g02qfc performs a multiple linear quantile regression, returning the parameter estimates and associated confidence limits based on an assumption of Normal, independent, identically distributed errors. g02qfc is a simplified version of g02qgc.

2 Specification

#include <nag.h>

void	g02qfc (Integer n, Integer m, const double x[], const double y[], Integer ntau, const double tau[], double df, double b[], double bl[], double bu[], Integer info[], NagError fail)

The function may be called by the names: g02qfc, nag_correg_quantile_linreg_easy or nag_regsn_quant_linear_iid.

3 Description

Given a vector of

n

observed values,

y = {y_{i} : i = 1, 2, \dots, n}

, an

n \times p

design matrix

X

, a column vector,

x

, of length

p

holding the

i

th row of

X

and a quantile

τ \in (0, 1)

, g02qfc estimates the

p

-element vector

β

as the solution to

\underset{β \in ℝ^{p}}{minimize} \sum_{i = 1}^{n} ρ_{τ} (y_{i} - x_{i}^{T} β)

(1)

where

ρ_{τ}

is the piecewise linear loss function

ρ_{τ} (z) = z (τ - I (z < 0))

, and

I (z < 0)

is an indicator function taking the value

1

z < 0

and

0

otherwise.

g02qfc assumes Normal, independent, identically distributed (IID) errors and calculates the asymptotic covariance matrix from

Σ = \frac{τ (1 - τ)}{n} {(s (τ))}^{2} {(X^{T} X)}^{- 1}

where

s

is the sparsity function, which is estimated from the residuals,

r_{i} = y_{i} - x_{i}^{T} \hat{β}

(see Koenker (2005)).

Given an estimate of the covariance matrix,

\hat{Σ}

, lower,

{\hat{β}}_{L}

, and upper,

{\hat{β}}_{U}

, limits for a

95 %

confidence interval are calculated for each of the

p

parameters, via

{\hat{β}}_{L i} = {\hat{β}}_{i} - t_{n - p, 0.975} \sqrt{{\hat{Σ}}_{i i}}, {\hat{β}}_{U i} = {\hat{β}}_{i} + t_{n - p, 0.975} \sqrt{{\hat{Σ}}_{i i}}

where

t_{n - p, 0.975}

is the

97.5

percentile of the Student's

t

distribution with

n - k

degrees of freedom, where

k

is the rank of the cross-product matrix

X^{T} X

Further details of the algorithms used by g02qfc can be found in the documentation for g02qgc.

4 References

Koenker R (2005) Quantile Regression Econometric Society Monographs, Cambridge University Press, New York

5 Arguments

1: $n$ – Integer Input

On entry:

n

, the number of observations in the dataset.

Constraint:

n \geq 2

2: $m$ – Integer Input

On entry:

p

, the number of variates in the model.

Constraint:

1 \leq m < n

3: $x [n \times m]$ – const double Input

Note: where

X (i, j)

appears in this document, it refers to the array element

x [(i - 1) \times m + j - 1]

On entry:

X

, the design matrix, with the

i

th value for the

j

th variate supplied in

X (i, j)

, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, m

4: $y [n]$ – const double Input

On entry:

y

, the observations on the dependent variable.

5: $ntau$ – Integer Input

On entry: the number of quantiles of interest.

Constraint:

ntau \geq 1

6: $tau [ntau]$ – const double Input

On entry: the vector of quantiles of interest. A separate model is fitted to each quantile.

Constraint:

\sqrt{ε} < tau [l - 1] < 1 - \sqrt{ε}

where

ε

is the machine precision returned by X02AJC, for

l = 1, 2, \dots, ntau

7: $df$ – double * Output

On exit: the degrees of freedom given by

n - k

, where

n

is the number of observations and

k

is the rank of the cross-product matrix

X^{T} X

8: $b [m \times ntau]$ – double Output

Note: where

B (j, l)

appears in this document, it refers to the array element

b [(l - 1) \times m + j - 1]

On exit:

\hat{β}

, the estimates of the parameters of the regression model, with

B (j, l)

containing the coefficient for the variable in column

j

of X, estimated for

τ = tau [l - 1]

9: $bl [m \times ntau]$ – double Output

Note: where

BL (j, l)

appears in this document, it refers to the array element

bl [(l - 1) \times m + j - 1]

On exit:

{\hat{β}}_{L}

, the lower limit of a

95 %

confidence interval for

\hat{β}

, with

BL (j, l)

holding the lower limit associated with

B (j, l)

10: $bu [m \times ntau]$ – double Output

Note: where

BU (j, l)

appears in this document, it refers to the array element

bu [(l - 1) \times m + j - 1]

On exit:

{\hat{β}}_{U}

, the upper limit of a

95 %

confidence interval for

\hat{β}

, with

BU (j, l)

holding the upper limit associated with

B (j, l)

11: $info [ntau]$ – Integer Output

On exit:

info [l]

holds additional information concerning the model fitting and confidence limit calculations when

τ = tau [l]

Code	Warning
$0$	Model fitted and confidence limits calculated successfully.
$1$	The function did not converge whilst calculating the parameter estimates. The returned values are based on the estimate at the last iteration.
$2$	A singular matrix was encountered during the optimization. The model was not fitted for this value of $τ$ .
$8$	The function did not converge whilst calculating the confidence limits. The returned limits are based on the estimate at the last iteration.
$16$	Confidence limits for this value of $τ$ 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: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 2$ .

On entry, $ntau = ⟨ value ⟩$ .
Constraint: $ntau \geq 1$ .
NE_INT_2: On entry, $m = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $1 \leq m < n$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL_ARRAY: On entry, $tau [⟨ value ⟩] = ⟨ value ⟩$ .
Constraint: $\sqrt{ε} < tau [l - 1] < 1 - \sqrt{ε}$ where $ε$ is the machine precision returned by X02AJC, for all ntau.
NW_POTENTIAL_PROBLEM: A potential problem occurred whilst fitting the model(s).
Additional information has been returned in info.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g02qfc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g02qfc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

Calling g02qfc is equivalent to calling g02qgc with

$order = Nag_RowMajor$ , $intcpt = Nag_NoIntercept$ ,
no weights supplied, i.e., wt set to NULL,
$pddat = m$ ,
setting each element of isx to $1$ ,
$ip = m$ ,
$Interval Method = IID$ , and
$Significance Level = 0.95$ .

10 Example

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.

g02qf: FL CL CPP AD PY MB

NAG CL Interfaceg02qfc (quantile_​linreg_​easy)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
g02qfc (quantile_linreg_easy)