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In trying to do time series analysis, I almost regularly stumble upon unit root and cointegration tests. The design of most these tests is based on a null of unit root (for both linear and non-linear models) and the statistic's distribution is derived using concepts/properties of Brownian motion, Functional CLT, etc. See this for example, and this question for my motivation.

Since I do not have a strong background in Stochastic Processes, I am looking for resources (hopefully a book) which covers relevant topics from stochastic processes and asymptotics so that I can make sense of the derivations in papers on unit root tests.

I tried to check in books on Stochastic Processes but their coverage is wider than what I am looking for. I did found two interesting links: this and this, which have a rather focused use of stochastic process concepts for unit roots. I am looking for similar resources - preferably a book.

Dayne
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    1/2 While you are learning about tests *of* unit root (i.e. $\text{H}_{0}\text{: time series has/have unit root}$) also take the time to learn about tests *of* stationarity (i.e. $\text{H}_{0}\text{: time series is/are stationary}$), such as the [Kwiatkowski-Phillips-Schmidt-Shin test](http://wexler.free.fr/library/files/kwiatowski%20(1992)%20testing%20the%20null%20hypothesis%20of%20stationarity%20against%20the%20alternative%20of%20a%20unit%20root.pdf) for single time series, and [Hadri's test](https://onlinelibrary.wiley.com/doi/abs/10.1111/1368-423X.00043) for multiple time series. – Alexis Mar 01 '21 at 19:11
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    2/2 Combining inference from tests of unit root with inference from tests of stationarity help guard against [confirmation bias](https://en.wikipedia.org/wiki/Confirmation_bias) in modeling decisions around the stationarity, weak stationarity, or non-stationarity (unit rootness) of a time series. – Alexis Mar 01 '21 at 19:12
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    @Alexis: Thanks for the references and I couldn't agree more. Also, i feel it's important to know about other stationarity tests becausr most unit root tests are designed to test for one type of non-stationarity and practitioners often miss other types. To share my experience, I have often seen that people use ADF on seasonal series and getting stationarity as output, which is clearly wrong. See [this](https://stats.stackexchange.com/q/504861/239481) for example. – Dayne Mar 02 '21 at 00:24
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    Thank you for that comment, I look forward to readingthe link! There is a good need for periodic review articles to lay out the considerations (e.g., size of $N$, $T$, and $N$ v $T$, seasonality, etc.), and present the current state of the art. – Alexis Mar 02 '21 at 17:17

2 Answers2

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While both books have "cointegration/cointegrated" in their titles, they do discuss unit roots, too, as that is a prerequisite for cointegration analysis which you seem to be interested in as well.

Richard Hardy
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  • Thanks Richard! I checked the table of contents for both these books and found some chapters of interest. Though these books cover VAR modeling for cointegration in more details they are still relevant as sooner or later I will have to take a deep breath and start learning VAR approach (equivalently, reduced rank regression?) to cointegration also. But accepting @ChristophHanck's answer as they appear to be immediately relevant to my requirement. – Dayne Mar 01 '21 at 10:28
  • @Dayne, thanks for your positive feedback! – Richard Hardy Mar 01 '21 at 11:45
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In addition to the references by Richard Hardy, the following may be helpful:

Bierens, Unit Roots, Ch. 29 in "A Companion to Theoretical Econometrics", https://onlinelibrary.wiley.com/doi/10.1002/9780470996249.ch30

A Primer for Unit Root Testing (Palgrave Texts in Econometrics) Hardcover by K. Patterson https://www.amazon.de/Primer-Testing-Palgrave-Texts-Econometrics/dp/1403902046

Christoph Hanck
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    Based on your comment from other question, you can probably also include 'Chapter 7: Functional Central Limit Theory and Applications of "Asymptotic Theory for Econometricians" by Halbert White'. – Dayne Mar 02 '21 at 11:44