Error term interpretation in the Cox PH model I am preparing a presentation on Survival Analysis models, with specific focus on the Cox model.
Suppose I am in the simple bivariate survival analysis case (with $x_1$ and $x_2$).
The Cox model functional form for the hazard rate is:
$$
h(t| x_1, x_2) = h_0(t) \exp(x_1 \beta_1 + x_2 \beta_2).
$$
My question is: if you had to explain to graduate students with a relatively good understanding of Econometrics/Statistics how the error term enters in the functional form, how would you do? 
 A: This is a very old question, but still doens't have an answer.
The simple answer to the question is that in its original formulation (both in discrete time and later in continuous time) the Cox model does not have an error term.
This implies that all sources of individual heterogeneity are captured by the vector of observable characteristics, $X$.
This is clearly extremely restrictive as we know that in reality exist many sources of variation in the hazard that are not observable. Lancaster (1979) generalized the Cox model to include unobserved heteorgeneity (frailty) by introducing a multiplicative error term (and assuming a parametric form for it). 
$$
h(t| x, v) = h_0(t) \exp(X \beta)\cdot v
$$
This mixed proportional hazard model has been modified in several ways over the years. Econometricians, in particular, don't like functional form assumptions, and a large literature have gone in the direction of relaxig the assumptions relative to the specification of the error term $v$ and as far as possible also keeping $h_0(t)$ semi- or non-parametrically identified. 
See Hausman and Woutersen (2014) introduction for an excellent review with references to key related papers. 
The paper itself provides a nice example of a recent result in this area of econometrics. The mixed proportional hazard is uniquely identified from the data without resorting to any functional form assumptions on $v$ nor $h_0(t)$. The main assumptions to achieve this are the PH assumption and the existence of time-varying exgoenous regressors (i.e. uncorrelated with $v$), whose variation is used to identify all other model components.
The use of time-varying covariates for indentifications dates back to older works by Heckman and Honore' (you can find references to their work in the Hausman and Woutersen (2014) paper as well).
Note that a causal interpretation of the right-hand-side regressors is challenging, 
since it's typically hard to believe in the independence between right-hand side regressors and $v$ (as this is a fundamentally untestable assumption).
A: You seem to be confusing the Cox proportional hazards model with a linear Gaussian model.  The Cox model has no need for an error term.  A consequence of this is that an omitted variable can destroy $\beta$s that are in the model, unlike normal regression when a terms is omitted that is orthogonal to other terms.
