Figure 2. Scatter Plot between X and Y
To be more convenient, we use the following notation to calculate the k th order of auto-correlation coefficient 3 .
∑ ( 𝑖 𝑖=𝑘+1
− ̅ )( 𝑖−𝑘
− ̅ )
𝑘
=
= 1, 2, 3 …
∑ ( 𝑖 𝑖=1
− ̅ ) 2
Therefore the above obtained coefficient is r 1 autocorrelation coefficient. Similarly we obtain r 2
which is the first order of
=0.0805, r 3
=0.0002 and r 4
=-
0.073 by letting 𝑖
= 𝑖+2
, 𝑖
= 𝑖+3
, 𝑖
= 𝑖+4
respectively.
To test whether sample correlation coefficient is significantly different from zero, we conduct the following test. The standard error ( SEr ) of the autocorrelation coefficient (r k ) is given:
2 𝑘−1
1 + 2 ∑ 𝑘 𝑖=1
𝑆𝐸( ) = √
The null and alternative hypothesis are shown below to test whether the r k
order auto-
correlation coefficient is significantly different from zero.
𝐻 0 : ≠ 0 The t statistic can be calculated (t statistic can be found in S4) − 𝑘 𝑆( ) For the up to 4 th order of auto-correlation, we obtain : 𝑘 = 0 . 𝐻 1 = 𝑘
1 290
−0.0440 0.0579
1
= −0.0440,
𝑆(1) = √
= 0.0587, =
= −0.749
3 Hanke, J. E., Reitsch, A. G. and Wichern, D. (2001) Business forecasting. United States: Prentice Hall
3
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