Money, Velocity, and Three-Stage Least Squares

Why should anyone care about velocity?

Imagine the Federal Reserve credits banks with an additional trillion dollars of reserves. If households increase savings and businesses postpone investment, much of that money remains idle. The money supply has increased, but consumer demand has not. Economists refer to this situation as "pushing on a string." Monetary policy can make funds available, but it cannot force people to borrow and spend. Inflation arises when money begins to circulate through the economy.

The RainbowStats Script

The full example can be run in RainbowStats here: Run the RainbowStats script.

VELOCITY = PCE / M2SL

set_title(
    Line_Chart(LIST(VELOCITY, M2V)),
    MONTHLY_VELOCITY_PROXY
)

r1 = REGRESSION(LOGDIFF(LIST(CPIAUCSL, PPIACO, VELOCITY)))

r2 = REGRESSION(LOGDIFF(LIST(PCE, CPIAUCSL, PAYEMS)))

r3 = REGRESSION(LOGDIFF(LIST(FEDFUNDS, CPIAUCSL, UNRATE, VELOCITY)))

s = SIMU_REGRESSION(r1, r2, r3)

t = THREE_STAGE_REGRESSION(
        s,
        LOGDIFF(LIST(PPIACO, PAYEMS, UNRATE, VELOCITY))
)

SLIDESHOW(t)

Series Used

The Three Equations

With the data synchronized to monthly observations, the example estimates three equations.

1. Inflation Equation

The first regression models monthly changes in consumer prices as a function of producer price inflation and the monthly consumption-to-money ratio.

CPI inflation = f(PPI inflation, monthly velocity proxy)

2. Consumption Equation

The second regression models monthly changes in consumer spending as a function of consumer prices and employment.

PCE growth = f(CPI inflation, payroll employment growth)

3. Policy Rate Equation

The third regression models monthly changes in the federal funds rate as a function of inflation, unemployment, and the velocity proxy.

Federal funds rate change = f(CPI inflation, unemployment, monthly velocity proxy)

Why Three-Stage Least Squares?

The three regressions are stored in the SIMU_REGRESSION operator. That object calculates the residual correlation and covariance matrix for the system. In other words, it asks whether the separate regressions contain hidden relationships in their residuals.

The THREE_STAGE_REGRESSION operator then takes the simultaneous regression system and an instrument list. It identifies endogenous right-hand-side variables, replaces them with fitted values from the first-stage regressions, and estimates the resulting system using the SUR/FGLS machinery.

What the Results Suggest

The strongest result is in the inflation equation. Producer prices are the dominant short-run driver of consumer prices. That is not surprising: changes in input and wholesale prices often pass through to consumer prices.

The velocity proxy is also positive and statistically significant, though its coefficient is much smaller than the producer price coefficient. This is the interesting monetary result. It suggests that the simple quantity of money is not the whole story. Money has to be spent, not merely created.

This explains why M2 growth alone can be a weak monthly predictor of inflation. During periods of uncertainty, households may save extra cash, businesses may postpone investment, and banks may hold reserves rather than expand credit. In that environment, central banks may be pushing on a string: liquidity is available, but private spending does not immediately follow.

Conclusion

This example does not claim that the PCE/M2 ratio is the official velocity of money. It is a monthly proxy for money utilization in the consumer economy. The proxy is useful because it behaves similarly to the official quarterly M2 velocity series while allowing monthly econometric analysis.

This study suggests that money creation alone is not enough to explain monthly inflation. Producer prices remain the dominant short-run driver of CPI, but a monthly consumption-to-money ratio adds statistically significant explanatory power. The results are consistent with the idea that money must circulate before it affects prices. Future work could compare this monthly proxy with other measures of spending intensity and investigate its performance during different monetary policy regimes.

The evidence points to a balanced conclusion: producer prices remain the primary short-run driver of monthly CPI inflation, but the consumption-to-money ratio adds information. Money creation alone is not enough. Inflation pressure rises when money circulates through spending.

Three-Stage Least Squares is a useful framework for this kind of question because inflation, spending, labor markets, and policy rates are not isolated variables. They form a system.