ACF - Autocorrelation Function

• ACF, also called correlogram, helps to determine q parameter
• ACF is defined as if time series is stationary

MA Model

$$ MA(q):Y_{t}=\mu+\phi_1\varepsilon_{t-1}+\phi_2\varepsilon_{t-2}+...+\phi_q\varepsilon_{t-q}+\varepsilon_t $$

Auto-covariance & Auto-correlation

• Covariance

$$ cov(X,Y)=E\{(X-\mu_{X})(Y-\mu_{Y})\} $$

• Correlation

$$ \frac{cov(X,Y)}{\sigma_{X}\sigma_{Y}} $$

• Autocovariance

$$ cov(X_{t},X_{t+k}) $$

• Autocorrelation

$$ \frac{{cov(X_t,X_{t-k})}}{\sigma_{X}(t)\sigma_{X}(t-k)} $$

ACF

$$ \rho_k=corr(X_t,X_{t-k}) $$

ACF Ploting & Determination

• Ignore $t_{0}$
• Assign q to be the maximum non-zero lag