LO 3.3.2.E

Learning Objective: Recognize and apply the formula for the autocorrelation function.

Review:

Autocorrelation function

For a stationary time series {Yt} with constant expectation and time independent covariance

The autocorrelation function ACF for this time series is defined by

$<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mi>k</mi></msub><mo>=</mo><mfrac><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mfenced><mrow><msub><mi>X</mi><mi>t</mi></msub><mo>,</mo><mo> </mo><msub><mi>X</mi><mrow><mi>t</mi><mo>-</mo><mi>k</mi></mrow></msub></mrow></mfenced></mrow><msqrt><mi>V</mi><mi>a</mi><mi>r</mi><mfenced><msub><mi>X</mi><mi>t</mi></msub></mfenced><mi>V</mi><mi>a</mi><mi>r</mi><mfenced><msub><mi>X</mi><mrow><mi>t</mi><mo>-</mo><mi>k</mi></mrow></msub></mfenced></msqrt></mfrac><mo>=</mo><mfrac><msub><mi>γ</mi><mi>k</mi></msub><msub><mi>γ</mi><mn>0</mn></msub></mfrac><mo> </mo><mo> </mo><mo> </mo><mtext>for the lag  </mtext><mi>k</mi><mo>≥</mo><mn>0</mn><mo> </mo><mtext>and </mtext><mo> </mo><msub><mi>ρ</mi><mrow><mo>-</mo><mi>k</mi></mrow></msub><mo>=</mo><msub><mi>ρ</mi><mi>k</mi></msub></math>$

For a non-stationary time series {Yt} the covariance is not independent of t

The autocorrelation function ACF for this time series is

$<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>k</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>=</mo><msqrt><mfrac><mrow><mi>t</mi><mo>-</mo><mi>k</mi></mrow><mi>t</mi></mfrac></msqrt><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi>i</mi><mi>f</mi><mo> </mo><mi>t</mi><mtext> is large relative to</mtext><mo> </mo><mi>k</mi><mo>,</mo><mo> </mo><mtext>then</mtext><mo> </mo><msub><mi>ρ</mi><mrow><mi>k</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>≈</mo><mn>1</mn></math>$