Anon

202 The Basics of financial economeTrics

In Table 10.2, we present the results of formal tests of nonstationarity.
The lag length for the augmented Dickey-Fuller test was determined by the
Bayesian criterion. The null hypothesis is that the S&P 500 stock index (div-
idends) contains a unit root; the alternative is that it does not. For both the
augmented Dickey-Fuller and the Phillips-Perron tests, the results indicate
that the S&P 500 index is nonstationary and the changes in that index are
stationary. The results for the dividends are mixed. The augmented Dickey-
Fuller statistic supports the presence of a unit root in dividends, while the
Phillips-Perron statistic does not. Since both the autocorrelation function
and the augmented Dickey-Fuller statistic conclude there is a unit root pro-
cess, we shall presume that the dividend series is nonstationary. In sum, our
analysis suggests that the S&P 500 index and dividend series each contain a
stochastic trend in the levels, but not in their first differences.
In Step 2 of the Engle-Granger conintegration test, whether the S&P
500 index and dividends are cointegrated is tested. This is accomplished
by estimating the long-run equilibrium relation by regressing the loga-
rithm (log) of the S&P 500 index on the log of the dividends. We use the
logarithms of both variables to help smooth the series. The results using
monthly data are reported in Table 10.3 for both the 1962–1982 and
1962–2006 periods. We pay little attention to the high t-statistic on the

table 10.2 Stationarity Test for the S&P 500 Index and Dividends, 1962–2006

Variable

Augumented

Dickey-Fuller Phillips-Perron

Critical Value of Test

Statistics at 1%, 5%,

10% Significance

S&P 500 1.22 1.12 −3.44 (1%)

Δ S&P 500 −19.07 −19.35 −2.87 (5%)

Dividends 1.52 4.64 −2.56 (10%)

Δ Dividends −2.13 −31.68

Notes

■ (^) Null hypothesis: Variable is nonstationary.
■ (^) The lag length for the ADF test was determined by the Bayesian criterion. For the
S&P 500 index and its first difference, the lag length was 1. For the dividends and
its first difference, the lag lengths were 12 and 11, respectively.
table 10.3 Cointegration Regression: S&P 500 and Dividends Log S&P
500 = a + b log dividends + zt
Period Constant Coefficient of Dividends t-Stat Dividends
1962–1982 4.035 0.404 17.85
1962–2006 2.871 1.336 68.54