# Cochrane on Return Predictability

Being a lover of Sir Arthur Conan Doyle's work, I picked up a copy of Cochrane’s 2008 paper, The Dog That Did Not Bark: A Defense of Return Predictability and read:

If returns are not predictable, dividend growth must be predictable, to generate the observed variation in divided yields. I find that the absence of dividend growth predictability gives stronger evidence than does the presence of return predictability

Given

$$\frac{DIV_1}{P_0} = r – g,$$

with expected dividend $$DIV_1$$, current share price $$P_0$$, return $$r$$ and dividend growth rate $$g$$, is Cochrane saying that since there is no statistical evidence that dividend yield $$DIV_1/P_0$$ is correlated with dividend growth rate $$g$$ when there should be, that there is a link (predictability) between dividend yield and returns?

• How comfortable, fluent are you with time-series terminology? Stationarity, ergodicity, etc...? (I'm gauging how to write my answer.) – Matthew Gunn Jun 29 '17 at 21:23
• I am familiar with basic statistical concepts, regression analysis, sequences, recurrences, limits and so on. Unfortunately, I have not had the pleasure of making the acquaintance of either of the two gentlemen you mentioned. – Anthony de Freitas Jun 29 '17 at 22:43

• Let $P_t$ be the price of the overall market index at the end of quarter $t$
• Let $D_t$ be the dividend for the overall market in quarter $t$
• Let $X_t = \frac{D_t}{P_t}$ be the dividend to price ratio.

Two key concepts in time-series statistics are stationarity and ergodicity.

If the dividends to price ratio is a stationary, ergodic process, then dividend to price ratio can't wander off arbitrarily far in some direction and stay there forever. Speaking informally, stationarity and ergodicity imply that if $X_t$ gets extremely low or extremely high, it'll tend to come back to normal (eventually).

Cochrane is an excellent writer, and you may be better off reading his explanations, but I'll take a shot at an abbreviated version for basic intuition.

### If $\frac{D_t}{P_t}$ is unusually high at time $t$, how does it return to normal?

If $\frac{D_t}{P_t}$ is a stationary ergodic process, then if $\frac{D_t}{P_t}$ is high, you can forecast that it will probably decline.

• For $\frac{D_t}{P_t}$ to come down, either $D_t$ has to decrease or $P_t$ has to increase. A high $\frac{D_t}{P_t}$ therefore implies either:
• Dividends $D_t$ decrease in the future or
• Prices $P_t$ increase in the future (i.e. there are high returns).

Cochrane gives evidence that high dividends to price ratios don't forecast changes in dividends, rather, they forecast higher returns.

Cochrane's "The Dog that Did Not Bark" argument is that since high $\frac{D_t}{P_t}$ has to forecast dividends or returns, $\frac{D_t}{P_t}$ not forecasting dividends is evidence that $\frac{D_t}{P_t}$ does forecast returns.

• Thank you, Matthew, for this exceedingly clear explanation. :) – Anthony de Freitas Jun 30 '17 at 14:07

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: $$pd_t = k + \sum_{j=0}^\infty\rho^j\Delta d_{t+1+j} - \sum_{j=0}^\infty\rho^j r_{t+1+j}$$ 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.

• Good that you mention Campbell and Shiller. These people were very important to stock return predictability research. Maybe more important (but this is controversial) than Cochrane. – Alex C Jul 3 '17 at 0:47
• Of course, they are more important! What Cochrane did is not very different from what Fama&Bliss have done for bonds in 1988: marshallinside.usc.edu/dietrich/… . But without the Campbell-Shiller decomposition it would have impossible to translate the bond intuition into stocks! – fni Jul 3 '17 at 17:38