Optimal Forecasting with Observation Costs and Imperfect Signals∗ Luigi Paciello EIEF
February 24, 2011 Abstract We study the forecasting problem of an agent in the presence of both observation costs in the spirit of Caballero (1989), Reis (2006) and imperfect signals in the spirit of Woodford (2001), Sims (2003). In each period the agent makes forecasting decision on the basis of available information. Before making the forecasting decision, the agent observes imperfect signals on the state of the economy at no cost. Conditional on the precision of available information, the agent optimally decides whether to collect additional costly information on the state, an activity which we refer to as “review”. We study how the agent’s choices map into several statistics: the frequency of information acquisition, forecast uncertainty, the distribution of forecasts. We provide an analytical characterization of the agent’s decisions and a mapping from the structural parameters to the observable statistics. Combined with micro level data from surveys of professional forecasters, the predictions of the model allow to quantify the importance of infrequent “reviews” relatively to continually observed imperfect signals for the reduction of uncertainty.
Preliminary draft. F.Alvarez and F.Lippi contributed to the development of the model of this paper. I thank seminar participants at EIEF. ∗
1
Introduction
We study the forecasting problem of an agent that observes costless but imperfect signals of the state in each period, and chooses when to pay a fixed cost to perfectly observe the state so to further reduce her uncertainty. The effects of each of these two sources of uncertainty reduction, i.e. imperfect signals and costly “reviews”, have been thoroughly analyzed in the literature in a variety of contexts. An obvious example of their application is the literature on optimal price setting decision, or the literature on optimal forecasting decisions.1 The novelty of this paper is to present an analytical characterization of the optimal informationgathering policy in a model that incorporates both imperfect signals and costly “reviews”, as well as an analytical characterization of its implications for several observable statistics. We apply the results of the paper regarding the mapping between structural parameters and observable statistics to recent ECB survey evidence on the inflation and GDP growth forecasting decisions, and measure the relative contribution of costly “reviews” and imperfect signals into the reduction of forecast uncertainty. We document several facts from the ECB Survey of Professional Forecasters concerning the individual and aggregate dynamics in forecasters’ uncertainty about inflation and GDP growth forecasts. First, in any given period, there is substantial heterogeneity in individual uncertainty across agents. Second, individual uncertainty varies over time, and there is no systematic information advantage by same agents. Third, dynamics in individual uncertainty over time are mostly individual specific. Fourth, dispersion in forecasts is not a good proxy for the average level of uncertainty. Motivated by these empirical findings, we study a model that accounts for the cross-agent heterogeneity in uncertainty at each point in time, as well as for the time dynamics of uncertainty for a given forecaster. Our model embeds the two “polar” cases models of information arrival through imperfect signals only (e.g. Woodford (2001); Sims (2003)) and infrequent 1
See Reis (2006b,a), Woodford (2009), Sims (2003), Mankiw, Reis, and Wolfers (2004). See Alvarez, Lippi, and Paciello (2009) for a model that combines information frictions with adjustment frictions.
1
costly “reviews” only(e.g. Caballero (1989); Reis (2006b)). These models produce different policy rules: frequent imperfect signals yield continuously changing forecasts and imply constant uncertainty as the precision of new information arrival is constant, while the costly “reviews” yield sticky forecasts and implies time varying uncertainty. Since these rules have different implications for the distribution of uncertainty across agents, and the distribution of uncertainty about inflation forecast is important for monetary policy, identifying the main source of information acquisition is important. The agent minimizes the expected discounted value of a per-period loss function plus the expected discounted sum of the fixed costs incurred. The firm’s instantaneous loss function is the forecast square error, B(ˆ x − x)2 , where xˆ is the best forecast, x the “target” variable, and the parameter B depends on the curvature of the agent’s loss function with respect to the error in forecasting. The target x follows a generalization of the OrnsteinUhlenbeck process where the long-run mean characterizing this process at each point in time, x∗ , is stochastic and evolves according to a brownian motion. This process has been shown to account very well for dynamics in inflation (e.g. Stock and Watson (2007)) and nominal interest rates (e.g. Hull and White (1990)). There are two state variables in the forecasting problem: the current value of the target x, and its long-run mean x∗ , which also varies over time. Consistently with the structure of surveys of professional forecasters, we assume that the agent perfectly observes the realization of the target variable x in each period. However, the agent is imperfectly informed on the realization of x∗ , which is relevant for the forecast of future values of the target x. In each period, each agent observes at no cost two different signals conveying information about x∗ : (i) the realization of the target x; (ii) the realization of a private signal about x∗ characterized by iid noise across agents. The latter is in the spirit of the rational inattention literature, where errors in processing available information are individual specific. All agents have access to the same information acquisition technologies, therefore the precision of information conveyed by signals is the same across agents. However, conditional on the realization of x and of the private signal
2
on x∗ in each period, the agent decides whether to perfectly observe the realization of x∗ by paying the fixed cost θ, an activity we refer to as “review”. The fixed cost of each review θ is random and i.i.d. over time and across agents. As a consequence, agents will differ in their timing of the reviews. Heterogeneity in the timing of the reviews causes heterogeneity in the precision of information available to agents at a given point in time. The optimal control problem of the agent can be described as the problem of determining when it is optimal to have a review and reduce uncertainty about the realization of x∗ . We show that, conditional on the cost of a review, θ, the optimal policy of when to perfectly observe the state x∗ is time dependent. The time between reviews increases with the fixed cost of a review, and increases with the precision of information arrival through signals. In addition we show that forecast uncertainty is an increasing and concave function of the time elapsed since the last review. The precision of information arrival through signals determines the degree of concavity of such a function, with more precise information implying higher concavity. Moreover, we show that heterogeneity in the timing of ‘reviews” complements with heterogeneity in the realizations of private signals in determining forecast dispersion. In fact, the “weight” that an agent attaches to its own signal in determining her best forecast depends on the time elapsed since the last review. The larger the time elapsed, the larger the weight attached. We show that dispersion in forecasts arise not only because agents face different realizations of private signals, but also because agents attach different weights to the same signal. We derive implications of our model for observable statistics. In particular, we are interested in deriving statistics that are informative about the relative importance of reviews and signals for uncertainty reduction. Everything else being equal, we show that agents obtaining more information through reviews relatively to signals face larger variability in forecast uncertainty over time. We show that the coefficient of variation in uncertainty depends only on one parameter, λ, representing a standardized measure of frequency of reviews. In particular, λ equals the average frequency of reviews scaled by a measure of signals precision. For a given
3
frequency of reviews, the more precise signals are, the smaller the standardized measure of frequency λ. The larger λ, the larger the proportion of information arriving through reviews relatively to signals, and the larger the coefficient of variation of uncertainty. More generally, we show that the shape of the cumulative distribution function (CDF) associated to the invariant distribution of individual uncertainty is completely determined by the parameter λ. As λ goes to zero our model predicts a strictly convex CDF of uncertainty, while for higher λ our model predicts a strictly concave CDF. Therefore, the larger λ, the more concave the shape of the CDF of uncertainty. We apply our theoretical results to Euro-area inflation and GDP growth forecasts collected by the ECB survey of professional forecasters. We interpret x as the logarithm of the aggregate price level (GDP), and x∗ as the logarithm of the long-run price level (GDP).2 We use the predictions of our theory to measure the value of λ from the survey data. In particular, we derive the empirical CDF of forecast uncertainty corresponding to the definition of uncertainty in our model. We find that the empirical CDF of forecast uncertainty is characterized by a relatively high degree of concavity. We find that our model can account very well for the empirical CDF of uncertainty. We estimate the value of λ by maximizing the log-likelihood of observed inflation and GDP growth forecast uncertainty. Our estimates of λ range between 3.7 and 4.2, implying that agents obtain 90-95% of their information about the realizations of the long-run level of the target, x∗ , through reviews, and only 5-10% from signals. The driver of this result is the high variability of forecast uncertainty, which our model can explain through a relatively high proportion of information about the long-run level of the target, x∗ , arriving from reviews instead of signals. These results imply that modeling infrequent information acquisition is key to capture microeconomic evidence about forecasting decisions. However, these results do not necessarily imply that we can fully capture dynamics in forecasts through a model where agents obtain information only through 2
In the appendix we show how the process for price level/GDP dynamics that we consider can be obtained as the outcome of a general equilibrium model in an economy similar to the one considered by Woodford (2001).
