NAG Library Routine Document

G02QGF

Note: this routine uses optional parameters to define choices in the problem specification and in the details of the algorithm. If you wish to use default settings for all of the optional parameters, you need only read Sections 1 to 9 of this document. If, however, you wish to reset some or all of the settings please refer to Section 10 for a detailed description of the algorithm, to Section 11 for a detailed description of the specification of the optional parameters and to Section 12 for a detailed description of the monitoring information produced by the routine.

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1 Purpose

G02QGF performs a multiple linear quantile regression. Parameter estimates and, if required, confidence limits, covariance matrices and residuals are calculated. G02QGF may be used to perform a weighted quantile regression. A simplified interface for G02QGF is provided by G02QFF.

2 Specification

SUBROUTINE G02QGF (

SORDER, INTCPT, WEIGHT, N, M, DAT, LDDAT, ISX, IP, Y, WT, NTAU, TAU, DF, B, BL, BU, CH, RES, IOPTS, OPTS, STATE, INFO, IFAIL)

INTEGER	SORDER, N, M, LDDAT, ISX(M), IP, NTAU, IOPTS(), STATE(), INFO(NTAU), IFAIL
REAL (KIND=nag_wp)	DAT(LDDAT,), Y(N), WT(), TAU(NTAU), DF, B(IP,NTAU), BL(IP,), BU(IP,), CH(IP,IP,), RES(N,), OPTS(*)
CHARACTER(1)	INTCPT, WEIGHT

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)

, G02QGF 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. Weights can be incorporated by replacing

X

and

y

with

W X

and

W y

respectively, where

W

is an

n \times n

diagonal matrix. Observations with zero weights can either be included or excluded from the analysis; this is in contrast to least squares regression where such observations do not contribute to the objective function and are therefore always dropped.

G02QGF uses the interior point algorithm of Portnoy and Koenker (1997), described briefly in Section 10, to obtain the parameter estimates

\hat{β}

, for a given value of

τ

Under the assumption of Normally distributed errors, Koenker (2005) shows that the limiting covariance matrix of

\hat{β} - β

has the form

Σ = \frac{τ (1 - τ)}{n} {H_{n}}^{- 1} J_{n} {H_{n}}^{- 1}

where

J_{n} = n^{- 1} \sum_{i = 1}^{n} x_{i} x_{i}^{T}

and

H_{n}

is a function of

τ

, as described below. Given an estimate of the covariance matrix,

\hat{Σ}

, lower (

{\hat{β}}_{L}

) and upper (

{\hat{β}}_{U}

) limits for an

(100 \times α) %

confidence interval can be calculated for each of the

p

parameters, via

{\hat{β}}_{L i} = {\hat{β}}_{i} - t_{n - p, (1 + α) / 2} \sqrt{{\hat{Σ}}_{i i}}, {\hat{β}}_{U i} = {\hat{β}}_{i} + t_{n - p, (1 + α) / 2} \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

Four methods for estimating the covariance matrix,

Σ

, are available:

(i)

Independent, identically distributed (IID) errors

Under an assumption of IID errors the asymptotic relationship for

Σ

simplifies to

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

where

s

is the sparsity function. G02QGF estimates

s (τ)

from the residuals,

r_{i} = y_{i} - x_{i}^{T} \hat{β}

and a bandwidth

h_{n}

(ii)

Powell Sandwich

Powell (1991) suggested estimating the matrix

H_{n}

by a kernel estimator of the form

{\hat{H}}_{n} = {(n c_{n})}^{- 1} \sum_{i = 1}^{n} K (\frac{r_{i}}{c_{n}}) x_{i} x_{i}^{T}

where

K

is a kernel function and

c_{n}

satisfies

\lim_{n \to \infty} c_{n} \to 0

and

\lim_{n \to \infty} \sqrt{n} c_{n} \to \infty

. When the Powell method is chosen, G02QGF uses a Gaussian kernel (i.e.,

K = ϕ

) and sets

c_{n} = \min (σ_{r}, (q_{r 3} - q_{r 1}) / 1.34) \times (Φ^{- 1} (τ + h_{n}) - Φ^{- 1} (τ - h_{n}))

where

h_{n}

is a bandwidth,

σ_{r}, q_{r 1}

and

q_{r 3}

are, respectively, the standard deviation and the

25 %

and

75 %

quantiles for the residuals,

r_{i}

(iii)

Hendricks–Koenker Sandwich

Koenker (2005) suggested estimating the matrix

H_{n}

using

{\hat{H}}_{n} = n^{- 1} \sum_{i = 1}^{n} [\frac{2 h_{n}}{x_{i}^{T} (\hat{β} (τ + h_{n}) - \hat{β} (τ - h_{n}))}] x_{i} x_{i}^{T}

where

h_{n}

is a bandwidth and

\hat{β} (τ + h_{n})

denotes the parameter estimates obtained from a quantile regression using the

(τ + h_{n})

th quantile. Similarly with

\hat{β} (τ - h_{n})

(iv)

Bootstrap

The last method uses bootstrapping to either estimate a covariance matrix or obtain confidence intervals for the parameter estimates directly. This method therefore does not assume Normally distributed errors. Samples of size

n

are taken from the paired data

\{y_{i}, x_{i}\}

(i.e., the independent and dependent variables are sampled together). A quantile regression is then fitted to each sample resulting in a series of bootstrap estimates for the model parameters,

β

. A covariance matrix can then be calculated directly from this series of values. Alternatively, confidence limits,

{\hat{β}}_{L}

and

{\hat{β}}_{U}

, can be obtained directly from the

(1 - α) / 2

and

(1 + α) / 2

sample quantiles of the bootstrap estimates.

