Time Seriesintermediate

Trending Time Series

Many economic series trend over time, drifting steadily up or down because of growth, inflation, or population. The danger is a spurious relationship: two unrelated series that both trend upward will look strongly correlated even though neither causes the other. The fix is to control for the trend directly, usually by adding a time index $t$ as a regressor, which is equivalent to detrending each variable before relating them. A common model is $y_t = \beta_0 + \beta_1 t + \beta_2 x_t + u_t$ , where including $t$ lets the slope on $x$ reflect the comovement after the shared trend is removed.

Why it matters

If GDP and the number of internet users both rise every year, a regression will find them tightly linked, but that link is mostly just "time passing". Adding a time trend $t$ to the regression soaks up the part of each variable that is pure drift, so the remaining coefficient measures genuine comovement rather than two escalators rising side by side.

Formulas

Linear time trend model

y_t = \beta_0 + \beta_1 t + \beta_2 x_t + u_t

The index t = 1, 2, 3, … captures steady drift; the coefficient on x is its effect net of the trend.

Exponential (constant-growth) trend

\log(y_t) = \beta_0 + \beta_1 t + u_t

In a log-linear trend, the slope on t approximates the average per-period growth rate of y.

Worked examples

Scenario

Housing investment and housing prices both rise over your sample, and you want the price effect free of the common trend.

Solution

Create a trend and include it: `gen t = _n` then `regress invest price t, robust`. The coefficient on `price` now reflects the relationship after detrending. Compare it with `regress invest price, robust` to see how much of the raw association was just shared drift.

NoteAdding t to the regression gives the same slope as regressing detrended y on detrended x (the Frisch-Waugh logic).

Common mistakes

✗A high $R^2$ between two trending series proves they are related. Shared trends inflate $R^2$ and can produce a spurious association with no causal link.
✗Detrending throws away the relationship of interest. Including a time trend removes only the common drift, leaving the genuine comovement you want to measure.
✗Adding a time trend and detrending each variable give different answers. By the Frisch-Waugh result they give the same slope on $x$ .
✗Trends only matter for the dependent variable. If either $y$ or $x$ trends, an uncontrolled trend can bias the estimated relationship.

Revision bullets

•Trending series drift steadily over time
•Two trending series can look related but be spurious
•Add a time trend $t$ as a regressor to control for drift
•Including $t$ is equivalent to detrending the variables
•A log-linear trend coefficient approximates a growth rate

Quick check

Two economic series both trend strongly upward and show a high correlation. The most likely concern is:

Including a linear time trend $t$ in the regression of $y_t$ on $x_t$ is equivalent to:

Connected topics

Dummies TS Data Seasonality Unit Roots Cointegration

Sources

Wooldridge (2019), §10.5
Wooldridge, Jeffrey M. Introductory Econometrics: A Modern Approach. 7th ed. Cengage, 2019.
Covers trending time series, spurious regression from shared trends, and detrending via a time index.
Granger & Newbold (1974)
Granger, C.W.J., and P. Newbold. Spurious Regressions in Econometrics. Journal of Econometrics 2 (1974): 111-120.
Classic demonstration that trending and integrated series produce misleading regression results.

How to cite this page

Dr. Phil's Quant Lab. (2026). Trending Time Series. Derivatives Atlas. https://phucnguyenvan.com/concept/efm-trends

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