Time series, stationarity and autocorrelation

Every chapter so far has assumed observations are independent draws. Put them in time order and that assumption fails in specific, measurable ways — and the failures are not nuisances, they are the most useful empirical facts in the subject.

Stationarity

A series is stationary if its statistical properties do not change over time: constant mean, constant variance, and a covariance between two points that depends only on the gap between them.

Prices are not stationary. A price series wanders, has no mean to revert to, and its variance grows with the horizon. Almost every statistical tool assumes stationarity, so applying them to prices produces confident nonsense.

Returns are approximately stationary, which is the real reason finance models returns rather than prices. Differencing — moving from level to change — is what makes the data usable, and chapter 7's log returns are the differencing of choice.

Spurious regression

Two independent series that both trend will regress against each other with a large $R^2$ and a highly significant slope. Nothing connects them. The trend in each is explaining the trend in the other.

This is not a rare pathology — it is the default outcome of regressing any two non-stationary series. Ice cream sales against Nifty levels will "work". So will a strategy's cumulative equity curve against anything else that rose.

The defence is mechanical: difference first, regress second. Regress returns on returns, never levels on levels. If a relationship survives differencing it may be real; if it only exists in levels, it was the trend.

Autocorrelation

The correlation of a series with its own lagged values:

$\rho_k = \text{Corr}(r_t,\ r_{t-k})$

Two findings, and they point in opposite directions.

Returns are close to unpredictable

For liquid markets, the autocorrelation of returns at almost every lag is near zero. Yesterday's direction says essentially nothing about today's. This is the weak form of market efficiency as an empirical claim rather than a theory, and it is the quantitative reason chapter 9 of the Technical analysis subject reaches the conclusion it does.

Small positive autocorrelation does appear in illiquid stocks, but that is largely stale prices — a share that did not trade carries yesterday's close into today — rather than a tradable pattern.

Volatility is highly predictable

Now take absolute or squared returns. Their autocorrelation is strongly positive and decays slowly over weeks.

This is volatility clustering: large moves follow large moves, of either sign. Calm periods are calm; turbulent periods stay turbulent.

Both facts at once: you cannot predict whether tomorrow is up or down, and you can predict quite well how far it will move. Direction is unpredictable, magnitude is not.

What clustering breaks

The $\sqrt{t}$ rule. Chapter 3 annualised volatility by $\sigma\sqrt{252}$, which assumes independence across days. With clustering, the real multi-day variance is larger than the formula during turbulent stretches and smaller during calm ones. A single annualised number describes neither regime.

Confidence intervals. Chapter 9 noted that dependence reduces the effective sample size. Clustering is that dependence, so the honest interval is wider than computed.

Risk models. Any model using a single constant $\sigma$ is using an average of two different states, and the number is wrong in both.

The practical consequence is that risk should be measured conditionally — what is volatility now, given recent volatility — rather than as a historical constant. That is the whole motivation for the GARCH family of models: let today's variance depend on yesterday's variance and yesterday's shock.

Regime change

Beyond clustering is the harder problem: the data-generating process itself changes. Regulation changes, market structure changes, participants change.

SEBI's market-wide circuit breakers are an explicit discontinuity written into the process: at a 10%, 15% or 20% index move, trading halts. A model fitted across such an event has fitted two different worlds and averaged them.

There is no statistical fix. The honest responses are to prefer recent data where the regime is more likely intact, to test whether conclusions survive on subsamples, and to treat any long-sample parameter as a blend rather than a measurement.

The point

Prices are non-stationary and must be differenced into returns before any statistical tool is applied — regressing two trending series produces a beautiful, meaningless fit. Returns have almost no autocorrelation, so direction is close to unpredictable; absolute returns have a great deal, so magnitude is quite predictable. That clustering breaks the square-root-of-time rule and widens every confidence interval, and regime change breaks the rest.