سیاست بهینه پولی در یک اقتصاد رژیم سوئیچینگ: پاسخ به تغییرات ناگهانی در پویایی نرخ ارز

This paper examines the trade-offs that a central bank faces when the exchange rate can experience sustained deviations from fundamentals and occasionally collapse. The economy is modelled as switching randomly between different regimes according to time-invariant transition probabilities. We compute both the optimal regime-switching control rule for this economy and optimised linear Taylor rules, in the two cases where the transition probabilities are known with certainty and where they are uncertain. The simple algorithms used in the computation are also of independent interest as tools for the study of monetary policy under general forms of (asymmetric) additive and multiplicative uncertainty. An interesting finding is that policies based on robust (minmax) values of the transition probabilities are usually more conservative.

A common concern amongst central bankers is that the true or perceived existence of financial imbalances or asset price misalignments could at some point in time lead to sudden and large adjustments in asset prices, with potentially adverse consequences for inflation and output stability. For instance, one of the major risks that has worried some members of the Bank of England's Monetary Policy Committee (MPC) in recent years has been the possibility that sterling could suddenly fall by a material amount.1 Other risks routinely debated by actual policymakers, including oil price hikes or abrupt changes in key econometric relationships, may also be asymmetric.2 Nevertheless, modelling of asymmetric risks is not very common in the monetary policy literature, possibly because of the lack of readily applicable technical tools.3 In this paper we examine the trade-offs that the policymaker faces when the exchange rate can experience sustained deviations from fundamentals and occasionally collapse. To do so we use a simple algorithm which has rarely been applied in the economics literature. Our analysis is based on the small open economy model of Ball (1999), which comprises a demand equation, a Phillips curve and an equation linking the real exchange to the real interest rate. We modify this model to incorporate regime switching in the exchange rate. In one regime, which we call the bubble regime, any shock can lead the exchange rate to deviate increasingly from fundamentals. In the other regime, which we call the no-bubble regime, the exchange rate displays transitory fluctuations around its fundamental value. The evolution over time of these two regimes is described by a Markov chain so that the times at which the boom begins and ends are stochastic. Moreover, the size of the correction in the exchange rate, which occurs when the economy switches from the bubble to the no-bubble regime, is endogenous for it depends on the lagged exchange rate as well as the policy instrument (plus any additive shocks). We compute the optimal control rule for such a regime-switching economy, which is itself regime-switching, and compare it to the responses implied by an optimised linear Taylor rule. Optimal monetary policy in any given regime, whether implemented through the optimal regime-switching rule or through a (suboptimal) optimised regime-invariant rule, crucially depends on probabilities of moving from one regime to the other.4 The algorithm for computing the optimal regime-switching policy is a modification of the standard linear quadratic regulator problem, in which the constraint is given by a Markov regime-switching vector autoregression (with any finite number of observable regimes) rather than a stationary vector autoregression. The regime-switching model belongs to a class of models that have been studied in the engineering literature at least since Aoki's contribution (1967).5 This formulation is sufficiently general to allow the modelling of a large range of different asymmetric (and symmetric) risks, either concerning changes in the economy's dynamics or additive disturbances, and to accommodate different models. The solution algorithm is also sufficiently simple to be amenable to further interesting developments. For these reasons the algorithm's applicability extends beyond the particular application considered in this paper and can be considered as a general tool for the study of multiplicative and additive uncertainty in monetary policy.6 The small open economy model used in the analysis is meant to capture the main effects of monetary policy in the simplest possible way. In particular, it does not incorporate rational or other forms of forward-looking expectations. Nevertheless, this simple model is close in spirit to the larger macroeconometric models that recently were or are still in use at several central banks. It is not unusual that in versions of these models the uncovered parity condition is, due to its empirical failure, replaced by reduced forms which are good empirical approximations of actual exchange rate behaviour.7 The introduction of regime-switching in the exchange rate equation in our model is meant to capture the complex behaviour of financial market agents, which arguably it would not be possible to characterise explicitly in current state-of-the-art fully microfounded general equilibrium models.8 Thus, the application in this paper can also be thought of as an example of how a policymaker can incorporate judgemental information about a potential misalignment and the uncertainties associated with it into her macromodel and work out the best policy response based on that judgement. Most importantly, there are clear advantages in terms of intuition and transparency of working with a backward-looking model. The trade-offs which a policymaker faces can often be more clearly analysed without the additional layer of complexity represented by any expectation-formation mechanism.9 Besides, we believe that the main insights and conclusions in this paper are likely to carry over to a forward-looking model.10 The recent literature on monetary policy and asset prices – e.g. Bernanke and Gertler, 2000 and Bernanke and Gertler, 2001, Cecchetti et al., 2000 and Cecchetti et al., 2003, Batini and Nelson (2000), Filardo (2001), Tetlow (2003) – looks at whether simple rules should give weight to asset prices, usually over their predictive power for inflation and output. These papers derive their conclusions mainly from simulating some model under different time-invariant linear reaction functions (optimised or not) and ranking them according to the computed losses. However, the presence of a non-linearity (e.g. a bubble or misalignment) makes an otherwise linear model non-linear and calls in principle for a non-linear reaction function. In this paper we compute a regime-dependent (and hence time-variant) policy rule as the solution of an optimal control problem. Bordo and Jeanne (2002) have also pointed out that in reality the optimal monetary rule is unlikely to take the form of a linear time-invariant rule, even if augmented by a linear term in asset prices.11 To prove their point, they consider a stylised New Keynesian model of the economy in which monetary policy can affect the probability of a credit crunch. This model is assumed to have only three periods and is solved by backward induction. The optimal interest rate is shown to be a function of the probability that private agents attach to being in a ‘new economy’. In our application we compute optimal policy for an infinite horizon and show how the optimal rule's feedback coefficients depend on the regime-switching transition probabilities. Unlike Bordo and Jeanne, however, while policy affects the size of the misalignment, it does not affect the probabilities. Assuming that the probabilities are exogenous is not an unreasonable assumption if one considers the high degree of uncertainty concerning both the knowledge of the stochastic properties of an asset price and their relationship with monetary policy.12 Given such uncertainty, we also complement the computation of optimal policy with the welfare analysis of the incorrect assumptions about the transition probabilities.13 It is important to note that the optimal regime-switching rule is computed in this paper under the assumption that the regime (bubble or no-bubble) is observable by the policymaker, albeit with a delay. Ex post identification of a regime is not implausible in several situations but more generally it might be regarded as a limitation. If the regime is not identifiable, even with a delay, then one obvious alternative for the policymaker is to adopt a regime-invariant policy rule optimised to take into account the regime-switching nature of the economy. For this reason, we also include an analysis of optimised regime-invariant rules and see how they compare with the optimal regime-switching rule. In particular, we consider a Taylor rule which includes a response to the exchange rate as well as a simple Taylor rule which does not.14 We briefly anticipate the main results which emerge from our application. The analysis of the optimal regime-switching control (ORSC) rule shows an intuitive link in the bubble regime between the response coefficient associated with the exchange rate and the (unconditional) expected duration of a bubble: when the bubble is expected to last for at least 2 years, the optimal interest rate is negatively correlated with real exchange rate fluctuations and becomes more responsive as the expected duration of the bubble lengthens (an increase in the exchange rate being an appreciation). In the no-bubble regime there is an intuitive link between the response to the exchange rate and the probability of the bubble emerging: for lower probabilities the interest rate is positively correlated with exchange rate fluctuations (reflecting the likely transitory nature of exchange rate movements) and becomes less responsive as the probability of a bubble increases; for higher probabilities the interest rate responds negatively and becomes more reactive to exchange rate fluctuations as the probability rises further (reflecting the likely onset of a bubble). Another characteristic of the optimal regime-switching interest rate rule is that in both regimes the interest rate is for the most part less responsive to inflation and output fluctuations than in the absence of regime uncertainty, with the degree of caution increasing as both transition probabilities approach their intermediate values of a half. As stressed above, the difficulty or impossibility in identifying the regime might in practice lead the policymaker to follow a regime-invariant rule. A natural question then is how costly the latter rule would be relative to the optimal one or, equivalently, how costly it would be not to be able to observe the regime. In this regard, we find that the losses from adopting the latter rule are small relative to the optimal regime-switching rule except when the bubble has both a low probability and a long expected duration. Indeed, as both transition probabilities approach View the MathML source12 the optimal regime-switching rule becomes ever more similar and eventually converge to the optimised regime-invariant rule. The differences between the two rules tends to be large when both transition probabilities are low. The optimised regime-invariant rule (which includes the exchange rate) also shows intuitive links between the response to the exchange rate and the transition probabilities, with the obvious difference that in this case the optimal responses are heavily affected by both probabilities. In addition, policy is also found to be for the most part less responsive to output and inflation fluctuations. By contrast, an optimised simple Taylor rule (without the exchange rate) entails stronger responses to inflation and output fluctuations than in the absence of regime uncertainty. For bubbles of high expected duration the responses increase with the probability of the bubble arising. These stronger responses possibly compensate for the lack of a negatively correlated response to the exchange rate which is found to be appropriate in the unconstrained regime-invariant rule. Finally, a key result of the paper concerns the assumptions that the policymaker makes about the (unknown) transition probabilities. These probabilities could be highly uncertain since historical experience might provide little or no help in quantifying them. We find that there are robust (minmax) values of the probabilities not falling on the boundaries of the feasible set of values. These robust values generally correspond to more muted policy responses. This result is interesting as in the robust control literature uncertainty is often found to lead to more reactive policy responses than under certainty equivalence.15 The paper is organised as follows. Section 2 describes the quadratic optimal control problem with regime shifts and its solution. It also explains how to evaluate regime-invariant and simple rules in a regime-switching economy. Section 3 describes the model used in the application. Section 4 analyses the responses of monetary policy to movements in output, inflation, and the lagged exchange rate. Section 5 examines the choice of the optimal assumptions about the uncertain transition probabilities. Section 6 concludes indicating possible future avenues for research.

