Is it appropriate to estimate a random slope without estimating the overall mean slope? I am trying to estimate whether there are differences in how individuals in different cities (my grouping variable) respond to a few predictor variables. So, in practice, I am interested in learning about the $\beta$s from each city. But, I want to use random slopes so that these estimates are "shrunk" towards some group mean.
I am fitting my model using stan_glmer from the stanarm package, which accepts input in the same way as glmer from lme4. 
So, question #1. If I don't care about the grand slope, is it ok to specify the model like this:
stan_glmer(Y ~ (V1 + V2 + V3|city))

Or do I need to include V1, V2, V3 as "fixed effects" first? e.g.
stan_glmer(Y ~ V1 + V2 + V3 + (V1 + V2 + V3|city))

Secondly, if the second model is the correct option (which I somewhat suspect), then how do I interpret the output? In order to get an estimate for the slope of V1 in city 1, do I add the overall $\beta_{v1}$ to city 1's $\beta_{v1}$? How do you account for error in that case?
Thanks!
 A: Fitting random slopes with the population-level slope fixed to zero is not out of the question - it's not mathematically or statistically ill-posed - but it's a rather weird model that would require some extra justification. Why would you expect that the average slope across cities would be exactly zero (which is what is implied by the model that omits the fixed effect)? The only cases where I've seen fitting such models make sense are 


*

*as a(n) (admittedly silly) null model, for doing a likelihood-ratio test of the significance of the population-level slope [not relevant in your case as you're using Bayesian methods]

*in cases where the effect is zero based on the experimental design, e.g. when samples are randomly assigned to test and treatment conditions in a pre-treatment condition (this would be eliminating a fixed effect of treatment in the "before" period, not a fixed slope, but the idea is similar).


If you have a fixed-effect slope and among-city variation in the slope, you do indeed need to add the population-level slope to the individual-city slope deviation, and use the posterior distribution of the sum for inference - I don't know exactly how this is done in rstanarm (related) ... the tidybayes package might help.
