equities - 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?

Answer

• 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.