Housing is not the business cycle

There is a widespread belief that the housing market is closely linked to the business cycle. In particular, housing starts are a major component of the leading indicator index, meaning that housing start statistics tends to precede whatever happens to the business cycle. This belief is grounded on the analysis of national data. What about the local level?

Andra Ghent and Michael Owyang compare permits issued with labor market data in 26 US metropolitan areas. They find, after controlling for national effects, that the local housing markets are very poor predictors for local labor markets. The critical aspect here is to control for interest rates.

I learn from this that housing as a leading indicator should be considered with a grain of salt. But it is not quite clear to me what this means in terms of our understanding of the economy. The measure here is the number of permits issued. Is the reason this does not explain subsequent labor market outcomes that fluctuations in interest rates are really driving the execution of those permits, and thus the labor market?