Professional survey forecasts and expectations in DSGE models
What this paper finds — and why it matters
This paper asks whether Survey of Professional Forecasters (SPF) data can be efficiently integrated into medium-scale DSGE models, and whether models with imperfectly rational expectations based on Adaptive Learning (AL) outperform the standard Rational Expectations (RE) hypothesis when survey forecasts are used as observables. The authors work with quarterly US data spanning 1981q2–2019q2, using the Philadelphia Fed Real-Time Data Set (first and second releases) alongside SPF nowcasts for inflation, consumption, investment, and output growth. The SPF nowcast is defined as a prediction formed in the middle of period t+1 for period t+1 given information for period t, making it a suitable proxy for the model-based expectation E_t y_{t+1}.
The core methodological contribution is a re-specification of structural shocks into persistent (AR) and transitory (i.i.d.) components. For the risk premium, investment-specific technology, government spending, and markup shocks, each shock is decomposed into two independent innovations, yielding 12 total structural innovations. A reduced-form VAR exercise motivates this: SPF nowcast innovations explain 19–33% of the 5-year forecast error variance of the macro variables and 44–71% of the variance of the nowcasts themselves. The 1-quarter RMSFE of the baseline RE model without SPF is 1.10 for inflation, 1.26 for consumption, 1.19 for investment, and 1.26 for GDP — all significantly exceeding the SPF RMSFEs of 0.21, 0.43, 1.49, and 0.35.
Log marginal likelihood improves monotonically as shocks are progressively re-specified: baseline RE (–577.37), RE with two-component markups (RE_mu, –536.63), adding real shocks stepwise (–473.29, –410.84), and finally all shocks (RE_all, –385.07). RE_all matches or beats SPF 1-quarter forecast accuracy (RMSFE ratio to SPF of 1.00 for inflation and investment; beats SPF for consumption growth), and Diebold-Mariano tests show no significant difference from SPF up to 5 quarters ahead. The paper further shows that once this two-component structure is imposed, exogenous sentiment shocks become unnecessary: RE_all (–385.07) outperforms RES_all (–388.17), and the RE model with all real shocks re-specified but without sentiment decisively dominates.
Three AL belief specifications are then estimated: MSVflex (full RE information set with an independently and rapidly updating constant, posterior autocorrelation 0.9937 — nearly a random walk), RBflex (restricted information set augmented with shock innovations, with meaningful time-variation of belief coefficients at rho_AL = 0.87), and HBflex (agents switch between MSV and RB based on past forecasting performance; average RB weight 0.34, weight sensitivity delta = 4.77). All AL models outperform RE_all: MSVflex (–381.38), HBflex (–355.09), RBflex (–351.59), with RB and HB yielding the largest gains particularly during and after the Great Financial Crisis.
AL models address three specific RE limitations. First, trend breaks: the ALM constant tracks persistent deviations, with ALM constants for consumption and investment successfully picking up rising macroeconomic trends in earlier sub-periods, yielding superior long-term forecasts. Second, time-varying transmission: the RB model generates cyclical volatility that stays lower in normal times and rises during distress, reducing reliance on large persistent investment-technology shocks relative to RE. Third, predictability of forecast errors: the RE model’s investment forecast inherits the SPF underreaction (b-coefficient 0.72, p < 0.001), while RBflex and HBflex reduce this to 0.17 and 0.34 respectively, both statistically insignificant.
On an extended sample including the Covid recession, the RBflex model underperforms because its restricted information set cannot handle abrupt complex dynamics; MSVflex and HBflex continue to perform well, with the MSV regime dominating in the HB model during Covid and post-Covid periods. Scope conditions: the dataset is US, 1981q2–2019q2 for baseline estimation; the predictability (underreaction) problem is confirmed only for investment SPF, not for inflation, consumption, or GDP growth in this sample.
Q1: What is the SPF nowcast, and why do the authors treat it as a proxy for model-based expectations? The SPF nowcast is defined as a prediction formed in the middle of quarter t+1 for the value of a variable in quarter t+1, conditional on information available through quarter t. Because agents are assumed to make decisions for period t and form expectations for t+1 based on information through t, this timing aligns precisely with the model-based conditional expectation E_t y_{t+1}. The authors use first-release data (r1) and the SPF nowcast (f0) both published in the course of t+1 as measurement variables, with the Kalman filter recovering implied structural shocks.
