule framework is then discussed in the third section and optimal polices in the fourth section. In the fifth section a vector error correction representation of the models is developed and examples given. The next section brings the results together and a conclusion follows. An appendix describes the optimal control problem in a little more detail.

A simple open economy model of output and inflation

This section outlines a wage-price system that generates a reduced form Phillips' curve for an open economy. In addition, a real exchange rate equation is required and the model is closed by an equation for the output gap. Prices are set by a simple markup on wages, taking relative real wages into account rather than money wages. The reduced form of the system yields a simple Phillips' curve, outlined next.

The model has much in common with a number of others in the literature. It originates in Blake and Westaway (1996) and was used by Batini and Haldane (1999) with minor modifications which are retained. It is similar to that described by Ball (1999), who most notably uses a different exchange rate equation, which has important implications for the interpretation of optimal monetary policy.

Wages and prices

It is assumed that domestic prices are set by:

[[p.sup.d].sub.t] = 1/2([w.sub.t] + [w.sub.t-1]) (5)

which is a straight markup on costs, with the markup normalized to zero. The markup is a function of assumed costs in so far as wages are set in alternate periods by a bargaining process. Consumer prices are set by:

[[p.sup.c].sub.t] = [phi][[p.sup.d].sub.t] + (1 - [phi])[e.sub.t] (6)

with e the nominal exchange rate and [phi] reflects the degree of openness. The wage contract is set as in Fuhrer and Moore (1995), depending on consumer prices as:

[w.sub.t] - [[p.sup.c].sub.t] = [[chi].sub.0]([E.sub.t][w.sub.t+1] - [E.sub.t][[p.sup.c].sub.t=1] + (1 - [[chi].sub.0])([w.sub.t-1] - [[p.sup.c].sub.t-1]) + [[chi].sub.1][y.sub.t] (7)

as a relative bargain, where [y.sub.t] is output. Using (7) in (5) gives:

[[2p.sup.d].sub.t] = [[p.sup.c].sub.t] + [[p.sup.c].sub.t-1] + [[chi].sub.0]([E.sub.t][w.sub.t+1] + [E.sub.t-1][w.sub.t]) + (1 - [[chi].sub.0])([w.sub.t-1] + [w.sub.t-2]) - [[chi].sub.0]([E.sub.t][[p.sup.c].sub.t+1] + [E.sub.t-1][[p.sup.c].sub.t]) -(1 - [[chi].sub.0])([[p.sup.c].sub.t-1] + [[p.sup.c].sub.t-2]) + [[chi].sub.1]([y.sub.t] + [y.sub.t-1]). (8)

Use (6) one period ahead and one period lagged to derive:

[[2p.sup.d].sub.t] = [[p.sup.c].sub.t] + [[p.sup.c].sub.t-1] + [2[chi].sub.0][E.sub.t][[p.sup.d].sub.t+1] 2(1 - [[chi].sub.0])[[p.sup.d].sub.t-1] - [[chi].sub.0]([E.sub.t][[p.sup.c].sub.t+1] + [E.sub.t-1][[p.sup.c].sub.t]) - (1 - [[chi].sub.0])([[p.sup.c].sub.t-1] + [[p.sup.c].sub.t-2] + [[chi].sub.1]([y.sub.t] + [y.sub.t-1]). (9)

From (6):

[[p.sup.d].sub.t] = [[p.sup.c].sub.t] - [micro]]/2 [c.sub.t] (10)

where [c.sub.t] = [e.sub.t] - [[p.sup.c].sub.t] the real exchange rate, and [micro]] = 2(1-[phi]/[phi]). Using (10) to eliminate [[p.sup.d].sub.t] in (9) and using [[pi].sub.t] = [delta][[p.sup.c].sub.t] gives:

[[pi].sub.t] = [[chi].sub.0][E.sub.t][[pi].sub.t+1] + (1 - [[chi].sub.0])[[pi].sub.t-1] + [[chi].sub.1]([y.sub.t] + [y.sub.t-1]) -[micro]([[chi].sub.0][E.sub.t][c.sub.t+1] - [c.sub.t] + (1 - [[chi].sub.0])[c.sub.t-1]) + [[epsilon].sub.t] (11)

where it can shown that [[xi].sub.t] may be expressed as:

[[xi].sub.t] = -[[chi].sub.0]([[pi].sub.t] - [E.sub.t-1][[pi].sub.t]) + [micro][[chi].sub.0]([c.sub.t] - [E.sub.t-1][c.sub.t]). (12)

This last term is usually subsumed into any stochastic part of the model, which indicates a moving average error. This is mostly ignored in the literature cited above and for simplicity the same is done here, as it has no impact on the equilibrium outcomes, but can affect the dynamics.

