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Explain what mean reversion is and calculate the level a series reverts to.

FRM® Part I · Quantitative Analysis · Stationary Time Series · FRM1.QA.10.g

The idea

Mean reversion means a stationary series is pulled back toward a fixed long-run level, so deviations from it fade over time rather than persisting. For an AR(1) model with intercept δ and coefficient φ, the level it reverts to is the unconditional mean, μ = δ ÷ (1 − φ). Rewriting the model in deviations from μ shows that a gap from μ shrinks by the factor φ each period, so when φ is positive, a larger φ means slower reversion. For an AR(p) model the long-run level is μ = δ ÷ (1 − φ₁ − φ₂ − … − φp), using the signed coefficients. In an ARMA model the MA coefficients do not enter the mean. The common mistake is to treat the intercept δ as the long-run level. It equals the mean only when φ is zero.

Formulas

Long-run level of a stationary AR(p)

μ = δ ÷ (1 − φ₁ − φ₂ − … − φp)

δ
the intercept
φ₁, φ₂
the first and second AR coefficients, with their signs
φp
the last AR coefficient, at lag p
p
the AR order

Signed coefficients: a negative φ adds to the denominator. MA coefficients never enter the mean of an ARMA model.

A practice question

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

Mean reversion in a stationary time series means that:

  1. the series returns to its starting value after a fixed number of periodsWhy not: Reversion is toward the long-run mean, not the initial value, and the speed is gradual rather than fixed in time.
  2. deviations from a fixed long-run level tend to fade over time - correct
  3. future values are always above the long-run mean after a declineWhy not: Reversion pulls toward the mean from either side; it does not push values past it.
  4. the shocks to the series are negatively autocorrelatedWhy not: Mean reversion describes the level of the series; the shocks are white noise with no autocorrelation.

A stationary series has a fixed unconditional mean. Mean reversion is the tendency of values away from that mean to be pulled back toward it, with the gap shrinking each period.

A harder one

A series follows y_t = 2 + 0.8 y_(t-1) + e_t. It currently stands at 14. The expected value of the series 3 periods ahead is closest to:

  1. 13.20Why not: Gives the one-period-ahead forecast, shrinking the gap to the mean only once.
  2. 11.64Why not: Raises phi to the fourth power instead of the third, shrinking the gap one period too many.
  3. 12.05 - correct
  4. 10.00Why not: Assumes the series has fully reverted to its mean, but the gap only shrinks by 20 percent per period.

mu = 2 / 0.2 = 10. The gap is 4 and shrinks by 0.8 per period: 4 x 0.8³ = 2.048, so the forecast is 10 + 2.048 = 12.05.

The hardest tier

An AR(3) process is y_t = 0.6 + 0.5 y_(t-1) - 0.2 y_(t-2) + 0.4 y_(t-3) + e_t. The level the series reverts to is:

  1. 2.0 - correct
  2. -6.0Why not: Subtracts the absolute value of the negative coefficient: 1 - 0.5 - 0.2 - 0.4 = -0.1 instead of 1 - 0.5 + 0.2 - 0.4.
  3. 0.86Why not: Ignores the third lag, using 1 - 0.5 + 0.2 = 0.7 in the denominator.
  4. 0.35Why not: Adds the sum of the coefficients to 1 (1.7 in the denominator) instead of subtracting it.

Sum of coefficients = 0.5 - 0.2 + 0.4 = 0.7. mu = 0.6 / (1 - 0.7) = 0.6 / 0.3 = 2.0. The negative coefficient adds to the denominator.

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