<p>Weak stationarity and weak dependence are complementary conditions.</p>
<p>A weakly stationary time series <span class="math-container">$y_t$</span> has an underlying statistical process which is time-invariant.</p>
<p>This is characterised by three conditions:</p>
<ol>
<li><span class="math-container">$$E(y_t)= \mu$$</span></li>
<li><span class="math-container">$$E((y_t-\mu)^2)=\sigma^2 &lt;\infty$$</span></li>
<li><span class="math-container">$$Cov(y_t,y_{t+s})=Cov(y_t,y_{t-s})$$</span></li>
</ol>
<p>Condition (1) indicates that a weakly stationary series has an underlying (or long-run equilibrium) unconditional average.</p>
<p>Condition (2) indicates that the unconditional variance of a weakly stationary series does not change through time and remains within a certain range/finite.</p>
<p>Condition (3) indicates that a weakly stationary series has stable (symmetrical) dynamics between periods, which only depend on the number of periods, rather than time itself.</p>
<p>That is to say, Condition (3) states that a weakly stationary process we can assume a stable relationship between certain periods, regardless of whether it's the year 2000 or the year 3000. Time plays no role, only the relationship across periods.</p>
<p><strong>Weak Dependency:</strong></p>
<p>Weak dependency ensures mean reversion back to an underlying process.</p>
<p>Weak dependency states that the correlation between <span class="math-container">$y_t$</span> and future periods <span class="math-container">$y_{t+h}$</span>, tends to zero sufficiently quickly. And with periods increasing to infinity, dependency should tend to zero.</p>
<p><span class="math-container">$$Corr(y_t,y_{t+h})\rightarrow 0, h\rightarrow\infty$$</span></p>
<p><strong>Importance in the Context of Time Series</strong></p>
<p>With time series data, our central objective remains to estimate the magnitude of change in Y terms of linear unidirectional causation from X.</p>
<p>That is to say, our goal is to take a sample time series and fit a line which best explains the change in Y given X characteristic of the underlying population.</p>
<p>Yet, in the time series context, conducting estimation representative of the underlying population is undermined by a failure of i.i.d conditions</p>
<p>An evolving (non-stationary) time series, entails that observations will not be drawn from an identical distribution and parameter estimates will not convey a constant linear (time-invariant) relationship on Y given X;</p>
<p>Furthermore, given that time series data is necessarily non-independently distributed (i.e. current realisations dependent on past realisations), then again parameter estimates will no longer convey information on the underlying population.</p>
<p>It follows that two mutually dependent properties must be upheld to ensure the i.i.d conditions known as:</p>
<ul>
<li><p>Sufficiently fast weak dependency – random variables should exhibit
finite memory/asymptotic independence;</p>
</li>
<li><p>Weak stationarity – the underlying statistical properties (mean,
variance, and covariance) should be time invariant as if drawn from
an identical distribution.</p>
</li>
</ul>
<p>Note that both conditions are mutually dependent, given that sufficiently quick finite memory is a requirement for a process to exhibit mean-reversion back to some underlying population.</p>
<p>Moreover, the combination of weak dependency and weak stationarity ensure that a property known as ergodic stationarity is upheld.</p>
<p>Essentially, ergodic stationarity implies that with a quick return to an identical distribution, then realisations are representatives of the greater process, and averaging is relevant.</p>
<p>Further, weak stationarity is important in terms of the exogeneity condition to control for a causal relationship, given that if contemporaneous exogeneity holds in one period; then by weak stationarity, it should hold for all periods.</p>
<p>Also, ergodic stationarity ensure that the process mimics a random independent sample required to uphold LoLN/consistent estimation/CLT are valid.</p>
<p>Finally, weak stationarity overall avoids the problem of so-called spurious regression in which two mutually trending series are wrongly identified to have a causal relationship.</p>

