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Risk-adjusted LQG optimal control with perfect and imperfect observation of the economy is used to obtain prudent Taylor rules for monetary policies and cautious Kalman filters. A prudent central bank adjusts the nominal interest rate more aggressively to changes in the inflation gap, especially if the volatility of cost-push shocks is large. If the interest rate impacts the output gap after a lag, the interest also responds to the output gap, especially with strong persistence in aggregate demand. Prudence pushes up this reaction coefficient as well. If data are poor and appear with a lag, a prudent central bank responds less strongly to new measurements of the output gap. However, prudence attenuates this policy reaction and biases the prediction of the output gap upwards, particularly if output targeting is important. Finally, prudence requires an extra upward (downward) bias in its estimate of the output gap before it feeds into the policy rule if inflation is above (below) target. This reinforces nominal interest rate reactions. A general lesson is that prudent predictions are neither efficient nor unbiased.

One of the most precious commodities of a modern capitalist economy is a stable price level or at least a low inflation rate. Central banks also try to avoid unemployment and steer towards near-zero output gaps. Such flexible inflation targeting gives rise to Taylor (1993) rules, which indicate how much the nominal interest rate should react to the inflation gap and output gap. But a central bank should also operate cautiously and prudently: A prudent man (or perhaps, I should say a prudent Bayesian) carries an umbrella even when the forecast says there is only a small chance of rain. If there is no rain, he suffers the small inconvenience of carrying the umbrella. But if he does not bring the umbrella and it does rain, he may suffer the much larger inconvenience of being caught in a downpour. The prudent central bank should behave similarly, accepting a high probability of a small adverse outcome in order to avoid the small risk of a very serious bad outcome (Feldstein, 2003). Feldstein argues that Greenspan’s policy of lowering the federal interest rate at that time was prudent, because of the asymmetric nature of the risk faced. The potential upturn could lose steam and there was a risk of deflation, while an unnecessarily strong stimulus would do little harm. Most research on optimal monetary policy is based on the certainty equivalence principle, which says that uncertainty can be ignored. In calculating optimal interest rate rules future disturbances are set to their expected values. This approach is only valid under special conditions (i.e., linear models, quadratic preferences, normally distributed errors). It abstracts from prudence and thus bears little relation to the practice of central banking. It is surprising that there is hardly any research on the behavior of prudent central banks. Most of the macroeconomics literature adopts a certainty-equivalent linear-quadratic-Gaussian framework (e.g., Svensson, 1997, Rudebusch and Svensson, 1999, Judd and Rudebusch, 1998, Rotemberg and Woodford, 1997, Rotemberg and Woodford, 1999, Woodford, 2001 and Woodford, 2003b). However, a recent approach explicitly recognizes that statistical properties and order of the processes driving the modeling disturbances are not known and derives robust (min–max) rules that perform well under different views of the world (e.g., Onatski and Stock, 2000, Giannoni and Woodford, 2003, Onatski and Williams, 2003 and Hansen and Sargent, 2008; Leitemo and Söderström, 2008a and Leitemo and Söderström, 2008b). Another approach is to employ model averaging in a Bayesian context (Brock et al., 2003). Yet another approach advocates room for judgement of central bankers in deriving optimal monetary policy rules (Svensson, 2003). Our approach is complementary. We study precautionary central banks and derive the resulting closed-loop monetary policy rules analytically. Central banks minimize the expected value of an exponential transformation of a quadratic welfare loss in terms of output and inflation, which allow for a constant Arrow–Pratt measure of absolute temporal risk aversion and also for prudence in the optimal policy rules. This leads to linear policy rules with reaction coefficients that depend on the covariance matrices of the stochastic process driving the modeling disturbances (cf., Jacobson, 1973, Speyer et al., 1974 and Whittle, 1981). Effectively, a prudent policy maker downplays the power of its instruments if the volatility of shocks hitting the economy is large. Monetary policy rules must recognize that national accounts consist of poor quality data with measurement errors and observation lags, especially for output data. Typically, ‘flash’ estimates of GDP appear quickly and are subsequently substantially revised. Measurement