Maybe I am a little bit late to the party, but I want to give a shot. As in Campbell and Shiller, start from the identity $R_{t+1}\equiv\frac{P_{t+1}+D_{t+1}}{P_t}$ where $R_{t+1}$ is the gross return between time $t$ and $t+1$, and $P_t$ is the price at time $t$. Rearrange the relationship as $R_{t+1} =\frac{D_{t+1}}{D_t}\frac{\left(1+\frac{P_{t+1}}{D_{t+1}}\right)}{\frac{P_t}{D_t}}$. Take logs of both sides and obtain:
$$r_{t+1}=\Delta d_{t+1}-pd_t+log\left(1+e^{pd_{t+1}}\right)\approx k+\Delta d_{t+1}-pd_t+\rho pd_{t+1}$$ where $r_{t+1}=logR_{t+1}$, $\Delta d_{t+1}=log \left(\frac{D_{t+1}}{D_t}\right)$, $pd_t=log\left(\frac{P_t}{D_t}\right)$ and $k$ and $\rho$ are constants coming from the Taylor expansion of $log\left(1+e^x\right)\approx k+\rho x$.For more details check here.
You can rearrange and iterate forward to obtain:
\begin{equation}
pd_t = k + \sum_{j=0}^\infty\rho^j\Delta d_{t+1+j} - \sum_{j=0}^\infty\rho^j r_{t+1+j}
\end{equation}
From here you can compute
$$var\left(pd_t\right)=Cov\left(pd_t,\sum_{j=0}^\infty\rho^j\Delta d_{t+1+j}\right)-Cov\left(pd_t,\sum_{j=0}^\infty\rho^j r_{t+1+j}\right)$$ and therefore the key relationship $$1=\frac{Cov\left(pd_t,\sum_{j=0}^\infty\rho^j\Delta d_{t+1+j}\right)}{var\left(pd_t\right)}-\frac{Cov\left(pd_t,\sum_{j=0}^\infty\rho^j r_{t+1+j}\right)}{var\left(pd_t\right)}=\beta_{d}-\beta_r$$ where $\beta_d$ is the OLS estimate of a regression of al future dividend growths on $pd_t$ and $\beta_r$ of all log-returns on $pd_t$.
Cochrane's starting point is exactly $1=\beta_d-\beta_r$. Any hypothesis regarding $\beta_r$, i.e. whether returns are predictable using the price-dividend ratio, generates a "twin" hypothesis regarding $\beta_d$, i.e. whether dividend growth can be predicted using the price-dividend ratio!
So for instance, let's assume that returns are indeed NOT predictable, i.e. $\beta_r=0$, then we whould expect $\beta_d=1+\beta_r=1$. A test of no return predictability implies a joint test of $\beta_r=0$ AND $\beta_d=1$.
Cochrane does exactly this. He simulates jointly $\beta_r$ and $\beta_d$ and shows that under the null of $\beta_r=0$ the coefficient of $\beta_d$ is much more extreme than what we observe in the data. Quoting the paper:
In sum, the lack of dividend forecastability in the data gives far stronger
statistical evidence against the null than does the presence of return forecastability.[...]Under the
unforecastable-return null, we see return forecast coefficients as large or
larger than those in the data about 20% of the time and a t-statistic as large
as that seen in the data about 10% of the time. However, I find that the
absence of dividend–growth forecastability offers much more significant
evidence against the null. The best overall number is a 1–2% probability
value (last row of Table 5)—dividend growth fails to be forecastable in only
1–2% of the samples generated under the null. The important evidence, as
in Sherlock Holmes’s famous case, is the dog that does not bark.
