Returns the variance-covariance matrix of the parameters estimates (including breakpoints) of a fitted stepmented model object.
# S3 method for stepmented
vcov(object, k=NULL, zero.cor=TRUE, type=c("cdf", "none", "abs"), ...)The full matrix of the estimated covariances between the parameter estimates, including the breakpoints.
a fitted model object of class "stepmented", returned by any stepmented method
The power of n for the smooth approximation. Simulation evidence suggests k in \([-1, -1/2]\); with \(k=-1/2\) providing somewhat 'conservative' standard errors especially at small sample sizes. In general, the larger \(k\), the smaller \(n^{-k}\), and the smaller the jumpoint standard error.
If TRUE, the covariances between the jumpoints and the remaining linear coefficients are set to zero (as theory states).
How the covariance matrix should be computed. If "none", the usual asymptotic covariance matrix for the linear coefficients only (under homoskedasticity and assuming known the jumpoints) is returned; if "cdf", the standard normal cdf is used to approximate the indicator function (see details); "abs" is yet another approximation (currently unimplemented).
additional arguments.
Vito Muggeo
The function, including the value of \(k\), must be considered at preliminary stage. Currently the value of \(k\) appears to overestimate slightly the true \(\hat\psi\) variability.
The full covariance matrix is based on the smooth approximation
$$I(x>\psi)\approx \Phi((x-\psi)/n^{k})$$
via the sandwich formula using the empirical information matrix and assuming \(x \in [0,1]\). \(\Phi(\cdot)\) is the standard Normal cdf, and \(k\) is the argument k. When k=NULL (default), it is computed via
$$k=-(0.6 + 0.5 \ \log(snr)/\sqrt snr - (|\hat\psi-0.5|/n)^{1/2})$$
where \(snr\) is the signal-to-noise ratio corresponding to the estimated changepoint \(\hat\psi\) (in the range (0,1)). The above formula comes from extensive simulation studies under different scenarios: Seo and Linton (2007) discuss using the normal cdf to smooth out the indicator function by suggesting \(\log(n)/n^{1/2}\) as bandwidth; we found such suggestion does not perform well in practice.
Seo MH, Linton O (2007) A smoothed least squares estimator for threshold regression models, J of Econometrics, 141: 704-735
stepmented