Statistical inference under model misspecification I have a general methodological question. It might have been answered before, but I am not able to locate the relevant thread. I will appreciate pointers to possible duplicates.
(Here is an excellent one, but with no answer. This is also similar in spirit, even with an answer, but the latter is too specific from my perspective. This is also close, discovered after posting the question.)

The theme is, how to do valid statistical inference when the model formulated before seeing the data fails to adequately describe the data generating process. The question is very general, but I will offer a particular example to illustrate the point. However, I expect the answers to focus on the general methodological question rather than nitpicking on the details of the particular example.

Consider a concrete example: in a time series setting, I assume the data generating process to be
$$
y_t=\beta_0 + \beta_1 x_t+u_t \tag{1}
$$
with $u_t \sim i.i.N(0,\sigma_u^2)$. I aim to test the subject-matter hypothesis that $\frac{dy}{dx}=1$. I cast this in terms of model $(1)$ to obtain a workable statistical counterpart of my subject-matter hypothesis, and this is
$$
H_0\colon \ \beta_1=1.
$$
So far, so good. But when I observe the data, I discover that the model does not adequately describe the data. Let us say, there is a linear trend, so that the true data generating process is
$$
y_t=\gamma_0 + \gamma_1 x_t+\gamma_2 t + v_t \tag{2}
$$ 
with $v_t \sim i.i.N(0,\sigma_v^2)$.
How can I do valid statistical inference on my subject-matter hypothesis $\frac{dy}{dx}=1$?


*

*If I use the original model, its assumptions are violated and the estimator of $\beta_1$ does not have the nice distribution it otherwise would. Therefore, I cannot test the hypothesis using the $t$-test.

*If, having seen the data, I switch from model $(1)$ to $(2)$ and change my statistical hypothesis from $H_0\colon \ \beta_1=1$ to $H'_0\colon \ \gamma_1=1$, model assumptions are satisfied and I get a well-behaved estimator of $\gamma_1$ and can test $H'_0$ with no difficulty using the $t$-test.
However, the switch from $(1)$ to $(2)$ is informed by the data set on which I wish to test the hypothesis. This makes the estimator distribution (and thus also inference) conditional on the change in the underlying model, which is due to the observed data. Clearly, the introduction of such conditioning is not satisfactory.
Is there a good way out? (If not frequentist, then maybe some Bayesian alternative?)
 A: The way out is literally out of sample test, a true one. Not the one where you split sample into training and hold out like in crossvalidation, but the true prediction. This works very well in natural sciences. In fact it's the only way it works. You build a theory on some data, then you're expected to come up with a prediction of something that was not observed yet. Obviously, this doesn't work in most social (so called) sciences such as economics.
In the industry this works as in sciences. For instance, if the trading algorithm doesn't work, you're going to lose money, eventually, and then you abandon it. Cross validation and training data sets are used extensively in development and making a decision to deploy the algorithm, but after it's in production it's all about making money or losing. Very simple out of sample test.
A: You could define a "combined procedure" and investigate its characteristics. Let's say you start from a simple model and allow for one, two or three more complex (or nonparametric) models to be fitted in case that the simple model doesn't fit. You need to specify a formal rule according to which you decide not to fit the simple model but one of the others (and which one). You also need to have tests for your hypothesis of interest to be applied under all the involved models (parametric or nonparametric).
With such a setup you can simulate the characteristics, i.e., with what percentage your null hypothesis is finally rejected in case it is true, and in case of several deviations  of interest. Also you can simulate from all involved models, and look at things such as conditional level and conditional power given that data came from model X, Y, or Z, or given that the model misspecification test procedure selected model X, Y, or Z. 
You may find that model selection doesn't do much harm in the sense that the achieved level is still very close to the level you were after, and the power is OK if not excellent. Or you may find that data-dependent model selection really screws things up; it'll depend on the details (if your model selection procedure is very reliable, chances are level and power won't be affected very strongly). 
Now this isn't quite the same as specifying one model and then looking at the data and deciding "oh, I need another", but it's probably as close as you can get to investigating what would be the characteristics of such an approach. It's not trivial because you need to make a number of choices to get this going.
General remark: I think it is misleading to classify applied statistical methodology binarily into "valid" and "invalid". Nothing is ever 100% valid because model assumptions never hold precisely in practice. On the other hand, although you may find valid (!) reasons for calling something "invalid", if one investigates the characteristics of the supposedly invalid approach in depth, one may find out that it still works fairly well.
