Measurement Models

This section introduces measurement models for quantitative as well as qualitative fore-casts. In these state-space models, the responses of a day are seen as draws from the population distribution of expectations, which changes from day to day. The goal is to estimate its shape on a given day.

The state-space modeling approach is a natural choice for this problem: First, within its framework, I can easily formalize that the responses on a specific day are noisy signals about the population distribution of forecasts. Second, it allows us to account for variation in the signal precision arising from a changing number of responses per day. Third, the approach naturally overcomes the hurdle of the missing values I observe when no survey responses are collected. Fourth, I can easily augment the model with additional measurement variables such as financial asset prices and relate them to the dynamics in the population distribution of forecasts.

1.2.1 Quantitative Forecasts

Denoting the point expectation of respondent i stated on day t by y_it, and the average across the responses collected on day t by y¯_t = 1/n_tP

iy_it, I set up the following state-space model

¯

y_t = µ_t+ε_t, ε_t^iid∼ N(0, σ_ε²/n_t), (1.1) µ_t+1 = µ_t+η_t+1, η_t+1 ^iid∼ N(0, σ_η²). (1.2) Given that σ²_ε and σ_η² are constant, interest lies in estimating how µ_t, the mean of the population distribution of forecasts, changes over time. Estimation is based on Maximum Likelihood and the Kalman Filter. Its details are outlined in Appendix 1.A.

I next discuss the model: The measurement equation (eq. 1.1) implicitly assumes that the individual responses on daytare draws from a distribution with meanµ_tand variance σ_ε², such that the average response has variance σ_ε²/nt, reflecting that the quality of the signal y¯_t for µ_t increases with the number of responses.

The implicit assumption about the individual responses is not particularly restrictive as it is compatible with individual responses that have the following structure

y_it =µ_t+u_i+ξ_it,

where µt is a common trend, ui is an individual random effect with ui

iid∼ (E[ui] = 0, V[u_i] = σ_u²), and ξ_it is an individual disturbance with ξ_it ^iid∼ (E[ξ_it] = 0, V[ξ_it] = σ²_ξ

it).

The random effect (u_i) reflects that some respondents are generally more or less optimistic about the future, thus permanently over- or under-predicting economic variables. The individual disturbances (ξ_it) captures that, despite the overlap between the information respondents have and the interpretations they derive from it, differences typically exist.¹ From these assumptions it immediately follows that the average response y¯_t has mean µ_t and variance σ_ε²/n_t = (σ_u² +σ²_ξ)/n_t, as I state in equation (1.1). Moreover, approximate normality of the measurement equation (eq. 1.1) arises if either (a) n_tis reasonably large according to the Lindeberg-Levy central limit theorem, or if (b) the distributions of the two disturbancesu_iandξ_itare normal themselves. Contrary to my assumption of a serially uncorrelated measurement error ε_t, a random effectu_i may induce correlation among any pair of measurement errorsy¯_t−µ_t = 1/n_tP

(u_i+ξ_it)andy¯_t+h−µ_t+h = 1/n_tP

(u_i+ξ_it+h), provided that there is significant overlap in the samples collected at t and at t+h, such that the same random effects are featured in both samples. As respondents reply (at most) once in a survey period, there can be no overlap for small values of h. If the value of h implies that I compare different survey waves, provided that respondents systematically respond on specific days of the survey period, the random effect may seriously invalidate the assumption of no serial correlation in the measurement errors. As this does not hold in the data I have at hand, I conjecture that the assumption is a reasonable simplification.

The state equation (eq. 1.2) has the structure of a random walk. It reflects that with no other information at hand, my best estimate of tomorrow’s state of expectations is today’s. Using the Kalman Filter, I will revise this estimate after observing tomorrow’s responses.² Later in this article, I will use regularly observed data such as the change in the price of certain financial assets to improve my estimate of the state of expectations.

1see the relatively recent strand of literature about forecast disagreement, e.g. Lahiri and Sheng (2008) or Mankiw, Reis, and Wolfers (2004).

2The random walk assumption does not imply that expectations are not influenced by other observable data and that I cannot learn from these data about the state of expectations.

