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Previous studies of non-certainty-equivalence with additive uncertainty have followed either Smets (1998) or Orphanides (1998). Smets (1998) points out that certainty-equivalence fails to hold when policymakers are constrained to respond to only a limited subset of the state variables of the system, so that the policy rule is a constrained optimum, rather than a global optimum. [This restriction makes sense, in particular, when the size of the state space is large, as in Orphanides et al. (2000), who work with the Federal Reserve Board's macro model.] Uncertainty about any of the variables in the policymakers' simple rule will then lead the optimal coefficients of the simple rule to change.⁵

Orphanides (1998) focuses on the fact that the data, such as the output gap and inflation rate, are observed only imperfectly in real time. By the certainty-equivalence principle, policymakers' optimal response in this case is to formulate best estimates of the output gap, E_t(y_t − y*_t), and inflation, E_t π_t, and act as if these estimates were known with certainty. Orphanides and others [Aoki (1999), Rudebusch (2001, 2002)] bring about non-certainty-equivalence in this framework by constraining policy to react to the actual real-time data, rather than to the best estimates above. This naturally raises the question as to why the actual real-time data are not (real-time) best estimates of the true values.

For example, if y*_t denotes the level of potential output, and y*_t|t the real-time estimate of the level of potential output, one would normally expect that

where η_t is a mean-zero random variable. In this case, y*_t|t = E_t y*_t, and so, the real-time data are the real-time best estimates! The constraint that policymakers react only to the real-time data is then not really a constraint at all, and the certainty-equivalence principle holds.

It is only because Orphanides formulates the real-time data problem as one of signal extraction, with

where y*_t (the true value underlying the data, or signal) and η_t (the noise) are orthogonal random variables, that the certainty-equivalence principle is circumvented. Note that in (4′), the real-time data y*_t|t has the property that y*_t|t ≠ E_t y*_t, so that it is no longer an unbiased estimate. When the problem is modified in this manner, an increase in the variance of η_t now does have an effect on the best estimate E_t y*_t—in particular, if y*_t and η_t are normally distributed around zero, then E_t y*_t =

y*_t|t in the univariate case. Although the optimal policy is still a certainty-equivalent function of policymakers' best estimate E_ty*_t, expressing the policy as a function of the real-time y*_t|t now leads to effects of additive uncertainty η_t on the coefficients in the optimal rule. It is in this respect that Orphanides (1998) finds an exception to the certainty-equivalence principle.

It should be clear from the above analysis that the use of real-time data per se has nothing to do with the non-certainty-equivalence demonstrated in Orphanides (1998). Instead, it is the introduction of a signal extraction problem into the policymakers' inference step that drives the result. The general implications of formulating policymakers' inference problem in this way is the aim of the present paper. In addition, the policies that are emphasized by Orphanides (1998), Orphanides et al. (2000), and Rudebusch (2001, 2002) are not fully optimal, so that it is difficult in those papers to distinguish between the effects of uncertainty itself and the effects of uncertainty interacting with a substantially constrained policy response function. In the present paper, all policy rules and all estimation are fully optimal.

A recent paper by Svensson and Woodford (2000) also considers several of the same issues as the present paper. There are two main contributions of the present paper that differentiate it from theirs. First, I show how the results of this paper are applicable to indicator variables that are estimated (such as the output gap), rather than simply indicator variables that are directly observed (output, unemployment, inflation, money growth, etc.). Second, I consider the effects of a structural break in uncertainty about an indicator variable, rather than an increase in the level of uncertainty about an indicator that stretches back into the infinite past. The latter is merely a thought experiment, whereas the former corresponds to situations faced by actual policymakers in practice, such as the (possible) structural break in U.S. productivity growth in the late 1990's.