4
reviews. In fact, while we show that infrequent reviews can account for the vast majority of information acquisition about the long-run level of the target, x∗ , agents do use these signals to obtain information about the other state variable, i.e. the current value of the target x. This implies, for instance, that forecasts change continuously rather than being sticky as predicted by model of information acquisition only through reviews. The latter is consistent with micro evidence. Another implication of these results is that forecast dispersion is mostly associated to heterogeneity in the timing of reviews, and it is not necessarily a good proxy for the average level of uncertainty. Moreover, given that most information about the long-run level of the target comes from reviews, instead of signals, shocks to the current value of the target have smaller impact on longer horizon forecasts. For instance, temporary shocks to inflation are unlikely to be confused with shocks to long-run inflation. The rest of the paper is organized as follows. In section 2 we describe the model while in section 3 we characterize its solution. In section 4 we derive several statistics about the invariant distribution of forecast uncertainty. In section 5 we derive the predictions of the model for dispersion in forecasts. Section 6 discusses an extension of our model that account for aggregate shocks to forecast uncertainty. Section 7 apply the theoretical results of the paper to data from the ECB Survey of Professional Forecasters. Section 8 concludes
1.1
Literature review
To be written.
2
The model
We consider a parsimonious model where an agent makes estimates of current and future dynamics of the target variable x(t), which evolves according to a generalization of the
5
OrnsteinUhlenbeck process, dx(t) = µ (x∗ (t) − x(t)) dt + φ dt + σ dW (t),
(1)
where µ > 0 is a parameter governing the speed of reversion of x(t) to x∗ (t), σ ∗ > 0 is the diffusion and dW (t) denotes a Wiener process, and φ is a constant drift. Dynamics in x∗ (t) follows a Wiener process with constant drift, dx∗ (t) = φ∗ dt + σ ∗ dW ∗ (t),
(2)
where dW ∗(t) is a Wiener process independent from dW (t) and φ∗ is a constant drift. The model in equations (1)-(2) is appealing as it has been shown to be flexible enough to account well for dynamics in macroeconomic variables such as inflation (e.g. Stock and Watson (2007)) and nominal interest rates (e.g. Hull and White (1990)).3 We assume that the agent perfectly observes the realization of the target variable x(t), but is imperfectly informed on the realizations of the state variable x∗ (t). There are two different sources of information on the realizations of x∗ (t). First, we assume that the agent perfectly observe the current and past realizations of the state {x∗ (t)} at endogenously determined times Tj , j=0, 1,....., where Tj+1 − Tj > 0. The observation in period Tj+1 entails a fixed cost θj , which is drawn randomly in Tj . Second, we assume that the agent observes a sequence of signals {yt } evolving according to ˜ (t). dy(t) = γ x∗ (t) dt + σ ˜ dW
(3)
Summing up, the information set of the agent in period t ∈ [Tj , Tj+1 ) is given by Ijt = 3
s=Tj s=t {x∗ (s)}s=−∞ , {x(s)}s=t , {y(s)} s=−∞ s=−∞ .
Section A shows how the process in equation (1) can be the process describing the dynamics of the price level in a new-Keynesian model similar to the one considered by Woodford (2001).
6
The economic interpretation of this information structure is along the lines of both Reis (2006b) and Sims (2003). Information on the realization of the state x∗ (t) is available but costly to acquire and process. agents run a review of their information set only sporadically because each review entails a fixed cost. In addition they process some information every period, which however reveals only imperfectly the realization of the state. The problem of the agent who cares about the mean square error in forecasts of x(t + hs ) for s = 1, 2, ...S is given by
V0 ≡ min E0 ∞ {Tj }j=0
"
∞ X
e−ρTj
θj + B
S X
ωs
s=0
j=0
Z
Tj+1
Tj
(4)
where, without loss of generality, we are starting at time t = 0 being an observation date, so that T0 = 0, and xˆ(t + hs ; t − Tj ) is the optimal forecast of average inflation over the next hj periods given information available at t to an agent that observed the state t − Tj periods ago, xˆ(t + hs ; t − Tj ) = E x(t + hs )|Ijt ; ωs are exogenous weights that the agent assigns to each forecast horizon.
3
Solving the model
Given agents face no adjustment cost, we can solve the model by setting the constant drift to zero, i.e. φ = φ∗ = 0, without loss of generality. In fact, all variables can be intended in deviation from a time trend. Given equations (1)-(2), starting at a generic period t0 we have ∗
−µ (t−t0 )
x(t) = x (t) + e
Proposition 1.
(xt0 −
x∗t0 )
+
Z
t
e−µ t0
(t−s)
(σ dW (s) − σ ∗ dW ∗ (s)) .
(5)
The best predictor of x(t + h) in period t given available information Ijt
7
!!#
e−ρ(t−Tj ) E (ˆ x(t + hs ; t − Tj ) − x(t + hs ))2 | It dt
j
is given by xˆ(t + h; t − Tj ) = 1 − e−µ
h
where xˆ∗ (t; t − Tj ) = E x∗ (t)|Ijt .
xˆ∗ (t; t − Tj ) + e−µ h x(t),
(6)
Optimal forecasting amounts to get the best estimate of x∗ (t). The larger the horizon
of forecast, the less inflation forecast depends on current realizations of inflation, the more inflation forecast depends on the realization of the long-run target. Next, we solve for the best estimate of x∗ (t) conditional on information available, Ijt . It is useful to define the vectors ˜ (t)]′ , of observables Y (t) ≡ [z(t), y(t)]′ and of temporary disturbances w(t) ≡ [dW (t), dW where the stochastic process z(t) evolves according to
dz(t) = dx(t) − µ x(t) dt. Notice that the processes x(t) and z(t) contain the same information about the unobservable process x∗ (t). The law of motion of Yt is given by 1
dY (t) = A x∗ (t) dt + Ω 2 w(t),
where
2
µ σ A= , Ω= γ 0
(7)
0 σ ˜2
Define the expected mean squared error in estimating current long-run inflation conditional on Ijt , Q(t − Tj ) ≡ E[(x∗ (t) − xˆ∗ (t; t − Tj ))2 |It j ]. Applying the Kalman-Bucy filter yields the
8
dynamics of xˆ∗ (t; t − Tj ), ∗
dˆ x (t; t − Tj ) = Q(t − Tj )
µ2 γ 2 + 2 σ2 σ ˜
γ ˜ (x(t) − xˆ (t; τ )) dt + dW (t) + dW (t) , σ σ ˜ ∗
µ
∗
(8) ∗ 2
dQ(τ ) =
where
1 σ ˆ
≡
µ2 σ2
+
(σ ) − γ2 σ ˜2
12
Q(τ ) σ ˆ
2 !
dτ
(9)
, xˆ∗ (t; 0) = x∗ (t) and Q(0) = 0.
The solution to the ODE in equation (9), is given by
Proposition 2.
∗
Q(τ ) = σ σˆ q q(x) ≡
σ∗ τ σˆ
ex − e−x . ex + e−x
(10) (11)
By solving the SDE in equation (8), it follows that t
∗ σ (t − s) dW ∗ (s) + xˆ (t; τ ) = x (t − τ ) + σ 1−β σ ˆ t−τ ∗ Z t ∗ σ σ µ γ ˜ ∗ +σ σ ˆ β (t − s) q (s − t + τ ) dW (s) + dW (s) . σ ˆ σ ˆ σ σ ˜ t−τ ∗
∗
where β(x) ≡
∗
1
1 x 1 −x e +2e 2
Z
(12)
.
