Developing econometric and data analysis skills: Championing the crossnumber puzzle
Steve Cook
s.cook@swan.ac.uk
Swansea University
and Duncan Watson
duncan.watson@uea.ac.uk
University of East Anglia
Published October 2024
1. Introduction
Cook and Watson (2023a) have recently championed the use of the crossword as a tool to promote the active learning of econometrics. Drawing on Mayer (2004, 2021), they argue that crosswords offer the flexibility to incorporate various forms of classroom activities differing in their behavioural and cognitive content. For example, when solutions to crossword clues require supplying a definition or filling a gap in a statement, this provides a more behavioural form of activity which may support icebreaking or recapping exercises. In contrast, when crossword clues are devised in a way that requires the synthesis or application of information to be completed, there is a greater focus on cognitive activity.
In this paper, we draw motivation from the analysis of Cook and Watson (2023a) to illustrate the benefits of the crossnumber puzzle (CNP), see Cuthbertson (2014). CNPs, which can be seen as a ‘numerical’ analogue to crosswords, are proposed here as a means of developing econometric and data analysis skills. This is achieved by requiring that the answers to CNPs be generated through empirical tasks using provided data. In this paper, we present an illustrative CNP serving this purpose along with an associated data set.
The tasks underlying the derivation of answers for the CNP have a deliberate two-part nature. First, the tasks require empirical analysis to be undertaken in a correct manner to generate output. Second, the resulting output must be interpreted correctly to identify the specific component that serves as the answer to input in the CNP. Therefore, completion of the CNP is designed to require both a mastery of methods and the ability to interpret the results arising from their application. While CNPs can obviously span a range of topics, the specific illustrative CNP presented here draws upon unit root and cointegration analysis for its tasks and answers.
2. Some rules for this puzzle
Before presenting our illustrative CNP puzzle and the tasks required for its completion, a few rules need to be outlined. All required answers for this CNP are numerical and should be provided to three decimal places. When inputting answers, decimal points and minus signs should be treated like digits and placed in their own separate box. For example, if a particular task resulted in −1.234567 as an answer, these rules mean it would be entered into the CNP as follows:
| − | 1 | . | 2 | 3 | 5 |
As a final point, the CNP presented here employs ‘blocking’ rather than ‘barring’ to separate answers (see Cuthbertson, 2014). Therefore, the format follows that of a conventional crossword, with some cells remaining empty, in contrast to a format where all cells are filled with answers and separated by vertical or horizontal bars.
3. Example CNP
The CNP presented here requires the analysis of the data set contained in the EViews workfile CNP.wf1.[1] This data set includes three artificially generated series denoted as {X,Y,Z} over a hypothetical sample period denoted as 1975Q1 to 2023Q4. The required CNP grid and its tasks are presented below. Note that, in all but one case, the tasks should be completed using the full sample period available. Additionally, where lag optimisation is required, the Schwert rule should be used to determine the maximum lag length considered.
| 1 | 2 | ||||||
| 3 | 4 | ||||||
| 5 | |||||||
| 6 | |||||||
Across
- Apply the Engle-Granger method to Y and Z employing a constant term in your static cointegrating regressions and using the AIC to determine lag lengths in the second stage testing equations. The answer for this task is the calculated Engle-Granger τ-statistic obtained when Z is the dependent variable in first stage cointegrating regression.
- Apply an ADF test to the first difference of Z. Select deterministics appropriately for your ADF testing equation and augment your ADF testing equation using four lags of the dependent variable. The answer for this task is the p-value associated with the calculated ADF test statistic.
- Change the sample to run from 1980,1 to 2023,4 and apply the GLS-ADF test to Z. Include an intercept and trend term as deterministics in your analysis and use the modified AIC to determine the degree of augmentation of the testing equation. The answer for this task is the value of the calculated GLS-ADF test statistic.
- Apply the Johansen procedure to the series X, Y and Z using option 3a in EViews 13 and employing 3 lags in your VECM. The answer for this task is the value of the Trace statistic calculated using the middle and smallest eigenvalues.
Down
- Apply the Engle-Granger method to X and Y employing a constant term in your static cointegrating regressions and using the SIC to determine lag lengths in the second stage testing equations. The answer for this task is the calculated Engle-Granger τ-statistic obtained when X is the dependent variable in first stage cointegrating regression.
- Apply the Johansen procedure to the series X, Y and Z using option 3a in EViews 13 and employing 3 lags in your VECM. The answer for this task is the largest eigenvalue obtained from your analysis.
- Apply an ADF test to the series Y including an intercept and trend term in your testing equation and determining its degree of augmentation using the AIC. The answer for this task is the value of the estimated coefficient on the second lagged difference term.
