Charta is on Google Play: practise on the go. Get the app
Charta

Give definitions for the autocorrelation function and for the autocovariance function.

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

The idea

The autocovariance function measures how a stationary series moves with its own past. Autocovariance = Cov(yt, yt−k). It is a function of the lag k only, and it is written γ(k). At lag zero it equals the variance, so γ(0) = Var(yt), and it is symmetric, so γ(k) = γ(−k). The autocorrelation function rescales it to a unit-free number. Autocorrelation = γ(k) ÷ γ(0). So ρ(0) = 1 and every ρ(k) lies between −1 and 1. Because the autocovariance carries the data's units twice over, changing from percent to decimals multiplies it by 0.0001 but leaves the autocorrelation unchanged. A frequent slip is dividing by γ(1) or by the lag-k variance instead of γ(0). Expect exam items that convert between γ and ρ or use them in the variance of a sum.

Formulas

Autocorrelation at lag k

ρ(k) = γ(k) ÷ γ(0)

γ(k)
the autocovariance at lag k
γ(0)
the variance of the series, the autocovariance at lag zero

Covariance-stationary series. The trap is dividing by γ(1) or some other lag instead of γ(0).

A practice question

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

Which expression defines the autocorrelation function of a covariance-stationary series?

  1. rho(k) = gamma(k) / gamma(1)Why not: Divides by the lag-one autocovariance; the normaliser must be the variance gamma(0) so that rho(0) = 1.
  2. rho(k) = gamma(k) x gamma(0)Why not: Multiplies instead of dividing, producing a number with squared units that is not bounded by one.
  3. rho(k) = gamma(k) / gamma(0) - correct
  4. rho(k) = gamma(k) / gamma(k-1)Why not: Divides by the previous autocovariance, which gives a successive ratio rather than a correlation with the variance as base.

The autocorrelation at lag k is the autocovariance at lag k scaled by the variance, rho(k) = gamma(k) / gamma(0). This makes it unit-free and bounded between -1 and 1.

A harder one

A stationary series has gamma(0) = 0.04, gamma(1) = 0.03 and gamma(2) = 0.012. The autocorrelation at lag 2 is closest to:

  1. 0.30 - correct
  2. 0.40Why not: Divides gamma(2) by gamma(1) = 0.03 instead of by the variance gamma(0) = 0.04.
  3. 0.012Why not: Reports the autocovariance itself rather than scaling it by the variance.
  4. 0.56Why not: Squares rho(1) = 0.75, assuming an AR(1) pattern; the question gives gamma(2) directly.

rho(2) = gamma(2) / gamma(0) = 0.012 / 0.04 = 0.30. The lag-1 value (0.75) is a distraction, and rho(1)² would only apply under an AR(1) assumption.

The hardest tier

A stationary return series has variance 9, lag-1 autocorrelation -0.4 and lag-2 autocorrelation 0.1 (all in squared percent units). The variance of the two-day sum y_t + y_(t-1) is:

  1. 18.0Why not: Adds the two variances and ignores the covariance term, as if consecutive returns were uncorrelated.
  2. 10.8 - correct
  3. 25.2Why not: Subtracts the negative covariance term, adding 7.2 to 18 instead of adding the negative value 2 x gamma(1) = -7.2.
  4. 14.4Why not: Includes the covariance only once (9 + 9 - 3.6) instead of twice.

gamma(1) = rho(1) x gamma(0) = -0.4 x 9 = -3.6. Var(y_t + y_(t-1)) = 2 gamma(0) + 2 gamma(1) = 18 - 7.2 = 10.8. The lag-2 value is not needed.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →

More in Stationary Time Series