# Why joint null test have less power when there are more coefficients involve?

From a description of DiD method of Borusyak,2020 , I saw that

pretrends(integer): if some value k>0 is specified, the command will performs a test for parallel trends, by a separate regression on nontreated observations only: of the outcome on the dummies for 1,...,k periods before treatment, in addition to all the FE and controls. The coefficients are reported as pre1,...,prek. The Wald statistic, pvalue, and degrees-of-freedom as reported in e(pre_chi2), e(pre_p), and e(pre_df) resp.

• Use a reasonable number of pre-trends, do not use all of the available ones unless you have a really large never-treated group. With too many pre-trend coefficients, the power of the joint test will be lower.

I am wondering how to explain the bold sentence. In another word, why joint null test has less power when there are more coefficients involve?

Each parameter $$k$$ you estimate comes at a similar cost, it costs a datapoint (also referred to as degree of freedom) which you fix and thus no longer provides any useful information to you for other statistical quantities you may want to estimate. The power of the joint test depends on the number of independent data points $$n-k$$ you have to 'spend'. The larger your sample size $$n$$, the more accurate your test (power). The more pre-trend coefficients you need to account for, the larger $$k$$ is and thus the smaller $$n-k$$ is.