[bashtage]

…ld test

f_statistic_robust of FamaMacBethResults, the Wald test that the parameters
other than the constant are zero, compared W / k with an F distribution with k
and T - 1 degrees of freedom (k and nobs - nvar in 7.0 and earlier), where k is
the number of tested parameters and T the number of periods whose
cross-section is estimated. The parameters are the average of the T
cross-sectional estimates, and their covariance is estimated from the same T
estimates, so that W is Hotelling's T-squared statistic and W / k is
(T - 1) / (T - k) times an F(k, T - k) variable. Neither F distribution is
right when k is larger than 1. With one parameter, F(1, T - 1) is the squared
t(T - 1) distribution and is right. Joint tests of several regressors reject a
true null hypothesis too often, and more with fewer periods.

The reported statistic is now F = (T - k) / (k (T - 1)) W and the p-value is
that of an F(k, T - k) distribution:

• It is exact when the T estimates are independent draws from the same normal
  distribution and the covariance is the standard covariance of their average,
  which is cov_type "unadjusted", "robust" and their aliases. With the kernel
  covariance it is an approximation, and the documentation says so.
• With one tested parameter F is the squared t-statistic and the p-value is that
  of the t(T - 1) test, as before.
• T is the number of periods whose cross-section is estimated, as for the p-values
  of the parameters, so periods that are dropped are not counted. It is
  df_resid + 1, and the denominator degrees of freedom are T - k.
• With debiased=False the statistic is W and the distribution is chi-squared
  with k degrees of freedom, as before.
• If debiased is True and T is not larger than k, the covariance of the
  parameters has rank at most T - 1 and is singular, and the statistic was the
  result of inverting it. It is an InvalidTestStatistic that says so.

In simulations of panels of 40 entities in which the slopes change from period to
period and average to zero (the null hypothesis), the 5% tests rejected in these
shares of 20,000 samples (the standard error is 0.15 percentage points). With the
same regressors in every period the T estimates are exactly independent and
normal, and with new regressors in every period they are not exactly:

                              same regressors           new regressors
                              7.0   T - 1   now         7.0   T - 1   now
k = 1, T = 10                 8.3%  5.2%    5.2%        7.6%  4.9%    4.9%
k = 1, T = 20                 6.5%  5.0%    5.0%        6.7%  5.2%    5.2%
k = 3, T = 10                 19.7% 10.6%   5.4%        19.7% 10.5%   5.0%
k = 3, T = 20                 11.0% 7.2%    4.9%        11.2% 7.5%    5.3%
k = 5, T = 10                 40.6% 24.1%   5.0%        41.2% 24.8%   5.1%
k = 5, T = 20                 17.9% 11.1%   4.9%        18.3% 11.3%   5.1%

"7.0" is F(k, nobs - nvar), "T - 1" is F(k, T - 1) of the previous commit, and
"now" is the statistic of this commit. In the wage panel (T = 8, 4,360
observations), the test of union and married has p-value 1e-62 in 7.0 and 1.9e-6
in the previous commit, and is F = 126.5 with the F(2, 6) distribution and
p-value 1.2e-5 now; the test of black, hisp, exper, educ and union has p-value
3.1e-5 now, where it was 1.3e-11, and 0 to machine precision in 7.0.

The docstrings of FamaMacBeth, FamaMacBeth.fit and the results say what the
statistic is, its reference distribution and why, and cite Hotelling (1931).

Tests (test_fama_macbeth.py has 182, 56 of which fail on the previous commit):

• The statistic, the degrees of freedom and the p-value equal a hand computation
  from the T cross-sectional estimates with scipy.stats.f(k, T - k), for k = 1, 2,
  3 and 5, T = 8, 10 and 20, the standard covariance and the kernel covariance
  with 0 and 2 lags.
• They equal the one-sample Hotelling test of statsmodels, test_mvmean, for the
  slopes, including its T-squared statistic, for k = 3 and 5 and T = 10 and 20.
• With k = 1 the statistic is the squared t-statistic and the p-value that of the
  t(T - 1) test, with the standard and the kernel covariance, for T = 4, 12, 30.
• With 4 of 25 periods dropped T is 21, and the statistic has 19 denominator
  degrees of freedom. Weighted cross-sections, a model without a constant, and
  the summary are tested too.
• debiased=False gives W and chi2(k) for k = 1 and 3.
• T <= k gives an InvalidTestStatistic, T = k + 1 a valid one with one
  denominator degree of freedom, and a model with only a constant is invalid as
  before.
• Independent normal estimates give the size of 5% for k = 3 and 5 and T = 10 and
  20, and the F(k, T - 1) test its much larger shares (1,000 samples each).

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 99.62%. Comparing base (1782b62) to head (6717c59).
⚠️ Report is 107 commits behind head on main.

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@@            Coverage Diff             @@
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- Partials       60       66       +6     

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