The remainder of the paper proceeds as follows. Section 2 develops the relationship between signal extraction and non-certainty-equivalence in a simple descriptive model of the economy, under both naive and rational expectations. Section 3 extends these results to the general linear-quadratic-Gaussian framework and proves the coefficient attenuation result for the general case. Section 4 extends the basic model forward in time to allow for the dynamic propagation of uncertainty, and proves that the results of Section 2 are robust to this extension. Section 5 discusses the results and concludes.

BASIC MODEL

Policymakers have preferences over inflation and output of the form

where π* denotes policymakers' long-run target for the inflation rate and y* denotes the level of “potential” output consistent with long-run equilibrium.

For the purposes of this section, the unobserved “signal” variable X_t is taken to be a scalar. The interpretation in this case is that the true underlying state of the economy is scalar, or that X_t is an index of inflationary pressures or “excess demand.” X_t is assumed to evolve according to

with the output gap and inflation being observable functions of this unobserved state:

Here, r* denotes the “natural” rate of interest, consistent with long-run equilibrium in the model; and α, β, and φ are known positive parameters with φ < 1. The stochastic disturbances ε_t, η_t, and ν_t are independent of each other, over time, and of current and prior values of X, y, and π; and are normally distributed with constant variances σ_ε², σ_η², and σ_ν², respectively. The variable X_t (and its past values) are never observed by policymakers and must be inferred from previous observations of output and inflation. Equations (6b) and (6c) represent the signal extraction aspect of the problem, with η_t and ν_t denoting noise disturbances that are orthogonal to the underlying signal X_t.

One may assume that y* is observed with certainty or, alternatively, that it is stochastic.

where

.

Economic agents' prior expectation of the inflation rate, π_t^e, can be either a rational expectation (π_t^e ≡ E_t π_t, where E_t denotes the expectation at the beginning of period t, before shocks are realized), or a naive expectation (π_t^e ≡π_t−1), without altering the results below (see the solution in the Appendix).¹⁰

The timing of policymakers' observations and actions is as follows. At the beginning of period t, policymakers update their beliefs about X_t−1 based on observations of y_t−1, π_t−1, and the earlier choice of r_t−1. On the basis of these updated beliefs, policymakers then choose a value of r_t that minimizes the expected loss function (5). Shocks to the economy (ε_t, η_t, and ν_t) are then realized and the values y_t and π_t are observed. Thus, policymakers' information set at time t is

where E₀X₀ and Var₀ X₀ denote the mean and variance of policymakers' prior (time 0) distribution on X₀, which is assumed to be Gaussian.

Policymakers update beliefs about X_t−1 via Kalman filtering. Because (ε_t, η_t, ν_t) is multivariate normally distributed, this is the optimal inference procedure (in the sense of minimizing the mean-squared error of the estimate), and is equivalent to Bayesian updating.

The optimal solution to policymakers' problem (5), given the structure of the economy (6) and information set (7), is

where a and b are positive constants determined by the parameters of the system, given in the Appendix. Note that a and b are completely invariant to σ_ε², σ_η², σ_ν², and Var_t X_t−1 (policymakers' time t prior variance on X_t−1, derived recursively from Var₀ X₀ by the Kalman filtering algorithm). In this respect, the linear-quadratic problem with signal extraction displays certainty-equivalence.

In forming the optimal estimate E_t X_t−1, however, policymakers do respond to the amount of uncertainty in the problem. Their prior (time t − 1, i.e., before values of y_t−1 and π_t−1 are observed) distribution on (X_t−1, y_t−1 − y*, π_t−1 − π_t−1^e) is given by

where I have let σ_x² denote Var_t−1 X_t−1, policymakers' prior, time t − 1, variance on X_t−1. Their posterior distribution on X_t−1, after observing y_t−1 and π_t−1, then has mean

where Δ ≡(σ_x²+σ_η²) (β²σ_x² + σ_ν²) − (βσ_x²)² = σ_x² σ_ν² + β²σ_x²σ_η² + σ_η² σ_ν². Equation (9) is analogous to the simpler formula for signal extraction with one observable variable, E_t X_t−1 = E_t−1 X_t−1 + [σ_x²/(σ_x²+σ_η²)] [(y_t−1−y*) − E_t−1 (y_t−1−y*)], with additional terms in the coefficients that take into account the covariance between output and inflation.

