advanced-algorithmic-trading

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Figure 10.6: Correlogram of First Order Differenced Daily Logarithmic Returns of AMZN Closing
Prices.
[,1] [,2]
[1,] 4.59499e-04 1.19519e-05
[2,] 1.19519e-05 4.59499e-04
Fitting the ar autoregressive model to the first order differenced series of log prices produces
an AR(2) model, with ˆα 1 = −0.0278 and ˆα 2 = −0.0687. I have also output the aysmptotic
variance so that we can calculate standard errors for the parameters and produce confidence
intervals. We want to see whether zero is part of the 95% confidence interval, as if it is, it
reduces our confidence that we have a true underlying AR(2) process for the AMZN series.
To calculate the confidence intervals at the 95% level for each parameter, we use the following
commands. We take the square root of the first element of the asymptotic variance matrix to
produce a standard error, then create confidence intervals by multiplying it by -1.96 and 1.96
respectively, for the 95% level:
> -0.0278 + c(-1.96, 1.96)*sqrt(4.59e-4)
[1] -0.0697916 0.0141916
> -0.0687 + c(-1.96, 1.96)*sqrt(4.59e-4)
[1] -0.1106916 -0.0267084
Note that this becomes more straightforward when using the arima function, but we will wait
until the next chapter before introducing it properly.
Thus we can see that for α 1 zero is contained within the confidence interval, while for α 2 zero
is not contained in the confidence interval. Hence we should be very careful in thinking that we
really have an underlying generative AR(2) model for AMZN.