Time-Varying Cointegration
US Economic Indicators and Structural Break Analysis
Abstract
This study investigates the time-varying cointegration relationships among key U.S. macroeconomic indicators and identifies structural breaks in their long-run equilibrium. Using monthly data from 1990 to 2024, we apply standard unit root and cointegration tests (e.g., Johansen, Engle-Granger), followed by structural break detection methods including the Bai-Perron test and rolling Johansen estimation. The study extends the analysis to fractional and threshold cointegration frameworks to capture long-memory properties and regime-dependent adjustments. Empirical findings suggest that cointegration relationships among economic variables are not stable over time and tend to shift around major macroeconomic shocks. This result highlights the importance of accounting for structural nonstationarities in economic modeling and long-term forecasting.
Keywords
Long-Run Equilibrium, Cointegration, Time-Varying Parameter, Structural Breaks, Regime Shift,
1 Research Overview
This study investigates whether key U.S. economic indicators exhibit time-varying cointegration and how structural breaks affect their long-run relationships. The analysis aims to:
- Identify which variables are cointegrated in the long term.
- Track how these relationships evolve over time.
- Detect and interpret structural breaks in equilibrium relationships.
- Extend beyond traditional models by using fractional and threshold cointegration frameworks.
2 Preliminaries
- Spurious correlation vs. Cointegration: High correlation between non-stationary variables may be misleading unless the variables are cointegrated.
- Cointegration: A linear combination of I(1) variables that is stationary (I(0)) implies a stable long-term equilibrium.
- Interpretation: Variables sharing a cointegration relationship tend to move together over time, despite individual stochastic trends.
- Empirical Rule: In long-term data (e.g., >10 years), a correlation coefficient > 0.7 between I(1) series may suggest stable equilibrium.
- Nelson & Plosser (1982): U.S. macroeconomic series often follow stochastic trends, cautioning against naive regression without testing for cointegration.
3 Research Questions
- Which economic indicators form long-term relationships?
- How have these relationships changed over time?
- Do identified structural breaks correspond to major economic shocks (e.g., 2008, COVID-19, inflation)?
4 Data and Preprocessing
Period: January 1990 – December 2024 (34 years)
Frequency: Monthly
| Category | Variable | Source | Start Year | Notes |
|---|---|---|---|---|
| Equity | SPY, NDX | Yahoo Finance | 1993, 1985 | Daily/Monthly |
| Currency | DXY | FRED/Yahoo | 1973 | Daily/Monthly |
| Bonds | Fed Funds Rate, 10Y Treasury Yield | FRED | 1954, 1953 | Monthly |
| Money Supply | M2 | FRED | 1959 | Monthly |
| Commodities | Gold Price | Yahoo | 1975 | Daily/Monthly |
| Inflation | CPI | FRED | 1947 | Monthly |
| Consumption | Consumer Sentiment | FRED | 1978 | Monthly |
| Investment | Real GPDIC1 | FRED | 1960 | Originally quarterly; interpolated monthly |
5 Methodology
5.1 Testing for Cointegration
5.1.1 Step 1: Unit Root Testing
- Ensure variables are I(1)
- Methods:
- ADF Test
- Phillips-Perron Test
- ADF-GLS (ERS)
- KPSS
5.1.2 Step 2: Cointegration Existence
- Apply only if variables are I(1)
- Methods:
- Johansen Test (multivariate)
- Engle-Granger Test (pairwise)
- If cointegration fails: consider VAR or short-run models
5.2 Detecting Structural Breaks
5.2.1 Step 1: Break Detection in Traditional Cointegration
- Methods:
- Bai-Perron Test
- Quandt-Andrews Test
- Rolling Johansen Test
- CUSUM & CUSUMSQ Tests
5.2.2 Step 2: Qualitative Mapping to Events
| Breakpoint | Likely Cause |
|---|---|
| 2008-Q3 | Global Financial Crisis |
| 2011-Q3 | European Debt Crisis |
| 2020-Q1 | COVID-19 Shock |
| 2022-Q1 | Inflation & Fed Rate Hikes |
Overlay structural breaks with macroeconomic shocks, policy shifts, and global market events.
5.3 Fractional Cointegration Extension
5.3.1 Step 1: Testing
- Estimate fractional differencing order (\(d\)) via:
- GPH Test
- Robinson Test
5.3.2 Step 2: Detecting Breaks
- Use methods for long-memory models:
- Rolling estimates of \(d\)
- Wavelet-based structural break detection
- Rolling Hurst exponent analysis
5.4 Comparison of Breakpoints (Traditional vs. Fractional)
- Common breakpoints strengthen the validity of structural shifts.
- Traditional: discrete shifts; Fractional: gradual long-memory transitions.
5.5 Threshold Cointegration Models
5.5.1 Step 1: Apply TECM
- Estimate threshold level (\(\gamma\))
- Model: \[ \Delta Y_t = \begin{cases} \alpha_1 (Y_{t-1} - \beta X_{t-1}) + \epsilon_t, & \text{if } |Y_{t-1} - \beta X_{t-1}| > \gamma \\ \alpha_2 (Y_{t-1} - \beta X_{t-1}) + \epsilon_t, & \text{otherwise} \end{cases} \]
5.5.2 Step 2: Interpret Regime-dependent Adjustments
- Use Sup-Wald test for significance
- Evaluate asymmetry in adjustment speeds (\(\alpha_1 \ne \alpha_2\))
5.6 Threshold Fractional Cointegration (TFECM)
5.6.1 Step 1: Estimation
- Combine fractional differencing with threshold effects: \[ \Delta Y_t = \begin{cases} (1 - L)^{d_1} X_t + \epsilon_t, & \text{if } |X_t - \beta Y_t| > \gamma \\ (1 - L)^{d_2} Y_t + \eta_t, & \text{otherwise} \end{cases} \]
5.6.2 Step 2: Interpretation
- Captures memory-driven and threshold-based nonlinearity
- Use Sup LM test for threshold significance
6 Summary Evaluation
Strengths: - Systematic, step-by-step progression from standard to advanced models - Combination of linear and nonlinear, short-memory and long-memory models - Identifies persistent shifts and gradual regime changes
Challenges: - Data-driven thresholds may introduce bias - Advanced methods require substantial computational resources - Fractional and nonlinear models need theoretical grounding for interpretation
Conclusion: This framework offers a rigorous, flexible, and empirically grounded approach to studying evolving long-run relationships in macroeconomic data, with wide applications in investment strategy, economic forecasting, and policy evaluation.