An N-shaped relationship between GDP and suicide?

The Kuznets Curve is the hypothesised inverted-U-shaped relationship between inequality and development (or income per capita, or GDP per capita) (see this post for more). It implies that countries at low levels of development have low inequality, countries at middle levels of development have high inequality, and countries at high levels of development have low inequality. A similar inverted-U-shape has been hypothesised for subjective wellbeing or happiness (although this has yet to be definitely demonstrated empirically - see this post for more). So, I was interested to read this 2018 article, with the title "A suicidal Kuznets curve?", by Nikolaos Antonakakis (Webster Vienna Private University) and Alan Collins (Nottingham Trent University), and published in the journal Economics Letters (ungated earlier version here).

Antonakakis and Collins used cross-country data on suicide rates and per capita GDP for 73 countries over the period from 1990 to 2010, controlling for the unemployment rate and demographic and social characteristics of the population. They found that:

We observe that, generally, the coefficients of per capita income, including squared and cubic counterparts are positive, negative and positive, respectively, across males of all ages... Yet, they are only significant for the 25–34 (at the 10% level), 35–54 (at the 5% level) and 55–74 (at the 10% level) age groups... This is suggestive of the existence of an N-shaped Suicidal Kuznets curve in the case of the aforementioned age groups of the male population... Turning to the female population results... an N-shaped SKC is identified for females in the 55–74 age group...

Here is where things turn a bit weird. Why were Antonakakis and Collins testing for an N-shaped relationship in the first place? As far as I know, there is no theoretical reason why there would be an N-shaped relationship here. For clarity, the N-shaped relationship suggests that the suicide rate is low at low levels of GDP per capita, increases with GDP per capita to a peak at some level ($7727 for males 15-24 years, or about the level of Laos or Morocco), then decreases to a trough at some level of GDP per capita ($22,726 for males 15-24 years, or about the level of Mauritius or Argentina), before increasing again. People in Macao and Luxembourg (GDP per capita of approximately $127,000 and $114,000 respectively) are in real trouble!

Aside from the general weirdness of this result from a theoretical perspective, there's problems with the control variables as well. Antonakakis and Collins use 'demographic controls', including the fertility rate and life expectancy. Fertility seems like a weird thing to control for, but it turns out that there is a correlation between fertility and suicide at the country level (see here, where the authors interpret fertility as a measure of social cohesion). However, since suicide affects life expectancy, having life expectancy as a control variable effectively gets the causality backwards. Also, having both unemployment and GDP per capita in the analysis might be problematic because of endogeneity. These latter two issues could generate some bias and weirdness in the results, and may explain the N-shaped result.

This is clearly a study that is crying out for a replication with more attention to theory and to the use of appropriate control variables.