Interpretation of lasso coefficients So I have generally been using linear regression for the analysis which I have been doing at work, but have recently been introduced to Lasso/Elastic/Ridge methods. I understand Lasso penalises coefficients, but I wanted to know what this has on the meaningfulness of these coefficients. As an example
Y~3A+2.5B+error
We can say that a single unit increase in A causes a 3 times unit increase in Y for a linear model. Now when I run the Lasso, I get different coefficients values. Am I still interpreting them in the same way as I have in the above? So If my new model is
Y~1.5A+0.7B-2C
Would I still be able to say an increase in one unit A causes a 1.5 increase in Y? 
 A: No, this is surely incorrect. Abstracting from all sorts of issues when forming causal statements (e.g. endogeneity) in the linear regression model (as mentioned in the comment correlation is not causation), LASSO, like any other penalized estimator induces bias due to the penalty term that is applied to regression coefficients. A bit more formally, suppose
$$ y \in \mathbf{R}^{n\times 1} \text{ response} \qquad X_{k} \in \mathbf{R}^{n\times 1} \text{ where } k\in\{1, \dots, p\} \text{ $k$th covariate }$$
And you have LASSO regression
$$ \hat\beta \in \underset{\beta\in \mathbf{R}^{p}}{ \text{arg min}}\frac{1}{n}\|y-X\beta\|^2_2 + 2\lambda \|\beta\|_1, \quad \lambda > 0.$$
It is easy to see that for $\lambda \neq 0$, $\hat\beta$ is biased. For the linear model, you can correct the bias induced by penalization by adding the following quantity
$$ \check\beta = \hat \beta + \hat\Theta X^{\top}(y-X\hat\beta)/n  $$
where $\hat\Theta$ is the estimated precision matrix. $\check\beta$ is shown (e.g. [1]) to be asymptotically normal (under certain conditions and $\lambda$ choices), so you can compute p-values for each coefficient. See the reference for more details. An R package called $\texttt{hdi}$ implements the procedure.
Hope this helps!
Reference (among several other references)
[1] Van de Geer, S., Bühlmann, P., Ritov, Y. A., & Dezeure, R. (2014). On asymptotically optimal confidence regions and tests for high-dimensional models. Annals of Statistics, 42(3), 1166-1202. https://projecteuclid.org/download/pdfview_1/euclid.aos/1403276911
A: Yes, the interpretation of the coefficients stays the same. You have to remember than "All models are wrong, but some are useful”. If the cross-validated accuracy is higher for the model with a penalty than for the standard linear model, then the model is a more accurate representation of the data.
