# The Dog That Did Not Bark?

I've been reading Cochrane's 2006 paper "The Dog that did not bark: A Defense of Return Predictability", but i am still struggling to understand what the dog was, and why it wasn't barking?

If anyone could shed some brief intuition that would be appreciated. Perhaps my knowledge of existing literature, in order to identify the 'dog' is lacking.

• Google "the Adventure of Silver Blaze" for the Sherlock Holmes mystery. A prize racehorse goes missing. The absence of the bark suggests the thief was known... "Is there any point to which you would wish to draw my attention?' 'To the curious incident of the dog in the night-time.' 'The dog did nothing in the night-time.' 'That was the curious incident,' remarked Sherlock Holmes.” Jan 8, 2020 at 2:43

As it was pointed above the phrase is taken from Sherlock Holme's novel. It describes the case when the dog should have bark, but didn't. Now if we come to the Cochrane paper. He introduces the system of equations ($$r_{t+1}$$ - returns, $$\Delta d_{t+1}$$ - dividend growth and $$d_t - p_t$$ - dividend-price ratio): $$r_{t+1} = a_r + \beta_r(d_t - p_t) + \epsilon^r_{t+1}, \\ \Delta d_{t+1} = a_d + \beta_d(d_t - p_t) + \epsilon^d_{t+1}, \\ d_{t+1} - p_{t+1} = a_{dp} + \phi(d_t - p_t) + \epsilon^{dp}_{t+1}.$$
He argues that if you just test $$H_0: \beta_r = 0$$, basically it tests the predictability of returns and you won't find significance. However, if you jointly test $$H_0: \beta_r = 0\land \beta_d = \rho\phi - 1$$ (I'll explain later from where it comes from). This new null gives your more "power" to reject the null (although Cochrane is wrong in the definition of power since $$power = \mathbb{P}[ \text{reject H}_0 | \text{H}_A]$$, but now it's not that important). To construct this null he uses Campbell-Shiller 1988 log-linearization for the returns to obtain:
$$r_{t+1} \approx \kappa + \rho(p_{t+1} - d_{t+1}) + \Delta d_{t+1} - (p_t - d_t),$$ where $$\kappa$$ - cosntant and $$\rho$$ - point of log-linearization. From this equation and previous system we can form the following identities:
$$\beta_r = 1 + \beta_d - \rho\phi, \\ a_r = \kappa + a_d - \rho a_{dp}, \\ \epsilon^r_{t+1} = \epsilon^d_{t+1} - \rho\epsilon^{dp}_{t+1}.$$
And now comes the most important part. In order to have the $$\beta_r = 0$$ we have to have $$\beta_d = \rho\phi - 1 \approx -0.1$$, but this is supported much less by the data and we adress the absence of this coefficient in the data. And here $$\hat{\beta}_d = 0$$ (estimated in the data) represents the dog that didn't bark!