CC BY-SA 4.0
2021-03-02T21:48:10.557Z
Perhaps "complementary conditions" could be rephrased? "Complementary" could be read (in the set theory/probability sense) as "exhaustive and mutually exclusive", so that  every process is either one or the other (but not both). Though I don't have an immediate suggestion to improve it.
Silverfish
CC BY-SA 4.0
2021-03-02T22:59:35.140Z
@Silverfish. Completely valid feedback on the terminology. I am of course open to a replacement, given that 'complementary' was used to reflect common parlance.
EB3112
CC BY-SA 4.0
2021-03-03T09:37:46.910Z
@EB3112 thank you so much for the reply. Very much clearer now
CWK
CC BY-SA 4.0
2021-03-03T13:14:52.460Z
I'm afraid re "complementary" that the best alternative I can think of off-hand is "related" which I don't think captures all you intended, but on the other hand isn't open to the same kind of confusion either...
Silverfish
CC BY-SA 4.0
2021-03-03T14:05:41.083Z
To be more precise, condition 3. should hold for every $s$.
Richard Hardy
CC BY-SA 4.0
2021-09-01T14:23:31.017Z
@EB3112: +1 what is a textbook where what you wrote about is best described? I haven't seen this explained well anywhere.
ColorStatistics
CC BY-SA 4.0
2021-09-01T17:44:59.997Z
Hi @ColorStatistics. Unfortunately, I can't think of a single textbook which best describes it. Instead, I was in a similar position to the question's author a few years ago, and I kept digging around to find more about these properties. However, I would argue that Martin Nyboe Tabor from Copenhagen Uni has some very good lectures. https://www.youtube.com/c/MortenNyboeTabor/videos
EB3112
CC BY-SA 4.0
2021-09-01T17:48:58.987Z
@ColorStatistics. The overarching point I was trying to get across in the post, is that time series variables (such as GDP), often do not change from period-to-period drastically. Instead, they are highly persistent (such as a unit root process). For example, a unit root AR (1) $y_t = y_{t-1} +e_t$ means that the LHS and RHS are equal, only except for a random error. This means that the information we have from the past (despite it being very persistent), is a poor indicator of the future.
EB3112
CC BY-SA 4.0
2021-03-02T11:13:42.713Z
2021-03-02T11:57:10.053Z
2021-03-02T11:57:10.053Z
EB3112
<p>I am doing an undergraduate course in econometrics where we are using the text Introduction to Econometrics by Dougherty. While going through time series, it was mentioned that one of the necessary assumption to justify OLS, is weakly dependence, which then states that this concept will be covered in more detail in next chapter, which after completing is primarily about stationarity and no mention about weak dependence. I am now wondering if weak dependence and stationarity are similar or related?</p>
<p>In a particular textbook, Introduction to Econometrics by Wooldridge, he states that weak dependence and stationarity are different concepts. While he states that weak dependence is a necessary assumption for OLS, stationarity is not but however needed for the Central Limit Theory to hold. And Wooldridge also defines an integrated process as highly dependent process that can be transformed into weakly dependent. Whereas Dougherty (and Gujarati) links integrated process with stationarity.</p>
<p>Could someone clarify the concepts of stationarity vs weak dependence and how they fit into OLS and also the definition of integrated process.</p>

CC BY-SA 4.0
2021-03-02T11:18:24.080Z
(+1): Some very related questions from CV: (1) https://stats.stackexchange.com/questions/325015/how-are-ergodicity-and-weak-dependence-related?rq=1; (2) https://stats.stackexchange.com/questions/439076/does-a-martingale-difference-sequence-mds-imply-strong-mixing?rq=1
Dayne
<econometrics><time-series><nonexpert>
Stationarity vs weak dependence
CWK
# Stationarity vs weak dependence

(Asked by: username_0 on 2021-03-02)

I am doing an undergraduate course in econometrics where we are using the text Introduction to Econometrics by Dougherty. While going through time series, it was mentioned that one of the necessary assumption to justify OLS, is weakly dependence, which then states that this concept will be covered in more detail in next chapter, which after completing is primarily about stationarity and no mention about weak dependence. I am now wondering if weak dependence and stationarity are similar or related?

In a particular textbook, Introduction to Econometrics by Wooldridge, he states that weak dependence and stationarity are different concepts. While he states that weak dependence is a necessary assumption for OLS, stationarity is not but however needed for the Central Limit Theory to hold. And Wooldridge also defines an integrated process as highly dependent process that can be transformed into weakly dependent. Whereas Dougherty (and Gujarati) links integrated process with stationarity.

Could someone clarify the concepts of stationarity vs weak dependence and how they fit into OLS and also the definition of integrated process.

**Tags:** econometrics, time-series, nonexpert

> **Comments:**
> **username_1**: (+1): Some very related questions from CV: (1) https://stats.stackexchange.com/questions/325015/how-are-ergodicity-and-weak-dependence-related?rq=1; (2) https://stats.stackexchange.com/questions/439076/does-a-martingale-difference-sequence-mds-imply-strong-mixing?rq=1  

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## 1 answers

### Answer by username_2 on 2021-03-02. Score: 6 | Accepted answer

Weak stationarity and weak dependence are complementary conditions.

A weakly stationary time series $y\_t$ has an underlying statistical process which is time-invariant.