Given the normality assumption, the formula for the best predictor E_t X_t−1 is the theoretical regression/projection

where

This yields

which is the expression given in (9).

Grouping terms in (9) yields

where the E_t−1 X_t−2 term can be cascaded backward and expressed as a function of lagged observations of the output gap, the inflation surprise, and interest rates, to the point where the original distribution on X₀ is negligible, as it is multiplied by a large power of φ.

Note that, even though policymakers' optimal reaction function (8) is certainty-equivalent in terms of the estimate E_t X_t−1, when the optimal policy is expressed as a function of present and past indicator variables (the output gap, inflation surprise, and interest rates), certainty-equivalence in the reduced form no longer holds. The variances of the additive disturbance terms enter into the coefficients of (10).

For example, consider the effects of a sudden increase in uncertainty surrounding potential output y* on equation (10). As shown in note 8, this can be regarded as an increase in σ_η², holding σ_ν² and σ_x² constant. As σ_η² increases, the quantity Δ increases, but less than proportionately. This implies that the coefficient on y_t−1 − y* in (10) decreases in magnitude, so that policymakers place less weight on the noisy output gap in forming their inference about the underlying state of the economy. Thus, we have an example of coefficient attenuation on the noisy or uncertain variable. In addition, the coefficients on each of the other variables in equation (10) increase in absolute value: Policymakers place more weight on those variables about which they are relatively more certain. In this sense, policymakers become “less proactive and more reactive,” responding less forcefully to the current output gap, and more forcefully to the current inflation surprise, and past output gaps and inflation surprises, because these variables provide more reliable information about the current state of the economy.¹³

This “less proactive” result could be emphasized by setting up the model with an additional lag in equation (6c),

so that current output is more closely related to future than to current inflation. This is the timing used by Svensson (1997); however, it complicates the analysis by making the signal extraction problem a function of two lags of the variable X_t instead of only one, which is why this approach was not taken here.

These conclusions are not idiosyncratic to an increase in the variance of the additive disturbance η_t. For example, an increase in σ_ν², instead of σ_η², leads to a decrease in the coefficient on (π_t−1 − π_t−1^e) in equation (10), and an increase in the coefficients on the other variables in that equation. Exactly analogous results (lower coefficient on the noisy variable, higher coefficients on the others) obtain in response to an increase in σ_x², the policymakers' prior variance on X_t−1. Thus, the result described above is robust, and derives not from any special assumptions surrounding the model, but rather from the general principle that in linear regression, or statistical projection, less weight is given to observations that have higher variance.

Obviously, as with linear regression and statistical projection, the covariances between the different variables matter for the coefficients in (10). So far, I have abstracted away from this problem by assuming that the disturbances are orthogonal to each other, and to policymakers' prior for the variable X_t−1, but I show below for the general case, with arbitrary covariances, that the coefficient attenuation result still holds: An increase in the uncertainty surrounding a given variable causes policymakers to assign less weight to that variable in forming their best estimate of the underlying state of the economy, E_t X_t−1. Moreover, the amplification of the coefficients on inflation and its lags in (10) is also quite robust, and holds for models more general than that of the present section. For example, Section 4 proves this result for the case in which increased uncertainty about potential output extends backward any number of periods, is correlated across time, and is correlated with policymakers' priors about the unobserved state of the economy X_t−1, in a manner that is consistent with policymakers learning about potential output over time.

Signal Extraction vs. Imperfect Observation of State Variables

Note that the above results hinge crucially on setting up model (6) as one involving signal extraction, rather than one simply involving imperfect observation of state variables. For example, replacing (6b) with

or, with an uncertain potential output,

where y* ≡ ŷ* + ζ and

, the non-certainty-equivalence results described above are completely eliminated. In both (6b^[dagger]) and (6b^[Dagger]), the policymakers' optimal estimate of the underlying state of the economy, E_t X_t−1, is simply (y_t−1 − y*) in the first case, and (y_t−1 − ŷ*) in the second. Plugged into the certainty-equivalent structural response in equation (8), the reduced-form policy response thus retains the certainty-equivalence property.

