Why don't Time Series books talk about exogeneity like in classical linear regression models?

Question

Consider the following linear regression: $$y_t = \beta_0 + \beta_1 x_{t} + u_t$$ Typically, we need to assume (assuming a random sample):

\begin{equation}\label{I}\tag{I} E[u_t]=0, \quad cov(u_t ,x_t)=0 \end{equation} to conclude that the OLS estimator of $\beta= (\beta_0, \beta_1)$ is unbiased. It's also possible to show consistence. We can also show that $E[u_t|x_t]= 0$ implies (\ref{I}).

Now, consider an $AR(1)$ process (Here we don't have a random sample. We can have dependence): $$y_t = \mu + \alpha y_{t-1}+ \epsilon_t$$ Note that making $x_t := y_{t-1}$, we are in the a linear regression context. But I can't understand why classic time series books don't talk about exogeneity assumption $E[u_t|x_t]=$ or even (\ref{I}). They just say that $(u_t)$ is a white noise process.

Answer 1

1. Saying $u_t$ is white notes implies that $E[u_t,x_t]=0$, and this will hold even if hypothetically there is $x_t$ that explains $y_t$ as you can have still a special cases where omitted variable is uncorrelated with error term (even though extremely rare).

2. The reasons why endogeneity is not discussed in basic time series are:

• Time series are typically taught after cross-sectional analysis so typically writers assume you already know the basics.

• Time series starts with univariate time series models as you for example show with your AR(1) example. In univariate time series models endogeneity is not an issue as rarely there is reverse relationship between past and future realization of the same variable.

• Time series is often used for forecasting and in forecasting model you don't care about bias as you have no interest in knowing what true relationships are. Running naive OLS might have forecasting value even if it offers not information about causal relationships.

However, most textbooks will at some point discuss endogeneity in time series context. If your textbook has chapter on Vector-Autoregression (VAR) I am 100% sure it will discuss endogeneity.