Short Term Correlation
The Correlation can be calculated by the following steps:
1) At first determine the trend values by the moving average method.
2) Deduct from the actual values the corresponding trend values obtained in step1; this would give the short-term fluctuation. Denote these short-term fluctuation by the symbol x for X series and y for the Y series.
3) Now square the short- term fluctuations of X series and obtain the total Σx2.
4) Then square the short-term fluctuation of Y series and obtain the total Σ y2.
5) After that multiply x with y for each value and obtain the total Σxy.
6) Now apply the formula
R = (Σx y)/(Σx2 xΣy2)
Here x denotes the deviation of X series from the moving average and not from the arithmetic mean. Similarly, Y denotes deviation of y series from moving average and not from the arithmetic mean.
Thus we find that the only difference between correlation and correlation in short-term changes is that, in the former we take deviations from arithmetic mean, and in the later we take deviations from the trend values.
Note:
It must be carefully noted that x and y in the above formula are different from the x and y of the Pearson formula
Illustration:
Calculate the Karl Pearson's coefficient of correlation of the short-term oscillations for the index of supply and price of certain commodity given here
|
Year
|
Index of supply
|
Index of price
|
Year
|
Index of supply
|
Index of price
|
|
1995
|
91
|
117
|
2003
|
104
|
77
|
|
1996
|
98
|
97
|
2004
|
98
|
93
|
|
1997
|
95
|
102
|
2005
|
100
|
89
|
|
1998
|
92
|
108
|
2006
|
108
|
83
|
|
1999
|
93
|
105
|
2007
|
116
|
78
|
|
2000
|
96
|
96
|
2008
|
114
|
84
|
|
2001
|
102
|
77
|
2009
|
111
|
93
|
|
2002
|
107
|
68
|
|
|
|
Take 5-year moving average and ignore the decimals while computing the average.
R = [(Σxy)/(Σx2xΣy2)]
Solution:
The calculation of the coefficient of correlation of short-term oscillations for the indexes of supply and price are:
|
Year
|
Index of supply
|
5- yearly moving average X
|
Deviation a actual values form moving X
|
Square of deviations X2
|
Index of price
|
5-yearly moving average Y
|
Deviation of actual values form moving y
|
Square of deviation Y2
|
Product of deviations Xy
|
|
1995
|
91
|
-
|
-
|
-
|
117
|
-
|
-
|
-
|
-
|
|
1996
|
98
|
-
|
-
|
-
|
97
|
-
|
-
|
-
|
-
|
|
1997
|
95
|
93.8
|
+1.2
|
1.44
|
102
|
105.8
|
-3.8
|
14.44
|
-4.56
|
|
1998
|
92
|
94.8
|
-2.8
|
7.84
|
108
|
101.6
|
+6.4
|
40.96
|
-17.92
|
|
1999
|
93
|
95.6
|
-2.6
|
6.76
|
105
|
97.6
|
+7.4
|
54.76
|
-19.24
|
|
2000
|
96
|
98.0
|
-2.0
|
4.00
|
96
|
90.8
|
+5.2
|
27.04
|
-10.40
|
|
2001
|
102
|
100.4
|
+1.6
|
2.56
|
77
|
84.6
|
-7.6
|
57.56
|
-12.16
|
|
2002
|
107
|
101.4
|
+5.6
|
31.36
|
68
|
82.2
|
-14.2
|
201.64
|
-79.52
|
|
2003
|
104
|
102.2
|
+1.8
|
3.24
|
77
|
80.0
|
-3.8
|
14.44
|
-6.84
|
|
2004
|
98
|
103.4
|
-5.4
|
29.16
|
93
|
82.0
|
+11.0
|
121.00
|
-59.40
|
|
2005
|
100
|
105.2
|
-5.2
|
27.04
|
89
|
84.0
|
+5.0
|
25.00
|
-26.00
|
|
2006
|
108
|
107.2
|
+0.8
|
0.64
|
83
|
85.4
|
-2.4
|
5.76
|
-1.92
|
|
2007
|
116
|
109.8
|
+6.2
|
38.44
|
78
|
85.4
|
-7.4
|
54.76
|
-45.88
|
|
2008
|
114
|
-
|
-
|
-
|
84
|
-
|
-
|
-
|
-
|
|
2009
|
111
|
-
|
-
|
-
|
93
|
-
|
-
|
-
|
-
|
|
|
|
|
|
Σx2 = 152.84
|
|
|
|
Σy2 = 617.56
|
Σxy = -283.84
|
Σ xy = - 283.84 Σ x2 = 152.48, Σ y2 = 617.56
Substituting the values r = - 283.84/ 152.84x 617.56
Log r = log 283.84 - ½ [log 152.84 + log 617.56] = 2.4531 - ½ [2.1831 + 2.7907]
=2.4531 - ½ [4.9738]
=2.4537 - 2.4869
Log r = - 0.0338 = 1.9662.
R = AL 1.9662 = 0.925
R = 0.925.