The current paper has discussed a method for analysing how policy should respond when the exchange rate experiences sustained booms followed by occasional corrections towards fundamentals. It would be interesting to extend the analysis to models with a richer dynamics than the one used in this paper. Furthermore, the algorithm used to compute the ORSC rule has some limitations. One limitation is that it applies to backward-looking models. Extending it to models that allow for forward-looking agents involves non-trivial technical issues and is clearly desirable.67 Another limitation is the assumption that the policymaker is able – unlike in the Bayesian model averaging approach68 – to identify the regime, albeit with a delay. So the policymaker's uncertainty is always about how the model will evolve in subsequent periods. This is a plausible assumption in some circumstances (e.g. an asset price collapse can be observed) but not in all (e.g. permanent improvement in productivity is not visible except with several years delay). For this reason we also looked at the performance of regime-invariant rules whose implementation does not require the policymaker to classify the regime. A further possible extension is to make the state of the world a latent variable as well as to require the policymaker to learn about the transition probabilities. A policymaker who does not know the regime could learn about it and estimate the transition probability matrix of the Markov chain by application of the Bayes rule. Optimal policy would then be computed taking into account the uncertainty about the current regime as well as the uncertainty of a future switch to a different regime. The type of learning we envisage is passive because in each period the policymaker would assume that the transition probabilities would not change in all subsequent periods, while in fact they will be changing.