13.9 Case studies

13.9.1A 100% recession forecast (October 2022)#

Context In 2022 US inflation was high and the Federal Reserve was raising rates quickly. Recession forecasts were everywhere.

Timeline and numbers

• 17 Oct 2022: Bloomberg reported that its economists' model put the probability of a US recession within 12 months at 100%, up from 65% in the previous update (Bloomberg). The model used 13 macroeconomic and financial indicators over horizons from one month to two years. Probabilities for shorter horizons were reported as 73% (11 months) and 25% (10 months) (ThinkAdvisor; Advisor Perspectives).
• In the same period a Bloomberg survey of 42 economists put the 12-month probability at 60% (reported by InsuranceNewsNet).
• Outcome: the NBER business-cycle chronology lists February 2020 as the most recent peak and April 2020 as the most recent trough, with no later peak (NBER, retrieved 2026-11-06). The BEA's advance estimate reported real GDP growth of 2.5% in 2023 (from the 2022 annual level to the 2023 annual level).
• Andrew Gelman later called the 100% figure a calibration error (blog post, 22 Oct 2023).

Analysis with this module's tools On this single question, the 100% forecast scores Brier 1.0 and log score $+\infty$ (clipped at $10^{-15}$, 34.5). The 60% survey scores Brier $0.36$ and log score $-\ln0.4 = 0.92$. One question does not establish skill (13.8), but no amount of later accuracy offsets an infinite log loss, and that is the point of a strictly proper unbounded rule: a model that outputs certainty about a macroeconomic event is mis-specified, whatever its inputs. Mechanically, a logistic or probit model produces 100% (after rounding) when a linear index of indicators moves far into one tail; the inputs (an inverted yield curve, tight financial conditions) were extreme by historical standards, and the model extrapolated beyond the range in which it had been calibrated.

13.9.2Equity-premium prediction as a forecasting contest#

Context By the early 2000s the finance literature had accumulated many variables that predicted US equity returns in sample: dividend–price and earnings–price ratios, book-to-market, term and default spreads, inflation, investment–capital ratio and others.

Timeline and findings

• 2004: Goyal and Welch circulate the analysis as NBER working paper w10483.
• 2008: Welch and Goyal (RFS 21(4)) report that, out of sample, none of the variables would have helped an investor beat the then-prevailing historical mean, and that the models were unstable.
• 2008: Campbell and Thompson (RFS 21(4)), same issue, show that sign restrictions on slopes and non-negative premium forecasts let many predictors beat the mean, with small but economically meaningful $R^2_{OS}$.
• 2024: Goyal, Welch and Zafirov (RFS 37(11)) extend to 29 newer variables; more than a third lose in-sample significance, half of the survivors fail out of sample, a few work in both.

Analysis This is 13.3 in full. In-sample $R^2$ is inflated by Stambaugh bias (Proposition 13.9) and by the selection of variables that happened to work (M04 · 4.6). Out of sample, estimation error in $\hat\beta_t$ can make the forecast worse than the mean even when the true $\beta$ is non-zero, and Campbell–Thompson's restrictions act as shrinkage that reduces that error. Proposition 13.7 explains why the two papers can both be right: an $R^2_{OS}$ of 0.5% monthly is statistically fragile but, if real, raises a mean-variance investor's expected excess return by about 20% (worked example in 13.3).

13.9.3The Good Judgment Project#

Context After intelligence failures in the 2000s, IARPA funded the ACE tournament to test whether aggregated, trained forecasting could improve on intelligence analysis.

Facts GJP's first year began with 2,246 participants forecasting 199 geopolitical outcomes, scored by Brier score (Mellers et al. 2014; APS Observer summary). Probability training, teaming and tracking the best forecasters each improved accuracy, through both calibration and resolution in the sense of (13.36).

Analysis The training GJP used is the content of 13.8: base rates, reference classes, decomposition, frequent small updates, and awareness of overconfidence. Teaming improves resolution by pooling information. Tracking works because Brier scores identify skill only over many questions, and selecting on a long record reduces the chance that "superforecasters" are lucky. Note the selection issue: picking the top 2% after year 1 guarantees some regression to the mean in year 2. Mellers et al. (2014) report that the elite teams were far more accurate than the other groups in year 2, which is evidence of skill rather than luck; how much of the year-1 margin survived is quantified in Tetlock and Gardner (2015).

13.9.4GARCH versus implied volatility in India#

Context India VIX is NSE's volatility index, computed from Nifty option quotes and quoted as an annualised percentage for a 30-day horizon, in a methodology modelled on CBOE's VIX (methodology details not re-verified for this module; read NSE's India VIX documentation). It is a natural benchmark for GARCH.

Evidence (abstract-level) Pandey (2003, IIMA) found conditional models forecast five-day and monthly Nifty realised volatility poorly. Rajvanshi, Santra and Basu (IIMC WPS 797) found that adding India VIX to GARCH-family models improved forecasts of Nifty 50 realised volatility measured from 5-minute returns, against both plain GARCH and implied volatility alone. Other Indian studies (e.g. Banerjee and Kumar, IIMC WPS 688, 2011) compare the two directly.

Analysis Proposition 13.12 says GARCH forecasts revert geometrically to $\bar\sigma^2$, with the same speed whatever is scheduled. Option prices embed known events (budget day, election results, RBI policy) and the variance risk premium (M09). Implied volatility is therefore a biased (too high, on average) but informative forecast; GARCH is unbiased if correctly specified but blind to the calendar. Combining them as regressors in the variance equation, or as a forecast combination, uses both.