It follows from equation (10) that our model encompasses the special cases extensively studied in the literature, i.e. the models of information acquisition through observation only, i.e. lim σ∗ → 0 Q(τ ) = (σ ∗ )2 τ , and through signals only,i.e. limτ →∞ Q(τ ) = σ ∗ σ ˆ . In the former, σ ˆ
uncertainty in the estimate of the state variable x∗ increases linearly as a function of time elapsed from a review. In the latter, uncertainty is constant.4 ˜ (s) It follows from equation (12) that the coefficient in front of ”temporary shocks”, i.e. dW and dW (s), is strictly increasing in time elapsed from the last review, while the coefficient 4
This is true after observing an infinite sequence of signals.
9
in front of ”permanent shocks”, i.e. dW ∗ (s), is strictly decreasing in time elapsed from the last review. In fact, the function β(·) is strictly decreasing, and captures the learning about x∗ (t) that takes place through mean reversion of x(t) to x∗ (t); the function q(·) is strictly increasing and captures the weight that the agent assigns to signals as a function of time elapsed since the last review. Therefore, the more time has elapsed since a review, the larger is the impact of current temporary shocks relatively to permanent shocks in the estimate of x(t)∗ . The impact of past temporary shocks on the estimate of x(t)∗ dies out at a rate µ, while the impact of past permanent shocks increases at a rate µ.
3.1
Optimal choice of time to next observation
Notice that the mean square error determining the loss function in equation (4) can be decomposed in two pieces, one depending on the information acquisition choices of the agent, and the other independent of it and only depending on the exogenous forecast horizon. Therefore, the problem of an agent that cares about hs , s = 1, 2, 3, ...., S horizons of forecasts can be written as
τˆj = arg min τ ≥0
P S
−µhs s=0 ωs 1 − e
2
B
Rτ 0
e−ρt Q(t)dt + e−ρτ θj
(1 − e−ρτ )/ρ
.
(13)
We let ρ → 0, so that the optimal τˆj is given by an implicit function Proposition 3.
Let ρ → 0, θ˜j ≡
θj (σ∗ )2 B
reviews solves the following equation σ∗
σ ∗ e σˆ τˆj ∗ σ ˆ e σσˆ
τˆj τˆj
σ∗
− e− σˆ
τˆj
− σσˆ τˆj ∗
+e
− log
PS
s=0
equation (14) is function of two factors,
ωs
(1−e−µhs )
2
> 0. The optimal time between
1 σ∗ τˆj 1 σ∗ e σˆ + e− σˆ τˆj 2 2
σ∗ τˆ σ ˆ j
and
σ∗ 2 ˜ θj . σ ˆ
=
σ∗ σ ˆ
2
(14)
The former is the time elapsed
from the last review scaled by a measure of signal precision. The larger 10
θ˜j .
σ∗ , σ ˆ
the more precise
Figure 1: Optimal choice of time to next observation 4.5
4
3.5
2.5
σ∗ σˆ
τˆ
3
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
³
σ∗ σ ˆ
´2
0.6
0.7
0.8
0.9
1
˜ θ
signals, the more information arrives through signals for a given τˆj . The term on the right hand side of equation (14) captures the cost-to-benefit ratio of a review scaled by a measure of signal precision. The more precise signals, the higher the cost-to-benefit ratio of a review, the larger the optimal time between reviews τˆj . The term on the left hand side is strictly increasing in σσˆ τˆj , and it is equal to zero at σσˆ τˆj = 0. This implies that for any value ∗ 2 of σσˆ θ˜j > 0, there will be a unique value of τˆj > 0 that solves equation (14) provided ∗ ∗ ∂ τˆ ∂ τˆ that σσˆ > 0. The function τˆj ≡ τ σσˆ , θ˜j is such that: (i) ∂ σj∗ ≥ 0; (ii) ∂ θ˜j ≥ 0, (iii) j σ ˆ q lim σ∗ →∞ τˆj = ∞, (iv) lim σ∗ →0 τˆj = 2θ˜j , (v) limθj →∞ τˆj = 0, (vi) limθj →∞ τˆj = ∞. ∗
σ ˆ
4
∗
σ ˆ
Forecast uncertainty
We derive the measure of uncertainty in forecast faced by an agent. Before we develop some useful notation. First, recall that the unconditional expectation of the mean square error in the estimate of the state x∗t by an agent in vintage τ is given by Q(τ ), as defined in equation (10). Let α ≡
σ∗ σ ˆ
τ be the normalized time elapsed since the last review. The
variable α takes values in [0, ∞). The larger α(τ ), the more information is acquired through signals relatively to reviews. Two special cases are worth mentioning. When α = 0 we are in 11
the Reis (2006b) case, i.e. τ > 0 and
σ∗ σ ˆ
= 0, while when α → ∞ we are in the case considered
by Woodford (2001); Sims (2003), i.e. τ → ∞ and of the average time between reviews, and λ ≡
1 E[α]
σ∗ σ ˆ
ˆ≡ > 0. Finally, let λ
1 E[τ ]
be the inverse
be the inverse of the average normalized
time between reviews. The parameter λ has a straightforward economic interpretation has it is equal to the average frequency of reviews scaled by a measure of signal precision. The more precise signals, the smaller the normalized frequency of reviews. It follows from the definition of α(·) and q(·) in equation (10) that the forecast error, et (t + h; τ ) ≡ x(t + h) − xˆ(t + h; τ ), is normally distributed with mean zero and variance υh2 (α), and υh2 (α)
σ ˆ −µh 2 2 ≡ 1−e q(α) + ϕh (σ ∗ )2 , σ∗
(15)
where the first term captures the uncertainty coming from imperfect information on the state πt∗ , while the second term captures uncertainty due to unpredictable future shocks. Notice that ϕ2h is exogenous to the agent’s decision, and it is given by ϕ2h
1 − e−µ = 1−2 µh
h
1 − e−2µ + 2µ h
h
σ 2 1 − e−2µ h h+ h. σ∗ 2µ h
(16)
The variable υh2 (α) is the measure of forecast uncertainty at horizon h. Next we derive the invariant distribution of forecast uncertainty.
4.1
Forecast uncertainty with constant cost-benefit of a review
˜ Under a In this section we study the special case of a constant cost-benefit of a review, θ. ˜ the time between consecutive reviews, τˆj = Tj+1 − Tj = τˆ, is constant. Therefore, constant θ, the distribution of time between reviews is degenerate. The unconditional distribution of associated to the event of a review taking place in the next ∆ ∈ [0, τˆ] periods is uniform with density given by τ1ˆ . In the case of an economy populated by identical agents, characterized by the state variable given by the time elapsed since the last review, τ , the invariant stationary distribution over τ ∈ [0, τˆ] is uniform. Notice that in this case λ = 12
σ ˆ /ˆ τ. σ∗
Next, we derive
the invariant distribution of a standardized measure of uncertainty. Let u ≡ q(α) be the standardized measure of uncertainty. The invariant
Proposition 4.
CDF of u, denoted by and G(u; λ) is equivalently given by
e1/λ −e−1/λ e1/λ +e−1/λ
1 if u > e1/λ −e−1/λ −1 G(u; λ) = λ q (u) if u ∈ [0, e1/λ +e−1/λ ] 0 otherwise
, G(u; λ) =
1 λ 2
ln
if u >
1+u
1−u
0
e1/λ −e−1/λ e1/λ +e−1/λ
if u ∈ [0, e1/λ +e−1/λ ] . otherwise (17) e1/λ −e−1/λ
where q −1 (·) is the inverse function associated to q(·) defined in equation (10). Figure 2: Cumulative distribution of uncertainty: non-stochastic cost of a review 1
0.9
0.8
0.7
G(·; λ)
0.6
0.5
0.4
0.3
λ = 5.0
0.2
λ = 2.0 λ = 0.5
0.1
0
0
0.1
0.2
0.3
0.4
0.5
u
0.6
The standardized measure of uncertainty u ≡ q(α) =
0.7
0.8
0.9
1
2 (α)−(σ ∗ )2 ϕ2 υh h
has a straightforward 2 (1−e−µh ) σ∗ σˆ interpretation. It is given by the mean square error in the estimate of the current state scaled by the maximum possible level of uncertainty that an agent can face when she never reviewed ˆ . Proposition 4 implies that the shape of the distribution the state, i.e. limτ →∞ υ02 ( σσˆ τ ) = σ ∗ σ ∗
of standardized uncertainty depends on λ. Figure 2 shows that the CDF of uncertainty is convex. The smaller λ the higher the convexity. In the extreme case where λ → ∞, the CDF of u is degenerate, with the distribution having all its mass at u = 0. The case λ → ∞ corresponds to the case of an agent who obtains information only through reviews. In fact, 13
one can show that lim σ∗ →0 υh2 ( σσˆ τ ) = (σ ∗ )2 τ . This implies that the maximum possible level ∗
σ ˆ
of uncertainty that an agent can face when she never reviewed the state is going to infinity, and therefore u = 0. At the other extreme, when λ = 0, we are in the case of an agent who obtains information only through signals. In this case the probability distribution is degenerate, with all the mass being at u = 1, as the agent never reviews the state. Proposition 5.