Note that this analysis implies that it is not real-time data per se that justify caution on the part of policymakers in Orphanides (1998) and Rudebusch (2001). If the real-time data are unbiased forecasts of the true values, analogous to (6b^[dagger]) or (6b^[Dagger]), then certainty-equivalence holds, and the amount of uncertainty surrounding these real-time estimates (σ_η²) is completely irrelevant for optimal policy. Certainty equivalence only fails to hold in their framework if the real-time data are realizations of true values plus noise (and thus are biased estimates of the true values), so that estimation of the true values involves a signal extraction problem, as in (6b).

This naturally raises the question as to whether the real-time data are better modeled as rational estimates or as noise-contaminated observations. Orphanides (1998, 2001) presents figures demonstrating that the output gap, in particular, has been badly mismeasured by policymakers in real time. However, both the size of these errors and their serial correlation were not evident until several years after the fact, and so, it is not clear that these real-time estimates of the output gap were not rational at the time. To take the position that policymakers were deliberately irrational in their real-time estimates seems unwarranted without a more rigorous analysis to support this point of view.

Rigorous analysis of the performance of real-time data and official forecasts has been undertaken by a number of authors, albeit with data that are more readily observable than the output gap, such as real GDP and inflation. Mankiw and Shapiro (1986) analyze whether the real-time real GNP data produced by the BEA are better modeled as a rational forecast or as a realization with noise of the “true” (i.e., “final release”) value. They find that the real-time data appear to be unbiased and efficient rational forecasts. McNees (1995) looks at the official forecasts of real GNP/GDP and inflation published by the CBO and Federal Reserve System (as presented in “Humphrey-Hawkins” reports) and finds that they perform at least as well, if not better, than private-sector forecasts in terms of mean-squared error. Romer and Romer (2000) find that not only are the Federal Reserve Board's internal “Greenbook” forecasts of output and inflation unbiased and efficient, they completely dominate private-sector forecasts, in the sense that the private-sector forecast should be thrown out entirely if the Board's forecasts were to be made public.

These results might at first seem to contradict Rudebusch's (2001) finding of a significant, irrational “noise” component in the real-time inflation data (as measured by either the GNP/GDP deflator or fixed-weight price index). However, as in Orphanides (1998, 2001), Rudebusch's “final” data are from the perspective of the late 1990's, and thus include definitional revisions to GNP and changes in base year. In contrast, all of the papers cited above take particular care to evaluate the performance of the forecasts with respect to a final measure of the statistic on a definitionally consistent basis. For example, it seems unfair to evaluate the rationality of policymakers' 1970 estimate of real GNP and inflation using today's estimates of 1970 GDP in chain-weighted 1996 dollars, yet this is exactly what Orphanides and Rudebusch do. Thus, it is likely that the “noise” found by Rudebusch (2001) reflects nothing other than definitional changes in the data rather than deviations from rationality in policymakers' estimates.¹⁴

The case for biased real-time data is thus somewhat uncompelling from an empirical as well as a theoretical standpoint. This implies that models incorporating only real-time data uncertainty should be certainty-equivalent. However, framing policymakers' inference problem about the state of the economy as one of signal extraction more generally, as is done in this paper, is still quite plausible. The interpretation of policymakers' estimation process in this case is one of an unobserved state of the economy that must be inferred using (possibly a large number of) economic indicators. For example, Alan Greenspan of the Federal Reserve Board is renowned for looking at a vast array of economic indicators in an attempt to infer the current state of the economy and its future course. When policymakers face a signal extraction problem such as this, a strong case for caution in the face of uncertainty can still be made.

