In absence of a constant, no need for $Cov(Y_i,\epsilon_i)$ just $E(Y_i\epsilon_i)$ We have the following simple linear model:
$C_i=kY_i+\epsilon_i$, where $\epsilon_i$ is the error term. 
In a book I'm reading, the author states that due to the absence of a constant, we do not need to check $Cov(Y_i,\epsilon_i)$ just $E(Y_i\epsilon_i)$ to determine whether the regressor is predetermined/uncorrelated with the error term.
Why is that?
Any help would be appreciated.
 A: "Predetermined" is the term used in econometrics to allow for correlation of a regressor on past errors, but not with current ones. Technically this is indeed an assumption of orthogonality.
In the specific example of the question, "predetermined income of period $t$" means that current income is not affected by any shock ($u_t$) that may occur at period $t$, but it allows for the possibility that current income has been affected by shocks that occurred in the previous periods.  
This is a relaxation of "strict exogeneity", that increases the realism of assumptions -since it permits to have causal or association links along the arrow of time (which is intuitive), at the price of losing the unbiasedness property. At the same time it permits the estimator to remain a consistent one.  
As regards the effects of an absent constant term in relation to whether the regressor is predetermined or not, I think Hayashi attempts here to avoid confusion since up to that point in the chapter he talks about "covariance", and also to link with the discussion in ch. 2 about why the inclusion of a constant term makes the assumption of a zero-mean error essentially a tautology (any non-zero mean of the error is included in the constant term). But in the specific example he uses, the assumption is that the error term is zero-mean, even though the constant term is absent, in which case "orthogonality" and zero-covariance coincide, since if $E(u_t) = 0$ we have
$${\rm Cov}(Y_t, u_t) = E(Y_tu_t) - E(Y_t)E(u_t) = E(Y_tu_t)$$
Note that the desired OLS estimator property of consistency does not require zero covariance but orthogonality. I.e. even if $E(u_t) \neq 0$ but we assume that $E(Y_tu_t) = 0$, the OLS estimator will be consistent, even though we may have that ${\rm Cov}(Y_t, u_t) \neq 0$. So, to return to Hayashi, he is actually saying "Since in almost all cases we assume a zero mean error, due the presence of a constant term, we are used to talk about predetermined regressors in terms of zero-covariance. But if the error term is not assumed to be zero-mean, then the covariance may be non-zero. But what we need is orthogonality, not zero-covariance, so when the constant term is absent, remember to discuss orthogonality, not covariance".
