The gist of the article is as follows. Naive regression-based estimates, cited in a number of studies, have suggested that coaching turnover can lead to positive results on team performance. These “simple econometric models” were not able to take into account that the team playing before the coaching change was different than the team playing after the coaching change.
One method of accounting for the correlation (or endogeneity) issues in a naive model is to use an instrumental variable, most commonly applied through 2-stage least squares. The instrument used in this study is the number of matches to be played by the team before the end of the season. I’m not sure this is the strongest of instruments, but it does seem to meet the requirements to be used as an instrument (see page 9 of the pdf for a good description). The 2-stage least squares estimates suggest that there is no benefit of the coaching change.
I love the idea of using an instrument in this case, but as (I think) generally tends to happen, one downside of this approach is that the standard errors on the coefficients of interest (in this case, coaching change indicators) are inflated after the 2-stage least squares procedure is implemented. In fact, in this example, the standard errors more than double. That said, the coefficient estimates regress towards 0, compared to the standard approach which did not use an instrument.
Is there a better instrument out there? Applied statisticians – especially those interested in causal inference – spend much of there day thinking about this! The impact of this approach is highlighted in this soccer article.