Output and real exchange rate

The assumption of a non-stochastic model is maintained for specifying the rest of the model. A simple IS curve is given by:

[y.sub.t] = [[alpha].sub.1][y.sub.t-1] - [[alpha].sub.2]([i.sub.t] - [E.sub.t][[pi].sub.t+1]) + [[alpha].sub.3][c.sub.t] (13)

where the output gap is affected by the real interest rate and the real exchange rate, but has some persistence, and the open arbitrage real exchange rate is:

[E.sub.t][c.sub.t+1] = [c.sub.t] + [i.sub.t] - [E.sub.t][[pi].sub.t+1] (14)

where i is the nominal interest rate. Disturbance terms could be added to both.

State-space representation

The model is collected into state-space form for convenience, enabling the state variables to be identified better. It is:

[y.sub.t] - [[alpha].sub.2][E.sub.t][[pi].sub.t+1] = [[alpha].sub.1][y.sub.t-1] + [[alpha].sub.3][c.sub.t] - [[alpha].sub.2][i.sub.t] (15)

[E.sub.t][c.sub.t+1] + [E.sub.t][[pi].sub.t+1] = [c.sub.t] + [i.sub.t] (16)

[[chi].sub.1][y.sub.t] - [micro][[chi].sub.0][E.sub.t][c.sub.t+1] + [[chi].sub.0][E.sub.t][[pi].sub.t+1] = -[[chi].sub.1][y.sub.t-1] - [micro][c.sub.t] +[micro](1 - [[chi].sub.0])[c.sub.t-1] - (1-[[chi].sub.0])[[pi].sub.t-1] + [[pi].sub.t] (17)

where essentially only the real variables are retained except for the inflation rate.

It is worth noting that this model, whilst similar to many found in the literature, is not derived from a rigorous set of microfoundations. It is used not because it represents any particular economy although versions of it have been calibrated roughly to United Kingdom data (e.g.Batini and Haldane, 1999; Batini and Nelson, 2000). It has several features in common with the National Institute's domestic macroeconometric model (NiDEM) but many differences. However, the inflation targeting characteristics of the model are plausible in that the effects of changes in the target and the stochastic properties are quite similar to NiDEM. It is also very similar to many of the models developed from microfoundations discussed in the introduction.

Model solution

The model (15)-(17) can be solved using the method proposed by Blanchard and Kahn (1980), outlined now. For a given model, written:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

a reduced form needs to be derived for the policy options to be comparable. The rational solution requires the jump variables to be expressed as a linear function of the state, so that:

[x.sub.t] = -N[z.sub.t-1]. (19)

The solution is usually found from the diagonalisation of the transition matrix A. as described by Blanchard and Kahn (1980), but is suffices to note that N satisfies:

N[A.sub.12]N-N[A.sub.11] + [A.sub.22]N-[A.sub.12] = 0 (20)

which can be easily verified by substitution. The reduced form of the model is then:

[z.sub.t] = ([A.sub.11] - [A.sub.12]N)[z.sub.t-1] (21)

For (15)-(17) this is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

constitutes the state vector. The jump states, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

are eliminated using (19). Thus the model is in reduced form a first order (VAR) system. The state may, of course, not be minimal in the sense that there are redundancies -- variables that could be eliminated in determining the time path of the system. One will be demonstrated for the numerical example below. Thus the variable count from a necessary set up of the model in its structural form does not necessarily tell us what the effective state is for the model under control and with expectations closed by a rule such as (19).