The median level of uncertainty is given by υh2 (ˆ τ /2), while the average
uncertainty is given by E[υh2 (α)]
∗ 2
= (σ )
−µh 2
1−e
λ log
1 1/λ 1 e + e−1/λ 2 2
+ ϕ2h ,
(18)
and the standard deviation in uncertainty is given by
σ[υh2 (α)]
∗ 2
= (σ )
s Z −µh 2 1−e λ 1/λ
1/λ
0
2
(q(x)) dx − log
1 1/λ 1 e + e−1/λ 2 2
2
.
(19)
From equation (18)-equation (19) we have that the coefficient of variation in uncertainty σ[υh2 (α)] E[υh2 (α)] − (σ ∗ )2 ϕ2h
v u u ≡ m (λ) = t
R 1/λ
(q(x))2 dx 1 2 − 1 ∈ 0, √ , 3 log 12 e1/λ + 12 e−1/λ 1/λ
0
(20)
The function m(·) is strictly increasing in its argument, and is equal to zero when λ = 0, and to
√1 3
when λ → ∞. The smaller λ, the larger the normalized time between reviews, the
more information is acquired through signals relatively to reviews, the smaller the dispersion in uncertainty relatively to its mean.
4.2
Forecast uncertainty with stochastic cost-benefit of a review
˜ is stochastic. In this section we consider the case where the cost-benefit of running a review, θ, In particular we assume that θ˜ is i.i.d. over time and across agents, distributed with a generic 14
density f (·) over the support [0, ∞). As a consequence, the time between consecutive reviews that solves equation (14), τˆj = Tj+1 − Tj , is itself stochastic and i.i.d. over time and across agents. We use the proof in Reis (2006b) to argue that the invariant distribution of time elapsed between consecutive reviews, τˆj , is exponential,
P r(ˆ τj ≤ x) = ˆ = where λ
1 . E(ˆ τj )
Z
x
ˆ ˆ −λs λe ds
0
It is convenient to work with the probability distribution of normalized
time between consecutive reviews, α ˆ j = σσˆ τˆj , which is distributed according to P r(α ˆj ≤ Rx ˆ x) ≡ Ψ(x; λ) = 0 λe−λs ds where λ = σσˆ∗ λ. ∗
Proposition 6.
Let u ≡ q(α) be the standardized measure of uncertainty. The invariant
CDF of u, denoted by Ψ(u; λ) is equivalently given by
1 −λT (u) Ψ(u; λ) = 1−e 0
if u > 1
1 R u −λT (s) e if u ∈ [0, 1] , Ψ(u; λ) = 0 λ 1−s2 ds otherwise 0
if u > 1
if u ∈ [0, 1] , otherwise (21)
where T (·) = q −1 (·) is the inverse function associated to q(·) defined in equation (10). The shape of the CDF in Figure 3 depends on λ, and ranges from strictly concave to strictly convex shapes. The higher λ, the larger the degree of concavity. As in the case of constant cost of reviews, in the extreme case when λ = 0 the distribution is degenerate with all mass at u = 1. In fact in this case all information arrives through signals. In the other extreme where λ → ∞, the distribution is degenerate at u = 1, as the agent reviews the state continuously. The reason why the shape of the CDF changes so much as a function of λ is straightforward. There are two contrasting forces determining the shape of the CDF. ˆ the higher the precision of signals, On one side, for a given average frequency of reviews, λ, the more concave is the shape of the function q(·) with respect to time elapsed since the last
15
Figure 3: Cumulative distribution of uncertainty: stochastic cost of a review 1
0.9
0.8
0.7
Ψ(·; λ)
0.6
0.5
0.4
λ = 5.0
0.3
0.2
λ = 2.0
0.1
λ = 0.5
0
0
0.1
0.2
0.3
0.4
0.5
u
0.6
0.7
0.8
0.9
1
review. This means that the inverse function of time elapsed since the last review is a more convex function of uncertainty. Therefore, for a given invariant distribution of time elapsed between reviews, we have a more convex CDF of uncertainty. On the other side, for given precision of the signals, an increase in the average frequency of reviews affects directly the invariant distribution of time elapsed between reviews by making it more concave. Notice that this second channel was not present in the case of constant review cost θ, as there the distribution of time elapsed between reviews was uniform, and therefore always characterized by a linear shape. It follows from Proposition 6 that coefficient of variation in uncertainty is given by v uR ∞ 2 −λx u t R0 (q(x)) λe dx ≡ F (λ) = − 1 ∈ [0, 1]. ∞ −λx dx 2 E[υh2 (α)] − (σ ∗ )2 ϕ2h q(x)λe 0 σ[υh2 (α)]
(22)
The function F(·) is strictly increasing in its argument, as documented in the left panel of Figure 4. The larger λ, the smaller the average normalized time between reviews, the less information is acquired through signals relatively to reviews, the smaller the dispersion in uncertainty relatively to its mean. Therefore, the main implication of the model is that when infrequent information acquisition is a relatively more important source of information 16
Figure 4: Coefficient of variation in uncertainty and fraction of uncertainty reduction due to reviews 1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
ξ(·)
F(·)
1
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0
1
2
3
4
5
6
7
8
0.1
9
0
1
2
λ
3
4
5
6
7
8
9
λ
than signals, the variability in uncertainty experienced by an agent over time relatively to its mean is larger. Next, we develop a statistic determining the fraction of uncertainty reduction due to ∗ h i Q( σ τ ) reviews on average. Let ξ ≡ E (σ∗σˆ)2 τ = E q(α) . Notice that q(α) is strictly decreasing α α and concave in α. The larger α, the larger the share of uncertainty reduction that takes place through signals. It turns out that
ξ(λ) =
Z
0
∞
q(x) −λx λe dx, x
(23)
only depends on λ and is strictly increasing in its argument, as documented in the right panel of Figure 4.
17
5
Dispersion in best point forecasts
In this section we derive a measure of average dispersion in agents’ forecast. Dispersion in agents’ forecast arises from two sources: (i) the different timing of reviews; (ii) the different realization of signals yt . Define the variable δ(t + h; τ ) ≡ xˆ(t + h; τ ) − x¯(t + h; τ ), representing the deviation of an agent forecast from the mean forecast within the vintage τ , x¯(t + h; τ ). The invariant distribution of δ(·; ·) is Gaussian with zero mean and variance ωh (τ ) given by The unconditional variance of δ(·; ·) is given by −µh 2
ωh (τ ) = 1 − e
σ ˆ (σ ) ∗ σ ∗ 2
γ2 σ ˜2 γ2 σ ˜2
+
µ2 σ2
D (τ ) ,
(24)
where the function D(·) is defined in the appendix and is strictly increasing in τ . The variance term ωh (τ ) is: (i) increasing in the horizon of forecast, h; (ii) increasing in the variance of fundamental shocks, σ ∗ ; (iii) increasing in the ratio of the precision of private to public signals, σσ˜ ; (iv) increasing in the time elapsed since the last review, τ ; (v) decreasing in the overall precision of signals, i.e. increasing in σ ˆ. Define the variable κ(t+h; τ ) ≡ x¯(t+h; τ )− x¯(t+h), representing the deviation of vintage τ mean forecast from the aggregate mean forecast. Consider the case of i.i.d. stochastic costbenefit of reviews. The unconditional distribution of κ(t + h; τ ) is Gaussian with zero mean and variance vh (τ )) given by 2 σ ˆ vh (τ ) = 1 − e−µh (σ ∗ )2 ∗ B σ
τ ; λ,
µ2 σ2 γ2 σ ˜2
+
µ2 σ2
!
