Linear model with both additive and multiplicative effects In linear regression, the independent variables have an additive effect on the response (level-level regression):
$y=\beta_0+\beta_1x+\epsilon$
In a log-level regression, the independent variables have an additive effect on the log-transformed response and a multiplicative effect on the original untransformed response:
$log(y)=\beta_0+\beta_1x+\epsilon$
Suppose that I know for each predictor if it has an additive or multiplicative effect on the response and that I need to estimate these effects through ordinary least squares. How can I specify the formula of the model so that I estimate correctly these effects?
 A: You can use Linear Regression to model any linear/non-linear relationship using basis expansion (slides from Elements of Statistical Learning). In your case you could probably exponentiate some of the variables, but it might be preferable to use an automatic method, such as Multivariate Adaptive Regression Splines, that still provides interpretable results.
A: I don't believe it's possible in general to do this with ordinary least squares, since OLS is at heart a trick to calculate $\hat \beta$ in $E[y] = \bf{X}\hat\beta$ using matrix division.
It can be done more generally, though.
I think the tricky bit is figuring out exactly what you mean by each predictor having an additive or multiplicative effect on the response. For example, with two predictors, do you mean:
$$
y = (\beta_0 \times \beta_2 x_2) + \beta_1 x_1 ?\\
y = (\beta_0 + \beta_1 x_1) \times \beta_2 x_2 ?\\
y = \beta_0 + (\beta_1 x_1 \times \beta_2 x_2) ? \\
$$
...and there's probably others as well. Of these, the first  (multiplication before applying the additive effects) is the simplest to estimate, as it has fewer high-order multiplicative terms, and is more likely to correspond to the model you intended.
Unfortunately, even this isn't simple to estimate, since the predictions with $\beta_0 = 2, \beta_2 = 2$ → $y = (2 \times 2 \times x_2) + \beta_1 x_1$
are the same as those when
$\beta_0 = 1, \beta_2 = 4$ → $y = (1 \times 4 \times x_2) + \beta_1 x_1$.
The best way around this is to use
a Bayesian estimation tool like Stan
to set reasonable priors on your model parameters
(for example that the multiplicative effect, $\beta_2$, should be close to 1),
and find the best parameter estimates that are consistent with those priors.