Forward targeting

In this section we make the distinction between forecast targeting based on a forecast conditional on the implemented policy and based on a forecast conditional on a fixed nominal interest rate.

Feeding back on the endogenous forecast: ex post targeting

A practical implication of endogenous forecast targeting is that the state-space model needs to be augmented for the extra leads. It is convenient to think of the jump states being augmented to give:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where each lead of the target variable adds a jump state. Provided the furthest lead is the one fed back on, there is no problem for finding a solution except as the lead gets too far distant and there is no saddlepath solution. However, we emphasise that the forecast values depend only on the current state, as it is the forecast at time t, and that it is still the case that [x.sub.t] = -N[z.sub.t-1]. It must now be that:

[i.sub.t] = -[F.sub.0][s.sub.t] - [F.sub.1][s.sub.t+1]

so that

A = [(I + [BF.sub.1]).sup.-1](A - [BF.sub.0])

for example. N depends on the coefficients of the model and the coefficients of the policy rule including any parameter k for the lead.

Conditioning on a constant policy: ex ante targeting Write the state-space model as:

[s.sub.t+1] = [As.sub.t] + [Bi.sub.t] (23)

where [s.sub.t] = [[z'.sub.t]([E.sub.t][x.sub.t+1])']'. If we wish to forecast k periods ahead, it seems natural that we can use this model recursively, giving predictions:

[s.sub.t+k] = [A.sup.k][s.sub.t] + [[[Sigma].sup.k+1].sub.l=0] [A.sup.l][Bi.sub.t+k-l-1]. (24)

As the model is a rational expectations one, then the final solution is more complicated: the closed form solution including expectations of future instrument values from k to [infinity] needs to be derived using (as before) Blanchard and Kahn (1980). This can be simplified if the interest rate is assumed fixed at a particular level when making the forecast. Assume that [i.sub.t+k] = [i.sub.t-1] for k [greater than or equal to] 0. Then let [s.sub.t+k] be the forecast of [s.sub.t+k] on the basis of such an 'unchanged' policy, i.e.:

[s.sub.t+k] = [A.sup.k+1][s.sub.t-1] + ([[[sigma].sup.k+1].sub.l=0] [A.sup.l])[Bi.sub.t-1]. (25)

If monetary policy is set using [i.sub.t] = [[gamma]i.sub.t-1] + (1 - [gamma])([F.sub.0][s.sub.t] + [F.sub.1][s.sub.t+k]) then together with the unchanged policy assumption of the forecast we can write:

[i.sub.t] = (F + KTB)[i.sub.t-1] + [KA.sup.k][s.sub.t] (26)

where [gamma] = [[[sigma].sup.k+1].sub.l=0][A.sup.l]. The state variables now evolve as:

[s.sub.t+1] = (A + [BKA.sup.k])[s.sub.t] + B(F + KTB)[i.sub.t-1] (27)

Together (26) and (27) gives:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (28)

This represents the model under control. We now compare and contrast the two forward targeting regimes in the next section.

Ex ante and ex post

There is, then, a considerable distinction between what constitutes the ex ante and the ex post forecast used in policy rules.

The ex post forecast-based policy is that studied by Clarida, Gali and Gertler (1998) and Batini and Haldane (1999). There are two aspects worth discussing here. Firstly, any such rule must be able to be re-expressed as a function of the state -- after all, the forecast itself is just that. This is investigated below. Expressing the policy rule as a forecast rule is only worthwhile if the rule is more easily interpretable in that form or perhaps it is more parsimonious. Secondly, there is a natural limit past which k, the forecast horizon, cannot be extended. For example, if the policy rule relies on deviations of the target variable from target value, then if the policy rule is successful in controlling inflation, say, there will be no deviation from target. If there is no deviation, there can be no policy reaction. Indeed, if it is to be made to work in some circumstances, the policy rule may have to be actively destabilising to give sufficient signal to the feedback rule. Thus k has to be shorter than the effective horizon of the implemented policy, the horizon past which disturbances are rejected. This restriction is absent in general from the ex ante forecast based policy.