Coefficient Attenuation, Simple Rules, and Robust Control

The implications of this paper contrast in an interesting way with those from the literature on “simple rules” and “robust control.” In particular, the signal extraction framework of the preceding section found that an increase in uncertainty surrounding a given indicator variable should be met with an attenuation in policymakers' response coefficient to that variable, and an amplification of their response coefficients on all other indicators, about which uncertainty has not changed.

The literature on “simple rules,” in contrast, generally finds that policymakers should attenuate their response coefficients on all variables in their reaction function, even if the increase in uncertainty surrounds only a single variable.¹⁵

For example, Smets (1998) restricts policymakers to rules involving only one lag of output, the four-quarter average inflation rate, and one lag of the interest rate as arguments, and finds that the optimal coefficients on all of these variables are attenuated by an increase in uncertainty surrounding the output gap. Orphanides et al. (2000), investigating a similarly constrained class of rules within the Federal Reserve Board's FRB/US model, also find that attenuating the coefficients on every variable in the rule is the best response to increased uncertainty surrounding the output gap.¹⁶

Finally, the emerging literature on robust control arrives at just the opposite conclusion: Policymakers ought to respond more aggressively to every variable in their reaction function when faced with model uncertainty. This literature, typified by Onatski and Stock (2000), chooses coefficients of a policy rule to minimize the maximum loss over all possible values for a given parameter within a given range; thus, the policymaker is guaranteed not to make mistakes that are extremely costly for parameters within this range. This approach is clearly very different from the maximization-of-expected-value approach that I have taken here, and so, it is not surprising that the results differ. Intuitively, their findings are driven by the fact that a bad draw on the effectiveness of the policy tool (the parameter α in my model) can result in very large losses if the rule's responsiveness is not sufficiently great. However, it is not clear that an increase in additive uncertainty about potential output would lead to the same conclusions.

SIGNAL EXTRACTION IN THE GENERAL LQG FRAMEWORK

The analysis of the preceding section can be generalized to a standard linear-quadratic-Gaussian framework [e.g., Bertsekas (1987)]. In this case, the underlying state of the economy X_t is permitted to be a vector, and X_t evolves according to a linear function of one lag of itself and a vector of policy instruments r_t:

where A and B are known matrices of the appropriate dimensions. Any constants can be incorporated by defining one component of X_t to be a vector of ones. I denote the observable variables of the system by Z_t. These may be a subset of the variables in X_t, noisy realizations of a linear function of variables in X_t, or some combination of the two. Thus,

where C is a known matrix of appropriate dimension, with every observable relationship among the elements of X_t corresponding to a row in (12). The noise vector η_t may have some components that are always zero, corresponding to elements of X_t that are actually observed. Other components of X_t, which are not directly observed, must be inferred from observations of Z_t.

The stochastic disturbances ε_t and η_t are assumed to be independent of each other, over time, of current and past values of r, X, and Z, and are (multivariate) normally distributed with constant variance-covariance matrices Σ_ε and Σ_η, respectively.¹⁹

Policymakers minimize a quadratic loss function:

where D is a positive semidefinite matrix. Note that this specification does not preclude policymakers' preferences from depending on observables Z, since X can be expanded to include elements of Z as needed. Past values of r can also be incorporated into X and Z.

Policymakers choose a value for the vector of instruments r_t at the beginning of each period t, conditional on all information available through the end of period t−1. After r_t is chosen, the shocks ε_t and η_t are realized, and the value of the vector Z_t is observed. Policymakers' information set at the beginning of period t is thus

where E₀X₀ and Var₀ X₀ denote the mean and variance of policymakers' prior (time 0) distribution on X₀, which is assumed to be normal.

Policymakers update beliefs about X_t via Kalman filtering, which is the optimal inference procedure, given the assumptions of normality above. Letting Σ_s|t denote Var_t X_s, the variance of X_s conditional on information available at the beginning of period t, we have the recursive equations

Note that the variance Σ_t|t evolves deterministically over time, as is typical in the LQG framework. In particular, the variances of policymakers' future estimates are unaffected by their choice of the current instrument r_t.

