Variance-Covariance Matrix for a Fitted Stepmented Model

Returns the variance-covariance matrix of the parameters estimates (including breakpoints) of a fitted stepmented model object.

Usage

# S3 method for class 'stepmented'
vcov(object, k=NULL, zero.cor=TRUE, type=c("cdf", "none", "abs"), ...)

Arguments

object: a fitted model object of class "stepmented", returned by any stepmented method
k: 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.
zero.cor: If TRUE, the covariances between the jumpoints and the remaining linear coefficients are set to zero (as theory states).
type: 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.

Details

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.

Value

The full matrix of the estimated covariances between the parameter estimates, including the breakpoints.

References

Seo MH, Linton O (2007) A smoothed least squares estimator for threshold regression models, J of Econometrics, 141: 704-735

Author

Vito Muggeo

Note

If the fit object has been called by stepmented(.., var.psi=TRUE), then vcov.stepmented will return object$vcov, unless the power k differs from -2/3.

Warning

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.

Examples

##see ?stepmented