My baseline model can be extended in several directions: One interesting avenue is to relax the assumption that the measurement error (ε_t) is homoscedastic and thus capture that the disagreement of expectations may change over time. Another is to relax the homoscedasticity of the state disturbance (η_t), e.g. by allowing that shocks may either come from a business-as-usual distribution with moderate variance or from an major-event distribution with a large variance.

1.2.1.1 Adding another measurement variable

Recognizing that surveys do not collect survey responses all the time, we may want to use additional data that are observed on a regular basis to estimate changes in the latent distribution of survey forecasts. The additional data I have in mind are variables that indirectly measure (market) expectations. For example, long-term bond yields impound expectations about future short-term nominal and real interest rates and about inflation.

To benefit from such indirect measures of expectations (denoted by x_t), I augment the measurement equation (eq. 1.1) in the following way:

"

and assume that the measurement error ε_x,t has variance σ²_x and that it is serially uncor-related and uncoruncor-related at all leads and lags with both the measurement errorε_tand the state disturbance η_t.

In equation (1.3), I assume that x_t is related to the day-over-day change in µ_t, which implies that xt must be stationary. Non-stationary financial asset prices will thus have to be transformed to stationarity, either by taking the day-over-day price change or the day-over-day return. If I were to assume that x_t loads on the level of µ_t, the unit root in µ_t (see eq. 1.2) would require that x_t and y¯_t are cointegrated - an overly restrictive assumption.

The way I link the additional measurement to the state variable is identical to Ghysels and Wright (2009). The idea is that the same economic news that impact on economic expectations are likely to be “impounded in financial asset prices” (Ghysels and Wright, 2009, p.504), such that the additional variable provides a noisy signal about the change in the state variable.

1.2.2 Qualitative Forecasts

Suppose instead of point forecasts, as in the case of the empirical application, I observe qualitative forecasts. Individual responses are either “will rise”, “will stay the same”, or

“will fall” and can thus be seen as draws from a multinomial distribution. Denoting by y¯_t a two-element column vector holding the shares of “will rise” and “will fall” responses on day t³, I set up the following non-linear state space model:

¯

whereη_t^i.i.d.∼ N(0,Σ_η),andε_t is a serially independent disturbance vector with mean zero and variance

Interest lies in estimating the path of π_t, the vector of the population probabilities for a “will rise” and a “will fall” response. This is a dual estimation problem as it involves the estimation of both the model parameters and the path of the state variables. Pro-vided with candidate model parameters, the Unscented Kalman Filter (e.g. Wan and Van Der Merwe, 2000) estimates the path of the state variables, which is in turn used to compute the model likelihood. Parameter estimates are obtained by maximizing the like-lihood with respect to the model parameters (maximum likelike-lihood estimation). Details are given in Appendix 1.B.

To understand the model, note that the mean and variance of the measurement equation (1.4) follow from the assumption that the responses are independent and identically dis-tributed draws from a multinomial distribution with π_R,t being the probability of a “will rise” and π_F,t being the probability of a a “will fall” response.

Finally, it is interesting to note that the state-space model specified by equations (1.4-1.7) implies that the vector of log-odds ratios with elementsln(πR,t/πS,t)andln(πF,t/πS,t), evolves according to the bivariate random walk specified in equation (1.6).

3Note, that in equation (1.4), I have not included the share of “stay the same” responses, since it is linearly determined from the two other shares as 1−1/ntP

i

1(y_it=’rise’)+1(y_it=’fall’)

. A similar argument holds for the probability of a “stay the same” response.

1.2.2.1 Adding another measurement variable

As for the model for point forecasts of section 1.2.1, we may want to use additional daily variables to estimate changes in the latent distribution of survey forecasts. I augment the measurement equation (1.4) in the following way:

"

and assume that the second measurement’s error ε_x,t has variance σ_x², and that it is serially independent and independent at all leads and lags from the measurement error vector ε_t and the state disturbance vector η_t. Similar to the model for point forecasts with an additional measurement variable, x_t is related to the day-over-day change in the difference of the two state variables. Thus, xt must be stationary.