A macro factor with no premium on impact and 0.22 with a t of 2.35 after two years of summed shocks demands skepticism about the smoothing. Franceschini argues that slow information diffusion explains the delay. The premia are insignificant for contemporaneous shocks, then become significant in rolling sums over multi-year horizons at roughly 2% annually. Citing Eberhart, Maxwell and Siddique, he interprets the pattern as investor underreaction. The abstract and conclusion both carry the 2% headline. Both leave out the unpriced contemporaneous shock.
Returns enter late in the construction. First come an estimated cointegration vector, a forecast coefficient and a Cholesky-identified VAR.
Building the signal
The long-run risk mechanism is familiar. Under Epstein-Zin preferences, investors dislike revisions to expected consumption growth far into the future. A shock that changes consumption growth for a decade can therefore command a premium far beyond what its variance would suggest. Kung and Schmid placed R&D dynamics at the start of that chain: R&D advances the technological frontier, productivity growth inherits the persistence, and consumption follows. Franceschini builds the state variable implied by the theory, then tests whether markets price it.
He calls the state variable 'effective R&D'. It scales R&D expenditure by the stock of ideas available to build on and by the range of products across which new ideas are diluted. In logs, the measure becomes a linear combination. Since productivity growth is stationary while R&D, ideas and variety trend, that combination must be a cointegration residual.
Franceschini estimates the residual through Fully Modified OLS, using the error-correction term from a regression of log private R&D on log TFP and log employment. TFP stands in for the stock of ideas; employment represents product variety. Across 309 quarterly observations, the baseline coefficient on TFP is 3.526 with a standard error of 0.439. Employment receives a coefficient of 0.909 with a standard error of 0.336.
This residual is gross effective R&D. Substituting TFP for ideas pulls the non-R&D part of productivity into the measure, which is why Franceschini labels it gross. He recovers net effective R&D recursively, using the R&D coefficient from a one-quarter forecast of TFP growth to remove that component. In the baseline with the Ludvigson-Ng control factors, the coefficient is 1.549% with a HAC standard error of 0.285 and an R-squared of 12.4%. One standard deviation of gross effective R&D maps to about 0.8% annualized extra TFP growth.
Every stage forces an implementation choice: the R&D, TFP and employment series, gross versus net, and the controls used in the recursion. Franceschini reports the alternatives. He writes that no correlation among the error correction terms falls below 0.86, although Table 10 reports 0.721 between the total-R&D and raw-TFP residuals. Integrated Modified OLS is the clear outlier, with correlations of 0.377 to 0.511 against the other estimates. Franceschini flags its instability and reports a condition number of 1.1 times ten to the eighth.
Changing private R&D to total R&D also reverses the employment coefficient, from +0.909 to -0.354, and removes its significance. Franceschini suggests that public R&D may operate through a different production function. Product variety is one of the two features separating his measure from Kung and Schmid's, and it fails this substitution. The other feature, a flexible spillover term, survives. Franceschini attributes most of the improvement over Kung and Schmid to that flexibility, arguing that their measure behaves as it does because the strength of knowledge spillovers from past innovations is fixed.
Short memory, long response
Gross effective R&D behaves like the persistent state required by standard long-run risk. Its AR(1) is 0.96, with a half-life bounded between 2.6 and 21.0 years. Net effective R&D looks very different. Its AR(1) is 0.70, the 95% bounds on its half-life run from 0.3 to 0.8 years, its ADF statistic is -3.95, and its KPSS statistic is 0.09. The series is comfortably stationary and carries little persistence.
Yet its shocks continue moving macro quantities for a decade.
The baseline two-variable VAR uses 305 observations and three lags. The TFP growth equation has an R-squared of 5.3%, while the largest companion root is 0.92. The 95% band around the response to an effective R&D shock includes zero only when the horizon nears ten years. Local projections extend the result. For gross shocks, cumulative productivity and cumulative consumption growth both respond significantly at the ten-year horizon, and productivity has an even larger effect at twenty. Consumption responds at fifteen to twenty years as well, though only under the LN control set.
Franceschini openly separates this mechanism from Kung and Schmid's. Persistence within effective R&D itself does not carry the propagation in his account. Feedback with the broader macro system does.
The choice between gross and net is the vulnerable point. In the VAR, gross effective R&D produces a Granger-causality F of 5.3 with p below 0.01. The net measure produces 1.7 with a p-value of 18%. Franceschini reports the gap, attributes it to the shorter sample and added measurement error, and says the 18% p-value still offers some evidence of predictive content.
The strongest persistence evidence therefore comes from gross effective R&D. This is the error correction term from a regression where TFP substitutes for the stock of ideas. It carries the non-R&D productivity component by construction, as Franceschini acknowledges when defining the measure.
He also reports diagnostics that work against the preferred interpretation. Heteroskedasticity LM tests on the effective R&D residuals are significant at 1% in four of five specifications. The setup warns that an unobserved external component could create residual serial correlation and invalidate identification. Autocorrelation LM tests at 1, 8, 16 and 40 lags are insignificant. The baseline statistic is 169.4 at 40 lags, the most relevant check for the Cholesky identification.
