Time Series Analysis Explained: Stationarity, Unit Roots and Cointegration

Understanding time series analysis — simply put

A time series is a variable observed in order over time: quarterly GDP, monthly inflation, daily exchange rates. Time series analysis is the set of methods for describing, modeling and forecasting such data. What sets it apart from ordinary regression is that observations are not independent: this quarter's value carries information about the next, and that dependence is both the opportunity (forecasting) and the danger (false relationships).

Why can two unrelated series look perfectly correlated?

Regress one trending series on another, say real GDP on the number of mobile phone subscriptions, and OLS will report a large R² and a highly significant slope even if the two have nothing to do with each other. Both simply drift upward over time. This is spurious regression, and it is the reason time series work begins with one question: is the series stationary?

A series is (weakly) stationary when its mean, variance and autocovariances do not change over time. Shocks to a stationary series fade; shocks to a non-stationary one can persist forever.

What does non-stationary data actually look like?

The level of U.S. real GDP trends upward with no tendency to return to a fixed mean. Its growth rate, the percentage change from a year earlier, fluctuates around a stable average. The same economy, transformed, moves from non-stationary to stationary.

The workhorse model of persistence is the first-order autoregression:

When φ = 1 the series has a unit root and becomes a random walk: every shock is permanent. The half-life of a shock in a stationary AR(1) is ln(0.5) / ln(φ); with φ = 0.9 it is about 6.6 periods, with φ = 0.5 just one.

How do economists test for a unit root?

The augmented Dickey–Fuller test rewrites the AR(1) in differences and tests whether the coefficient on the lagged level is zero:

The test statistic does not follow the usual t-distribution under the null, so it is compared with Dickey–Fuller critical values, which are more negative. Its power is low in short samples: failing to reject a unit root is weak evidence that one exists.

Can two non-stationary series still have a real relationship?

Yes, if they are cointegrated. Consumption and income each trend upward, but a household cannot spend far more or far less than it earns for long, so the gap between them is stationary. When a linear combination of non-stationary series is stationary, the series share a long-run equilibrium, and an error-correction model captures how they return to it:

Cointegration is what separates a genuine long-run relationship from the spurious correlation of two unrelated trends, and it underpins much of empirical macroeconomics, from money demand to purchasing power parity.

The takeaway

Time series data remember their past. Check stationarity before running regressions, difference trending series or model their cointegration explicitly, and treat high R² values between trending series with suspicion.

Further reading

• OLS Regression Explained — internal
• Difference-in-Differences Explained — internal
• FRED: Real Gross Domestic Product (GDPC1) — external reference