The ex ante forecast is that described by the Bank of England's fan chart diagram (Britton, Fisher and Whitley, 1998), and as such has a seemingly straightforward interpretation. However, there are problems with it. For the model we describe, the expectational equilibrium is indeterminate for a fixed nominal interest rate; there are insufficient unstable roots to use the Blanchard-Kahn method. This implies in terms of the outlined solution that N is not unique.

But (28) does not actually rely on there being an ex ante expectational equilibrium. Monetary policy conditioned on an ex ante fixed interest rate yields an interesting interpretation ex post. Despite the policy rule being absent when the monetary authority sets current monetary policy through the forecast, the expectational equilibrium is closed by the private sector using the rule through (28). This formulation of the policy rule indicates the difficulty in separating the impact of expectations on current interest rate setting.

Notice too, that (26) implies considerable restrictions on the coefficient values that can be used on the 'ex ante' feedback rule to yield sensible looking policies, certainly if the ex ante policy rule is to look like a sensible Taylor rule. Experiments with the model outlined in this paper have yielded only small regions of stability and very poor control performance for Taylor-type ex ante forecast based rules. Typically they exhibit highly cyclical behaviour characteristic of borderline stable policies, and as such cannot be recommended, and we do not pursue them further.

It should also be noted that presenting a forecast based on some ex ante monetary policy is not entirely contentless. Although it is difficult to imagine a feedback rule that used that forecast directly, conditioning on an announced policy will inform agents of the state of the economy. This would be enhanced if the forecast were conditioned on intended policy.

Optimal policy

Any definition of optimality needs to make clear the objective function to be optimised. An appropriate perperiod cost function reflecting concerns with both inflation and output is:

[C.sub.t] = [omega]([[pi].sub.t] - [[pi].sup.*]) + (1 - [omega])[y.sub.t] + [varphi][delta][i.sub.t] (29)

which includes an additional term penalising changes in the interest rate to incorporate interest rate smoothing into the policymaker's objective function. [3] This last term appears in Batini and Nelson (2000) and Woodford (1999). Discounted and summed from t = 0, ... [infinity], (29) is:

[V.sub.0] = [[[sigma].sup.[infinity]].sub.t=0][[rho].sup.t][C.sub.t]. (30)

The optimisation of such an objective function subject to a linear model is a standard problem. The solution is sketched in the Appendix. Given the solution of appropriate matrix Ricatti equations the optimal policy can be represented as a feedback rule on the predetermined states and co-states.

It must also be possible that the objective function can be minimised given any parametric policy rule. There is a substantial complication: such a rule, if not the unrestricted fully optimal rule, is not certainty equivalent. This carries the implication that there is usually no one parameterisation of an arbitrary rule that minimises the value of the objective function independent of the initial values of the state. Optimisation subject to a given shock (or variance-covariance matrix of shocks) yields what is often referred to as an 'optimal simple rule'. In reality this is the optimal simple rule for a given set of initial conditions and expected rather than realised shocks.

In the next section we develop an encompassing representation of the policy rules, based on error correction. We then illustrate the various policy rules discussed using numerical examples.

A VECM control representation

Error-correction representations of decision models have long been popular, and since Nickell (1985) and Salmon (1982), are often used to represent agent behaviour characterised as the solution to inter-temporal optimisation problems. [4] The feedback-based simple rules policy framework was first set out in seminal work by Phillips (1954) and Phillips (1957). The rule has three components of Proportional, Integral and Derivative coefficients and takes the form:

[delta][i.sub.t] = [[delta].sub.p][delta]([i.sup.*] - [i.sub.t]) + [[delta].sub.I]([i.sup.*] - [i.sub.t]) + [[delta].sub.D][[delta].sup.2]([i.sup.*] - [i.sub.t]) (31)

where [[delta].sub.P] is the proportional coefficient and so on. Turnovsky (1977), for example, casts this explicitly as a target-instrument problem as:

[delta][i.sub.t] = [[delta].sub.P][delta]([[psi].sup.*] -[[psi].sub.t])[[delta].sub.I]([[psi].sup.*] - [[psi].sub.t]) + [[delta].sub.D][[delta].sup.2]([[psi].sup.*] - [[psi].sub.t]) (32)

where [psi] is the target variable and [[psi.sup.*] is the target value. The difference between the two is similar to how problems can be formulated either using calculus of variations or using control theory. See Intriligator (1971) for examples of how different decision models can be cast in these forms.