,
(25)
where the function B(·) is defined in the appendix, and it is decreasing in the second argument and the third arguments, while it is non-monotonic in the first argument. Intuitively, agents that reviewed the earliest or the latest are the ones that on average have an information set that differs the most from the average. The larger the opportunity-cost of a review, the less frequent the reviews, and the more different the information set. The more informative
18
signals are, the smaller variance. Finally, the larger the more informative public signals are relatively to private signals, the larger the variance. We are now ready to derive an expression for forecast dispersion in this economy. The dispersion in h-horizon forecast is given by
Proposition 7.
2 σ ˆ w¯h2 = 1 − e−µh (σ ∗ )2 ∗ H σ
λ,
µ2 σ2 γ2 σ ˜2
+
µ2 σ2
!
,
(26)
where the function H(·) is decreasing in the first argument and second arguments.
Figure 5: Dispersion relatively to average uncertainty 1
ρ=0.0 ρ=0.5 ρ=1.0
0.9
H (λ; ρ)
0.8
0.7
0.6
0.5
0.4
0
1
2
3
4
5
6
7
8
9
10
λ
Note: ρ ≡
γ2 σ ˜2
µ2 σ2
2
µ +σ 2
.
More precise public signals relative to private increases forecast dispersion. Less information available to agents, either in the form of more costly reviews, i.e. higher θ˜ on average, or less informative signals, i.e. higher
σ ˆ , σ∗
increase forecast dispersion. These two frictions
are complementary. Figure 5 plots the function H(·) as a function of λ at different values of the ratio of public to private information,
µ2 σ2
2 γ2 + µ2 σ ˜2 σ
19
.
6
Modeling aggregate shocks to uncertainty
Empirical evidence suggests that uncertainty comoves with the business cycle. A straightforward extension of our model can allow for the possibility of aggregate shocks to uncertainty. In particular, suppose that β(t) is a stochastic process taking values in R++ and bounded above. Let σ ∗ (t) = σ0∗ β(t), σ(t) = σ0 β(t) and σ˜ (t) = σ ˜0 β(t), so that β(t) is a scaling factor affecting the diffusion parameters in the same way.5 Finally, let the distribution of θ˜j be invariant to β(t). The latter is equivalent to say that the fixed cost of each review is proportional to the level of uncertainty in the economy. This implies that the arrival rate of reviews is constant over time. It follows that aggregate shocks to uncertainty β(t) will act as parallel shifts to the invariant distribution of uncertainty, without having any impact on λ. In fact, we can rescale the measure of individual uncertainty in equation (15) by σ ∗ (t)2 , while the invariant distribution of α is unaffected by the level of σ ∗ (t)2 .
7
Application to the ECB survey of professional forecasters
In this section we apply the theory developed above to inflation and real GDP growth forecasts from the ECB survey of professional forecasters. In particular, let x(t) be the process for the logarithm of the aggregate price level (real GDP), and x∗ (t) be the process for the logarithm of the aggregate long-run price level (real GDP) in the case of inflation (real GDP growth) forecasts. Given the assumption that agents are perfectly informed on the realization of x(t), forecasting x(t + h) is equivalent to forecasting the growth rate between t and t + h, i.e. (x(t + h) − x(t))/h.6 5
A microfoundation for this assumption may be that agents have limited information processing capabilities, so that an increase in σ ∗ (t) is matched by an increase in σ(t) and σ ˜ (t) so that agents process the same amount of information per period. 6 See the appendix for an interpretation of the process for x(t) as the process for the price level in a general equilibrium model with imperfectly informed price setters.
20
7.1
Data description
We use survey data collected by the ECB on forecasts of inflation and GDP growth by a pool of nearly 90 professional forecasters. Professional forecasters are people who produce forecasts as part of their regular duties.7 Within the survey, each forecaster in each quarter is requested to report its forecast for inflation, GDP growth and unemployment at one calendar year ahead, as well as her probability distribution over a given grid of values. A clear advantage of the ECB survey data is that forecasters are asked to report their estimate over a constant horizon interval of time. The data is available at quarterly frequency from 1999:Q1 to 2010:Q4. We focus on the 1999:Q1-2007:Q4 sample. In each quarter, forecasters are given the latest available data on inflation and GDP growth, and are asked to report their best estimates of inflation and GDP growth at a horizon of twelve months from the month in which the last data was released. Forecasters are also asked to assign probabilities that the true value will fall into predetermined bins.
8
Before deriving the empirical measures of
uncertainty, we describe the basic properties of point forecasts. The left panel of Figure 6 plots median one-year ahead inflation forecast in each period, as well as the realized level of inflation. The average inflation over this period is 2.1%, while the standard deviation is 0.28%. There is substantial and very persistent error in inflation forecast between 2000 and 2006. The right panel of Figure 6 plots median one-year ahead GDP growth forecast in each period, as well as the realized level of GDP growth. The average GDP growth over this period is 2.2%, while the standard deviation is 1.1%.
7.2
Constructing measures of uncertainty
We apply the following procedure to inflation and gdp growth forecasts separately. First, we discard those observations where either forecasters do not report their best 12-months forecast or do not report the probabilistic assessment of the forecast target. We assume 7
Information on this dataset is available at http://www.ecb.int/stats/prices/indic/forecast/html/index.en.html. See Bowles et al. (2007) for a review of this data. 8 Bins have equal size of 0.5%, and range from -0.5% to 5.4%.
21
Figure 6: One year ahead forecast versus realized inflation/gdp growth Inflation
GDP Growth
2.5
4.5 Median One Year Ahead Inflation Forecast Realization of One Year Ahead Inflation
Median One Year Ahead GDP Forecast Realization of One Year Ahead GDP 4
3.5 2 3
2.5
2 1.5 1.5
1
1 1999
2000
2001
2002
2003
2004
2005
2006
0.5 1999
2007
2000
2001
2002
2003
2004
2005
2006
2007
Note: Variables are measured in % terms. The timeline refers to the time of the forecast. Data from the ECB survey of professional forecasters, described in Bowles et al. (2007).
that this information is missing randomly and in particular that it is not correlated with the level of uncertainty faced by the forecaster. Second, we discard those forecasters that do not fully respond to the survey in at least 8 of the 32 periods considered, so to increase the homogeneity of the pool of forecasters over time. After these operations, we are left with 59 forecasters, where the median forecaster answer the surveys in 24 of the 32 periods considered. The total number of observations is about 1,400. Finally, we estimate a measure of uncertainty for each forecaster i, at each survey data t. We do so by using the predictions of our model about the distribution of forecast errors. In particular, we construct the empirical CDF of forecast errors at each point in time from the reported distribution of the target and its best estimate. Then we fit a normal CDF with 2 zero mean and variance υi,t to the empirical CDF of forecast errors, and obtain a measure of
uncertainty for forecaster i in period t.9 9
In the cases in which forecasters report all the mass of forecast errors in one bin, this procedure may be 2 inaccurate. In such case, we use a different procedure to estimate υi,t . In particular, we find the level of υi,t that would imply that at least 90% of probability mass is in the bin reported by the forecaster.
22
Figure 7: Median uncertainty versus dispersion in forecasts Inflation
GDP Growth
0.22
0.35 Median Uncertainty Dispersion in Inflation Forecast
Median Uncertainty Dispersion in GDP Forecast
0.2 0.3 0.18 0.25
0.16
0.14 0.2 0.12 0.15 0.1
0.08
0.1
0.06 0.05 0.04
0.02 1999
2000
2001
2002
2003
2004
2005
2006
0 1999
2007
2000
2001
2002
2003
2004
2005
2006
2007
Note: Variables are measured in % terms. The timeline refers to the time of the forecast. Dispersion refers to the variance across individuals best estimates at a given point in time. Median uncertainty refers to the median value of estimated uncertainty across individuals at a given point in time.
