What Is a Time Series?

Most of statistics begins with a bag of measurements you could shake up without losing anything: the heights of a hundred people, the yields of fifty fields. Shuffle them and every average, every histogram, every test comes out the same. A time series is the opposite kind of data — shuffle it and you destroy it. The order is the information.

Formally, a time series is a sequence of observations x_1, x_2, \dots, x_T recorded in order at times t = 1, 2, \dots, T. A closing price each trading day, the temperature each hour, the number of births each month, the voltage across a nerve each millisecond. The single defining feature is serial dependence: the value now is statistically tied to the values just before it. That one fact is what breaks classical "independent observations" statistics and forces an entire subject of its own.

Four questions the subject answers

Everything in this course is aimed at one of four practical goals — and they are genuinely different goals, not four names for the same thing:

A weather model forecasts; a fraud system detects; an economist explains; an engineer controls. Different goals, but all rest on the same core idea: capture the dependence between a value and its own past.

What time series look like

The three curves below are stylized versions of the shapes you meet again and again. The rising line is a trending series (think atmospheric CO₂). The wavy one is seasonal — the same pattern repeating on a fixed period (retail sales peaking each December). The jagged one wanders with no trend and no season: a random walk, the archetype of an unpredictable path like a stock price.

Notice what they share: each point sits close to the one before it. That closeness — nearby times take nearby values — is serial dependence made visible, and measuring it precisely is the whole game.

Discrete time, and the sampling interval

We work almost entirely in discrete time: the index t steps through the integers, one tick per observation, whether the underlying process is truly discrete (daily rainfall totals) or a continuous signal sampled at a fixed rate (a temperature probe read every minute). The gap between observations — the sampling interval — is a modelling choice with teeth: sample too coarsely and you miss fast structure; the fastest wiggle you can even represent is set by how often you look, a fact spectral analysis will make exact.

A series can also be univariate (one number per time, like a single thermometer) or multivariate (a vector per time, like temperature and pressure and humidity together). Most of this course is univariate; the final module handles the vector case, where series also influence one another.

A tempting first instinct is to fit x_t = \beta_0 + \beta_1 t + \varepsilon_t by ordinary least squares and call it a day. Sometimes that captures the trend — but the standard errors it hands you are usually lies. OLS assumes the residuals are independent; in a time series they are almost never independent — a positive error today makes a positive error tomorrow more likely. That leftover autocorrelation means your effective sample size is far smaller than T, so OLS reports confidence intervals that are far too narrow and significance that isn't there. Time series methods exist precisely to model the dependence OLS pretends away — not as a decoration, but because ignoring it makes you confidently wrong.

In the 1920s the statistician Udny Yule was puzzling over the sunspot numbers — a famously regular-ish cycle of roughly eleven years. Everyone modelled such cycles as a hidden sine wave buried in noise. Yule's radical idea: maybe there is no fixed hidden sine at all. Maybe each year's value is just a weighted echo of the previous couple of years, plus a fresh random shock — a pendulum being kicked at random. That single reframing became the autoregressive model, the cornerstone of the whole field. The sunspots taught us that a series can look cyclic without any clock ticking underneath — the "cycle" can be manufactured entirely by memory and chance.