US Macro LabReal wages · Tax burden · Productivity  |  1947–2025 annual, from FRED

Historical series — w, t and a

w = log average real wage · t = total tax rate (share of GDP) · a = log labour productivity. Hover for values.

Equation 1 — (w − t) = h₀ + h₁·y

Equation 2 — a = a₀ + a₁·y(−1)

Estimation summary (OLS)

Ordinary least squares, homoskedastic standard errors. p-values use a normal approximation.

Interpret with care: OLS on trending macro series is descriptive, not causal; the "mixed" structure you have in mind (simultaneous equations) would call for 2SLS/3SLS or cointegration techniques.

Simulator

Move the sliders to see what the fitted equations predict. The wage prediction uses Equation 1 with your chosen tax rate; productivity uses Equation 2 with y(−1) = your y.

y — real GDP growth (log)
t — total tax rate
Predicted w − t (Eq. 1)
(w−t) = h₀ + h₁·y
Implied real wage level w
w = (w−t) + t, with your t
Predicted productivity a (Eq. 2)
a = a₀ + a₁·y(−1)
Context: historical y

Methodology & data definitions

Y — Real GDP
Quarterly sum of Q1–Q4, billions of constant 2017 dollars (FRED: GDPC1, BEA NIPA).
YN — Nominal GDP
Billions of current dollars (FRED: GDPA). Used only to convert state & local tax receipts into a share of GDP.
E — Employed persons
Total nonfarm payroll employment, seasonally adjusted; annual average of 12 months, thousands of persons (FRED: PAYEMS, BLS).
W — Wages & salaries paid
NIPA total wages and salaries paid, billions of current dollars (FRED: A034RC1A027NBEA).
CPI
Consumer Price Index, All Urban Consumers (CPI-U); annual average of January & December, 1982–84 = 100 (FRED: CPIAUCSL).
w — log average real wage
ln( (W / E) / CPI ). Average nominal wage per worker deflated by the CPI. Levels differ from "real hourly wage" but are consistent across years.
t — total tax rate
Federal receipts as % of GDP (FRED: FYFRGDA188S, decimal) + state & local current tax receipts (FRED: W070RC1Q027SBEA) ÷ nominal GDP. Covers federal, state and local.
a — log labour productivity
ln( Y / E ). Real GDP per worker in constant 2017 dollars.
y — growth term
ln(Yt) − ln(Yt−1), i.e. annual real GDP growth in logs; undefined for 1947 (no prior year).

Requested range was 1946–2025, but US national accounts begin in 1947, so the data start there. Regressions use 1948–2025 (Eq. 1, n = 78) and 1949–2025 (Eq. 2, n = 77).

First-difference (Δ) equations

A "hack" to sidestep cointegration: instead of regressing levels, regress year-to-year changes. This strips out the common long-run time trend so the slope isn't just picking up "time passes → everything grows". The intercepts (h₀₀, a₀₀) have no real meaning — only the slopes matter.

Δ(w − t)t = h₀₀ + h₁·Δyt

Δat = a₀₀ + a₁·Δyt−1

ΔEq. 1 — Δ(w − t) vs Δy

Each dot is one year's change in net wage against that year's change in growth.

ΔEq. 2 — Δa vs Δy(−1)

Each dot is one year's change in productivity against the prior year's change in growth.

First-difference estimation summary (OLS)

Same OLS machinery as the levels equations, run on first-differenced data. We expect both slopes to be positive.

The "scoreboard" — time-varying intercepts h₀(T) and a₀(T)

Using the slopes just estimated, we back out an intercept for every year: h₀(T) = (w−t)T − h₁·yT and a₀(T) = aT − a₁·yT−1. Read it as what each force contributes beyond pure economic momentum.

h₀(T) — labour / "teacher's unions" effect a₀(T) — technology / "AI" effect

A positive bar means the force pushed that year's outcome above what momentum alone would predict; a negative bar means it came in below. Each series is centred on its own mean so the two can share one axis.

Interpretation is Dr. Smithin's rough framing (unions vs. AI). The two series are measured in the units of their own variable (log wage / log productivity) and have been re-centred on their own mean for plotting only — the centring does not change any estimate.

How to read this tab

Why difference?
Over 78 years every macro variable drifts upward, so a levels regression can be fooled by "time itself". First differences (Δx = xt − xt−1) remove that shared drift and isolate the year-to-year relationship.
h₁ and a₁ (the slopes)
How strongly a change in economic acceleration moves net wages (h₁) and productivity (a₁). Both "should" be positive — faster growth lifts real pay and output per worker. The t-stat tells you if that's statistically reliable.
h₀(T) and a₀(T) (the scoreboard)
Once the slopes are fixed, each year has a leftover — the part of that year's wage / productivity not explained by momentum. Dr. Smithin reads large positive h₀(T) as strong institutional/labour-market forces, and large positive a₀(T) as strong technological progress.
Caveat
This is an OLS "hack", not full cointegration analysis (which would use Engle–Granger / Johansen tests). It's a clean first look; the rigorous version comes later.

Any-equation estimator

Pick any two embedded variables (1947–2025), optionally log them, lag the regressor, and estimate OLS on your chosen sample. Click Estimate.




Sample –

Result


Fitted relationship

Residuals over time

Multi-series chart

Select up to 4 variables to overlay. Series with very different scales are placed on a secondary axis automatically.

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Raw data table

Annual values, 1947–2025 (y is undefined for 1947). Export as CSV with the button below.