Figure 7 plots the median level of uncertainty across forecasters in each period, as well as the variance in best forecast over time, i.e. a measure of dispersion in forecasts. Median uncertainty is larger than dispersion in most periods, and more generally dispersion is not a perfect proxy for uncertainty. Moreover, the median level of uncertainty displays some variability over time, suggesting the presence of aggregate shocks to uncertainty. The results presented in the previous sections refer however to invariant distribution of uncertainty in absence of aggregate shocks. We use the results in Section 6 and adopt the following procedure to remove the aggregate shocks to uncertainty. First, from the definition of the standardized measure of uncertainty we have that
ui,t =
Let a ≡
ϕ2h σ ˆ −µ )2 (1−e σ∗
υt2 (αi ) − σ ∗ (t)2 σσˆ∗ (1 − e−µ )2
ϕ2h . σ ˆ −µ )2 (1 − e ∗ σ
, and z(t) ≡ σ ∗ (t)2 σσˆ∗ (1 − e−µ )2 . Second, notice that according to our
model the maximum and minimum level of uncertainty across individuals at a given point in 23
time is given by υ¯t2
∗
= σ (t)
2
σ ˆ −µ 2 2 (1 − e ) + ϕh ; υt2 = σ ∗ (t)2 ϕ2h . σ∗
From the equations above it follows that
a=
1 υ ¯t2 υt2
−1
, z(t) =
υ¯t2 . 1+a
Therefore, we compute a by taking the maximum realization of
υ ¯t2 υt2
over time, and construct
the variable z(t). Given a and z(t) we obtain the standardized measure of uncertainty, ui,t , for each forecaster at each point in time, which has the property of being identically distributed over time and across forecasters. For illustrative purposes, Figure 8 plots the dynamics of Figure 8: Examples of the dynamics in individual standardized uncertainty, ui,t 0.2
0.4 0.35
0.15
0.3 0.25
0.1
0.2 0.15
0.05
0.1 0.05
0 1998
2000
2002
2004
2006
0 1998
2008
0.8
2000
2002
2004
2006
2008
2000
2002
2004
2006
2008
1
0.7 0.8 0.6 0.5
0.6
0.4 0.4
0.3 0.2
0.2 0.1 0 1998
2000
2002
2004
2006
0 1998
2008
Note: Dynamics of uncertainty about GDP growth one year ahead for 4 forecasters reporting information in 31 of the 32 periods.
standardized uncertainty, ui,t , of 4 agents forecasting GDP growth at one year horizon. These are forecasters responding to the survey in 31 of the 32 periods considered. The figure shows that (i) each forecaster experiences quite a large amount of variability in forecast uncertainty 24
over time, (ii) variability in individual uncertainty is roughly uncorrelated across forecasters.
7.3
Identifying the relative importance of review to signals for uncertainty reduction
In this section we find the value of λ that best fits the empirical CDF of uncertainty u for inflation and GDP growth respectively. Figure 9 plots the empirical CDF (blue line) and the model implied CDF (red line) for inflation and GDP growth standardized measures of uncertainty. The empirical CDF is obtained by pooling together all the observations of uncertainty ui,t both across time and agents for inflation and GDP growth forecast respectively. The model implied CDF is given by equation (21) after estimating λ through maximum likelihood. Figure 9: Empirical versus model implied CDF of uncertainty Inflation
GDP Growth
1
1 Empirical CDF Model CDF
0.9
0.9
0.8
0.8
0.7
0.7
CDF, Ψ(·; λ)
CDF, Ψ(·; λ)
Empirical CDF Model CDF
0.6
0.5
0.4
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1
0
0.1
0.2
Normalized uncertainty, u
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized uncertainty, u
Note: Variables are measured in % terms.
The empirical CDF is well explained by the model both for inflation and GDP growth forecast. In fact, the two empirical CDF are quite similar. The first line of Table 1 reports the maximum likelihood estimates of λ. Similar values are 25
Table 1: Estimates of λ Inflation GDP Growth Maximum Likelihood Estimate of λ 3.7 (0.10) 4.2 (0.11) % of uncertainty reduction due to reviews, ξ(λ) 93% 95% Note: Standard errors in parenthesis.
obtained through an extremum estimator that chooses λ to minimize the distance between the empirical and model implied CDF. The second line of Table 1 reports the model implied fractions of uncertainty reduction that takes place through reviews relatively to signals. The model implies that almost all uncertainty reduction takes places through reviews.
8
Concluding remarks
To be written.
26
References Alvarez, Fernando E., Francesco Lippi, and Luigi Paciello. 2009. “Optimal price setting with information and menu costs.” Mimeo - EIEF. Bowles, Carlos, Roberta Friz, Veronique Genre, Geoff Kenny, Aidan Meyler, and Tuomas Rautanen. 2007. “The ECB survey of professional forecasters (SPF): A review after eight years experience.” Occasional Paper Series 59, European Central Bank. Caballero, R.J. 1989. “Time Dependent Rules, Aggregate Stickiness And Information Externalities.” Discussion Papers 198911, Columbia University. Hull, J. and A. White. 1990. “Pricing interest-rate-derivative securities.” Review of financial studies 3 (4):573. Mankiw, N. Gregory, Ricardo Reis, and Justin Wolfers. 2004. “Disagreement about Inflation Expectations.” In NBER Macroeconomics Annual 2003, Volume 18, NBER Chapters. National Bureau of Economic Research, Inc, 209–270. Reis, Ricardo. 2006a. “Inattentive consumers.” Journal of Monetary Economics 53 (8):1761– 1800. ———. 2006b. “Inattentive producers.” Review of Economic Studies 73 (3):793–821. Sims, Christopher A. 2003. “Implications of rational inattention.” Journal of Monetary Economics 50 (3):665–690. Stock, J.H. and M.W. Watson. 2007. “Why has US inflation become harder to forecast?” Journal of Money, Credit and Banking 39:3–33. Woodford, Michael. 2001. “Imperfect Common Knowledge and the Effects of Monetary Policy.” NBER Working Papers 8673, National Bureau of Economic Research, Inc. ———. 2009. “Information-Constrained State-Dependent Pricing.” Journal of Monetary Economics 56:s100–s124.
