<ol>
<li><p>Saying <span class="math-container">$u_t$</span> is white notes implies that <span class="math-container">$E[u_t,x_t]=0$</span>, and this will hold even if hypothetically there is <span class="math-container">$x_t$</span> that explains <span class="math-container">$y_t$</span> as you can have still a special cases where omitted variable is uncorrelated with error term (even though extremely rare).</p>
</li>
<li><p>The reasons why endogeneity is not discussed in basic time series are:</p>
</li>
</ol>
<ul>
<li><p>Time series are typically taught after cross-sectional analysis so typically writers assume you already know the basics.</p>
</li>
<li><p>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.</p>
</li>
<li><p>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.</p>
</li>
</ul>
<p>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.</p>

CC BY-SA 4.0
2023-03-14T14:35:35.020Z
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.
Richard Hardy
CC BY-SA 4.0
2023-03-14T14:36:48.257Z
(I understand the sympathy for causal analysis, but I think it is only fair and sensible to regard statistical/probabilistic analysis as scientific, too.)
Richard Hardy
CC BY-SA 4.0
2023-03-14T14:41:31.380Z
@RichardHardy thanks for the comment I edited my answer
1muflon1
CC BY-SA 4.0
2023-03-14T16:06:28.197Z
@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 $E[u_t x_t]=0$ ?
André Goulart
CC BY-SA 4.0
2023-03-14T17:02:39.497Z
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).
mark leeds
CC BY-SA 4.0
2023-03-14T17:16:47.967Z
@AndréGoulart 1. I am referring to x being able to explain y. You can always drop unrelated x from regression as for any x that cannot explain y you won't get cov(x,u)=/=0. If x is unrelated to y then always cov(x,u)=0.  2. No that is not what I meant to say white noise is rather by construction already created such that it will always be unrelated to x but if unless x can explain y then this is not an issue in first place
1muflon1
CC BY-SA 4.0
2023-03-14T18:06:25.350Z
@AndréGoulart, let me repeat myself: $\{u_t\}$ being white noise says nothing about the relationship between $\{u_t\}$ and other processes such as $\{x_t\}$ or $\{y_t\}$. The answer got this wrong.
Richard Hardy
CC BY-SA 4.0
2023-03-14T19:21:11.630Z
@RichardHardy but I didnt say that in my answer read it carefully, it is written in a way to mean even if there is relationship white noise means that any covariaties are uncollected. I am certainly not saying that white noise implies any sort of relationship. If the answer is not clear perhaps you could suggest how I could clarify it given the info in this comment
1muflon1
CC BY-SA 4.0
2023-03-14T19:51:18.077Z
I read your point 1, and I see no ambiguity in it; it is clearly incorrect. The term *white noise* characterizes $\{u_t\}$ alone; it does not characterize the relationship of $\{u_t\}$ to other variables. Your point 1 talks about that relationship. I noted this problem in my first comment.
Richard Hardy
CC BY-SA 4.0
2023-03-14T19:56:52.783Z
@RichardHardy but I am not saying it does. I will try to rewrite it to be more clear
1muflon1
CC BY-SA 4.0
2023-03-14T20:20:16.583Z
I still find point 1 to be incorrect, as you keep interpreting the term *white noise* as to refer to a relationship between $\{u_t\}$ and other variables/processes... $\{u_t\}$ can be white noise at the same time as $\text{Corr}(u_t,x_t)\neq 0$. Heck, $x_t$ could be a deterministic function of $u_t$, and even that would not remove the property of $\{u_t\}$ being white noise.
Richard Hardy
CC BY-SA 4.0
2023-03-14T20:36:20.113Z
@RichardHardy how exactly you define white noise? Because I was taught in my graduate econometrics courses (perhaps incorrectly or perhaps I am misremembering) that white noise is just sequence of random variables generated by some distribution (e.g. gaussian white noise is just sequence of numbers randomly drawn from N(0,sigma). This is also how the term was used in our textbook (Hamilton time series analysis e.g. see page 25). How can you then say that white noise can be completely deterministic function of x?
1muflon1
CC BY-SA 4.0
2023-03-15T07:30:12.567Z
I am saying $x_t$ may be a deterministic function of $u_t$ where I am taking $u_t$ as the primitive, not the other way around. Though if $\{x_t\}$ is itself white noise and $u_t=2x_t$, then $\{u_t\}$ is white noise, too. White noise is a process that has zero mean, constant variance and zero autocovariance at all lags except for the 0th lag. (It does not have to be i.i.d. nor normally distributed.) See e.g. Brockwell & Davis "Time Series: Theory and Methods" (2nd edition, 2009) p. 78 or Hamilton "Time Series Analysis" (1994) p. 47.
Richard Hardy
CC BY-SA 4.0
2023-03-14T13:16:24.017Z
2023-03-14T19:58:27.920Z
2023-03-14T19:58:27.920Z
1muflon1
<p>Consider the following linear regression:
<span class="math-container">$$y_t = \beta_0 + \beta_1 x_{t} + u_t$$</span>
Typically, we need to assume (assuming a random sample):</p>
<p><span class="math-container">\begin{equation}\label{I}\tag{I}
E[u_t]=0, \quad cov(u_t ,x_t)=0
\end{equation}</span>
to conclude that the OLS estimator of <span class="math-container">$\beta= (\beta_0, \beta_1)$</span> is unbiased. It's also possible to show consistence. We can also show that <span class="math-container">$E[u_t|x_t]= 0$</span> implies (\ref{I}).</p>
<p>Now, consider an <span class="math-container">$AR(1)$</span> process (Here we don't have a random sample. We can have dependence):
<span class="math-container">$$y_t = \mu + \alpha y_{t-1}+ \epsilon_t$$</span>
Note that making <span class="math-container">$x_t := y_{t-1}$</span>, we are in the a linear regression context. But I can't understand why classic time series books don't talk about exogeneity assumption <span class="math-container">$E[u_t|x_t]=$</span> or even (\ref{I}). They just say that <span class="math-container">$(u_t)$</span> is a white noise process.</p>

