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.


1 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.

  • $\begingroup$ Re 1., the notion that $\{u_t\}$ is white noise says nothing about the relationship between $\{u_t\}$ and other processes such as $\{x_t\}$ or $\{y_t\}$. It only defines how $\{u_t\}$ behaves over time. Re 2., true relationships can be statistical/probabilistic or causal. You seem to be implicitly neglecting/excluding the former sense of the term, and that might mislead some readers. You also seem to be implying that such relationships have zero scientific value, which I would disagree with. Causal analysis is not the only form of scientific analysis there is, within economics or outside. $\endgroup$ Mar 14, 2023 at 14:35
  • $\begingroup$ (I understand the sympathy for causal analysis, but I think it is only fair and sensible to regard statistical/probabilistic analysis as scientific, too.) $\endgroup$ Mar 14, 2023 at 14:36
  • $\begingroup$ @RichardHardy thanks for the comment I edited my answer $\endgroup$
    – 1muflon1
    Mar 14, 2023 at 14:41
  • $\begingroup$ @1muflon1 When you say that if $x_t$ explains $y_t$ what are you referring to? Would it be the case of $cov(x_t, y_t)\neq 0$? So, I would have to show that, under the assumption $cov(x_t, y_t)\neq 0$, $\{u_t\}$ white noise implies $E[u_t x_t]=0$ ? $\endgroup$ Mar 14, 2023 at 16:06
  • $\begingroup$ Hi: The problem with the AR(1) being estimated using OLS is that the response, $y_t$ is correlated with PAST $\epsilon_t$. So, the problem is one of $cov(y_t, past~ u_t)$ not equalling zero. The question of exogeneity usually comes up in systems of equations ( VARS) as 1muflon1 alluded to but it's not really relevant in the AR(1). $\endgroup$
    – mark leeds
    Mar 14, 2023 at 17:02

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