Testing Velocity Stability During Deflation

Hello! Welcome back to our course on deflation.

Introduction

In our last lesson, we critically examined the tools used to measure inflation, like the CPI and PCE, understanding their inherent biases. We learned that how we measure the price level ($P$) is a complex issue. Today, we turn to one of the most foundational concepts in macroeconomics that uses this price level: the Quantity Theory of Money (QTM).

This lesson directly addresses the learning outcome: "Empirically test the stability of the velocity of money (V) in the Quantity Theory of Money (MV=PY) using data from a major deflationary period."

For centuries, the QTM has been a cornerstone of monetary thought, suggesting a direct link between the money supply and the price level. However, this relationship hinges on a crucial, often-overlooked variable: the velocity of money ($V$). We will put this assumption to the test, using the Great Depression as our laboratory. Given your background in econometrics and statistics, we will go beyond a simple calculation and explore the sophisticated methods economists use to model velocity's seemingly erratic behavior.

This lesson should take approximately 60 minutes to complete.

Recap from Last Lesson:
We deconstructed price indices, learning that the CPI's fixed-basket (Laspeyres) formula leads to substitution bias, generally overstating inflation compared to the chain-weighted (Fisher) PCE index. You saw this systematic difference firsthand by plotting both series in Python.

1. The Quantity Theory and its Achilles' Heel: Velocity

You'll recall the equation of exchange from your economics studies, which underpins the Quantity Theory of Money:

$MV=PY$

Where:

• $M$ = The total money supply in an economy.
• $V$ = The velocity of money (the average number of times a unit of money is used in a transaction for goods and services in a given period).
• $P$ = The average price level of those goods and services.
• $Y$ = The real output of the economy (real GDP).

The equation itself is an identity—it must be true by definition. It becomes a theory when we make assumptions about its components. The monetarist school of thought, most famously associated with Milton Friedman, posits that for the QTM to be a useful guide for policy:

1. Velocity ($V$) is stable and predictable.
2. Real output ($Y$) tends toward its natural or potential level in the long run.

If these assumptions hold, then changes in the money supply ($M$) must lead to proportional changes in the price level ($P$). This implies a powerful role for central banks in controlling inflation.

However, a key critique, often associated with John Maynard Keynes, is that velocity ($V$) is not stable, especially during economic crises. In times of uncertainty, people and firms may choose to hoard cash rather than spend or invest it. This increase in "liquidity preference" causes velocity to fall, potentially severing the link between an expanding money supply and nominal income ($PY$).

Is velocity stable or not? This is not a philosophical question; it's an empirical one.

2. Calculating Velocity During the Great Depression

To test the stability of velocity, we first need to calculate it. Since the equation of exchange is an identity, we can rearrange it to solve for $V$, which is not directly observed:

$V=\frac{PY}{M}=\frac{\text{Nominal GDP}}{\text{Money Supply}}$

Let's calculate and plot the velocity of M2 (a broad measure of the money supply) for the United States during the Great Depression.

Coding Exercise: Charting the Collapse

The following Python script uses the FRED API to download annual data for Nominal GDP and M2 Money Stock, then calculates and plots M2 velocity. The NBER paper we will be looking at uses annual data, so we will do the same. FRED's M2 series is monthly, so we resample it to an annual average to match the annual GDP data.

import pandas as pd
import matplotlib.pyplot as plt
from fredapi import Fred

# --- Your FRED API Key ---
# Replace with your actual key
fred_key = 'YOUR_API_KEY_HERE' 
fred = Fred(api_key=fred_key)
plt.style.use('seaborn-v0_8-whitegrid')

# --- Download Data ---
start_date = '1929-01-01'
end_date = '1945-01-01'

# Download annual nominal GDP (Billions of Dollars)
gdp = fred.get_series('GDPA', observation_start=start_date)
gdp = gdp.loc[start_date:end_date]

# Download monthly M2 (Billions of Dollars) and resample to annual average
m2 = fred.get_series('M2SL', observation_start=start_date)
m2_annual = m2.resample('A').mean()
m2_annual = m2_annual.loc[start_date:end_date]

# --- Calculate Velocity ---
# Ensure both series are aligned by year
velocity = gdp / m2_annual
velocity.dropna(inplace=True)

# --- Create Plot ---
fig, ax = plt.subplots(figsize=(14, 8))

ax.plot(velocity.index, velocity, marker='o', linestyle='-', color='darkred')

ax.set_title('M2 Velocity During the Great Depression Era (1929-1941)', fontsize=16)
ax.set_xlabel('Year', fontsize=12)
ax.set_ylabel('Velocity (Nominal GDP / M2)', fontsize=12)
ax.set_xticks(velocity.index[::2]) # Show every other year for clarity
ax.grid(True)

plt.show()

# --- Analysis ---
print("M2 Velocity:")
print(velocity.round(2))

Analysis Question:
Run the script and examine the plot. What happened to the velocity of money between 1929 and 1933? Does this initial finding support the monetarist assumption of stable velocity during a major economic crisis?