Q2: How large is the informational content of SPF nowcasts in reduced-form analysis? A 7-variable Cholesky VAR places each SPF series last, so the survey innovation is orthogonal to standard macro variables by construction. The 5-year forecast error variance decompositions show SPF nowcast shocks explain 19% of inflation variance, 33% of consumption variance, 33% of investment variance, and 29% of GDP variance (Table 1). The nowcasts themselves are explained 44–71% by their own innovations. SPF nowcasts also substantially outperform the baseline RE model: the RE model without SPF produces RMSFE ratios of 1.10 for inflation, 1.26 for consumption, 1.19 for investment, and 1.26 for GDP relative to SPF (all statistically significant by Diebold-Mariano test).
Q3: What is the shock re-specification, and why is it necessary to exploit survey data? The Smets-Wouters (2007) ARMA(1,1) shock structure conflates the transitory and persistent innovation into a single disturbance, making it impossible for the Kalman filter to separately attribute high-frequency and low-frequency movements. The re-specification splits each shock b_t into a persistent component b_t^ar (driven by epsilon^bar with persistence rho_b) and an i.i.d. transitory component b_t^iid (driven by epsilon^biid), yielding 12 total structural innovations. This allows survey nowcasts — which are forward-looking — to identify the persistent component separately from the transitory one. Without this, marginal likelihood improvements are far smaller (RE: –577 vs. RE_all: –385).
Q4: Does re-specification of real shocks render exogenous sentiment shocks redundant? Yes. Models with standard real shock processes but exogenous sentiment shocks (RES: –477.88; RES_mu: –488.96) do fit substantially better than models without sentiment (RE: –577.37; RE_mu: –536.63), confirming Milani’s (2017) result. However, once the two-component real shock structure is introduced, RE_all (–385.07) outperforms RES_all (–388.17) and the estimated sentiment shocks become small and explain little of the business cycle. The fundamental shock re-specification subsumes what sentiment shocks were previously capturing.
Q5: How do AL models compare to RE in terms of model fit? All three AL models outperform RE_all: MSVflex (–381.38, improvement of 3.69 log-likelihood units), HBflex (–355.09, improvement of 29.98 units), RBflex (–351.59, improvement of 33.48 units). The RB and HB specifications, which assume more severe deviation from RE with restricted information sets and time-varying transmission, achieve the largest gains. The MSV improvement accumulates gradually, concentrating in the late 1990s and 2000s, while RB shows sustained improvement in the 1980s and mid-1990s and performs exceptionally well during and after the GFC.
Q6: How does the AL mechanism handle macroeconomic trend shifts? Under RE with fixed coefficients, expectations anchor around a constant steady state, so persistent deviations from trend generate systematic forecast errors. Under AL, the ALM constant mu_t in the Actual Law of Motion evolves over the business cycle. In the MSVflex model, the autocorrelation parameter for the constant is estimated at 0.9937 (posterior mean), making it nearly a random walk that can track long-lasting trends. ALM constants for consumption and investment in the MSV setup successfully pick up rising macroeconomic trends in earlier sub-periods, translating into superior longer-term forecast performance relative to RE.
Q7: How does the RB model generate time-varying volatility, and why does this matter for investment dynamics? In RBflex, as beliefs are revised via the Kalman filter, the sensitivity of expectations and realized variables to shocks changes over the business cycle. The model generates cyclical volatility that remains lower in normal times and rises during distress — a realistic pattern absent from RE models. Consequently, RB does not need to rely as heavily on large persistent risk premium and investment-specific technology shocks: average volatility of these processes in the RB model does not increase in the last sub-period and remains generally lower across the whole sample, in contrast to RE’s behavior during the GFC. The RB model also shows a 3-times-smaller estimated measurement error in the investment SPF equation relative to the AL specification without restricted beliefs.
Q8: What happens to predictability of model-based forecast errors under AL versus RE? Using the Coibion-Gorodnichenko (2015) regression of forecast errors on forecast revisions, the RE model’s investment forecast shows a b-coefficient of 0.72 (p < 0.001), inheriting the underreaction documented in SPF investment data (b = 0.49, p = 0.006). AL models break this inheritance: RBflex ALM b-coefficient for investment is 0.17 (not statistically significant) and HBflex is 0.34 (not statistically significant). AL models achieve this because they relax the RE constraint of internal consistency between agents’ and model forecasts, allowing the ALM to generate efficient forecasts even when agent PLMs display sluggish adjustment.
Q9: How do the models perform during the Covid recession? The RBflex model does not perform optimally on the extended sample including the Covid recession. The authors attribute this to the restricted information set in the RB PLM being insufficient to describe the abrupt, complex macroeconomic dynamics of the Covid crisis. The MSVflex and HBflex models continue to perform well. In the HBflex model, the MSV regime naturally dominates during the Covid and post-Covid periods, while the RB regime had been more prominent between recessions in the pre-Covid sample.
