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In preparation for my exam, I'm trying to interpret the residuals in order to understand if the time series has been modelled correctly. Otherwise, I have to suggest an improvement.


Here is the text:

The time series of interest is given by the daily log returns of Volkswagen stock, from 2015-01-02 to 2017-05-26. Figure 1 shows the data together with the estimated autocorrelation and partial autocorrelation functions. The time series has been modelled as a white noise:

Call:
arima(x = dlVOW.F, order = c(0, 0, 0))

Coefficients:
      intercept
         -4e-04
s.e.      9e-04

sigma^2 estimated as 0.0005142: log likelihood = 1441.81, aic = -2881.63

The diagnostics are the following:

Figure 1: Plot of the observed data (daily log returns of Volkswagen stock) Figure 1

Figure 2: Some residual diagnostics Figure 2

Figure 3: Mc Leod Li test on residuals Figure 3

Figure 1 shows a cluster of volatility on "Oct 01 2015" and a lit bit of it between "Jan 04 2016" and "Apr 01 2016".

This is also confirmed by the plot of the standardized residuals (Figure 2 - a) because the spread of the residuals, between the values "200" and "300", is different from the other ones, which are more centred at zero-mean.

On the other hand, the ACF of residuals looks white noise because only 2 lags are outside the bounds, so the model seems suitable. (Is correct this statement?)

The Ljung-Box test (Figure 2 - c) shows that 3 p-values (6% of 50 lags) are below the red line. Since the confidence interval is 95%, I reject the null hypothesis: there is a little bit of serial correlation.
(Is correct to reject the null hypothesis when only 6% of the p-values are below the red line?) (If the series shows some volatility, is still valid the Ljung Box test which is based on the homoskedasticity assumption?)

Finally, looking at Figure 3, I reject the null hypothesis of the Mc Leod Li test, meaning that there is autoregressive conditional heteroskedasticity in this time series (ARCH effect).

To sum up, I suggest, as a model improvement, an ARCH(p).


Was my interpreting wrong? what can you suggest me to improve this analysis?

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