When a Replication Works Almost Too Well

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A chart from the Federal Reserve Bank of New York stopped me. Over more than fifty years, the estimated natural rate of interest—R*—and the labor share of income appeared to travel together with remarkable precision. It was visually persuasive. Perhaps too persuasive.

The original Liberty Street Economics article, “The R*–Labor Share Nexus”, by Sophia Cho and John C. Williams, does not ignore this concern. The authors explicitly caution that unrelated time series can share long trends and remind readers that “correlation does not imply causation.” They therefore test for cointegration, add a time trend, compare different sample periods, and allow the model's constant to vary over time.

The authors' skepticism matters. Their paper does not argue that a striking chart alone proves a causal connection. It argues that theory, cointegration evidence, parameter stability, and forecasting performance make the relationship worthy of serious attention.

Replication with public data

We used the BLS nonfarm-business labor-share index from FRED (PRS85006173) and the New York Fed's latest-vintage HLW estimate of U.S. R*. The labor-share index was converted to natural logarithms, and the common quarterly sample was limited to 1970:Q1–2025:Q4.

Parallel Movement of the Labor Share and R*

U.S. labor-share index (2017=100) and HLW R*; 1970:Q1–2025:Q4

RainbowStats replication using FRED/BLS PRS85006173 and the New York Fed HLW U.S. R* series. Annual first-quarter observations are displayed for readability, with 2025:Q4 added as the endpoint; the estimation uses all quarterly observations. Unlike the New York Fed's percentage presentation, the public BLS series is shown as an index with 2017=100.

Chart=DUAL_LINE_CHART(LIST(LaborShare,RStar))
Chart=SET_TITLE_SUBTITLE(Chart,"Parallel Movement of the Labor Share and R*","U.S. labor-share index (2017=100) and HLW R*; 1970:Q1–2025:Q4")
Chart=PUBLICATION_CHART(Chart)

The New York Fed's adjustment model can be written as:

S_t = (1−ρ)S_t−1 + ραR*_t + ρθ + ε_t

A two-variable VAR(1) reproduced the published parameters surprisingly closely. Adding quarterly-average total capacity utilization (TCU) then tested whether ordinary cyclical conditions explained the apparent nexus.

Implied parameter	New York Fed	Two-variable VAR	VAR with TCU
Adjustment rate, ρ	0.222	0.217	0.213
Long-run R* coefficient, α	0.044	0.042	0.040
Spectral radius	—	0.995	0.991

The relationship survives the cyclical control. With TCU in the system, the implied long-run coefficient on R* is 0.040—still close to the article's 0.044. Capacity utilization reduces the estimate modestly; it does not make the nexus disappear.

From a VAR to a DSGE representation

RainbowStats turns the estimated VAR(1) into an editable linear DSGE representation. This is not a claim that a reduced-form VAR suddenly becomes a fully identified economic theory. It is a transparent bridge from estimated dynamics to a structural-equation workspace.

1. EstimateFit a VAR(1) to R*, quarterly TCU, and log labor share.

2. ConvertEach intercept and lag coefficient becomes an editable DSGE parameter.

3. IdentifyFactor the VAR residual covariance matrix using a Cholesky decomposition.

4. SolveCompile the equations and calculate dynamic shock responses.

VM=VAR_MODEL(Data,1)
DSGE=EXTRACT(VM,dsge)
DSGE=CONFIGURE(DSGE,"IMPULSE_HORIZON",20)
DSGE=COMPILE_DSGE(DSGE)
DSGE=SOLVE_DSGE(DSGE)
SLIDESHOW(VM,DSGE)

The extracted model contains three equations and three orthogonal innovations: EPS_RSTAR, EPS_TCU, and EPS_LABORSHARE. Its spectral radius is 0.9914. The model is technically stable, but it is very close to the unit circle—an important warning that persistent trends account for much of the apparent fit.

What happens after a TCU shock?

We ordered the variables RSTAR → TCU → LABORSHARE. Under that Cholesky ordering, a one-standard-deviation TCU innovation raises utilization by about 1.01 percentage points on impact. The utilization response then decays, while R* and log labor share respond gradually.

RainbowStats VAR-derived DSGE impulse responses to EPS_TCU. Each panel uses its own vertical scale so that the smaller R* and labor-share responses remain visible. The Cholesky ordering is RSTAR, TCU, LABORSHARE.

Quarter after shock	TCU response	R* response	Log labor-share response
0	1.0104	0.0000	0.001020
4	0.6596	0.0184	0.001220
8	0.4456	0.0284	0.001443
12	0.3086	0.0340	0.001594
20	0.1599	0.0386	0.001715

The pattern is economically suggestive: a positive utilization surprise fades, while the estimated responses of R* and labor share build slowly. But the ordering matters. It determines which variables may react contemporaneously to each orthogonal shock. Reversing the order is therefore a robustness test, not a cosmetic change.

Why “almost too well” is the point

The replication does not reveal an error. In fact, it reproduces the article's central coefficient with unusual accuracy and shows that the relationship survives the addition of TCU. Nor were the two series mechanically constructed from one another: labor share is not an input to the HLW R* model.

Still, both variables are highly persistent, R* is an estimated latent series, and our transition system sits close to a unit root. Those facts should make us more careful—not less—when moving from statistical fit to causal explanation. Cointegration and robustness tests are evidence in favor of a long-run relationship. They do not identify the precise economic mechanism that created it.

Our conclusion: replication establishes that the R*–labor-share relationship is present in the public data and is robust to a basic cyclical control. It does not establish that movements in R* independently cause movements in labor share. When a replication works almost too well, successful replication should begin the investigation—not end it.

Replication with public data

Parallel Movement of the Labor Share and R*

From a VAR to a DSGE representation

What happens after a TCU shock?

Why “almost too well” is the point

Sources and replication