Time-varying TFP
Stationarity and Structural Change
TFP (총요소생산성)
1 Introduction
The growth rate of Total Factor Productivity (TFP) is not a fixed constant; rather, it evolves over time and may be subject to long-term decline. While mainstream macroeconomic models frequently assume a constant TFP growth rate, this assumption is increasingly being questioned. Several structural factors—diminishing returns to R&D, demographic constraints, and sectoral shifts towards lower-productivity industries—suggest that aggregate TFP growth may continue to slow. However, TFP growth is not uniform across sectors. Certain industries, such as AI, biotechnology, and advanced energy technologies, exhibit significantly higher productivity gains, leading to growing sectoral disparities in innovation.
From an econometric perspective, a decline in TFP growth could alter its stationarity properties, weakening its stochastic trend and making it more likely to transition from an \(I(1)\) (non-stationary) stochastic process to an \(I(0)\) (stationary) stochastic process. If structural breaks exist in TFP growth, traditional time-series models based on constant-trend cointegration assumptions may fail. Thus, accurately modeling TFP trends requires dynamic econometric techniques that account for time-varying structural changes.
2 Main
2.1 Understanding TFP in Macroeconomic Models
TFP represents technological progress that enhances economic output beyond capital and labor inputs. In the Solow-Swan growth model, TFP follows an exponential trend, expressed as:
\[ A_t = A_0 e^{gt} \epsilon_t \]
where \(g\) is the growth rate and \(\epsilon_t\) captures short-term fluctuations. This implies that TFP is not strictly stationary; rather, its log form follows a unit root process with a deterministic trend:
\[ \log A_t = g t + u_t, \quad u_t \sim I(0) \]
Thus, mainstream macroeconomic models do not assume TFP is stationary in levels, but often assume its deviation from trend is stationary.
2.2 Empirical Evidence on TFP Stationarity
Empirical studies show:
- Long-Run TFP behaves as a non-stationary process (\(I(1)\)), meaning that it exhibits persistent growth and shocks that do not revert over time.
- Short-Run TFP Shocks (deviations from trend) are often stationary (\(I(0)\)) process.
If GDP and capital stock are also \(I(1)\), Solow’s production function suggests they should be cointegrated:
\[ \log Y_t - \alpha \log K_t - (1-\alpha) \log L_t - g t = u_t, \quad u_t \sim I(0) \]
Thus, GDP, capital, and labor may be cointegrated due to the common trend driven by TFP.
2.3 Is TFP Growth Declining in the Long Run?
Many studies suggest that TFP growth rates have declined since the mid-20th century.
U.S. TFP Growth Trends (Annual Growth Rates):
- 1950s–1970s: ~2.5% per year
- 1980s–1990s: ~1.5% per year
- 2000s–2020s: ~0.5%-1.0% per year
The productivity slowdown hypothesis suggests that long-term economic growth potential is becoming weaker than before.
2.4 Why Might TFP Growth Decline Over Time?
Several key factors contribute to the observed decline in TFP growth:
- The Low-Hanging Fruit Hypothesis (Gordon, 2016)
- Past technological revolutions (electricity, automobiles, antibiotics) were one-time events that provided massive boosts.
- Recent innovations (social media, fintech, AI) may not have the same economy-wide productivity effects.
- Past technological revolutions (electricity, automobiles, antibiotics) were one-time events that provided massive boosts.
- Declining R&D Efficiency (Bloom et al., 2020)
- “Ideas are getting harder to find.”
- R&D investment has increased, but the number of researchers needed to sustain TFP growth has risen exponentially.
- “Ideas are getting harder to find.”
- Demographic Constraints and Human Capital Decay
- Aging populations in advanced economies reduce labor force dynamism and innovation.
- Education quality improvements may have plateaued, reducing skilled labor supply.
- Aging populations in advanced economies reduce labor force dynamism and innovation.
- Sectoral Shift to Low-Productivity Industries
- Advanced economies have shifted from manufacturing to services (e.g., healthcare, education, public sector).
- Services typically have lower TFP growth, as they rely heavily on human labor and are difficult to automate.
- Advanced economies have shifted from manufacturing to services (e.g., healthcare, education, public sector).
- Increased Regulation and Market Distortions
- Growing regulations, tax policies, and political uncertainty increase inefficiencies, reducing overall productivity.
2.5 Could TFP Growth Accelerate Again?
While aggregate TFP growth may appear to be declining, there are huge disparities across sectors. Just as wealth concentration is increasing, TFP growth could also be highly concentrated within specific industries:
- AI and Automation
- AI-driven automation could drastically improve productivity, particularly in certain white-collar jobs.
- However, productivity gains may be highly concentrated in tech-driven industries, rather than the economy as a whole.
- AI-driven automation could drastically improve productivity, particularly in certain white-collar jobs.
- Sectoral Disparities in TFP Growth
- Within-market concentration: A small number of firms (e.g., winners in the winner-takes-all structure) are experiencing high TFP growth, while others stagnate.
- Cross-market concentration: Some sectors (AI, biotech) may experience massive productivity boosts, while others (traditional services, public sector) may stagnate.
- Within-market concentration: A small number of firms (e.g., winners in the winner-takes-all structure) are experiencing high TFP growth, while others stagnate.
- Energy Revolution and New Technological Frontiers
- Advances in nuclear fusion, renewables, and space-based energy production could dramatically shift TFP growth.
- However, large-scale adoption is slow due to political, regulatory, and financial constraints.
- Advances in nuclear fusion, renewables, and space-based energy production could dramatically shift TFP growth.