<regression><time-series><autoregressive>
Why don't Time Series books talk about exogeneity like in classical linear regression models?
André Goulart
# Why don't Time Series books talk about exogeneity like in classical linear regression models?

(Asked by: username_0 on 2023-03-14)

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.

**Tags:** regression, time-series, autoregressive

---

## 1 answers

### Answer by username_1 on 2023-03-14. Score: -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.

> **Comments:**
> **username_2**: 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.  
> **username_2**: (I understand the sympathy for causal analysis, but I think it is only fair and sensible to regard statistical/probabilistic analysis as scientific, too.)  
> **username_1**: @RichardHardy thanks for the comment I edited my answer  
> **username_0**: @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 $E[u_t x_t]=0$ ?  
> **username_3**: 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).  
> **username_1**: @AndréGoulart 1. I am referring to x being able to explain y. You can always drop unrelated x from regression as for any x that cannot explain y you won't get cov(x,u)=/=0. If x is unrelated to y then always cov(x,u)=0.  2. No that is not what I meant to say white noise is rather by construction already created such that it will always be unrelated to x but if unless x can explain y then this is not an issue in first place  
> **username_2**: @AndréGoulart, let me repeat myself: $\{u_t\}$ being white noise says nothing about the relationship between $\{u_t\}$ and other processes such as $\{x_t\}$ or $\{y_t\}$. The answer got this wrong.  
> **username_1**: @RichardHardy but I didnt say that in my answer read it carefully, it is written in a way to mean even if there is relationship white noise means that any covariaties are uncollected. I am certainly not saying that white noise implies any sort of relationship. If the answer is not clear perhaps you could suggest how I could clarify it given the info in this comment  
> **username_2**: I read your point 1, and I see no ambiguity in it; it is clearly incorrect. The term *white noise* characterizes $\{u_t\}$ alone; it does not characterize the relationship of $\{u_t\}$ to other variables. Your point 1 talks about that relationship. I noted this problem in my first comment.  
> **username_1**: @RichardHardy but I am not saying it does. I will try to rewrite it to be more clear  
> **username_2**: I still find point 1 to be incorrect, as you keep interpreting the term *white noise* as to refer to a relationship between $\{u_t\}$ and other variables/processes... $\{u_t\}$ can be white noise at the same time as $\text{Corr}(u_t,x_t)\neq 0$. Heck, $x_t$ could be a deterministic function of $u_t$, and even that would not remove the property of $\{u_t\}$ being white noise.  
> **username_1**: @RichardHardy how exactly you define white noise? Because I was taught in my graduate econometrics courses (perhaps incorrectly or perhaps I am misremembering) that white noise is just sequence of random variables generated by some distribution (e.g. gaussian white noise is just sequence of numbers randomly drawn from N(0,sigma). This is also how the term was used in our textbook (Hamilton time series analysis e.g. see page 25). How can you then say that white noise can be completely deterministic function of x?  
> **username_2**: I am saying $x_t$ may be a deterministic function of $u_t$ where I am taking $u_t$ as the primitive, not the other way around. Though if $\{x_t\}$ is itself white noise and $u_t=2x_t$, then $\{u_t\}$ is white noise, too. White noise is a process that has zero mean, constant variance and zero autocovariance at all lags except for the 0th lag. (It does not have to be i.i.d. nor normally distributed.) See e.g. Brockwell & Davis "Time Series: Theory and Methods" (2nd edition, 2009) p. 78 or Hamilton "Time Series Analysis" (1994) p. 47.  

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