Cointegration 201
In accordance with Step 1 of the four-step Engle-Granger conintegra-
tion test, it is necessary to establish the nonstationarity of the two vari-
ables—stock prices as proxied by the S&P 500 and dividends. To identify
nonstationarity, we will use both formal and informal methods. The first
informal test consists of analyzing the plot of the series shown in Figure
10.1. Neither series appears to exhibit mean reversion. The dividend series
wanders less from its mean than the stock price series. Nevertheless, neither
series appears stationary.
The second informal method involves examining the autocorrelation
function. Table 10.1 shows the autocorrelation function for 36 lags of the
S&P 500 index and the dividends for the 1962–2006 period using monthly
data. The autocorrelations for the early lags are quite close to 1. Furthermore,
the autocorrelation function exhibits a slow decay at higher lags. This provides
sufficient evidence to conclude that stock prices and dividends are nonstation-
ary. When we inspect the autocorrelation function of their first differences
(not shown in Table 10.1), the autocorrelation of the first lag is not close to
one. A fair conclusion is that the series are stationary in the first differences.
table 10.1 Autocorrelation Functions of the S&P 500 Index and Dividends
Lag Autocorrelation 1 2 3 4 5 6 7 8
S&P 500 Index 0.993 0.986 0.979 0.973 0.967 0.961 0.954 0.948
Dividends 0.991 0.983 0.974 0.966 0.958 0.979 0.941 0.933
Lag Autocorrelation 9 10 11 12 13 14 15 16
S&P 500 Index 0.940 0.933 0.926 0.918 0.911 0.903 0.896 0.889
Dividends 0.925 0.916 0.908 0.900 0.891 0.883 0.876 0.868
Lag Autocorrelation 17 18 19 20 21 22 23 24
S&P 500 Index 0.881 0.874 0.866 0.858 0.851 0.843 0.835 0.827
Dividends 0.860 0.852 0.845 0.837 0.830 0.822 0.815 0.808
Lag Autocorrelation 25 26 27 28 29 30 31 32
S&P 500 Index 0.819 0.811 0.804 0.796 0.789 0.782 0.775 0.768
Dividends 0.801 0.794 0.788 0.781 0.775 0.769 0.763 0.758
Lag Autocorrelation 33 34 35 36
S&P 500 Index 0.761 0.754 0.748 0.741
Dividends 0.753 0.747 0.743 0.738