This is characterised by three conditions:

1. $$E(y\_t)= \mu$$
2. $$E((y\_t-\mu)^2)=\sigma^2 <\infty$$
3. $$Cov(y\_t,y\_{t+s})=Cov(y\_t,y\_{t-s})$$

Condition (1) indicates that a weakly stationary series has an underlying (or long-run equilibrium) unconditional average.

Condition (2) indicates that the unconditional variance of a weakly stationary series does not change through time and remains within a certain range/finite.

Condition (3) indicates that a weakly stationary series has stable (symmetrical) dynamics between periods, which only depend on the number of periods, rather than time itself.

That is to say, Condition (3) states that a weakly stationary process we can assume a stable relationship between certain periods, regardless of whether it's the year 2000 or the year 3000. Time plays no role, only the relationship across periods.

**Weak Dependency:**

Weak dependency ensures mean reversion back to an underlying process.

Weak dependency states that the correlation between $y\_t$ and future periods $y\_{t+h}$, tends to zero sufficiently quickly. And with periods increasing to infinity, dependency should tend to zero.

$$Corr(y\_t,y\_{t+h})\rightarrow 0, h\rightarrow\infty$$

**Importance in the Context of Time Series**

With time series data, our central objective remains to estimate the magnitude of change in Y terms of linear unidirectional causation from X.

That is to say, our goal is to take a sample time series and fit a line which best explains the change in Y given X characteristic of the underlying population.

Yet, in the time series context, conducting estimation representative of the underlying population is undermined by a failure of i.i.d conditions

An evolving (non-stationary) time series, entails that observations will not be drawn from an identical distribution and parameter estimates will not convey a constant linear (time-invariant) relationship on Y given X;

Furthermore, given that time series data is necessarily non-independently distributed (i.e. current realisations dependent on past realisations), then again parameter estimates will no longer convey information on the underlying population.

It follows that two mutually dependent properties must be upheld to ensure the i.i.d conditions known as:

* Sufficiently fast weak dependency – random variables should exhibit
  finite memory/asymptotic independence;
* Weak stationarity – the underlying statistical properties (mean,
  variance, and covariance) should be time invariant as if drawn from
  an identical distribution.

Note that both conditions are mutually dependent, given that sufficiently quick finite memory is a requirement for a process to exhibit mean-reversion back to some underlying population.

Moreover, the combination of weak dependency and weak stationarity ensure that a property known as ergodic stationarity is upheld.

Essentially, ergodic stationarity implies that with a quick return to an identical distribution, then realisations are representatives of the greater process, and averaging is relevant.

Further, weak stationarity is important in terms of the exogeneity condition to control for a causal relationship, given that if contemporaneous exogeneity holds in one period; then by weak stationarity, it should hold for all periods.

Also, ergodic stationarity ensure that the process mimics a random independent sample required to uphold LoLN/consistent estimation/CLT are valid.

Finally, weak stationarity overall avoids the problem of so-called spurious regression in which two mutually trending series are wrongly identified to have a causal relationship.

> **Comments:**
> **username_3**: Perhaps "complementary conditions" could be rephrased? "Complementary" could be read (in the set theory/probability sense) as "exhaustive and mutually exclusive", so that  every process is either one or the other (but not both). Though I don't have an immediate suggestion to improve it.  
> **username_2**: @username_3. Completely valid feedback on the terminology. I am of course open to a replacement, given that 'complementary' was used to reflect common parlance.  
> **username_0**: @username_2 thank you so much for the reply. Very much clearer now  
> **username_3**: I'm afraid re "complementary" that the best alternative I can think of off-hand is "related" which I don't think captures all you intended, but on the other hand isn't open to the same kind of confusion either...  
> **username_4**: To be more precise, condition 3. should hold for every $s$.  
> **username_5**: @username_2: +1 what is a textbook where what you wrote about is best described? I haven't seen this explained well anywhere.  
> **username_2**: Hi @username_5. Unfortunately, I can't think of a single textbook which best describes it. Instead, I was in a similar position to the question's author a few years ago, and I kept digging around to find more about these properties. However, I would argue that Martin Nyboe Tabor from Copenhagen Uni has some very good lectures. https://www.youtube.com/c/MortenNyboeTabor/videos  
> **username_2**: @username_5. The overarching point I was trying to get across in the post, is that time series variables (such as GDP), often do not change from period-to-period drastically. Instead, they are highly persistent (such as a unit root process). For example, a unit root AR (1) $y_t = y_{t-1} +e_t$ means that the LHS and RHS are equal, only except for a random error. This means that the information we have from the past (despite it being very persistent), is a poor indicator of the future.  

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