# Endogeneity and the AR model?

an AR(p) process is given by:

$$y_t=\phi_0+\phi_1y_{t-1}+\phi_2y_{t-2}+...+\phi_{t-p}y_{t-p}+u_t$$

In such a model we have endogenous variables. My question is, why is this not an issue when dealing with time series data.

It is not necessarily true that lagged $y$ are endogenous. That depends entirely on the structure you assume for $u_t$. If $u_t$ is white noise, there is no endogeneity, in the sense that

$$E(u_t|\mathbf{y}) = 0$$

where $\mathbf{y}=\{y_{t-1}, \cdots , y_{t-p}\}$.

This is trivial to prove, once you realise that the above equality can also be expressed as

$$E(u_t y_{t-i})=0 \,\,\quad, i=\{1, \cdots, p\}$$

using the Law of Total Expectations.

Conversely, it is trivial to show that if $u_t$ follows an autoregressive process, the above conditions does not hold anymore, in which case there is endogeneity, and OLS estimates are inconsistent.

• That takes care of endogeneity with the error term and $y_t$, but wouldn't there be multicolinearity among the independent variables? – EconJohn Jul 6 '17 at 16:31
• @EconJohn There is, but that does not invalidate the endogeneity condition, which is what makes OLS inconsistent (and what you were asking for, as far as I understood). Multicolinearity by itself is not a problem (unless there is perfect colinearity, which seems weird in TS, unless the series is a constant, and so unfit for the model). Its actually part of the reason why TS usually gives so high $R^2$s (since dependent and independent are likely to be highly correlated), compared with cross-section. – luchonacho Jul 6 '17 at 16:40
• @EconJohn The existence of multicolinearity is the very reason why we use multiple regression (many explanatory variables) and not just simple regression (one explanatory variable). If you meant to refer to "severe-perfect" multicolinearity that creates technical issues as regards the execution of the LS estimator, this is a per-case issue. – Alecos Papadopoulos Jul 6 '17 at 18:18