3 Conclusion
3.1 Implications for Time Series Analysis
If the long-run TFP growth rate is declining, its stochastic trend may weaken, making it more likely to transition from a unit root (\(I(1)\)) process to a stationary (\(I(0)\)) process over time. This has important implications for empirical modeling and forecasting, particularly in cases where researchers assume a constant growth trend in TFP-driven models. To properly account for these shifts, researchers should:
- Use structural break tests (e.g., Bai-Perron) to detect shifts in TFP growth trends and assess whether different growth regimes exist.
- Apply rolling unit root tests to examine whether TFP has moved from an \(I(1)\) process toward an \(I(0)\) process over time.
- Incorporate time-varying cointegration approaches to avoid biased estimations that assume stable relationships in economic growth models.
3.2 Remarks
TFP growth is not uniform across industries; while aggregate TFP growth may be declining, specific sectors are experiencing rapid technological advancements. Just as wealth and market concentration have increased, TFP gains are increasingly concentrated within high-tech and innovation-driven industries. The result is a growing disparity between leading-edge industries (AI, biotech, energy) and traditional sectors (public services, healthcare, education), which exhibit slower productivity improvements.
As sectoral divergence in productivity widens, traditional macroeconomic models that assume homogeneous TFP growth may require revision. Future research should focus on developing heterogeneous growth models that account for cross-sectoral differences, while also employing more dynamic econometric techniques to capture the evolving nature of TFP trends.
4 Appendix
Covariance Stationarity, Integration Order, and Cointegration
Covariance stationarity and cointegration describe different statistical properties of time series data. Covariance stationarity applies to series where mean, variance, and autocovariance do not change over time, while cointegration describes cases where non-stationary \(I(1)\) time-series maintain a stable long-run equilibrium. This study explores how these concepts relate and how cointegration allows for non-stationary series to form stationary linear combinations.
4.1 Integration Order
The integration order of a time series determines how many times it must be differenced to become stationary. A series is:
- \(I(0)\) (stationary) process if it has a constant mean, variance, and autocovariance.
- \(I(1)\) (unit root) process if it is non-stationary but becomes stationary after first differencing.
- \(I(d)\) process if it requires \(d\) differences to become stationary.
Economic time series such as GDP, money supply, and asset prices often exhibit \(I(1)\) behavior, meaning they contain stochastic trends and require differencing to achieve stationarity.
4.2 Joint Covariance Stationarity vs. Cointegration
Joint covariance stationarity applies when each time-series maintains a constant mean, variance, and autocovariance over time. Cointegration, on the other hand, describes cases where two or more non-stationary \(I(1)\) time-series share a long-term equilibrium, forming a stationary linear combination.
4.2.1 Joint Covariance Stationary Series (Weak-Sense Stationarity)
A set of time series \(X_t\) and \(Y_t\) are jointly covariance stationary if they satisfy:
- Constant Mean: \(E[X_t] = \mu_X\), \(E[Y_t] = \mu_Y\) for all \(t\).
- Constant Variance: \(Var(X_t)\) and \(Var(Y_t)\) do not change over time.
- Autocovariance Depends Only on Lag: \(Cov(X_t, X_{t-h})\) and \(Cov(Y_t, Y_{t-h})\) depend only on the lag \(h\), not on \(t\).
If two time series are both weakly stationary, then any linear combination of them is also stationary.
4.2.2 Cointegrated Series
A set of time series \(X_t\) and \(Y_t\) are cointegrated if:
Each series is \(I(1)\) (non-stationary) process.
A linear combination exists that is \(I(0)\) (stationary) process:
\[ \beta_1 X_t + \beta_2 Y_t = u_t \]
where \(u_t\) is \(I(0)\) (stationary) process.
Thus, even though individual variables are non-stationary, their linear combination is stationary, implying a long-run equilibrium relationship.
4.2.3 The Relationship Between Covariance Stationarity and Cointegration
| Property | Joint Covariance Stationary Series | Cointegrated Series |
|---|---|---|
| Stationarity | Each series is stationary (I(0)) | Each series is non-stationary (I(1)), but a linear combination is stationary |
| Unit Root \(I(d)\) | I(0) for each series | I(1) for each series, but a specific linear combination is I(0) |
| Mean & Variance Stability | Mean & variance are constant over time | Individual series do not have stable mean & variance, but the combination does |
| Long-run Relationship | No long-term relationship constraint | A long-run equilibrium relationship exists |
4.2.4 Cointegrated Series Can Be Transformed into Covariance Stationary Series
If \(X_t\) and \(Y_t\) are cointegrated, their first differences \(\Delta X_t\), \(\Delta Y_t\) (or the residual \(u_t\)) are stationary.
- The error correction term \(u_t\) is stationary \(I(0)\) process, meaning it satisfies the covariance stationarity conditions.
4.2.5 Joint Covariance Stationary Series Are Not Cointegrated
If \(X_t\) and \(Y_t\) are both already \(I(0)\) (stationary), then any linear combination of them is also stationary.
- They cannot be cointegrated because cointegration only applies to non-stationary (\(I(1)\)) series.
- If all series are already covariance stationary, testing for cointegration is unnecessary.
4.3 Implications for Empirical Analysis
- Before testing for cointegration, check for stationarity. If all series are \(I(0)\), cointegration does not apply.
- If series are cointegrated, their residuals (error correction term) should be covariance stationary.
- Many macroeconomic variables (e.g., GDP & consumption, money supply & inflation) are cointegrated rather than purely covariance stationary.