advanced-algorithmic-trading

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Coefficients:
ma1 intercept
-0.7298 0.0486
s.e. 0.1008 0.0246
sigma^2 estimated as 0.7841: log likelihood = -130.11, aic = 266.23
ˆβ 1 = −0.730, which is a small underestimate of β 1 = −0.6.
confidence interval:
Finally, let us calculate the
> -0.730 + c(-1.96, 1.96)*0.1008
[1] -0.927568 -0.532432
We can see that the true parameter value of β 1 = −0.6 is contained within the 95% confidence
interval, providing us with evidence of a good model fit.
MA(3)
We can run through the same procedure for a MA(3) process. This time we should expect
significant peaks at k ∈ {1, 2, 3}, and insignificant peaks for k > 3.
We are going to use the following coefficients: β 1 = 0.6, β 2 = 0.4 and β 3 = 0.3. Let us
simulate a MA(3) process from this model. I have increased the number of random samples to
1000 in this simulation, which makes it easier to see the true autocorrelation structure at the
expense of making the original series harder to interpret:
> set.seed(3)
> x <- w <- rnorm(1000)
> for (t in 4:1000) x[t] <- w[t] + 0.6*w[t-1] + 0.4*w[t-2] + 0.3*w[t-3]
> layout(1:2)
> plot(x, type="l")
> acf(x)
The output is given in Figure 10.12.
As expected the first three peaks are significant. However, so is the fourth. But we can
legitimately suggest that this may be due to sampling bias as we expect to see 5% of the peaks
being significant beyond k = q.
We can now fit a MA(3) model to the data to try and estimate parameters:
> x.ma <- arima(x, order=c(0, 0, 3))
> x.ma
Call:
arima(x = x, order = c(0, 0, 3))
Coefficients:
ma1 ma2 ma3 intercept
0.5439 0.3450 0.2975 -0.0948
s.e. 0.0309 0.0349 0.0311 0.0704
sigma^2 estimated as 1.039: log likelihood = -1438.47, aic = 2886.95