Time Series Analysis

A stationary time series has constant mean, variance, and autocorrelation over time. Most financial price series are non-stationary (they trend). Quants often transform prices to log-returns, which are approximately stationary, before modeling.

Measures the correlation of a series with its own past values at different lags. High autocorrelation at lag 1 suggests momentum (trend-following signals). The ACF plot is used to identify the MA order in ARIMA models.

Measures the direct correlation between a series and its lagged values, removing the effect of intermediate lags. Used to identify the AR order in ARIMA models.

Regular, predictable patterns in data linked to the calendar. In equities: the "January effect," end-of-quarter rebalancing, options expiration cycles. Quants test for and model these to isolate true alpha from seasonal noise.

The Augmented Dickey-Fuller test checks whether a time series has a unit root (is non-stationary). A p-value below 0.05 rejects the null of a unit root, confirming stationarity. Required before fitting any ARIMA model.

The AR(p) model uses p lagged values of the series itself as predictors. An AR(1) model says: today's return is partly explained by yesterday's return multiplied by a coefficient φ. Positive φ = momentum; negative φ = mean reversion.

The MA(q) model uses q lagged forecast errors as predictors. Unlike a simple moving average of prices, the MA component in ARIMA uses past prediction errors to refine future forecasts.

If the series has a trend (non-stationary), differencing removes it: Δyt = yt − yt−1. A series that becomes stationary after one round of differencing is called I(1). Most stock price series are I(1); log-returns are I(0).

The Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) are used to select the optimal ARIMA(p,d,q) order. Lower AIC/BIC indicates a better balance between model fit and complexity — penalizing overfitting.

Used for forecasting interest rates, currency trends, and commodity prices. In equities, ARIMA residuals are analyzed to confirm that no predictable pattern remains — confirming the model has captured all linear structure in the data.

Empirical observation that large price moves (positive or negative) tend to cluster in time. GARCH formalizes this: the variance of today's return depends on past variance and past squared returns.

The most widely used specification. Today's variance σ²t = ω + α·ε²t−1 + β·σ²t−1. α measures the reaction of volatility to new shocks; β measures volatility persistence. (α + β) close to 1 indicates high volatility persistence.

Captures the leverage effect — negative returns tend to increase volatility more than positive returns of equal magnitude. EGARCH allows asymmetric volatility responses, more realistic for equity markets.

GARCH-based volatility forecasts are compared against implied volatility (IV) to identify rich or cheap options. If GARCH forecasts realized vol will be lower than current IV, options are overpriced — a signal to sell premium (the core of the wheel strategy).

GARCH models feed into VaR calculations — estimating the maximum loss a portfolio is expected to sustain over a given period at a given confidence level (e.g., 99% VaR). Regulatory standard for bank risk management.

A two-step test for cointegration. Step 1: Regress one series on the other (OLS). Step 2: Apply the ADF test to the residuals. If residuals are stationary, the two series are cointegrated and a statistical arbitrage relationship exists.

More powerful than Engle-Granger for multivariate cases. Tests for cointegration among multiple time series simultaneously, identifying the number of cointegrating relationships using trace and maximum eigenvalue statistics.

When two series are cointegrated, the ECM describes how they adjust back toward the long-run equilibrium after a short-term deviation. The error correction term is the key input to the mean-reversion trading signal.

Once cointegration is confirmed, quants compute the hedge ratio (from the cointegrating regression), calculate the spread (current deviation from equilibrium), normalize it to a Z-score, and trade when the Z-score exceeds ±2 standard deviations.