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Outline what a random walk is and what a unit root is.

FRM® Part I · Quantitative Analysis · Non-Stationary Time Series · FRM1.QA.11.c

The idea

A random walk is a series in which today's value equals yesterday's value plus an unpredictable white-noise shock: yt = yt−1 + εt. With a constant drift d added to each step, the random walk model becomes: yt = d + yt−1 + εt. The best forecast of any future value is the latest value, plus d for each step ahead. Shocks never fade. With shock variance σ², the variance of yt+h around the current value grows linearly with the horizon h: Variance = h × σ². The standard deviation therefore grows with the square root of the horizon: Standard deviation = σ × √h. A random walk is a special case of the AR(1) model, with the coefficient φ equal to 1: yt = φ × yt−1 + εt. A unit root means the characteristic equation of the autoregression has a root equal to 1. Equivalently, the AR coefficients sum to 1. Such a series is non-stationary, but its first difference is stationary. The usual mistake is to scale the standard deviation by h rather than by the square root of h.

Formulas

Expected value of a random walk with drift h steps ahead

E[y(T + h)] = y(T) + d × h

y(T)
the latest observed value
d
the drift added each step
h
the number of steps ahead

With no drift the forecast is just the latest value. Shocks never fade, so the forecast does not revert to any mean.

Standard deviation of a random walk h steps ahead

σ(h) = σ × √h

σ
the standard deviation of one step's shock
h
the number of steps ahead

Variance grows linearly as h × σ², so the standard deviation grows with the square root of h. Scaling σ by h rather than √h is the usual mistake.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which description defines a random walk?

  1. Each value is a constant fraction (below one) of the previous value plus a shockWhy not: This is a stationary AR(1) with phi < 1, which reverts to its mean.
  2. Each value is a deterministic linear function of time plus a shockWhy not: This is a deterministic trend model, which is trend-stationary rather than a random walk.
  3. Each value equals the previous value plus an unpredictable white-noise shock - correct
  4. Each value is a weighted average of the current and previous shocksWhy not: This is a moving-average process, which is stationary.

A random walk is y_t = y_(t-1) + e_t: the AR(1) coefficient is exactly 1, so shocks accumulate and never decay.

A harder one

The daily change in an index level is white noise with a standard deviation of 2.0 points, and the level follows a random walk with no drift. Given today's level, what is the standard deviation of the level 25 days from now?

  1. 10.0 points - correct
  2. 50.0 pointsWhy not: Scales the daily standard deviation by the number of days rather than its square root.
  3. 100.0 pointsWhy not: Reports the variance (25 x 4 = 100) as if it were the standard deviation.
  4. 2.0 pointsWhy not: Assumes the uncertainty does not accumulate, i.e. treats the level as stationary.

Shocks accumulate: variance = 25 x 4 = 100, so the standard deviation is the square root, 10.0.

The hardest tier

A series follows y_t = 1.3 y_(t-1) - 0.3 y_(t-2) + e_t. Which statement is correct?

  1. It is stationary, since the second coefficient, -0.3, is small in absolute valueWhy not: Judging stationarity from one coefficient's size is wrong; the roots of the characteristic equation decide it.
  2. It is stationary, since the two coefficients 1.3 and -0.3 sum to less than 2Why not: The relevant test is whether the coefficients sum to 1 (a root at 1), not whether the sum is below 2.
  3. It has a unit root, since the characteristic equation z² - 1.3 z + 0.3 = 0 has a root equal to 1 - correct
  4. It has an explosive root, since 1.3 is greater than 1Why not: Takes the first coefficient as the root; the roots are 1 and 0.3, so nothing is explosive.

z² - 1.3 z + 0.3 = (z - 1)(z - 0.3), with roots 1 and 0.3. The coefficients sum to 1.3 - 0.3 = 1, so there is a unit root; the series is a random walk with a stationary AR(1) component in its differences.

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