The empirical estimation with product market tightness

[YoshiShono]

This is what I am considering to conduct empirical estimation with product market tightness, and I would appreciate it if we could discuss it.

I am seeing the equation (16) on p.558 in Michaillat and Saez (2015, QJE). One of the points on this page is that we have to correct the SPC data, cu_t, to obtain ln(f(x)/(s+f(x))). Given the assumption that N_t is a slow-moving variable, the paper conducts an HP filter to see the cyclical dynamics of ln(f(x)/(s+f(x))).

Clearly, this is a very suitable way for this paper. On the other hand, when we conduct empirical analysis, such as local projection/VAR, we often want to use raw data. I mean that when we derive cyclical components from the data, we have to choose the smoothing methodology and parameters (for example, we have to choose smoothing parameters for the HP filter), and this procedure may generate a bias for the resulting data and there seems to exist some arbitrariness.

My idea is following. We have the data of cu_t and n_t. So, we can easily get a new variable x_t = ln(cu_t)-aln(n_t), that is, the first and second term of (16). Then, given the assumption that N_t is exogenous to economic shocks (essentially seems same as slow-moving assumption), the estimation result for x_t is the same as the estimation result of ln(f(x)/(s+f(x))) because 1 unit increase of x_t means 1 unit increase of ln(f(x)/(s+f(x))). (We can see the equation (16) as ln(f(x)/(s+f(x)))-aln(N_t) = ln(cu_t)-aln(n_t) = x_t.)

Of course, trying many kinds of filtering methods and checking the robustness may be one way, but I think that for the empirical estimation with time-series analysis, the methodology seems a little bit arbitral and some people may be concerned...

[pmichaillat]

Very interesting question. Let's discuss it in class today.