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Time-varying Cointegration

US Economic Indicators and Structural Break Analysis

Theory
미국 주요 거시경제 지표 간의 장기 균형 관계가 구조적 변화(Structural Break)에 따라 시간에 따라 어떻게 변화하는지 시변 공적분(Time-Varying Cointegration) 및 분수 공적분으로 분석한다.
Author

gitSAM

Published

March 16, 2025

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

  1. Which economic indicators form long-term relationships?
  2. How have these relationships changed over time?
  3. 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.