Lag Lead Correlation
The study of lag and leads has its special significance while studying the economics and business series. In the correlation of time series the investigator may find that there is a time gap before a cause and an effect relationship is established. For e.g. The supply of a commodity may increase nowadays, but it may not have an immediate effect on prices- it may take a few days or even months for prices to adjust to the increased supply. This difference in the period before a cues and effect relationship is established is called while computing the correlation. Their gap in time must be considered otherwise fallacious conclusions may be drawn. The pairing of items is adjusted according to the time lag.
Taking the new pairs of values, the correlation can be calculating in the same manner as discussed earlier:
Illustration:
The following are the monthly figures of advertising expenditure ad sales of a firm. It is normally found that the advertising expenditure has its impact on sales normally after 2 months. Allowing for this time the lag calculates coefficient of correlation.
|
Months
|
Advertising expenditure
|
Sales
|
Months
|
Advertising
|
Sales
|
|
Jan
|
50
|
1200
|
July
|
140
|
2400
|
|
Feb
|
60
|
1500
|
Aug
|
160
|
2600
|
|
March
|
70
|
1600
|
Sep
|
170
|
2800
|
|
April
|
90
|
2000
|
Oct
|
190
|
2900
|
|
May
|
120
|
2200
|
Nov.
|
200
|
3100
|
|
June
|
150
|
2500
|
Dec.
|
250
|
3900
|
Solution:
Allow for a time lager of 2 months link advertising the expenditure of January with sales for March and so on.
Calculation of correlation coefficient:
|
month
|
Advertising expenditure X
|
(X - X) / 10X
|
X2
|
Sales Y
|
(Y-Y) / 100 y
|
Y2
|
xy
|
|
Jan
|
50
|
-7
|
49
|
1600
|
-10
|
10
|
70
|
|
Feb.
|
60
|
-6
|
36
|
2000
|
-6
|
36
|
36
|
|
March
|
70
|
-5
|
25
|
2200
|
-4
|
16
|
20
|
|
April
|
90
|
-3
|
9
|
2500
|
-1
|
1
|
3
|
|
May
|
120
|
0
|
0
|
2400
|
-2
|
4
|
0
|
|
June
|
150
|
+3
|
9
|
2600
|
0
|
0
|
0
|
|
July
|
140
|
+2
|
4
|
2800
|
+2
|
4
|
4
|
|
Aug.
|
160
|
+4
|
16
|
2900
|
+3
|
9
|
12
|
|
Sep
|
170
|
+5
|
25
|
3100
|
+5
|
25
|
25
|
|
Oct.
|
190
|
+7
|
49
|
3900
|
+13
|
169
|
91
|
|
|
ΣX = 1,200
|
ΣX= 0
|
Σx2 = 222
|
Σy =26,000
|
Σy =0
|
Σy2=364
|
Σxy = 261
|
R = Σ Xy /Σx2 x Σy2
X = 1.200 / 10 = 120
Y = 26,000/10=2,600
Σxy = 261 Σx2 = 222, Σy2 = 364
R = 261/222x 364 = 261/284.267 = + 0.918.