Question

> basicStats(sp5)

sp5

nobs 792.000000

NAs 0.000000

Minimum -0.299400

Maximum 0.422200

1. Quartile -0.020650

3. Quartile 0.037225

Mean 0.006143

Median 0.008700

Sum 4.865300

SE Mean 0.002077

LCL Mean 0.002065

UCL Mean 0.010221

Variance 0.003417

Stdev 0.058459

Skewness 0.410561

Kurtosis 9.269204

> Box.test(sp5, 20, type=’Ljung’)

Box-Ljung test

data: sp5

X-squared = 2024.8, df = 20, p-value<-2.2e-16

>adf.test(sp500)

Augmented Dickey-Fuller Test data:

sp500_training

Dickey-Fuller = -1.7877, Lag order = 6, p-value = 0.6652

alternative hypothesis: stationary

a) Based on the output, is the mean of the log return significantly different from 0.02 at 5% level? Z0.025 = 1.96, Z0.05 = 1.65? Since degree of freedom is 792-1=791, you can assume that the test statistics is approximately normal?

b)Let µ be the mean of the log return. Test to see if µ is significantly different from 0 at 5% level.

c)Let ρi be the lag i autocorrelation of the return. We are testing if all the ρi = 0. What’s the test statistics, draw your conclusion

d) What conclusions can you draw based on the Augmented Dickey-Fuller Test result?

Answer 1

As t*<-1.96, we say that the mean is significantly different from 0.02.

As t*>1.96, we conclude the mean is significantly different from 0

c)Test stat=2024.8. As pvalue<2.2e-16 we conclude that the autocorrelations are not independent. They exhibit serial correlation.

d)We see that at lag 6, pvalue>0.05, we fail to reject null hypothesis. Hence, there is a unit root at lag=6 and hence is not stationary.