errors show up, because the raw data violate the national accounting identities. One can use subjective estimates of data reliability to adjust the data so that all accounting identities must be satisfied (e.g., Ploeg, 1982 and Barker et al., 1984). Subjective variances of the raw data are provided by national accountants and then reduced by imposing the accounting restrictions. Unfortunately, they are seldom used in applied econometrics or in deriving optimal policy rules. Here we allow for measurement errors and lags in output data (cf., Orphanides, 2000). If the central bank adjusts its interest rate in reaction to changes in output gaps, it presumably does this less intensively if substantial measurement errors and lags in output data cause a deterioration of the signal-to-noise ratio (cf., Rudebusch, 2001). Taylor rules also allow for reactions to changes in inflation. However, inflation data are more readily and accurately available than output data. We investigate how measurement errors and lags in output data affect the Taylor rule for the nominal interest rate. Pearlman, 1986 and Pearlman, 1992 demonstrated the use of the Kalman filter for predicting states of the economy in monetary models with forward-looking expectations. In backward-looking models and forward-looking models where policy makers and private agents have access to the same partial information sets, the Kalman filter calculations can be performed independently of deriving the optimal monetary policy rule. The separation of control and prediction is trickier in forward-looking monetary models with commitment and asymmetric information (cf., Svensson and Woodford, 2005). We analyze how a prudent central banker takes account of incoming unreliable output data and use this in the Taylor rule with the aid of a modified separation principle (Whittle, 1981). The prudent Kalman filter depends on welfare preferences and yields biased predictions. In particular, a prudent policy maker gives less weight to new observations with large standard errors and that are less relevant for welfare. Conversely, to avoid costly mistakes prudence requires more weight to faulty data that are relevant for welfare. In the umbrella example a prudent person assigns a larger subjective probability of rain than the objective probability of rain, especially if he or she dislikes rain a lot. Section 2 states the problem of risk-adjusted LQG control and prediction. Prudence implies that policy makers play a min–max game against nature. Policy makers hedge against undesirable outcomes by postulating that shocks damage its objectives even though statistically they do not hurt on average. The beauty is that this leads to linear feedback policy rules and a recursive scheme for prediction of state variables. The reader uninterested in these mathematical details can skip through Section 2. Section 3 shows how prudence affects the optimal inflation-output trade-off within the New Keynesian macroeconomic model (e.g, Galí, 2008) with commitment and no measurement errors and derives results closely related to Leitemo and Söderström (2008a). To focus on the intricacies of prudence in dynamic models, Section 4 considers short-run inflation-output trade-off in a macroeconomic model with an accelerationist Phillips curve. Considering first the case of no measurement errors and lags, we show that the optimal nominal interest rate of a prudent central bank reacts more aggressively to the inflation gap, especially if cost-push disturbances are volatile. We then derive a prudent Taylor rule for the case where the real interest rate impacts aggregate demand after one period and national income is measured without error. We demonstrate that the optimal interest rate again responds more aggressively to the output gap if prudence and volatility of cost-push shocks are large and also if there is substantial persistence in aggregate demand. We also show that more weight to output targeting weakens policy responses of the central bank, particularly if triggered by changes in the inflation gap. Section 5 analyzes the role of measurement errors for optimal monetary policy in the accelerationist model. It shows that the reactions of the nominal interest rate to the measured output gap are less vigorous, especially if incoming data are relatively unreliable. We also show that a prudent central bank attenuates these policy reactions and furthermore biases its estimate of the output gap upwards. This makes reactions of the central bank to the output gap more aggressive, particularly if cost-push shocks are volatile and output targeting is important. Finally, we show that a prudent central bank introduces an extra upward (downward) bias in its estimate of the output gap to be fed into the policy rule if inflation is above (below) target. The nominal interest rate reactions become more aggressive. Section 6 concludes and suggests area for further research.

