If advertising and revenue rise together, is that really an advertising effect?
Monthly revenue depends not only on current advertising but also on previous revenue, long-run trend, and seasonality. Ignoring time order can leave strong autocorrelation in the residuals.
Time-series regression includes past information and time structure so that a shared trend is not automatically mistaken for a substantive relationship.
Key question
How should past values, trend, and seasonality enter a regression that explains the current outcome?
What makes time-series regression different?
Dynamic regression example
yₜ = β₀ + β₁xₜ + φyₜ₋₁ + εₜ
| Term | Revenue example |
|---|---|
| xₜ | current advertising spending |
| yₜ₋₁ | last month’s revenue |
| t | long-run time trend |
| seasonal indicators | recurring monthly or quarterly differences |
Trend and season
yₜA series usually carries a slow drift and a repeating shape at the same time.
Residuals that remember
corr(eₜ, eₜ₋₁)Errors of the same sign arrive in runs, so ordinary standard errors mislead.
The lagged term
φ yₜ₋₁Last period's outcome enters as a predictor and carries the effect forward.
Why are trends and spurious regression dangerous?
This is spurious regression. A high R² and small p-value are not convincing when the series are nonstationary and the residuals retain trend or autocorrelation.
Stationarity
A stationary series has a reasonably stable mean, variance, and autocorrelation structure over time. Plot the series and examine unit-root or stationarity diagnostics before deciding how to model it.
What do lagged variables represent?
| Term | Interpretation |
|---|---|
| yₜ₋₁ | persistence from the previous outcome |
| xₜ₋₁ | a delayed predictor effect |
| t | a deterministic linear trend |
| month indicators | seasonality |
Persistence
If φ=0.70, a higher value last month tends to carry substantially into the current month, provided the dynamic model is stable.
Too many lags reduce the usable sample and can create multicollinearity, so use information criteria, domain knowledge, and residual diagnostics together.
When should we use a trend or a difference?
| Problem | Typical response |
|---|---|
| Deterministic linear trend | include a time trend |
| Stochastic trend / unit root | difference the series |
| Seasonality | seasonal indicators or seasonal differencing |
| Residual autocorrelation | lag terms or ARIMA errors |
| Stable long-run relationship among nonstationary series | cointegration and an error-correction model |
Interpretation after differencing
A coefficient in a differenced model explains relationships between period-to-period changes, not differences in the original levels.
Change trend, lag, and seasonality
Adjust the time structure and compare the level series, first differences, and lag-1 autocorrelation.
Series form
Level
Current-series ACF(1)
0.83
Observations
36
ACF(1) here is calculated from the displayed series, not from fitted-model residuals.
What to watch
Time-series regression must explain not only relationships among variables but also the remaining structure across time.