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Outline autocorrelation and partial autocorrelation as measured on a sample.

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

The idea

Sample statistics estimate the theoretical functions from data. The sample autocorrelation at lag k is ACF = Σ(y(t) − ȳ)(y(t − k) − ȳ) ÷ Σ(y(t) − ȳ)². The numerator sums over the available pairs, and the denominator sums over all T observations. The partial autocorrelation at lag k is the correlation between y(t) and y(t − k) after removing the effect of the lags in between. Equivalently it is the coefficient on y(t − k) in a regression of y(t) on lags 1 to k. At lag 1 the two coincide. At lag 2 the partial autocorrelation can be found from the first two autocorrelations as PACF = (ρ2 − ρ1²) ÷ (1 − ρ1²). Under white noise the sample autocorrelations are approximately normal, with a standard error given by SE = 1 ÷ √T. Values outside plus or minus 1.96 × SE are significant at 5 percent. Patterns guide identification. An AR(p) has a partial autocorrelation that cuts off after p, and an MA(q) has an autocorrelation that cuts off after q.

Formulas

Partial autocorrelation at lag 2

PACF(2) = (ρ₂ − ρ₁²) ÷ (1 − ρ₁²)

ρ₁
the autocorrelation at lag 1
ρ₂
the autocorrelation at lag 2

Strips out the part of ρ₂ explained through lag 1. At lag 1 the partial and ordinary autocorrelations are equal.

A practice question

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

The partial autocorrelation at lag k measures:

  1. the total correlation between y_t and y_(t-k), including indirect effects through intermediate lagsWhy not: Describes the ordinary autocorrelation, which includes the indirect effects the partial version removes.
  2. the correlation between y_t and the shock at time t - kWhy not: Relates the series to a shock, not to its own lagged value; neither the ACF nor the PACF uses shocks.
  3. the average correlation over lags 1 to kWhy not: The PACF is the direct effect at lag k alone; it is not an average over lags.
  4. the correlation between y_t and y_(t-k) after removing the effect of the intermediate lags - correct

The PACF at lag k is the coefficient on y_(t-k) in a regression of y_t on lags 1 through k. It isolates the direct link at lag k, whereas the ACF also includes indirect transmission.

A harder one

The sample autocorrelations of a series at lags 1 and 2 are 0.60 and 0.50. The sample partial autocorrelation at lag 2, (rho2 - rho1²) / (1 - rho1²), is closest to:

  1. 0.50Why not: Reports the lag-2 autocorrelation as the partial autocorrelation, without removing the effect of lag 1.
  2. 0.22 - correct
  3. 0.14Why not: Computes rho2 - rho1² but forgets to divide by 1 - rho1².
  4. -0.10Why not: Subtracts rho1 from rho2 instead of rho1², mis-stating the formula.

(0.50 - 0.36) / (1 - 0.36) = 0.14 / 0.64 = 0.219. The partial autocorrelation strips out the part of rho2 explained through lag 1.

The hardest tier

The observations of a short series are 2, 4, 3, 5, 4, 6 (mean 4). Using the sample autocorrelation formula (sum of products of deviations for lag 1 divided by the sum of squared deviations over all observations), the lag-1 sample autocorrelation is:

  1. -0.17Why not: Divides by the sum of squared deviations of only the first five observations (6) rather than all six (10).
  2. 0.10Why not: Gets the sign wrong; the deviations -2, 0, -1, 1, 0, 2 give a negative sum of products of -1.
  3. -0.10 - correct
  4. -0.20Why not: Divides the sum of products by 5, giving an autocovariance-style number, rather than by the total sum of squares.

Deviations: -2, 0, -1, 1, 0, 2. Lag-1 products: 0, 0, -1, 0, 0 sum to -1. Sum of squares = 4 + 0 + 1 + 1 + 0 + 4 = 10. rho-hat(1) = -1 / 10 = -0.10.

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