Is this an Autoregression or OLS? Say I have a time series $y_t = \frac{1}{n} \sum_{i=1}^n x_t^{(i)}$. For example, $y_t$ can be the returns of the S&P 500 index at time $t$ and the $x^{(i)}_t$ is return of the $i^{th}$ company in the S&P 500 at time $t$ (notice this means $n = 500$). Now suppose instead of imposing a simple autoregressive structure on $y_t$, such as
$$y_t = \phi_0 + \phi_1 y_{t-1} + \phi_2 y_{t-2} \dots + e_t$$
for white noise process $e_t$, we instead have the following 
$$y_t = \phi_0 + \sum_{i=1}^n \phi^{(i)}_1 x^{(i)}_{t-1} + \sum_{i=1}^n \phi^{(i)}_2 x^{(i)}_{t-2} \dots + e_t$$
meaning each component series $x^{(i)}_t$ is allowed its own own parameter. My intuition tells me that, since the process $y_t$ is endogenous (in fact fully parametrized by) the $x^{(i)}_t$, this can't be considered a regression on exogenous regressors. So is this an autoregression? If it is an autoregression, what are the constraints that the $\phi$ parameters must satisfy for the model to be invertible?
Also, a follow up question: Suppose that $y_t = f\left(x_t^{(1)}, x_t^{(2)},\dots, x_t^{(n)}\right)$ where $f$ is not necessarily linear, but is invertible (meaning all the $x^{(i)}_t$ play a role in explaining $y_t$). Does the answer change?
 A: This type of model 
\begin{equation}
E[Y_t|, X_t, X_{t-1}, \ldots] = \beta_0 + \sum_t \beta_t X_t + \epsilon_t
\end{equation}
is technically considered a distributed lag model, at least as far as the fixed effects are concerned. Whether it's auto-regressive depends on the assumptions you make about $\epsilon_t$ conditional upon the history of $x_{s:s<t}$.
Autocorrelation means that errors are temporally correlated unconditional on any lagged effects. This temporal correlation usually follows such a format that $cor(\epsilon_{t},\epsilon_{s}) < cor(\epsilon_{t},\epsilon_{r})$ if $|t-s| < |t-r|$. Variograms are excellent visual tools to inspect the nature of the trend. Auto-regressive models use fixed lagged effects or random effects in an attempt to create independent errors. 
AR-1 autoregressive trends have a correlation of $cor(\epsilon_{t},\epsilon_{s}) = \rho^{|t-s|}$. It turns out that additionally adjusting for a single lagged effect (of $y$) suffices to produce conditionally independent errors. Or you can use a mixed model or GEE.
If, even after adjustment for the covariate history, you find residual errors which are temporally correlated, no amount of "history" of $x$ will suffice to produce conditionally independent errors. This is because the $x,y$ relationship is not a deterministic one. You will need to add a lagged outcome as a covariate to actually produce independent errors.
In the last expression of your question, you do not have autoregressive effects when the model is correctly specified. If you omit some components of $x$ in the model, a backdoor autoregressive effect is introduced because past $y$ is predicting unmeasured $x$.