- Apply an ADF test to the series X including an intercept and trend term in your testing equation and determining its degree of augmentation using the t-statistic rule at the 5% level of significance. The answer for this task is the value of the calculated ADF test statistic.
The previously mentioned two-part nature of tasks is apparent from the above examples. The tasks initially require the application of specific approaches (methods and tests) to generate output. Once this is done, the next challenge is to identify the specific element to input into the CNP from the generated output.
As an example of this, consider ‘3 Across’. The initial stage of this task involves several steps: first, the variable Z must be differenced; then, the differenced series is graphed to examine its nature; based on the graph, a decision is made to include an intercept in the subsequent unit root analysis; finally, the correct unit root test (the ADF test) is chosen from the available alternatives and applied appropriately. After the correct analysis is conducted, the p-value for the ADF test statistic must be extracted from the collection of results provided in the generated output.
Completing the tasks to derive answers for the CNP presents several challenges. Empirical skills are tested by requiring tasks to be carried out in a specific way (e.g. selecting tests and methods, lag optimisation, selecting deterministic terms), while data manipulation skills and the ability to interpret output are tested by the need to modify the sample period, transform a variable and identify specific elements of output from a vast array of generated information (e.g. calculated test statistics, eigenvalues and results relating to a specific normalisation).
To illustrate how specific results are identified from a collection of information, Table One presents the output generated when undertaking ‘6 Across’ and ‘2 Down’. The elements of output that serve as the answers for these questions are highlighted in blue and green, respectively, and are clearly just two among many values to consider.
Table One
| Hypothesized No. of CE(s) | Eigenvalue | Trace Statistic | 0.05 Critical Value | Prob.** Critical Value |
|---|---|---|---|---|
| None * | 0.205039 | 52.72319 | 29.79707 | 0.0000 |
| At most 1 | 0.043820 | 8.666539 | 15.49471 | 0.3971 |
| At most 2 | 0.000329 | 0.063244 | 3.841465 | 0.8014 |
Trace test indicates 1 cointegrating equation(s) at the 0.05 level * denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values | ||||
| Hypothesized No. of CE(s) | Eigenvalue | Max-Eigen Statistic | 0.05 Critical Value | Prob.** Critical Value |
|---|---|---|---|---|
| None * | 0.205039 | 44.05665 | 21.13162 | 0.0000 |
| At most 1 | 0.043820 | 8.603295 | 14.26460 | 0.3207 |
| At most 2 | 0.000329 | 0.063244 | 3.841465 | 0.8014 |
Max-eigenvalue test indicates 1 cointegrating equation(s) at the 0.05 level * denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values | ||||
The answers to the illustrative CNP are presented below.
| − | 1 | 3 | . | 2 | 3 | 0 | |
| 2 | . | ||||||
| . | 2 | ||||||
| 2 | 0 | . | 0 | 0 | 0 | ||
| 3 | . | 5 | |||||
| 2 | 2 | ||||||
| − | 0 | . | 8 | 0 | 7 | ||
| 1 | 4 | ||||||
| . | |||||||
| 9 | |||||||
| 8 | . | 6 | 6 | 7 | |||
| 7 |
4. Concluding comments
This paper has sought to illustrate the benefits of crossnumber puzzles as an interesting means of including activity in the teaching of econometrics. The specific example considered here has focused on the development of empirical skills, given the nature of the tasks required to complete the CNP. More specifically, the tasks are designed to test empirical skills by requiring a specific form of analysis to be undertaken correctly and the results it generates to be interpreted appropriately. While reference has been made to challenging or testing knowledge, CNPs clearly support the development of understanding and knowledge through the work undertaken.
The benefits of CNPs can be considered in relation to numerous pedagogical issues beyond a means of introducing active learning. The successful completion of a CNP like the one above can provide evidence of an ability to undertake tasks and work with empirical output. Consequently, the confidence gained can support the development of self-efficacy (Bandura, 1978; Rozgonjuk et al., 2020) and help address ‘anxiety towards quants’ (see, inter alia, Dreger and Aiken, 1957; Dowker et al., 2016; Cook and Watson, 2023b). Additionally, the flexibility offered by CNPs allows for the incorporation of tasks of varying complexity and their use at different stages of the learning process. As such, CNPs allow for problem-solving activities in a manner that supports the Expertise Reversal Effect (Kalyuga et al., 2003), where tasks can be adjusted according to a learner’s prior knowledge and experience.
Notes
[1] EViews 13 has been employed to undertake the empirical analysis presented in this paper.
References
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Rozgonjuk, D., Kraav, T., Mikkor, K., Orav-Puurand, K. and Täht, K. 2020. Mathematics anxiety among STEM and social sciences students: the roles of mathematics self-efficacy, and deep and surface approach to learning. International Journal of STEM Education 7, 46. https://doi.org/10.1186/s40594-020-00246-z
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