3.1. Foundational Concepts and Models

A time series is a sequence of observations ordered in time. They are often decomposed into four components:

1. Trend (T): A long-term movement.
2. Seasonal (S): A periodic pattern (e.g., day-of-the-week effects).
3. Irregular/Cyclical (I): Non-periodic fluctuations.
4. Error (U): A random disturbance term.

Key univariate models include:

• Autoregressive (AR) Model: The current value of a series depends linearly on its own past values (yₜ = c + ρyₜ₋₁ + εₜ).
• Moving Average (MA) Model: The current value depends on current and past random error terms (yₜ = c + εₜ + θεₜ₋₁).
• ARMA Model: A combination of AR and MA processes, providing a more parsimonious model for complex dynamics.
• Vector Autoregressive (VAR) Model: A multivariate model where several variables are modeled as a function of their own past values and the past values of the other variables in the system, capturing interdependencies.

3.2. Stationarity and Cointegration

• Stationary vs. Nonstationary: A time series is stationary if its mean, variance, and autocorrelation structure are constant over time. It exhibits mean-reversion. A nonstationary series has a time-dependent structure, often containing a stochastic trend or “unit root,” and can wander arbitrarily far from its mean. The simplest example is a random walk.
• Spurious Regression: Regressing one nonstationary series on another can produce statistically significant results (high R², significant t-statistics) even if the variables are unrelated.
• Cointegration: This concept applies when two or more nonstationary variables share a common stochastic trend. If a linear combination of these variables is stationary, they are said to be cointegrated. This implies a stable, long-run equilibrium relationship.
  • Error-Correction Model (ECM): If variables are cointegrated, their short-run dynamics can be modeled using an ECM, which describes how the variables adjust back to their long-run equilibrium after a short-term deviation.
• Testing for Cointegration:
  • Engle-Granger Test: A two-step method primarily for bivariate analysis.
  • Johansen-Juselius Test: A more robust, multivariate method based on a VAR framework that can identify multiple cointegrating relationships.
• Application: The relationship between the S&P 500 and its dividends was found to be cointegrated for the 1962-1982 period, but this relationship broke down in the subsequent period (1962-2006), consistent with the presence of a stock market bubble in the 1990s.

3.3. Volatility Modeling: ARCH and GARCH

Financial asset returns exhibit volatility clustering, where periods of high volatility are followed by more high volatility, and calm periods are followed by more calm. ARCH and GARCH models are designed to capture this time-varying, conditional variance.

• Autoregressive Conditional Heteroscedasticity (ARCH): The ARCH(m) model states that the conditional variance at time t is a linear function of the past m squared residuals (returns).
  • σ²ₜ = c + a₁R²ₜ₋₁ + … + aₘR²ₜ₋ₘ
• Generalized ARCH (GARCH): The GARCH(p,q) model is an extension where the conditional variance also depends on its own past values. This provides a more parsimonious model.
  • σ²ₜ = c + a₁R²ₜ₋₁ + … + aₙR²ₜ₋ₙ + b₁σ²ₜ₋₁ + … + bₘσ²ₜ₋ₘ
• Variants: Many extensions exist to capture stylized facts of financial returns, such as the leverage effect (negative news has a larger impact on volatility than positive news), which is addressed by models like EGARCH and TARCH. The ARCH-in-the-Mean (ARCH-M) model allows the conditional variance to directly influence the conditional mean return.