Prudence has been introduced into the LQG control and prediction framework. A prudent policy maker increases the shadow penalty on target variables to hedge against stochastic shocks that might hamper the achievement of desired values. This invalidates certainty equivalence, so one can no longer substitute future disturbances by their expected values. Instead, a prudent policy maker uses subjective, cautious estimates of future disturbances that depend on preferences in order to avoid costly mistakes. To cope with measurement errors, a risk-neutral policy maker uses the Kalman filter to revise BLUE-estimates of the states of the world as new information comes in. These predictions are independent of preferences and used in the optimal policy rules. A prudent policy maker uses inefficient predictions of the states of the world, since variances of the states are raised versus those of the incoming observations in fear of incorporating faulty information that may induce significant welfare losses. Large penalties for the targets and severe prudence imply large reductions in the relative precision of the measurements. Prudence also introduces a bias in the prediction of the states, so that the prudent Kalman filter no longer yields BLUE-predictions of the states. With prudence special care must be taken to couple the derivation of the optimal control rules and the prediction of the states. In line with Sargent (1999), we find that prudence yields more aggressive reactions of the nominal interest rates to the inflation gap, particularly if volatility of cost-push shocks is large. Craine (1979) and Söderström (2002) allow for parameter uncertainty in dynamics of inflation and also find, in contrast to Brainard (1967), more vigorous policy responses. To be safe a prudent central bank assigns a lower effectiveness of its monetary instrument. More prudent behavior is in the limit equivalent to strict inflation targeting. If the real interest rate affects aggregate demand with a lag, the nominal interest rate also reacts to the output gap, especially with substantial persistence in aggregate demand. Prudence and bigger volatility of cost-push disturbances imply stronger reactions to both the inflation and output gaps, even under strict inflation targeting. More weight to output targeting weakens policy responses of the central bank, particularly if caused by changes in the inflation gap. Earlier work shows that data uncertainty weakens monetary policy reactions (Rudebusch, 2001). We also find that reactions of the nominal interest rate to the measured output gap are less vigorous, especially if incoming output data are much more unreliable than the current output gap estimate. However, a prudent central bank attenuates its reactions to changes in the output gap much less, especially if output targeting is important. It also introduces as a precautionary measure an upward bias in its estimate of the output gap and thus in the nominal interest rate, again especially if output targeting is important. Both these elements make reactions of the central bank to the output gap more aggressive, particularly if shocks to inflation are volatile and output targeting is important. Finally, a prudent central bank introduces an extra upward (downward) bias in its estimate of the output gap before it feeds into the policy rule if inflation is above (below) target. This makes nominal interest rate reactions more aggressive. We have shown that prudence also induces more aggressive interest rate responses in the New Keynesian framework with rational expectations and commitment. It is possible to also derive prudent policy rules and prediction formulae to macroeconomic models with forward-looking expectations. This is crucial for a deeper understanding of prudent Taylor rules in environments where central banks have to cope with imperfect information. It also is worthwhile to investigate how prudence affects optimal fiscal and monetary policy when prices are sticky and government debt is used to smooth tax distortions; e.g., by introducing prudence in the analysis of Benigno and Woodford (2003). In then seems realistic to suppose that the central bank is more prudent than the fiscal authority. To conclude, we qualify our results in two important respects. Common sense of many practitioners dictates that prudence implies that the nominal interest rate should respond less strongly to changes in inflation and output gaps, while our analysis suggests more aggressive policy responses. There is no reason why prudence should imply passive behavior of the policy maker. In fact, it is more likely to lead to overzealous, even uptight control of the economy. Just as a prudent driver may react strongly to every bend in the road in order to avoid driving into the curb, a prudent central bank changes the interest rate more frequently. Practitioners may be more concerned with avoiding interest rate volatility and turmoil in financial markets than output targeting (e.g., Lippi and Neri, 2007 and Woodford, 2003a). Also, central bankers sometimes delay interest rate adjustments and prefer some unpredictability in order not to be seen to follow the pressure of market players and commentators. The second qualification is that neither the traditional nor the New Keynesian Phillips curves fully capture real world features such as credit constraints, equity constraints, bankruptcies and other market failures arising from imperfect information. Stiglitz and Greenwald (2003) point out that then the nominal interest rate affects aggregate demand and that monetary policy is associated with big allocative distortions and should be as much about supervision and regulation as the interest rate.