methods-vcov-confint: Variance-covariance Matrix for a Fitted 'stsm' Model Object

In stsm: Structural Time Series Models

Description

The method vcov computes the variance-covariance matrix of the parameters fitted in a structural time series model. This matrix is used to compute confidence intervals for those parameters returned by the coef.stsmFit method.

Usage

## S3 method for class 'stsmFit'
vcov(object, 
  type = c("hessian", "infomat", "OPG", "sandwich", "optimHessian"), ...)
## S3 method for class 'stsm'
vcov(object, 
  type = c("hessian", "infomat", "OPG", "sandwich"), 
  domain = c("frequency", "time"), ...)
## S3 method for class 'stsmFit'
confint(object, parm, level = 0.95, 
  type = c("vcov", "bootstrap"),
  vcov.type = c("hessian", "infomat", "OPG", "sandwich", "optimHessian"), 
  breps = 100, ...)

Arguments

object

a stsmFit list or object of class stsm.

type	a character. In vcov, it is the type of covariance matrix. In confit, it is the type of confidence intervals: based on the covariance matrix of the estimated parameters or on a bootstrap procedure.

domain

a character indicating whether the covariance matrix is obtained upon the frequency or time domain likelihood function.

parm	character indicating the name of the parameter on to obtain the confidence interval. If missing, all parameters are considered.

level

the confidence level.

vcov.type

a character indicating the type of covariance matrix. Ignored if type = "bootstrap".

breps

number of bootstrap replicates. Ignored if type "vcov".

...	additional arguments to be passed to the functions called in these methods. Currently ignored.

Details

The following estimators of the covariance matrix of parameter estimates are available (Davidson and MacKinnon (2004), Section 10.4):

• hessian: the inverse of the analytical Hessian.

• infomat: the inverse of the analytical expression for the information matrix.

• OPG: the inverse of the outer product of the analytical gradient. Also known as the BHHH estimator since it was proposed by Berndt, Hall, Hall and Hausman (1974). This method requires only first order derivatives. It tends to be less reliable in small samples.

• sandwich: the sandwich estimator defined as: H^{-1} (G'G) H^{-1}, where G is the gradient vector and H is the Hessian matrix. It requires more computations and may be unreliable in small samples. However, contrary to the previous methods, it is valid when the information matrix equality does not hold for example due to misspecification of the model.

• optimHessian: the inverse of the numerical Hessian returned by optim.

The natural input for the method method vcov is a stsmFit list returned by maxlik.fd or maxlik.td. However, vcov can be also applied directly on a stsm model object. This is useful for example when maximum likelihood parameter estimates are found by means of optim using the functions maxlik.fd.optim or maxlik.fd.optim. In that case, only the covariance matrix based on the numerical Hessian returned by optim would be available. Updating the slot pars of a stsm model object with the parameter values obtained from other algorithm is a convenient solution to obtain the covariance matrix based on other methods available in vcov.stsmFit.

For the time domain likelihood function the covariance matrix of the initial state vector is considered diagonal, P0cov = FALSE.

The analytical Hessian for the time domain version is not available, the information matrix is used instead.

By default, vcov.type = "infomat" for the time domain likelihood function and vcov.type = "hessian" for the frequency domain likelihood function.

Confidence intervals can either be computed upon the covariance matrix of the parameter estimates (type = "vcov") or by means of bootstrapping (type = "bootstrap"). The bootstrap approach takes advantage of the following result (Harvey (1989) eq. (4.3.25)):

4π I(λ_j) / g(λ_j) \sim χ^2_2, \hbox{ for } j\neq 0, n/2 \hbox{ (for n even)}

2π I(λ_j) / g(λ_j) \sim χ^2_1, \hbox{ for } j=0, n/2 \hbox{ (for n even)}

where I(λ_j) and g(λ_j) are respectively the periodogram and the spectral generating function at frequency λ_j. Upon this result, bootstrap replicates of the periodogram are generated and for each of them parameter estimates are obtained maximizing the spectral likelihood function. The quantiles of the bootstrapped parameter estimates are the confidence interval. Dahlhaus and Janas (1996) studied the properties of the frequency domain bootstrap which has been applied, among others, in Koopman and Wong (2006). An advantage of the bootstrap method is that it yields confidence intervals within the bounds of the parameters, i.e., positive variances. This procedure is computationally intensive and requires some time to run, especially for large breps.

Value

vcov.stsm, vcov.stsmFit	return the covariance matrix of the parameters of the model.
confint.stsmFit	returns a matrix containing confidence intervals for the parameters of the model.

References

Berndt, E. R., Hall, B. H., Hall, R. E. and Hausman, J. A. (1974). ‘Estimation and inference in nonlinear structural models’. Annals of Economic and Social Measurement, 3, pp. 653-65.

Dahlhaus, R. and Janas, D. (1996). ‘A Frequency Domain Bootstrap for Ratio Statistics in Time Series Analysis’. Annals of Statistics, 24(5), pp. 1934-1963.

Davidson, R. and MacKinnon, J. G. (2004). Section 10.4. Econometric Theory and Methods. Oxford University Press.

Koopman, S. J. and Wong, S. Y. (2006). ‘Extracting Business Cycles using Semi-Parametric Time-varying Spectra with Applications to US Macroeconomic Time Series’. Tinbergen Institute Discussion Papers, No. 2006-105/4. http://papers.tinbergen.nl/06105.pdf

See Also

maxlik.fd, maxlik.td, methods-stsmFit, stsm.

Examples

## Not run: 
data("llmseas")
# fit the local level plus seasonal model with default arguments
# using the Newton-Raphson algorithm
m <- stsm.model(model = "llm+seas", y = llmseas)
res <- maxlik.fd.scoring(m = m, information = "observed")
coef(res)
# confidence intervals for parameter estimates ...
# ... based on the covariance matrix of parameter estimates
# gives a warning since the lower limit of the confidence interval
# for parameter 'var2' was forced to be non-negative (fixed to 0)
civcov <- confint(res, type = "vcov", vcov.type = "hessian")
civcov
# ... based on bootstrapping the periodogram
# NOTE: this will take a while to run
set.seed(643)
ciboot <- confint(res, type = "bootstrap", breps = 100)
ciboot

## End(Not run)

Example output

     var1      var2      var3 
200.33269  11.08894 164.28570 
Warning message:
In confintStsmFitVcov(object, parm, level, vcov.type) :
  ‘confint’ did not account for the fact that some of the parameters lied
 outside the boundary of the parameter space.
         2.5%     97.5%
var1 76.80890 323.85649
var2  0.00000  22.65248
var3 66.29777 262.27362
          2.5%     97.5%
var1 84.584069 300.05447
var2  2.638915  25.68343
var3 81.044106 289.78798

stsm documentation built on May 2, 2019, 7:39 a.m.