A first difference is the change between consecutive observations. See why price changes differ from returns, how levels can be reconstructed, and when differencing helps stationarity.
First difference
A first difference is the change between consecutive observations of a time series:
Δyt=yt−yt−1.
For an asset price, this measures the move in price units. It can be a useful input to a model of short-run changes, but it is not automatically a percentage return or a stationary series. The transformation changes the question from “What is the level?” to “How much did it move since the previous observation?”
Four prices, three changes
Suppose an asset has these synthetic closing prices at equally spaced trading-session observations. Prices are USD per share; no dividends, fees or corporate actions occur in this example.
Observation
Price
First difference
t=0
100
Not available
t=1
103
103−100=+3
t=2
101
101−103=−2
t=3
106
106−101=+5
The initial observation has no earlier price in this sample, so its difference is missing, not zero. With n consecutive observed levels, there are n−1 adjacent differences. Each difference belongs to the interval ending at its timestamp.
Synthetic prices 100, 103, 101, 106 USD per share at equally spaced times yield changes plus 3, minus 2, plus 5 USD per share. The middle change is minus 2 USD per share, versus a simple price return of approximately minus 1.9417 percent.
Original synthetic observations. Differences retain USD-per-share units and are aligned to the ending observation. Lines between prices guide the eye; there is no observed difference at the starting point.
Price change, simple return and log return
The move from 103 to 101 illustrates three related calculations:
The last row is a log return because the input series is log-price. Taking a first difference does not itself apply a logarithm or divide by the earlier price. Simple price return requires a nonzero earlier price; the real-valued log expression requires positive prices. With cash distributions, a price-only calculation also differs from total return.
Differencing can help stationarity; it does not establish it
If the innovations are white noise with constant finite variance, these differences are covariance-stationary. That conclusion comes from the model assumptions, not from the subtraction alone. A process needing two differences can remain nonstationary after the first. Even a deterministic sequence yt=t2 gives Δyt=2t−1, which still trends with time.
Conversely, unnecessary differencing can introduce dependence. If yt=α+βt+ut with stationary ut, both removing the justified deterministic trend and taking differences can produce stationary series, but they produce different objects:
detrended level=ut,Δyt=β+ut−ut−1.
When ut is white noise with positive variance, the differenced noise has lag-one correlation −1/2. The negative dependence was introduced by the transformation. Detrending preserves deviations around the fitted or stipulated trend; differencing measures changes between dates. Choose the object the application needs, then investigate its stationarity.
For the random-walk equation and the practical choices behind unit-root testing, see Stationarity and Unit Roots (public preview; full lesson requires membership).
Keep the starting value and the clock
Differences alone lose the absolute starting level: adding the same constant to every level leaves all differences unchanged. Retaining the starting value restores the path by accumulation:
yt=y0+s=1∑tΔys.
Here, 100+3−2+5=106. Modelling changes therefore need not prevent reconstructing a level forecast, provided its starting level is retained.
The clock still matters. Adjacent records need not cover equal time spans if observations are missing. An overnight change and a multi-session change are not interchangeable merely because each occupies one row. Differencing is also distinct from a rate of change per unit time: it does not divide by elapsed time. The change ending at t becomes available only when yt is observed; a backtest cannot use it to justify an earlier decision.
Sources
Chris Brooks, Introductory Econometrics for Finance, second edition, Cambridge University Press, 2008, §7.1.2, pp. 322–323, and §7.1.3, pp. 326–327: differencing, deterministic trends and integration order.