Salmon (1982) in particular suggested that PID-type control mechanisms can be optimal for certain loss functions. Such a result is related to that of Woodford (1999), who considers the instrument smoothing problem. Both of these suggest that particular loss functions can be associated with a final form interpretable as a PID control rule. [5]

The vector version of (31) is straightforward to derive. We concentrate on the integral mechanism, in common with Blake and Westaway (1996). A general (necessarily first order) state-space model with only predetermined variables and under control is:

[[zeta].sub.t] = A[z.sub.t-1] (33)

which can clearly be rewritten as:

[delta][z.sub.t] = [pi][z.sub.t-1] (34)

where [pi] = A - I. This formulation is no longer in state-space; indeed, it looks much more like the transform of a first order VAR. If the state does not include lagged predetermined variables then they are equivalent. This is so for our example model, so the VAR transform is entirely appropriate. Further decompose this to:

[delta][z.sub.t] = [alpha][beta]'[z.sub.t-1] (35)

where [pi] = [alpha][beta]'.[beta] is chosen to represent the long-run target values of relationships. [alpha] is a diagonal matrix of weights determining how much disequilibrium in each long run affects current values for each equation. If the state does not include additional lags of variables (as with our reduced form model) then each equation has an error correction representation. Such an error correction representation generalises (31) by including the simultaneous disequilibrium from the other long-run relationships in the model relevant to that variable.

For our inflation-targeting model, the output gap and the real exchange rate are stationary variables; the inflation rate is necessarily stationary in an inflation targeting regime with effective monetary policy. As this is a state-space model different conditions to those normally applied to the VECM model apply to the number of target variables and error correction terms. [6] The error correction transformation may still make perfect sense and be properly interpretable algebraically: below it is shown to be the case for the model (15)-(17).

Alternative policy rules

We recap the proposed rules and nomenclature. The first is the simplest Taylor rule:

* The Taylor rule (TR) is:

[i.sub.t] = [[pi].sup.*] + [[kappa].sub.y][y.sub.t] + [[kappa].sub.[pi]]([[pi].sub.t] - [[pi].sup.*]).

Forward looking Taylor rules come in two forms, with and without output feedback:

* The Forward-Looking Taylor rule (FLTR) is:

[i.sub.t] = [[pi].sup.*] + [[kappa].sub.y][y.sub.t] + [[kappa].sub.[pi]]([E.sub.t][[pi].sub.t+k] - [[pi].sup.*])

* The Batini-Haldane Forward-Looking Taylor rule (BH-FLTR) is:

[i.sub.t] = [[pi].sub.t] + [[kappa].sub.[pi]]([E.sub.t][[pi].sub.t+k] - [[pi].sup.*]).

The version of the PID rule used in the literature cited before is simply:

* The Blake and Westaway (1996) PID rule is:

[i.sub.t] = [[pi].sub.t] + [[kappa].sub.[pi]]([E.sub.t][[pi].sub.t+k] - [[pi].sup.*]).

Finally, the class of rule based on the VECM is:

* The Error-Correcting Taylor rule (ECT):

[delta][i.sub.t] = -[eta]([i.sub.t-1] - [[pi].sup.*]) - [[kappa].sub.y][y.sub.t-1]

-[[kappa].sub.[pi]]([[pi].sub.t-1] - [[pi].sup.*]) + [[kappa].sub.c][c.sub.t-1])

where this has the characteristics of including all the state variables inside the error-correction mechanism (we give an example where [[kappa].sub.c] = 0 below), which essentially looks like a 'long-run Taylor rule'. The inclusion of the exchange rate in such a rule has been interpreted by Ball (1999) as reflecting a 'monetary conditions index' or MCI, where a weighted average of the interest rate and exchange rate determines the true monetary position. [7]

Each of these parametric policy rules and optimal policy rules can be combined with the model (together with any appropriate leads) and then closed by calculating N.