PROPOSITION 1

The optimal solution to policymakers' problem (13), subject to the law of motion (11), observation equation (12), and information set (14) is given by

where V is the “value” matrix, defined to be the unique negative semidefinite solution to the Riccati equation

Proof. See Bertsekas (1987, pp. 292–293). █

Equation (17) is certainty-equivalent with respect to the state variable X_t−1. However, as should be clear from the preceding section, certainty-equivalence generally will not hold with respect to the observable variables Z_t−1. The following proposition demonstrates this fact by proving the coefficient attenuation result from the preceding section for the general LQG framework. Note that by holding Σ_t−1|t−1 fixed in what follows, the model is consistent with the interpretation that a structural break in the degree of uncertainty surrounding the indicator variables has occurred in the previous period.

PROPOSITION 2

Suppose that the variance of the first component of η_t−1 in (12) is increased, in the sense that element (1,1) of Σ_η is increased while all other elements of Σ_η, and all elements of Σ_t−1|t−1 in (15) and (16), are held fixed. Then, policymakers' optimal response to observables, obtained by substituting (15) into (17), exhibits an attenuation in the response of all elements of policymakers' instrument r_t to the first component of Z_t−1.

Proof Recall that r_t is a vector of instruments; hence the proposition states that the optimal setting of each of these is attenuated with respect to the first component ofZ_t−1. This is intuitive because the ordering of the elements of r_t is arbitrary.

Let M denote the positive-definite matrix C Σ_t−1|t−1 C′ + Σ_η in equation (15), and partition M into

where M₁₁ is a scalar, M₂₁ a column vector, and M₁₂ = M₂₁′. Letting N denote M⁻¹ and partitioning N in accordance with M, we have

Let M₁₁ be multiplied by a factor λ > 1, corresponding to the increase in Σ_η. Then, |M| increases in magnitude because, expanding along the first row or column, |M| = M₁₁ |M₂₂| + S, where S is a sum of element-cofactor products not involving M₁₁, and M₁₁, |M₂₂|, and |M| are positive. Thus, N₁₁ is attenuated, and it follows from (19) that N₂₁ is attenuated in the same proportion, say, by the factor μ < 1. Thus, the first column of N is attenuated by the factor μ.

Inspection of (18) reveals that V is invariant to the change in Σ_η, and by the invariance of the other parameters in (15), (16), and (17), it follows that the first column of − (B′VB)⁻¹ B′V A Σ_t−1|t−1 C′ (CΣ_t−1|t−1C′ + Σ_η)⁻¹ is attenuated by the same factor μ. These are exactly the coefficients in question, completing the proof. █

One would like to be able to increase the covariances among the components of η_t as well, but unfortunately, completely general statements in this case cannot be made.

SIGNAL EXTRACTION DYNAMICS IN THE BASIC MODEL

The analysis of the preceding sections has been essentially static in nature, in that a structural break in uncertainty was known to have occurred in the previous period. It is not clear, then, that the results still apply if the increase in uncertainty occurs several periods earlier, particularly when we take into account the fact that uncertainty about potential output is typically serially correlated, and feeds through to increased uncertainty about subsequent estimates of X, the unobserved state of the economy.

The analysis of this section thus focuses on the dynamic effects of a structural break in uncertainty surrounding y*—that is, how optimal policy is affected as this uncertainty propagates forward through time. Again, this is in contrast to Svensson and Woodford (2000), who consider only counterfactual experiments in which uncertainty about an indicator is increased going back into the infinite past.

The model here is essentially the same as in Section 2. Policymakers' preferences have the form

where, for ease of notation, π_s now denotes the deviation of inflation from policymakers' target and y_s denotes the output gap, both at time s. The economy follows:

where X_t denotes the unobserved state of the economy, r_t the deviation of the real interest rate from its “natural” value, and π_t^e agents' expectation of inflation, as before.

