Deterministic and Stochastic Trend

Study guide: Exam MAS-II

A deterministic trend is a fixed function of time that the series returns to after a shock; a stochastic trend is a unit root, where each shock permanently moves the level. They look alike on a plot and demand opposite treatments — the distinction that objective D3 turns on.

$\text{deterministic: } Y_t = \alpha + \beta t + z_t, \quad z_t \text{ stationary}$

$\text{stochastic: } Y_t = \delta + Y_{t-1} + \varepsilon_t \quad (\text{[Random Walk](/wiki/concept/Random+Walk) with drift})$

	Deterministic (trend-stationary)	Stochastic (difference-stationary)
Effect of a shock	dies out — the series returns to the line	permanent — the level shifts forever
Correct treatment	regress on $t$ , model the residuals	difference the series
$\mathrm{Var}(Y_t)$	bounded	grows with $t$
Long-horizon forecast	the trend line	last value plus $h\delta$ , with a widening interval

Getting it wrong costs both ways. Differencing a trend-stationary series over-differences it, injecting a spurious negative MA(1) term. Detrending a unit-root series leaves the trend behind and the residuals still non-stationary
The formal test is a unit-root (Dickey–Fuller) test: $H_0$ is a stochastic trend
Forecast intervals differ sharply — a deterministic trend gives intervals of roughly constant width, a stochastic one gives intervals widening as $\sqrt{h}$
Spurious regression: two independent series with stochastic trends will show a highly significant regression and a high $R^2$ purely from the shared drift. Always check the residual ACF before believing a time-series regression

Which Trend Is It? A 60-quarter premium index trends upward. Regressing on $t$ gives $R^2 = 0.94$ but residuals with $\hat\rho_1 = 0.93$ decaying very slowly. First-differencing gives a series with $\hat\rho_1 = 0.11$ and nothing else significant. Which trend, and what is the model?

Answer Stochastic. The detrended residuals are still non-stationary — $\hat\rho_1 = 0.93$ with slow decay is a unit root that the regression did not remove. A true deterministic trend would leave stationary residuals. First differencing produces something close to White Noise, so the model is a random walk with drift, ARIMA(0,1,0) with a constant. The $R^2 = 0.94$ from the trend regression is the spurious-regression trap.