Math 2.0

Statistics Definition

Partial autocorrelation function

Builds on Autocorrelation functionStationary processOrthogonal projectionCorrelation coefficientAutoregressive model

Definition

Let (Xt)t∈ℤ(X_t)_{t \in \Z} be a real weakly stationary process with mean μ\mu and autocovariance γ\gamma, with γ(0)>0\gamma(0) > 0. Let L2L^2 be the space of square-integrable random variables. For t∈ℤt \in \Z and h≥2h \ge 2, let Pt,hYP_{t,h} Y be the orthogonal projection in L2L^2 of YY onto the span of 1,Xt+1,…,Xt+h−11, X_{t+1}, \dots, X_{t+h-1}. So Pt,hYP_{t,h} Y is the best linear predictor of YY from these variables. The partial autocorrelation function α\alpha is defined for h≥1h \ge 1 by:

  1. α(1)=ρ(1)\alpha(1) = \rho(1), where ρ\rho is the autocorrelation function.
  2. For h≥2h \ge 2, α(h)\alpha(h) is the correlation coefficient of Xt+h−Pt,hXt+hX_{t+h} - P_{t,h} X_{t+h} and Xt−Pt,hXtX_t - P_{t,h} X_t.

By stationarity, α(h)\alpha(h) does not depend on tt.

Remarks

Equivalently, α(h)=ϕhh\alpha(h) = \phi_{hh}, the last coefficient of the best linear predictor μ+∑j=1hϕhj(Xt+h−j−μ)\mu + \sum_{j=1}^{h} \phi_{hj} (X_{t+h-j} - \mu) of Xt+hX_{t+h} from Xt+h−1,…,XtX_{t+h-1}, \dots, X_t, provided the h×hh \times h matrix [γ(i−j)][\gamma(i - j)] is nonsingular. The sample partial autocorrelation is obtained by putting the sample autocovariances into these formulas, for instance by the Durbin–Levinson recursion.

Example: let Xt=ϕXt−1+εtX_t = \phi X_{t-1} + \varepsilon_t be a stationary autoregressive model of order 11 with ∣ϕ∣<1\abs{\phi} < 1 and uncorrelated noise εt\varepsilon_t. Then α(1)=ρ(1)=ϕ\alpha(1) = \rho(1) = \phi. For h≥2h \ge 2, the best linear predictor of Xt+hX_{t+h} from Xt+1,…,Xt+h−1X_{t+1}, \dots, X_{t+h-1} is ϕXt+h−1\phi X_{t+h-1}, and the residual εt+h\varepsilon_{t+h} is uncorrelated with Xt−Pt,hXtX_t - P_{t,h} X_t, so α(h)=0\alpha(h) = 0.

Non-example: a random walk is not weakly stationary, so clause 1 and clause 2 do not define a function of the lag alone.

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  1. Autocorrelation function concept
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  3. Correlation coefficient concept
  4. Orthogonal projection concept
  5. Stationary process concept
  6. Partial autocorrelation function definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Partial_autocorrelation_function
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DefinesPartial autocorrelation function
UsesAutocorrelation function, Stationary process, Orthogonal projection, Correlation coefficient, Autoregressive model

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