A stochastic trend accumulates random shocks. Learn why its effects persist, why a flat expected level can be nonstationary, and how differencing differs from detrending.
Stochastic trend
A stochastic trend is a nonstationary component of a time series that evolves through accumulated random shocks. In the standard unit-root model, a shock changes the level permanently rather than fading away. For a quant modelling an asset-price series, this means that a move away from an old average does not, by itself, create a reason to expect reversal.
“Trend” does not mean a smooth upward slope. A stochastic trend can wander up, down and sideways, even when its expected change is zero.
Assume a fixed initial level y0 and independent innovations with mean zero and constant finite variance σ2>0. Each innovation enters every subsequent level with coefficient one. Later shocks may offset an earlier shock in the total, but the model contains no restoring force pulling the series back to its pre-shock path.
Consider an index starting at 100 points. At equally spaced observations, stipulate realised shocks of +1, −2 and +3 points:
Observation
New shock (points)
Level (points)
t=0
—
100
t=1
+1
101
t=2
−2
99
t=3
+3
102
The first shock contributes +1 to all three later levels. Holding the other shocks fixed and replacing that first shock with zero would give later levels of 100, 98 and 101. This counterfactual isolates persistence; it does not say the observed index must keep rising after a positive shock.
Synthetic random-walk levels are 100, 101, 99 and 102 index points at times zero through three. First differences are plus 1, minus 2 and plus 3 points. The first shock contributes zero initially and plus 1 point at every later observation.
Original fixed illustration, not simulated or market data. The left panel accumulates the stipulated innovations; the right isolates the first innovation while holding the others fixed. All markers represent discrete observations; connecting lines guide the eye.
These four levels illustrate an assumed specification. They do not prove that a real series has a unit root, that its shocks are independent, or that it follows this model outside the sample.
Constant expected level, growing uncertainty
Under the stated assumptions,
E[yt]=y0,Var(yt)=tσ2.
The expected level is flat, yet the variance grows with elapsed observations. That alone rules out covariance stationarity. A stochastic trend therefore need not appear as a trend in the unconditional mean.
Adding a constant drift μ gives yt=y0+μt+∑s=1tεs. The deterministic component μt and the accumulated random component coexist. Subtracting the drift leaves the stochastic trend in place.
Differencing and detrending answer different questions
The example returns +1,−2,+3 points. Under the maintained innovation assumptions, these differences are stationary. The arithmetic alone does not establish those assumptions, and differencing is not a universal cure for nonstationarity.
By contrast, detrending subtracts a specified or fitted deterministic trend, such as a+bt. A line cannot generally remove the accumulated shocks of a unit-root process. Fitting one can make a short sample look centred without creating a stable long-run mean.
For a genuinely trend-stationary model yt=a+bt+ut with stationary ut, subtracting the true trend recovers ut. Differencing can also give a stationary series, but it produces b+ut−ut−1 and changes the noise dynamics. The choice concerns both the source of nonstationarity and the quantity being modelled.
The Stationarity and Unit Roots lesson develops the random-walk representation and practical test specifications (public preview; full lesson requires membership). Failing to reject a unit root is evidence to interpret under a test's assumptions, not proof of a random walk.
There is a further possibility in relative-value work: two series can share a stochastic trend that cancels in a suitable linear combination. That is the central idea behind cointegration. Similar-looking upward price charts alone do not establish that cancellation.
Sources
Chris Brooks, Introductory Econometrics for Finance, second edition, Cambridge University Press, 2008, §7.1.2, pp. 320–323: random walks, persistence of shocks, stochastic versus deterministic trends, and the different effects of differencing and detrending.