The transformed model

We chose the following values for the parameters of the model: [[alpha].sub.1] = 0.8, [[alpha].sub.1] = 0.5 and [[alpha].sub.1] = 0.2, with [[chi].sub.0] = 0.2$ and [[chi].sub.1] = 0.1. The only parameter we consider variation in is which is either 0.9 or 1, although the former is the main case. With [phi] = 1 then the exchange rate only affects inflation through demand, an assumption made by Batini and Haldane (1999).

The reduced form model under control can therefore be written as:

[delta][i.sub.t] = -[eta]([i.sub.t-1] - [[pi].sup.*]) - [[kappa].sub.y][y.sub.t-1]

-[[kappa].sub.[pi]]([[pi].sub.t-1] - [[pi].sup.*]) + [[kappa].sub.c][c.sub.t-1]) (36)

[delta][y.sub.t] = -[[eta].sub.y]([y.sub.t-1] - [[tau].sub.i]([i.sub.t-1] - [[pi].sup.*])

-[[tau].sub.[pi]]([[pi].sub.t-1] - [[pi].sup.*]) + [tau].sub.c][c.sub.t-1] (37)

[delta][[pi].sub.t] = [[eta].sub.[pi]](([[pi].sub.t-1] - [[pi].sup.*]) + [v.sub.y][y.sub.t-1]

-[v.sub.i]([i.sub.t-1] - [[pi],sup.*]) - [v.sub.c][c.sub.t-1] (38)

[delta][c.sub.t] = -[nu.sub.c]([c.sub.t-1] - [[iota].sub.[pi]]([[pi].sub.t-1] - [[pi].sup.*])

-[[iota].sub.y][y.sub.t-1] + [[iota].sub.i]([i.sub.t-1] - [[pi].sup.*])) (39)

which can be algebraically represented by (35).

Given the model coefficients, we then set [[[kappa].sub.[pi]]= 0.15 and [gamma] = 0.6 in the BH-FLTR, the simplest form, and allow the lead to vary. For a lead of 4, the complete reduced form model (35) is then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

These numbers are pretty much in general what would be expected, but note that the coefficient in the long-run part is positive for the inflation rate equation.

More generally, in Table 1 we vary the lead for the endogenous feedback. It is interesting how little the coefficients actually change. There are radical changes for shorter horizons than 2. The coefficients then look much closer to the original Taylor rule. Variations on the base case are given in Table 2.

The first notable result of the variations is that if we set [phi] to unity the exchange rate completely drops out of the feedback rule. The model behaves very much like a closed economy one and the reduced form feedback rule looks like an integrated version of the Taylor rule. A second result is that the FLTR variation, where there is additional feedback on the output gap, yields relatively little change in the coefficients. We return to this below. We also note that experiments with the exogenous forecast in the feedback rule were very unsuccessful: few of the parameterisations produced stability and none satisfactory control. Next we discuss the optimal control experiments.

Optimal control coefficients

For the optimal control experiment reported we set [omega] = 0.5, [omega] = 0.2 and [rho] = 0.99, so that inflation and output are given the same weight in the objective function, and there is a fairly high discount factor. The optimal policy generated is of the form:

[i.sub.i] = - F[[[pi].sup.*] [i.sub.t-1] [y.sub.t-1] [[pi].sub.t-1] [c.sub.t-1] [[[micro].sup.1].sub.t] [[[micro].sup.2].sub.t]]'

where the [[[micro].sup.n].sub.t] are predetermined co-state variables. It is straightforward to reparameterise these coefficients to yield a form exactly as our preferred error correction form. In Table 3 we report the coefficients in that form with the predetermined variables as part of the long run, although as they must go to zero in the long run perhaps we should leave them outside.

There is considerable similarity between the coefficients of the optimal rule and the various FLTR versions in this parameterisation. This is an important result. The FLTR (with or without output feedback) looks very much like an optimal rule. The main difference is that the output and inflation feedback coefficients are larger and, of course, there is no feedback on the predetermined co-states. From the starting values of the FLTR rules we have been unable to mimic these values exactly: from the starting point of our objective function we have been unable to do the reverse. However, they remain close, explaining the Batini and Haldane (1999) result that the (BH-)FLTR is close to optimal. A further interesting result (not reported here) is that this model exhibits considerable inflationary bias under consistent optimal control. This raises the possibility that the FLTR is too time-inconsistent to be sustainable, and is an avenue for future investigation.

