G02QFF performs a multiple linear quantile regression, returning the parameter estimates and associated confidence limits based on an assumption of Normal, independent, identically distributed errors. G02QFF is a simplified version of G02QGF.
Given a vector of observed values,
, an design matrix , a column vector, , of length holding the th row of and a quantile , G02QFF estimates the -element vector as the solution to
where is the piecewise linear loss function , and is an indicator function taking the value if and otherwise.
G02QFF assumes Normal, independent, identically distributed (IID) errors and calculates the asymptotic covariance matrix from
where is the sparsity function, which is estimated from the residuals,
(see Koenker (2005)).
Given an estimate of the covariance matrix, , lower, , and upper, , limits for a confidence interval are calculated for each of the parameters, via
where is the percentile of the Student's distribution with degrees of freedom, where is the rank of the cross-product matrix .
Further details of the algorithms used by G02QFF can be found in the documentation for G02QGF.
Koenker R (2005) Quantile Regression Econometric Society Monographs, Cambridge University Press, New York
1: – INTEGERInput
On entry: , the number of observations in the dataset.
2: – INTEGERInput
On entry: , the number of variates in the model.
3: – REAL (KIND=nag_wp) arrayInput
On entry: , the design matrix, with the
th value for the th variate supplied in , for and .
4: – REAL (KIND=nag_wp) arrayInput
On entry: , the observations on the dependent variable.
5: – INTEGERInput
On entry: the number of quantiles of interest.
6: – REAL (KIND=nag_wp) arrayInput
On entry: the vector of quantiles of interest. A separate model is fitted to each quantile.
where is the machine precision returned by X02AJF, for .
7: – REAL (KIND=nag_wp)Output
On exit: the degrees of freedom given by , where is the number of observations and is the rank of the cross-product matrix .
8: – REAL (KIND=nag_wp) arrayOutput
On exit: , the estimates of the parameters of the regression model, with containing the coefficient for the variable in column of X, estimated for .
9: – REAL (KIND=nag_wp) arrayOutput
On exit: , the lower limit of a confidence interval for , with holding the lower limit associated with .
10: – REAL (KIND=nag_wp) arrayOutput
On exit: , the upper limit of a confidence interval for , with holding the upper limit associated with .
11: – INTEGER arrayOutput
On exit: holds additional information concerning the model fitting and confidence limit calculations when .
Model fitted and confidence limits calculated successfully.
The routine did not converge whilst calculating the parameter estimates. The returned values are based on the estimate at the last iteration.
A singular matrix was encountered during the optimization. The model was not fitted for this value of .
The routine did not converge whilst calculating the confidence limits. The returned limits are based on the estimate at the last iteration.
Confidence limits for this value of could not be calculated. The returned upper and lower limits are set to a large positive and large negative value respectively.
It is possible for multiple warnings to be applicable to a single model. In these cases the value returned in INFO is the sum of the corresponding individual nonzero warning codes.
12: – INTEGERInput/Output
On entry: IFAIL must be set to , . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value is recommended. If the output of error messages is undesirable, then the value is recommended. Otherwise, if you are not familiar with this argument, the recommended value is . When the value is used it is essential to test the value of IFAIL on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6 Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
On entry, .
On entry, and .
On entry, .
On entry, is invalid.
A potential problem occurred whilst fitting the model(s). Additional information has been returned in INFO.
An unexpected error has been triggered by this routine. Please
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.
8 Parallelism and Performance
G02QFF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
G02QFF makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9 Further Comments
Calling G02QFF is equivalent to calling G02QGF with
A quantile regression model is fitted to Engels 1857 study of household expenditure on food. The model regresses the dependent variable, household food expenditure, against household income. An intercept is included in the model by augmenting the dataset with a column of ones.