The error term η is now allowed to be persistent:

where [theta] can be regarded as the degree of persistence of shocks to potential output. The normally distributed disturbances ε_t, ν_t, and ξ_t are assumed to be orthogonal to each other, across time, and to all other variables in the system.

The timing of policymakers' observations and actions is the same as in Section 2, with information set

Policymakers' optimal solution to (20), subject to (21) and (22), is given by

where a, b, and c are constants invariant to the uncertainty surrounding X and η (see the Appendix). In contrast to Section 2, policymakers now care about past values of η as well as X. Note that because y_t−1 = X_t−1 + η_t−1, we can rewrite (23) as

where

.

As values of y and π are observed, policymakers update their beliefs about X and η by Kalman filtering. Letting σ_Xt², σ_ηt², and σ_xηt denote Var_t X_t, Var_t η_t, and Cov_t (X_t,η_t), respectively, policymakers' best estimate of X_t−1 at time t is given by

where Δ_t ≡β² (σ_Xt² σ_ηt² − σ_xηt²) + σ_ν² (σ_Xt² + 2σ_xηt + σ_ηt²) for all t. Equation (25) was derived exactly as was policymakers' estimation equation (10) in Section 2. The E_t−1 X_t−1 term can be written as φ E_t−1 X_t−2 − α r_t−1 and cascaded backward, as before.

Given an exogenous increase in uncertainty about η_s at the beginning of period s < t, we must begin by tracing out its effects on subsequent values of σ_η², σ_x², and σ_xη. The interpretation of the exogeneity of the increase in σ_ηs² is that of a structural break in uncertainty surrounding potential output in period s. The case in which uncertainty about η increases exogenously in several periods s₁, s₂, …, s_k, although perhaps more interesting, is simply a positive linear combination of the effects given below, and thus does not need to be considered separately.

A straightforward computation [using equations (16) from the preceding section] shows that policymakers' variances evolve according to

For notational convenience, define

which is the key term in (26). Totally differentiating N_t with respect to σ_ηs², using (26), yields

which is a recursive sequence terminating with

The first equality in (29) follows from the assumption that the increase in uncertainty about η_s in period s is exogenous, whereas the covariance σ_xηs and variance σ_xs², which derive from uncertainty about η and X in prior periods, are held fixed.

Note that equations (28) and (29) imply d N_τ/ d σ_ηs² ≥ 0 for all τ ≥ s. This fact, together with the relations implied by (26), helps us assess the effects of the break in uncertainty about η_s on the coefficients in the period-t estimation equation (25).

For example, the coefficient on the most recent inflation surprise, (π_t−1 − π_t−1^e), unambiguously increases in (25). This follows from the fact that

and the coefficient on the inflation surprise in (25) is positive. Thus, the finding of an amplification of the coefficient on inflation in Section 2 is robust to extending the increase in uncertainty about the output gap backward any number of periods.

Similarly, the coefficient on the most recent output gap, y_t−1, is necessarily attenuated, under the assumption that [theta] > φ (which corresponds to assuming shocks to potential output are more persistent than movements in the output gap):

Thus, this finding from Section 2 is also robust, no matter when the structural break in uncertainty occurred.

The coefficient on the lagged estimate of the state variable, E_t−1 X_t−1, may either increase or decrease in (25), according to

which is positive if and only if ([theta]−φ) > β² φσ_ξ² / σ_ν².

Note that I use the terminology “not clear,” rather than “ambiguous,” here because the change in coefficients may be theoretically unambiguous, but computation of the analytical derivatives and a further sign check of the result would be required to ascertain this fact, and these computations quickly become very burdensome. For example, it can be shown analytically that the coefficient on the second lag of the inflation surprise, (π_t−2 − π_t−2^e), also necessarily increases for [theta]>φ (no matter what the value of β² φ σ_ξ² / σ_ν²), further corroborating the finding that policymakers should react more aggressively to inflation.