A time series is a sequence of observations indexed by time. Almost every important quantity in finance is a time series: prices, returns, volatilities, yields, spreads, volumes, fund flows, macroeconomic releases. The distinguishing feature is dependence — observations in a time series are correlated with their own past, so the i.i.d. machinery of standard statistics doesn't apply.
Stochastic processes
A stochastic process is a collection of random variables {X_t : t ∈ T} indexed by time. For discrete-time finance we usually take T = {0, 1, 2, ...}. A single realisation (one path through time) is what we actually observe; the underlying distribution governs all possible paths.
Strict vs weak stationarity
- Strict stationarity: the joint distribution of (X_t1, ..., X_tk) is the same as (X_{t1+h}, ..., X_{tk+h}) for any shift h and any indices. The full law is time-invariant.
- Weak (covariance) stationarity: E[X_t] = μ (constant mean), Var(X_t) = σ² (constant variance), Cov(X_t, X_s) depends only on |t - s|. The first two moments are time-invariant.
Strict stationarity implies weak stationarity (under finite second moments). For Gaussian processes, the two are equivalent. Most of our practical machinery requires only weak stationarity.
Why we model returns and not prices
Asset prices are not stationary — they trend, drift, have unit roots. Returns are typically (approximately) stationary. ARMA, GARCH, and almost all of classical time-series modelling assume stationarity. The choice to model returns (or log returns) is not arbitrary; it's the price you pay to use these tools.
Autocovariance and autocorrelation
γ(h) = Cov(X_t, X_{t-h}) = E[(X_t - μ)(X_{t-h} - μ)]ρ(h) = γ(h) / γ(0) = γ(h) / σ²
γ(h) is the autocovariance at lag h; ρ(h) is the autocorrelation function (ACF). These measure linear dependence in time. ρ(0) = 1 by definition. For a stationary process, ρ(h) → 0 as h → ∞.
Ergodicity
A stationary process is ergodic if time averages converge to ensemble averages: (1/T) Σ X_t → E[X_t] as T → ∞. Without ergodicity, observing a long path tells us nothing about the underlying distribution. Mild conditions on the decay of γ(h) — typically γ(h) → 0 — guarantee ergodicity for ACF and the mean.
Three building-block series
- White noise: i.i.d. with zero mean. ρ(h) = 0 for h ≠ 0. The 'pure noise' baseline.
- Random walk: X_t = X_{t-1} + ε_t with ε_t ~ white noise. Non-stationary (variance grows linearly with t). The textbook null for log prices.
- AR(1): X_t = φ X_{t-1} + ε_t. Stationary if |φ| < 1. ρ(h) = φ^h — geometric decay.
Visual ACF inspection
Plotting the sample ACF is the single most useful diagnostic in time-series analysis. Look for the ACF bars relative to the ±2/√T 'no-correlation' confidence bands. Slow decay = persistence (possibly non-stationary). Single tall spike = MA(1)-style structure. Alternating signs = mean-reverting behaviour.
Exercise
You have 1000 daily log returns on the NSE-20 index. The sample ACF shows ρ̂(1) = 0.05, ρ̂(2) = -0.02, ρ̂(3) = 0.01, with subsequent lags within ±2/√1000 ≈ ±0.063. (1) Are returns linearly dependent? (2) Now you compute the ACF of squared returns; all lags up to 50 are positive and outside the bands. What does this tell you?