Optimal, error-correcting and forward-looking policies

So far we have developed error-correction simple rules, forward-looking simple rules and optimal polices in a common framework for a simple open economy macro-model. This yields a number of related results.

The first, most obvious, result is that the optimal policy can be interpreted in terms of its feedback on the state in straightforward ways. Thus a policy can be often rewritten to correspond to an error-correction representation in exactly the way that familiar reparameterisations of econometric models are obtained. The second (and well known result) that is emphasised is that both the model and the objective function can have a substantial impact on what are the relevant state variables for the optimal feedback.

Thirdly, the forward-looking Taylor rule is encompassed by the VECM form used to express optimal control rules in that the reduced form representation of the models is very similar, with, of course, different coefficient values. The forward looking Taylor rules for this model are integral control rules: this is also the case for a reasonable class of optimal rules.

Fourthly, the distinction between ex ante and ex post forecasts is crucial. Neither is very satisfactory on its own as a description of policy: the forward-looking Taylor rule for this model, common to most of the literature, is a version of an integral control rule in reduced form, familiar from the error correction representation. The policy conclusion should be clear. It might be better to adopt a familiar and well understood control framework such as PID which does not rely on agents and policymakers sharing common perceptions of the economy to guarantee success. If it is possible to implement complex policy rules which use such information, it is better by definition to optimise directly rather than approximately and use the best possible rule.

Conclusions

This article discusses a number of proposed monetary policy rules in a unified framework. It takes as its starting point the Taylor rule proposal, and in particular focuses on forward-looking variants of it. These have become popular both in the academic literature and in the presentation of actual forecasts. An important caveat on the actual results is that we need to use a simple numerical open economy model to illustrate them. Similar models are widely used in the literature, and differences in models will yield different results. However, the principles will remain the same, and more sophisticated models will strengthen the argument.

The article emphasises the following:

* In determining what are the relevant state variables, it is crucial to know how policies actually work, developed in the paper by discussion of the rational expectations properties of the representative model.

* Notions of ex ante and ex post forecasts in policy rules are important, and the article describes exactly how they could be used.

* PID control rules are reviewed, in the context of Taylor rules and alternative proposals.

* Optimal policy rules as a criterion for assessing other policy prescriptions are developed by reviewing their parametric form.

* A VECM framework is shown to nest the other policy prescriptions.

* The policy conclusion reached is that ex post forward targeting may give the degree of freedom required to deliver close to optimal policies. Conversely, it has the same informational requirements as optimal policies, and is therefore not a simple alternative to optimal policy design.

What implications does this have for the conduct of monetary policy in the United Kingdom? The Bank of England takes the role of forecasts seriously: it presents forecasts through the fan chart in an ex ante framework, and has investigated the implications of ex post forward targeting. The evidence is that forward-looking Taylor rules are empirically more satisfactory at explaining central bank behaviour than simple ones. This article suggests that this is consistent with complex optimising behaviour by policymakers. It is also consistent with error correction based policies, as they all have similar representations. The lesson of this analysis is that forward-looking Taylor rules are not a simple effective alternative to formal optimisation. They are at least as complicated to implement as optimal rules. Forecast-based rules can be difficult to interpret without a clear understanding of the underlying forecast assumptions.

(*.) National Institute of Economic and Social Research. e-mail: a.blake@niesr.ac.uk. This article develops some themes and ideas which came about from conversations with Tibor Hledik, to whom I am grateful. I would like to thank Ray Barrell and Martin Weale for useful comments on an earlier draft.

NOTES

(1.) Nelson (2000), amongst others, investigates the United Kingdom, where regime shifts need to be included for any satisfactory fit to be found. By contrast, Kapetanios, Mitchell and Weale (2000) are able to find a stable relationship from 1985 but not one that is immediately recognisable as part of the Taylor 'family'

(2.) Below, in an important departure from this, we investigate conditioning on forecasts of the target variable conditioned on an arbitrary policy.