3. Deconstructing Instability: A Modern Econometric Approach

Your plot reveals that velocity was anything but stable. This is precisely the phenomenon observed by Anderson, Bordo, and Duca in their detailed study of money and velocity during financial crises.

Money and Velocity During Financial Crises (Disruptions of Financial Markets Circa Financial Crises)

Take a look at Figure 3 in the paper (reproduced below). The blue line on the left-hand chart, "M2 Velocity 1928-1938," shows the same collapse you just plotted.

Money and Velocity During Financial Crises
This chart from Anderson, Bordo, and Duca (2016) plots M2 velocity during the Great Depression (left) and the Great Recession (right), normalized to 1 at the start of each period. It empirically confirms the sharp decline in velocity during the early 1930s.

So, if velocity isn't stable, is it just unpredictable noise? Or is it driven by other economic forces in a predictable way? This is the central question the paper tackles. The authors argue that velocity's movements can be explained by a model that incorporates three key factors:

1. Opportunity Cost: The standard trade-off between holding money (which pays little to no interest) and holding other assets like Treasury bills.
2. Financial Innovation: Proxied by the transaction costs of moving money into substitutes like mutual funds. Lower costs make it easier to move money, affecting velocity.
3. Risk Premia: This is the crucial factor for crises. Measured by the spread between risky corporate bonds (Baa-rated) and safe Treasury bonds, this captures the "flight to quality." When risk premia spike, people demand more safe, liquid money, causing velocity to fall.

Given your background, you'll appreciate their methodology. They don't just run a simple OLS regression. They use a Vector Error Correction Model (VECM), which is appropriate for modeling I(1) non-stationary time series that are cointegrated. This framework allows them to model both the long-run equilibrium relationship between velocity and its determinants, and the short-run dynamics of how it adjusts back to that equilibrium after a shock.

Please read the following section of the paper to understand how they construct their model.

Money and Velocity During Financial Crises (The Empirical Model: Specification)
Reading time: ~10 minutes. Focus on the logic progressing from the simple model in equation (1) to the comprehensive model in equation (5), which includes load (transaction costs) and Baa10TR (risk premium). Note the final specification in equations (6a,b), which lays out the VECM structure.

The key takeaway from this section is that a "stable" relationship might exist, not for velocity itself, but for a function that determines velocity.

4. The Verdict: Is Velocity Stable After All?

The authors estimate their VECM and find compelling results. The model that includes the risk premium and financial innovation variables (Model 1 in their analysis) performs significantly better than simpler models.

Please review the section where they discuss their estimates.

Money and Velocity During Financial Crises (The Empirical Model: Estimates)
Reading time: ~5 minutes. You don't need to parse every detail of the regression table, but focus on the authors' summary of the results. They find that the coefficients on opportunity cost, stock fund loads, and the corporate-Treasury yield spread are all statistically significant with the expected signs. Crucially, they note that models omitting these variables (Models 4 and 5) produce insignificant coefficients and imply an implausibly slow speed of adjustment, indicating misspecification.

The conclusion is clear and profound:
The velocity of money is not constant. However, it is also not random. It is a predictable function of rational economic behavior. The collapse in velocity during the Great Depression was a direct and understandable consequence of the massive spike in perceived risk, which led to a widespread flight to the safety of holding money.

As the paper's overall conclusion states:

Money and Velocity During Financial Crises (Conclusion)
Reading time: ~2 minutes. This section summarizes the core finding: a model incorporating opportunity cost, transaction costs, and perceived risk can successfully track M2 velocity over 80 years, including through the two greatest financial crises.

Conclusion

Today, we've moved from a textbook definition of the Quantity Theory of Money to a sophisticated empirical test of its core assumption. You've seen with your own data analysis and a state-of-the-art academic paper that the stability of velocity is a myth, especially during the deflationary turmoil of the Great Depression.

Key Takeaways:

• The simple Quantity Theory of Money ($MV=PY$) relies on the critical assumption that velocity ($V$) is stable and predictable.
• Empirical data from the Great Depression shows this assumption is false. Velocity collapsed from 1929 to 1933, severing the direct link between the money supply and nominal GDP.
• This collapse was not random. Sophisticated econometric models (VECMs) show that velocity is a function of opportunity cost, financial innovation, and, most importantly during crises, risk premia.
• The drop in velocity was a rational "flight to quality" as households and firms hoarded cash in response to extreme uncertainty.

Implication for Policy: This has enormous implications for combating deflation. A central bank cannot simply increase the money supply ($M$) and expect nominal spending ($PY$) to rise. If the policy fails to reduce risk premia and restore confidence, the new money may be entirely hoarded, causing $V$ to fall and neutralizing the stimulus. This is a central challenge we will revisit when we analyze Japan's "lost decades."

Preview of the Next Lesson:

We have now examined the price level ($P$) and the quantity equation ($MV=PY$). Next, we will focus on another critical variable that links deflation to the real economy: the real interest rate. We will use the Fisher Equation to calculate its behavior during a deflationary period, which will lay the groundwork for understanding Irving Fisher's famous "debt-deflation" theory.