0.1 Autocovariance and Autocorrelation Function
For a time series \(\{X_t\}\) with \(\mathbb{E}X_t^2 < \infty\), the autocovariance function is \[ \gamma_X(t,s) = Cov(X_t, X_s) = \mathbb{E}[X_tX_s] - \mathbb{E}X_t\mathbb{E}X_s. \] If \(\{X_t\}\) is weakly stationary, \(\gamma_X\) depends only on the lag \(h=t-s\), written \(\gamma_X(h) = Cov(X_{t+h}, X_t)\).
\[ \rho_X(h) = \frac{\gamma_X(h)}{\gamma_X(0)} = Cor(X_{t+h}, X_t). \] Note \(\rho_X(0)=1\) and \(|\rho_X(h)|\leq 1\) for all \(h\) (Cauchy-Schwarz).
0.2 Sample Estimation
Given observations \(x_1, \dots, x_n\), the sample ACVF and sample ACF are \[ \hat\gamma(h) = \frac{1}{n}\sum_{t=1}^{n-h}(x_{t+h}-\bar x)(x_t - \bar x), \qquad \hat\rho(h) = \frac{\hat\gamma(h)}{\hat\gamma(0)}. \]
If \(\{X_t\}\) is stationary with i.i.d. innovations and \(n\) is large, \(\hat\rho_h := (\hat\rho(1),\dots,\hat\rho(h))'\) is approximately \(\mathcal{N}(\rho_h, n^{-1}W)\) with \[ w_{ij} = \sum_{k=1}^\infty \{\rho(k+i)+\rho(k-i)-2\rho(i)\rho(k)\}\{\rho(k+j)+\rho(k-j)-2\rho(j)\rho(k)\}. \] This gives the standard \(\pm 1.96/\sqrt{n}\) significance bounds used in ACF plots to test for white noise (also see Ljung-Box Test).
The Best Linear Predictor of a stationary series is expressed entirely in terms of \(\gamma_X\). The Fourier transform of \(\gamma_X\) is the spectral density — see Spectral Distributions and Periodogram and Spectral Estimation.
0.3 References
- Time Series Analysis
- [[pstat274-time-series-summary-notes]]