Question

The CEO of JP Morgan analyzed the data for exchange rates between Japanese Yen and US Dollars for the past 12 months and found the first three autocorrelation coefficients to be 0.75, 0.37 and 0.10 respectively (i.e. r1 = 0.75, r2 = 0.37 and r3 = 0.10). Based on his findings, he believes that the exchange rate can be predicted using lagged data. (a) Set up a hypothesis and test for the significance of r1 (i.e. H0 : r1 = 0). What is your conclusion? (b) Set up a joint hypothesis to test H0 : r1 = r2 = r3 = 0 and compute associated LBQ statistic. What is your conclusion on the hypothesis? (c) Based on your findings, what are your suggestions to the CEO?

Answer 1

For the 12 month data for exchange rates between Japanese Yen and US Dollars, the first three auto-correlation coefficients are:

r1 = 0.75

r2 = 0.37 and

r3 = 0.10

n=12

(a) H0 : r1 = 0

Ha : r1 is not equal to 0

The degree of freedom is n-2 = 12-2 = 10

r1=0.75

The t-statistic is: t = r1*sqrt(n-2)/sqrt(1-r1*r1)

Hence, t-statistic = 0.75*sqrt(10)/sqrt(1-0.75*0.75) = 3.585686

Using =tdist(3.585686,10,2) in excel, p-value evaluates to 0.00496, indicating that at 5% level of significance, actual p-value of 0.00496 is less than 0.05 and hence, null hypothesis H0 : r1 = 0 is rejected

Thus, statistically speaking, at 5% level of significance, the r1 of 0.75 is significantly different from 0

(b) H0 : r1 = r2 = r3 = 0

Ha : At least r1 or r2 or r3 is not equal to 0

The Ljung-Box test statistic is given by

Q=n(n+2)*sum(rho_k^2/(n-k)), the sum goes from k=1 to 3

Where,

rho_1=r1;

rho_2=r2;

rho_3=r3

It is one-tailed test

Hence,

Q = 12(12+2)*((0.75*0.75/(12-1))+(0.37*0.37/(12-2))+(0.10*0.10/(12-3))

Q=11.0774

The critical Chi-square at 3 degree of freedom is 0.351846 at 5% level of significance

Q=11.0774 is greater than 0.351846 at 5% level of significance, thereby indicating that null hypothesis H0 : r1 = r2 = r3 = 0 is rejected.

Thus, statistically speaking, at 5% level of significance, there is at least one auto-correlation that is statistically significantly different from 0.

(c) It shows that there is presence of auto-correlation in the data, the CEO should be advised to use first or second or third difference series of data for forecasting and use generalized least square method for correcting for auto-correlation