(3.) Despite a common misperception, direct instrument costs are not strictly required for the existence of an optimal policy for this model, and make only a small difference to the optimal policy without [phi] being large.

(4.) A further notable application of error correction models, usually (but not exclusively) derived from optimising behaviour such as consumption, has been the econometric implications of co-integrated systems (Engle and Granger, 1987). In a multivariate co-integrating framework Johansen (1988) suggested a way to both model multiple long-run relationships in a vector error correction model and to perform inference. The latter has little immediate importance and the former is a natural extension of the Granger representation theorem for co-integrated systems. A similar notation to Johansen (1988) is adopted below. There, long-run (indeed co-integrating) relationships are denoted by [beta]. For co-integrated systems the number of co-integrating vectors needs to be less than dim(s), a restriction not necessary in general.

(5.) Of course, Woodford (1999) is interested in sluggish adjustment rather than PID representations: such behaviour can be incorporated into PID responses.

(6.) For example, the allowable number of long-run relationships is not purely related to the number of states even for co-integrated systems.

(7.) Using MCIs to determine the correct policy stance -- reducing the nominal interest rate if the exchange rate is high for example -- in reality still leaves open what role the exchange rate plays in the generalised rule. It can be interpreted as a target, an instrument or an intermediate target with ease.

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Appendix. Optimal control for economic models

An alternative brief analysis is given in Woodford (1999). See the references there for more comprehensive surveys. The solutions are sketched here without proof.

The standard discounted quadratic loss function is given by:

[V.sub.0] = [[[sigma].sup.[infinity]].sub.t=0] [[rho].sup.t]([s'.sub.t]Q[s.sub.t] + [2s'.sub.t][Ui.sub.t] + [i'.sub.t][Ri.sub.t] (41)

which is the appropriate matrix generalisation of (30). A general linear state space model can be written:

[s.sub.t+1] = [As.sub.t] + [Bi.sub.t]. (42)

Comining the quadratic loss function with the linear model without rational expectations, then the optimal feedback rule is a linear function of the current state given by:

[i.sub.t] = -[Fs.sub.t]

where:

F = [(R + [rho]B'SB).sup.-1] (U' + [rho]B'SA) (44)

with:

S = Q-UF-F'U'+F'RF+[rho]A'SA (45)

where A = A-BF.

The matrix S has a number of interpretations, but if the optimal control is derived from from Lagrange multipliers such that the co-state variables are required to be on the saddlepath, then write:

[[micro].sub.t] = [Ss.sub.t] (46)

With rational expectations some adjustments must be made. If the [x.sub.t] variables are jump variables then the co-states associated with them are in turn predetermined for the optimal control. Thus:

[x.sub.t] = -[[S.sup.-1].sub.22][S.sub.21][z.sub.t] + [[S.sup.-1].sub.22][[[micro].sup.2].sub.t] (47)

so that the 'free' co-states are determined by:

[[[micro].sup.1].sub.t] = ([S.sub.11] - [S.sub.12][[S.sup.-1].sub.22][S.sub.21])[z.sub.t] + [S.sub.12][[S.sup.-1].sub.22][[[micro].sup.2].sub.t]. (48)

Define:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (49)

then the control rule can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (50)

so it turns out the model under control is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (51)

with the free variables given by (47) and the instruments by (50).

An optimal consistent policy is usually calculated by eliminating the feedback on either the jump variables or the associated co-states. This implies calculating the optimal control subject to either:

[x.sub.t] = - [N.sub.k] [z.sub.t] (52)

for a given [N.sub.k] or:

[x.sub.t] = -[J.sub.k][z.sub.t] - [K.sub.k][i.sub.t] (53)

for given [J.sub.k] and [K.sub.k]. In the latter case, [K.sub.k] is used in obtaining the first order conditions. In either case the procedure is to substitute out for [x.sub.t] using (52) or (53) and calculate the equivalent:

[i.sub.t] = -[F.sub.k][z.sub.t]

and then iterate through k to a fixed point.