Consider the daily log returns of EXXON-Mobil, S\&P composite index, and Apple stock used in Example 7.6. Fit a dynamically orthogonal component (DOC) model to the series. Write down the fitted model. Is the model adequate? Why?

Data from Example 7.6

Figure 7.17

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Example 7.6 As a second example, we consider the daily log returns, in per- centages, of EXXON-Mobil stock, S&P composite index, and Apple stock from September 4, 2007 to September 28, 2012 for 1280 observations. The simple returns are from CRSP. The left panel of Figure 7.16 shows the time plots of the daily log return series. As expected, the volatilities of the three assets were high during the 2008 financial crisis and the variability of the Apple stock returns appears to be higher than those of the other two series. Let z denote the daily log returns, in percentages. Analysis of individual series indicates some minor serial correla- tions in the Apple stock returns. However, for simplicity, we continue to assume that the mean equation of z is just a constant. This provides an estimate of the (d) -15 Inrtn 0 10 (b) -10 05 (c) 2008 2009 2010 2011 2012 Year -20 -5 2008 2009 2010 2011 2012 Year 2008 2009 2010 2011 2012 Year TTT 02 -4 eta_t A 2008 2009 2010 2011 2012 Year TTTTTTTT 2 1- -4 eta_t -4 0 4 2008 2009 2010 2011 2012 Year TTTTT 2008 2009 2010 2011 2012 Year