27
A
Interpretation of the target process as the process for the price level
Consider the simple new-Keynesian model studied by Woodford (2001). There is a continuum of identical firms, each selling its own variety and setting the price of its own good in a regime of monopolistic competition with constant elasticity of substitution across varieties, Λ. Each firm produces output through a linear technology in labor input and maximizes expected profits. In this framework each firm would set prices equal to a constant markup, Λ/(Λ − 1), over nominal marginal cost. Let the log of nominal marginal cost in deviation from the nonstochastic steady state, m(t), follow the process, dm(t) = φ∗ dt + σ ∗ dW ∗ (t), where W ∗ (t) is a Wiener process common to all firms, can be interpreted as aggregate productivity shocks, or aggregate nominal shocks. More generally, the process m(t) can be interpreted as the process for nominal spending, or money supply in standard new-Keynesian model. Then the optimal target price for each firm in deviation from the non-stochastic steady state, p∗i (t), also follows brownian motion with drift, dp∗i (t) = φ∗ dt + σ ∗ dW ∗(t). Assume that price setters are imperfectly informed about the realization of the process p∗i (t), but in each period observe a signal dsi (t) = p∗i (t)dt + σe dW (t), where W (t) is a Wiener process identical across firms and independent from W ∗ (t). Take a second order expansion of the profit function in a neighbor of the profit function evaluated at the optimal price p∗ (t). Then applying the Kalman-Bucy Filter the price optimally set by each firm conditional on the information available, pi (t) = E[p∗i (t)|sti ], follows the process dpi (t) =
σ∗ ∗ (pi (t) − pi (t)) dt + φ∗ dt + σ ∗ σe e(t), σe
(A-1)
Let µ = σσe , σ = σ ∗ σe , φ∗ = φ. Then by re-labeling x with p and x∗ with p∗ , and using pi (t) = p(t) in equation (A-1) we obtain equation (1). ∗
B
Derivation of optimal forecast 1
Rt
Define yt (τ ) = xˆ∗t (τ ) − x∗t , and zt (τ ) = e σˆ 2 t−τ Q(u+τ −t) du yt (τ ), where τ is the time elapsed from the last review. Then Rt 1 µ γ ˜ dzt (τ ) = e σˆ 2 t−τ Q(u+τ −t) du −σ ∗ dW ∗ (t) + Q(τ ) ( dW (t) + dW (t)) σ σ ˜ From zt−τ (0) = 0, it follows that Z t Rs 1 µ γ ˜ zt (τ ) = e σˆ 2 t−τ Q(u+τ −t) du −σ ∗ dW ∗ (s) + Q(s + τ − t) ( dW (s) + dW (s)) . σ σ ˜ t−τ Then
28
1
Rt
yt (τ ) = e− σˆ 2 t−τ Q(u+τ −t) du zt (τ ) = Z t Rt 1 µ γ ˜ = e− σˆ 2 s Q(u+τ −t) du −σ ∗ dW ∗ (s) + Q(s + τ − t) ( dW (s) + dW (s)) . σ σ ˜ t−τ Then xˆ∗t (τ ) = Z
t
Rt
γ ˜ µ −σ dW (s) + Q(s + τ − t) ( dW (s) + dW (s)) , σ σ ˜
Q(u+τ −t) du
Notice that equation (10) implies
1 σ ˆ2
=
B.1
x∗t
+
−
e t−τ
1 σ ˆ2
s
∗
Rs 0
∗
Q(u) du = log
1 2
σ∗
e σˆ
(s)
∗
+
1 − σσˆ (s) e 2
.
Derivation of uncertainty due to unforeseeable component
The unforeseeable part of x∗ (t + h) in period t is given by x(t + h) − x(t + h; t) = Z t+h Z −µ (t+h−s) ∗ =σ e dW (s) + σ t
(A-2) t+h t
1 − e−µ
(t+h−s)
it follows that ϕ2h
B.2
2
2
Z
h −2µ (h−s)
∗ 2
= E[(x(t + h) − xˆ(t + h; t)) ] = σ e ds + (σ ) 0 −2µ h 1 − e−2µ h 1 − e−µ h 21 − e ∗ 2 =σ + (σ ) h + −2 . 2µ 2µ µ
Z
dW ∗ (s) ,
h 0
1 − e−µ
(h−s) 2
(A-3)
ds = (A-4)
Derivation of unconditional variance in innovations to x(t)
The variance of innovations in x(t) is given by V ar(x(t + h) − x(t)) = V ar(x(t + h) − xˆ(t + h; t)) + V ar(ˆ x(t + h; t) − x(t)). The first term is given by ϕ2h . The second term is given by −µh 2
V ar(ˆ x(t + h; t) − x(t)) = 1 − e
∗
−µh 2
V ar(x (t) − x(t)) = 1 − e
29
(σ ∗ )2 (σ)2 + 2µ 2µ
.
It follows that (σ ∗ )2 (σ)2 E[(x(t + h) − x(t)) ] = + 1−e + = 2µ 2µ −µ h 1 − e−µ h ∗ 2 21 − e + (σ ) h − . =σ µ µ 2
C
−µh 2
ϕ2h
(A-5)
Dispersion
C.1
Variance of forecast deviation from the same-vintage-mean Z
1
2 xˆ(t + h; τ j ) − x¯(t + h; τ j ) di = (A-6) 0 2 ∗ Z t σ∗ (s−t+τ ) − σσˆ (s−t+τ ) σ ˆ e − e γ2 2 = 1 − e−µh (σ ∗ σ ˆ )2 ds ∗ ∗ ∗ σ∗ 1 σσˆ (t−s) 1 − σσˆ (t−s) (s−t+τ ) − σσˆ (s−t+τ ) σ ˜2 t−τ σ ˆ e + e e + e 2 2 j
ωh (τ ) =
(A-7)
= 1 − e−µh
2
−µh 2
= 1−e
(σ ∗ )2 γ 2 σ ˜2 ∗ 2
(σ )
γ2 σ ˜2
+
µ2 σ2
γ2 σ ˜2 γ2 σ ˜2
+
µ2 σ2
Z
t
Z
τ
β(
t−τ
β(
0
σ∗ σ∗ (t − s))2 q(s − t + τ ; )2 ds σ ˆ σˆ
σ∗ σ∗ (τ − s))2 q( s)2 ds σ ˆ σ ˆ
where β(x) ≡ Therefore: D (α(τ )) ≡ where α(τ ) =
C.2
σ∗ τ. σ ˆ
Z
α(τ ) 0
1 x e 2
1 , + 12 e−x
(β(s) q(α(τ ) − α(s)))2 dα(s),
It follows that ωh (τ ) = 1 − e−µh
2
(A-8) (A-9)
(A-10)
(A-11)
(σ ∗ )2 σσˆ∗ D (α(τ )).
Variance of deviation of vintage-mean from aggregate mean
First, notice that average forecast is given by v u µ2 Z ∞ u σ∗ σ∗ ∗ −µ h t σ2 ˆ x¯(t + h) = σ 1 − e β( s) q¯ s; λ, dW (t − s)− γ2 µ2 σ ˆ σ ˆ + 0 2 2 σ ˜ σ Z ∞ ∗ σ ˆ −σ ∗ (1 − e−µ h ) β( s) e−λs dW ∗ (t − s) + e−µ h x(t), σ ˆ 0
30
R∞ ˆ σ∗ ˆ ˆ −λτ where q¯ s; λ, σˆ = σσˆ∗ s q(τ − s)λe dτ . Then compute the deviation of mean forecast within vintage τ from aggregate mean: κ(t + h; τ ) ≡ x∗ (t) + x¯(t + h; τ ) − x¯(t + h) = v u 2 Z τ u σµ2 σ∗ σ∗ σ∗ ∗ −µ h t ˆ =σ 1−e dW (t − s) − β( s) q( (τ − s)) − q¯ s; λ, γ2 µ2 σ ˆ σ ˆ σ ˆ + 0 2 2 σ ˜ σ Z τ ∗ σ ˆ ∗ −µ h −λs −σ 1−e β( s) 1 − e dW ∗ (t − s) − σ ˆ 0 v u µ2 Z ∞ Z ∞ u σ2 σ∗ σ∗ ˆ ∗ −λs ∗ −µ h t ˆ −σ 1−e β( s) q ¯ s; λ, dW (t − s) + σ e dW ∗(t − s), 2 2 γ µ σ ˆ σ ˆ + τ σ ˜2 σ2 τ
where dW ∗ (t − s) and dW (t − s) are differentiated with respect to s. It follows that: ! µ2 ∗ σ 2 2 σ ˆ ˆ τ ; λ, , , vh (τ ) = 1 − e−µh (σ ∗ )2 B σ ˆ γ 22 + µ22 σ ˜ σ ! Z ! 2 2 µ2 µ2 2 τ ∗ ∗ ∗ ∗ σ σ σ σ ˆ 2 2 − λs σ σ ˆ ˆ ˆ τ ; λ, B ≡ β( s) q( (τ − s)) − q¯ s; λ, + 1−e ds+ γ2 µ2 σ ˆ γ 22 + µ22 σˆ σ ˆ σˆ + 0 σ ˜ σ σ ˜2 σ2 ! 2 2 Z ∞ µ2 2 ∗ σ∗ σ ˆ 2 − λs σ ˆ q¯ s; λ, + β( s) + e ds. µ2 γ2 σˆ σˆ + τ 2 2 σ ˜
σ
Equivalently we can write
B
α(τ ); λ,
γ2 σ ˜2
! µ2 σ ˆ 2 2 vh (τ ) = 1 − e−µh (σ ∗ )2 ∗ B α(τ ); λ, γ 2 σ µ2 , σ + σ2 σ ˜2 ! ! Z ατ µ2 µ2 2 2 2 2 2 σ σ ≡ (β(α(s))) (q(α(τ ) − α(s))) − q¯ (s; λ)) + 1 − e−λα(s) dα(s)+ 2 γ2 µ2 + µσ2 + 0 σ ˜2 σ2 ! Z ∞ µ2 2 2 2 2 −λα(s) σ + (β(α(s))) q (α(s); λ)) + e dα(s), 2 (¯ γ2 + µσ2 α(τ ) σ ˜2
where q¯ (α(s); λ) =
C.3
R∞
α(s)
q(α(τ ) − α(s))λe−λα(τ ) dα(τ ).