Net effective R&D brings another cost. Its recursive initial condition is unobservable, so Franceschini trims the sample until the decay weight falls below 0.01. Depending on the specification, only 151 to 226 observations remain.
Pricing arrives late
The pricing exercise covers 183 portfolios over 213 quarters. Franceschini projects the R&D shock onto 6, 14 and 22 principal components, then estimates premia with Giglio and Xiu's estimator to control for omitted factors. The asset pool combines 153 anomaly stock portfolios from Jensen, Kelly and Pedersen, 17 French industry portfolios, and 13 zero-coupon bond portfolios derived from Nelson-Siegel-Svensson fits to Gürkaynak, Sack and Wright yields. The sample runs from 1971 Q4 to 2023 Q4. Projection onto principal components of this cross-section provides the protection against omitted-variable bias needed for the factor estimates.
Contemporaneous shocks earn no premium. Across every specification and factor count, the t-statistics range from -1.30 to 1.35. With shocks summed over two years, the baseline using 14 factors produces 0.22 and a t of 2.35. At four years, the estimates are 0.48 with a t of 3.28 using 14 factors and 0.54 with a t of 2.80 using 22. These figures amount to roughly 2% annually.
Franceschini interprets the horizon pattern as underreaction to R&D news. The reading is plausible. It also leaves a factor whose pricing result depends on multi-year smoothing. Significance appears at two of three horizons and two of three factor counts, and the 45 reported premia receive no multiple-testing adjustment.
The principal-component diagnostics have wider relevance. Factors 7 through 14 account for 15% of the time-series variation in returns and 24% of the cross-sectional variation. The two criteria from Alessi, Barigozzi and Capasso select 6 and 14. Bai and Ng's three criteria return 39, 22 and one diverging estimate; the paper retains only the 22. For this cross-section, pricing appears to require more factors than the lower variance-based estimates imply.
Cash flows provide supporting evidence for the proposed mechanism. Using the Bansal, Dittmar and Lundblad design, dividend betas yield premia of 2.28 (t=2.31) in the extended pool and 2.00 (t=2.51) in the wide pool at one quarter. At two years, the corresponding estimates are 0.71 (t=2.14) and 0.60 (t=1.92). The legacy pool produces 1.16 with a t of 1.34.
Effective R&D in levels is never priced at the 5% level in the cash-flow test. Its largest estimate is 0.06 with a t of 1.68 in the extended pool. The pricing result therefore belongs specifically to the orthogonalized-shock construction, rather than the R&D measure in isolation. Franceschini qualifies his own interpretation: three of the six premia estimated for effective R&D in levels have t-statistics above 1.2, which he says indicates that some risk remains captured.
In the legacy pool, the consumption premium is insignificant at both horizons, with 0.68, t=1.60, and 0.26, t=1.42. Its cross-sectional R-squared nevertheless rises from 30.99% to 54.21%. The wider pools go the other way. Their R-squared falls from 24.85% to 3.06% and from 18.05% to 2.17%. The extended pool uses 2-by-3 characteristic sorts, and Franceschini flags the leverage sort as difficult to interpret and potentially a proxy for omitted characteristics.
The Kung-Schmid warning
Section 4.3 is worth the download by itself, although its figures sit in an appendix table. Anyone using aggregate R&D as a predictor should look closely at the updated Kung and Schmid R&D intensity series. Over 62 annual observations from 1963 to 2020, its AR(1) is 0.995 with a standard error of 0.063. The implied half-life has a lower bound of 40.2 years and an upper bound of infinity.
The unit-root tests disagree. An ADF statistic of -2.82 rejects a unit root at 10%, while a KPSS statistic of 1.29 rejects stationarity at 1%. Franceschini writes that "an implied half-life exceeding 40 years raises questions about its economic interpretation," and the plot favors the KPSS result. If the series is non-stationary while the tests conflict, predictive regressions using it face a greater danger of spurious results. Franceschini designed his own measure partly to solve this problem, and it succeeds on that count.
Outside a tradable test
We could not reproduce any of the paper's results. Constructing the factor requires an aggregate US private R&D expenditure series, Fernald's utilization-adjusted TFP excluding R&D capital, aggregate employment and per-capita consumption. None of those series are available through the macro data we can access. The 13 bond test portfolios are fitted yield curves rather than traded instruments. ETF proxies for either the factor or the test assets would change the specification.
The paper estimates a cross-sectional premium on portfolios. It reports no turnover, costs, Sharpe or drawdown. Its cointegration vector and VAR coefficients use the full sample, meaning the shocks supplied to the pricing tests differ from those available to an investor in real time. Franceschini makes no contrary claim. His question is whether innovation risk is priced. We made the reverse point about a variance-premium rotation whose legs cannot be spanned by any traded instrument in the correlation rotation note. Franceschini never presents his estimate as a strategy.
A real-time construction would change my view on the horizon dependence if it estimated the cointegration vector and forecast coefficient solely on a trailing window, then retained the same four-year sum. Survival of the t of 3.28 would make underreaction the convincing interpretation and justify building the required data pipeline. Failure would leave the multi-year smoothing as the source of the result.