Q10: What is the role of heterogeneous beliefs, and how do agents switch between PLMs? In HBflex, expectations are a weighted average of MSV and RB predictions with weights evolving as a function of past belief forecast errors. The weight sensitivity parameter is estimated at delta = 4.77, indicating weights are relatively sensitive to fitness. The average estimated weight on the RB PLM is 0.34 (MSV receives 0.66 on average). The RB weight tends to increase and reach its highest values between recessions, consistent with the restricted model being more parsimonious and useful in stable periods, while the fuller MSV model dominates in high-volatility episodes such as the Covid recession.
Q11: What are the out-of-sample forecasting results? The out-of-sample evaluation covers 2008q1–2019q2. The RB model outperforms the RE model in predicting investment and interest rate dynamics, and for investment it also outperforms professional forecasters during this period. At longer horizons (up to 5 quarters ahead), RE model forecasts are generally not statistically significantly different from SPF predictions once SPF nowcasts are included as observables, suggesting that observing the SPF data is sufficient to capture the most informative content from surveys for longer-horizon predictions.
Q12: What is the relationship to Milani (2017) and the prior literature on sentiment shocks? Milani (2017) found that exogenous sentiment shocks orthogonal to fundamentals were needed to fit SPF forecasts alongside an AL model and explained a significant portion of US business cycle fluctuations. The current paper shows this result is not robust to re-specifying fundamental shocks into persistent and transitory components: once the two-component structure is introduced, sentiment shocks become small and economically unimportant (RES_all at –388.17 versus RE_all at –385.07). What Milani attributed to sentiment was largely capturing the inability of single-innovation shocks to separately account for high-frequency and low-frequency variance.
SPF Nowcast as proxy for model expectations: The Survey of Professional Forecasters’ nowcast is defined as a prediction formed in the middle of quarter t+1 for the value of a variable in that same quarter, conditional on information available through quarter t. This timing makes it directly comparable to the model-based conditional expectation E_t y_{t+1}, so the SPF nowcast can be added to the DSGE model’s observable set with a straightforward measurement equation linking it to model expectations plus i.i.d. measurement error.
Shock re-specification into persistent and transitory components: Each structural shock (risk premium, investment-specific technology, government spending, and markup shocks) is decomposed into an AR(1) persistent component driven by epsilon^bar and an i.i.d. transitory component driven by epsilon^biid, replacing the ARMA(1,1) specification in Smets-Wouters (2007) that conflates both into a single innovation. This decomposition is the key technical device enabling survey data to separately identify low-frequency and high-frequency sources of volatility.
Adaptive Learning (AL): An expectation-formation mechanism in which agents do not know true model parameters and instead estimate linear forecasting models (PLMs) that are updated each period via a Kalman filter algorithm. This produces a time-varying Actual Law of Motion — transmission parameters mu_t, T_t, R_t all evolve with beliefs — enabling endogenous trend drift and time-varying shock responses absent from RE models with fixed coefficients.
Minimum State Variable (MSV) beliefs with flexible constant: An AL specification in which agents use the same endogenous state variables and shocks as in the RE solution but with the constant term updated at an independent, more rapid rate. The constant’s autocorrelation is estimated at 0.9937, making it nearly a random walk capable of tracking persistent macroeconomic trend deviations from the deterministic steady state.
Restricted Beliefs (RB): An AL specification in which each agent’s PLM uses a reduced information set — autoregressive terms of the forward-looking variable augmented with selected shock innovations — rather than the full RE state space. This more severe departure from RE yields the largest marginal-likelihood gain over RE_all, generates realistic cyclical volatility amplification, and produces a 3-times-smaller measurement error for investment SPF, but underperforms during the Covid recession due to the restricted set’s inability to handle abrupt complex dynamics.
Heterogeneous Beliefs (HB): An AL specification in which agents may switch between MSV and RB PLMs as a weighted average, with weights evolving as a function of past belief forecast errors. The average weight on RB is 0.34 and the weight sensitivity delta is estimated at 4.77; the RB weight tends to be highest between recessions and lowest during high-volatility episodes such as the Covid recession when the fuller MSV information set dominates.
FIRE predictability test (Coibion-Gorodnichenko regression): Under Full Information Rational Expectations, the regression of forecast errors on forecast revisions should yield a b-coefficient of zero. A positive and significant b indicates systematic underreaction to news. The paper confirms b = 0.49 (p = 0.006) for investment SPF — but not for inflation, consumption, or GDP — and shows the RE model inherits this inefficiency (b = 0.72, p < 0.001 for investment), while AL models reduce it to insignificance (RBflex: 0.17; HBflex: 0.34).