Derivation of dispersion
Aggregate dispersion is a weighted average of dispersion arising from the different vintages of forecasters. Z ∞ Z ∞ 2 −λα(τ ) wh = ωh (α(τ ))λe dα(τ ) + vh (α(τ ))λe−λα(τ ) dα(τ ). (A-12) 0
0
31
The first term is given by Z
∞
ω(α(τ ))λe−λα(τ ) dα(τ ) =
0
2 σˆ = 1 − e−µh (σ ∗ )2 ∗ σ −µh 2
= 1−e
γ2 σ ˜2 γ2 σ ˜2
σ ˆ (σ ) ∗ σ ∗ 2
µ2 σ2
+
+
∞
Z
∞
0
γ2 σ ˜2 γ2 σ ˜2
Z
µ2 σ2
Z
0
α(τ )
(β (α(s)) q(α(τ ) − α(s))))2 dα(s) 2
(β (α(s)))
0
Z
∞
α(s)
2
!
λe−λτ dα(τ )
−λα(τ )
(q(α(τ ) − α(s))) λe
dα(τ ) dα(s)
The second term is given by Z ∞ Z ∞ −λα(τ ) v(α(τ ))λe dα(τ ) = E (¯ xt (α(τ )) − x¯t )2 λe−λα(τ ) dα(τ ) = 0 0 Z ∞ ∗ 2 −λα(τ ) = E[(¯ xt (α(τ )) − xt )] λe dα(τ ) − E[(¯ xt − x∗t )2 ] = 0
−µh 2
= 1−e
−µh 2
− 1−e
−µh 2
+ 1−e
σˆ (σ ) ∗ σ ∗ 2
σ ˆ (σ ) ∗ σ ∗ 2
σˆ (σ ) ∗ σ ∗ 2
µ2 σ2
γ2 σ ˜2
+
µ2 σ2
µ2 σ2 γ2 σ ˜2
Z
0
+ ∞
µ2 σ2
Z
∞
2
(β (α(s)))
0
Z
0
∞
Z
∞
α(s)
2
(β(α(s)))
Z
∞
α(s)
2
−λα(τ )
(q(α(τ ) − α(s)))) λe −λα(τ )
q(α(τ ) − α(s)))λe
(β (α(s)))2 1 − e−λα(s) e−λα(s) dα(s)
dα(τ )
dα(τ ) dα(s) − 2
dα(s)+
It follows that ! Z Z ∞ µ2 ∞ −λα(s) 2 2 −λα(τ ) −λα(s) σ2 H λ, γ 2 µ2 ≡ β (α(s)) (q(α(τ ) − α(s))) λe dα(τ ) + 1 − e e dα(s)− + 0 α(s) 2 2 σ ˜ σ Z ∞ 2 Z ∞ µ2 2 −λα(τ ) σ2 (β (α(s))) − γ 2 µ2 q(α(τ ) − α(s))λe dα(τ ) dα(s). + 0 α(s) 2 2 σ ˜ σ
32
In the derivation above we have used the following results: Z
∞
ˆ
ˆ −λτ dτ = E[(¯ xt (τ ) − πt∗ )]2 λe 0 Z ∞ Z t ∗ σ ˆ −µh 2 ∗ 2 ˆ −λτ = 1−e (σ ) E[ β (τ − s) dW ∗(s)]2 λe dτ + σ ˆ 0 t−τ Z ∞ Z t ∗ µ2 σ ˆ −µh 2 ∗ 2 σ2 ˆ −λτ E[ β (τ − s) q(s + τ − t) dW (s)]2 λe dτ + 1−e (σ ) γ 2 µ2 σˆ + 0 t−τ 2 2 σ ˜ σ 2 Z ∞Z τ ∗ σ ˆ −µh 2 ∗ 2 ˆ −λτ = 1−e (σ ) β (τ − s) dsλe dτ + σ ˆ 0 0 Z ∞Z τ ∗ µ2 σ ˆ −µh 2 ∗ 2 σ2 ˆ −λτ + 1−e (σ ) γ 2 µ2 (β (τ − s) q(τ − s))2 dsλe dτ σ ˆ + σ2 0 0 σ ˜2 2 Z ∞ Z ∞ ∗ σ ˆ −λτ −µh 2 ∗ 2 ˆ (τ − s) = 1−e (σ ) β λe dτ ds+ σ ˆ s 0 Z ∞ Z ∞ ∗ µ2 σ ˆ −µh 2 ∗ 2 2 2 ˆ −λτ σ2 (σ ) γ 2 µ2 (β (τ − s) ) + 1−e (q(τ − s)) λe dτ ds σ ˆ + 0 s 2 2 σ ˜ Z σ∞ σ ˆ 2 = 1 − e−µh (σ ∗ )2 ∗ (β (α(τ ) − α(s))))2 e−λα(s) dα(s)+ σ 0 Z ∞ Z ∞ µ2 ˆ −µh 2 ∗ 2 σ 2 2 −λα(τ ) σ2 (β(α(s))) (q(α(τ ) − α(s))) λe dα(τ ) dα(s) + 1−e (σ ) ∗ γ 2 µ2 σ 2+ 2 0 α(s) σ ˜
σ
33
and Z E[
∞
ˆ ˆ −λτ (¯ xt (τ ) − πt∗ ) λe dτ ]2 = 0 Z ∞Z t ∗ σ ˆ −µh 2 ∗ 2 ˆ −λτ = 1−e (σ ) E[ β (τ − s) dW ∗(s)λe dτ ]2 + σˆ 0 t−τ Z ∞Z t ∗ µ2 σ ˆ −µh 2 ∗ 2 σ2 ˆ −λτ + 1−e (σ ) γ 2 µ2 E[ β (τ − s) q(s + τ − t) dW (s)λe dτ ]2 σ ˆ + 0 t−τ 2 σ ˜2 Z ∞σ Z ∞ ∗ σ ˆ −µh 2 ∗ 2 −λτ ˆ = 1−e (σ ) E[ β (τ − s) λe dτ dW ∗ (τ − s)]2 + σ ˆ 0 s Z ∞ Z ∞ ∗ µ2 σ ˆ −µh 2 ∗ 2 −λτ σ2 ˆ + 1−e (σ ) γ 2 µ2 E[ β (τ − s) q(τ − s)λe dτ dW (τ − s)]2 σ ˆ + 0 s σ ˜2 σ2 Z ∞ 2 Z ∞ ∗ σ ˆ −µh 2 ∗ 2 2 −λτ ˆ = 1−e (σ ) (β (τ − s) ) λe dτ ds+ σ ˆ 0 s Z ∞ 2 Z ∞ ∗ µ2 σ ˆ −µh 2 ∗ 2 2 −λτ σ2 ˆ + 1−e (σ ) γ 2 µ2 (β (τ − s) ) q(τ − s)λe dτ ds σ ˆ + 0 s 2 2 σ ˜ Z σ∞ 2 σ ˆ 2 = 1 − e−µh (σ ∗ )2 ∗ (β(α(s)))2 e−λα(s) dα(s)+ σ 0 Z ∞ 2 Z ∞ µ2 ˆ 2 −λα(τ ) −µh 2 ∗ 2 σ 2 (β(α(s))) q(α(τ ) − α(s))λe dα(τ ) dα(s) + 1−e (σ ) ∗ γ 2 σ µ2 σ 2+ 2 0 α(s) σ ˜
σ
34
