Unit root analysis

Steve Cook, Department of Economics, Swansea University
(s.cook@swan.ac.uk)
Published September 2020

Summary

The use of ‘replication’ in the teaching of unit root analysis is discussed. It is argued that the process of replicating published empirical results has many benefits, including: introducing research into teaching; prompting closer reading of the papers in which the results are contained; the development of quantitative skills; and providing the opportunity to address issues in both the self-efficacy and ‘anxiety towards quantitative modules’ literatures. While unit root analysis is the specific topic considered here, a replication-based approach to delivery and assessment can obviously be used far more widely. To illustrate this approach, two particular examples are presented with their associated data sets provided so the reader can undertake the work discussed.

Introduction

Unit root analysis is a familiar feature of econometrics modules at both undergraduate and postgraduate levels. When considering the empirical application of unit root tests for both delivery and assessment, lecturers are clearly faced with a number of options with regard to the data to be employed. While textbooks provide numerous datasets with which to demonstrate unit root testing, further alternative series are clearly available from a range of readily accessible databases. However, while these options have their own merits, an alternative approach is possible via the use of data employed in empirical research articles. It is this use of ‘replication’ that is the subject of this case study.

A replication-based approach to delivery has appeal for a number of reasons including its obvious introduction of research into teaching and learning activities. In particular, the replication of published findings permits the introduction of not just ‘research-led’ teaching but also the more interactive or engaging ‘research-tutored’ and ‘research-based’ teaching approaches (see Healey and Jenkins, 2009). In addition, working towards published empirical results also assists in the generation of the quantitative skills called for in the reports of, inter alia, Mansell (2015) and Mason et al. (2015) while addressing issues arising in the self-efficacy and ‘anxiety towards quants’ literatures by working towards a known ‘target’ outcome.

This case study presents two examples of the application of unit root tests to data from published research. The intention is to discuss the use of replication exercises in teaching and provide the data required for this to be adopted in connection with two specific papers. As a final point, while replication might be considered more in relation to delivery than assessment, it can help shape both formative and summative assessment materials as will be discussed later.

Example 1 : Nelson and Plosser (1982, Journal of Monetary Economics)

Nelson and Plosser (1982) is a seminal article on the topic of unit root analysis, with its consideration of 14 macroeconomic times providing a collection of examples for reconsideration. Importantly the data employed are freely available from a number of sources (for example, http://korora.econ.yale.edu/phillips/index.htm).

The EViews file NP.wf1 contains two of the series (the natural logs of employment and real GNP) examined by Nelson and Plosser (1982). To introduce students to not just empirical results but also undertake research and engage in research discussions, the results presented in the paper can be replicated. By placing themselves in the position of the authors, a variety of issues arise naturally and have to be considered to undertake this replication exercise. Consider the augmented Dickey-Fuller (ADF) test employed in the empirical analysis:

1 Delta yt = dt + rho yt-1 + i=1 k lambda i Delta yt-i + vt

where yt denotes the series of interest and dt denotes the selected deterministic term to be employed. Replication of the results presented in Nelson and Plosser (1982) will require close consultation of the paper so that the correct choice of deterministic terms and degree of augmentation (k) are employed, and the various elements of the testing equation and means for determining inferences are understood. This reinforces understanding of the test by working with, and considering options available for, the various elements of its structure. Table One below presents the results provided in Table 5 of Nelson and Plosser (1982) for the two series in the NP.wf1 data set. A task to both develop and demonstrate understanding of unit root testing is then available via replication of the results below. The challenge is therefore to draw upon understanding of the ADF test and the analysis undertaken by Nelson and Plosser (1982) to reproduce these results and consider the resulting inferences drawn in light of those presented in the paper.

Table One: ADF test results

Series

ADF test statistic

Real GNP

-2.99

Employment

-2.66

Example 2: Cook (2009, Journal of Applied Econometrics)

While concerned with the introduction of a new unit root test incorporating the volatility, Cook (2009) provides an application of the Dickey-Fuller GLS (DF-GLS) test to inflation rate series for a number of OECD economies. As the data employed are available via the data archive associated with the journal (http://qed.econ.queensu.ca/jae/), this study, like Nelson and Plosser (1982), provides a source of results for replication. Again, when considering the relevant testing equation, decisions concerning the optimisation of the lag length and deterministic terms need to be considered to replicate the results in the paper. In addition, given the series provided in the data archive are price series, the inflation series required for the empirical analysis need to be created and this itself prompts further reading of the paper. Given the testing equation below:

Delta y td = gamma y t-1 d + i=1 k delta i Delta yt-i d + et

where ytd denotes the transformed version of the series of interest following appropriate demeaning/detrending, the replication of results in the paper require understanding of the creation of the original series, the relevant method of demeaning/detrending and the determination of the lag length. As noted above, this requires students to place themselves in the position of the author and choose the relevant options from a number available. The data set JAE.wf1 contains two of the series considered in the paper with Table Two below providing the associated results presented in Cook (2019, Table 1). Therefore attempted replication of these results will prompt or direct reading of the paper in which they are presented and also test understanding of the structure and application of the DF-GLS test.

Table Two: DF-GLS test results

Series

DF-GLS test statistic

Italy

-0.944

Spain

-2.160

The above discussion has illustrated the use of replication as a means of supporting the delivery of teaching which, inter alia, involves the introduction of research along with the provision of specific results to both challenge understanding and provide outcomes which can be worked towards. However, this replication-based approach can also be used as the basis for both formative and summative assessment. While formative assessment might involve adoption of a similar approach, summative assessment can be structured in a manner that requires initial replication of results ahead of subsequent extended analysis. To illustrate this ‘extended’ element using one of the examples provided above, a replication exercise could be undertaken for the Nelson-Plosser study before the work is revisited to consider the application of alternative tests, the consideration of structural change, analysis using other long-run series which could both start earlier and finish later to involve very long sample periods, discussion of research into sample spans and data frequency and so on.

References

Cook, S. 2009. A re-examination of the stationarity of inflation. Journal of Applied Econometrics, 24: 1047-1053. DOI 10.1002/jae.1098

Healey, M. and Jenkins, A. 2009. Developing undergraduate research and inquiry. York: Higher Education Academy. ISBN 978-1-905788-99-6.

Mansell, W. 2015. Count us in: Quantitative skills for a new generation. London: British Academy.

Mason, G., Nathan, M. and Rosso, A. 2015. State of the Nation: A review of evidence on the supply and demand of quantitative skills. London: The British Academy and NIESR.

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