Further details of the algorithms used to calculate the covariance matrices can be found in Section 10.

All three asymptotic estimates of the covariance matrix require a bandwidth,

h_{n}

. Two alternative methods for determining this are provided:

(i)

Sheather–Hall

h_{n} = {(\frac{1.5 {(Φ^{- 1} (α_{b}) ϕ (Φ^{- 1} (τ)))}^{2}}{n (2 Φ^{- 1} (τ) + 1)})}^{\frac{1}{3}}

for a user-supplied value

α_{b}

(ii)

Bofinger

h_{n} = {(\frac{4.5 {(ϕ (Φ^{- 1} (τ)))}^{4}}{n {(2 Φ^{- 1} (τ) + 1)}^{2}})}^{\frac{1}{5}}

G02QGF allows optional arguments to be supplied via the IOPTS and OPTS arrays (see Section 11 for details of the available options). Prior to calling G02QGF the optional parameter arrays, IOPTS and OPTS must be initialized by calling G02ZKF with OPTSTR set to

Initialize = G02QGF

(see Section 11 for details on the available options). If bootstrap confidence limits are required (

Interval Method = BOOTSTRAP XY

) then one of the random number initialization routines G05KFF (for a repeatable analysis) or G05KGF (for an unrepeatable analysis) must also have been previously called.

4 References

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

Mehrotra S (1992) On the implementation of a primal-dual interior point method SIAM J. Optim. 2 575–601

Nocedal J and Wright S J (1999) Numerical Optimization Springer Series in Operations Research, Springer, New York

Portnoy S and Koenker R (1997) The Gaussian hare and the Laplacian tortoise: computability of squared-error versus absolute error estimators Statistical Science 4 279–300

Powell J L (1991) Estimation of monotonic regression models under quantile restrictions Nonparametric and Semiparametric Methods in Econometrics Cambridge University Press, Cambridge

5 Parameters

1: SORDER – INTEGERInput

On entry: determines the storage order of variates supplied in DAT.

Constraint:

SORDER = 1

2

2: INTCPT – CHARACTER(1)Input

On entry: indicates whether an intercept will be included in the model. The intercept is included by adding a column of ones as the first column in the design matrix,

X

$INTCPT ='Y'$: An intercept will be included in the model.
$INTCPT ='N'$: An intercept will not be included in the model.

Constraint:

INTCPT ='N'

'Y'

3: WEIGHT – CHARACTER(1)Input

On entry: indicates if weights are to be used.

$WEIGHT ='W'$: A weighted regression model is fitted to the data using weights supplied in array WT.
$WEIGHT ='U'$: An unweighted regression model is fitted to the data and array WT is not referenced.

Constraint:

WEIGHT ='U'

'W'

4: N – INTEGERInput

On entry: the total number of observations in the dataset. If no weights are supplied, or no zero weights are supplied or observations with zero weights are included in the model then

N = n

. Otherwise

N = n +

the number of observations with zero weights.

Constraint:

N \geq 2

5: M – INTEGERInput

On entry:

m

, the total number of variates in the dataset.

Constraint:

M \geq 0

6: DAT(LDDAT, $*$ ) – REAL (KIND=nag_wp) arrayInput

Note: the second dimension of the array DAT must be at least

M

SORDER = 1

and at least

N

SORDER = 2

On entry: the

i

th value for the

j

th variate, for

i = 1, 2, \dots, N

and

j = 1, 2, \dots, M

, must be supplied in

$DAT (i, j)$ if $SORDER = 1$ , and
$DAT (j, i)$ if $SORDER = 2$ .

The design matrix

X

is constructed from DAT, ISX and INTCPT.

7: LDDAT – INTEGERInput

On entry: the first dimension of the array DAT as declared in the (sub)program from which G02QGF is called.

Constraints:

if $SORDER = 1$ , $LDDAT \geq N$ ;
otherwise $LDDAT \geq M$ .

8: ISX(M) – INTEGER arrayInput

On entry: indicates which independent variables are to be included in the model.

$ISX (j) = 0$: The $j$ th variate, supplied in DAT, is not included in the regression model.
$ISX (j) = 1$: The $j$ th variate, supplied in DAT, is included in the regression model.

Constraints:

$ISX (j) = 0$ or $1$ , for $j = 1, 2, \dots, M$ ;
if $INTCPT ='Y'$ , exactly $IP - 1$ values of ISX must be set to $1$ ;
if $INTCPT ='N'$ , exactly IP values of ISX must be set to $1$ .

9: IP – INTEGERInput

On entry:

p

, the number of independent variables in the model, including the intercept, see INTCPT, if present.

Constraints:

$1 \leq IP < N$ ;
if $INTCPT ='Y'$ , $1 \leq IP \leq M + 1$ ;
if $INTCPT ='N'$ , $1 \leq IP \leq M$ .

10: Y(N) – REAL (KIND=nag_wp) arrayInput

On entry:

y

, observations on the dependent variable.

11: WT( $*$ ) – REAL (KIND=nag_wp) arrayInput

Note: the dimension of the array WT must be at least

N

WEIGHT ='W'

On entry: if

WEIGHT ='W'

, WT must contain the diagonal elements of the weight matrix

W

. Otherwise WT is not referenced.

When

$Drop Zero Weights = YES$: If $WT (i) = 0.0$ , the $i$ th observation is not included in the model, in which case the effective number of observations, $n$ , is the number of observations with nonzero weights. If $Return Residuals = YES$ , the values of RES will be set to zero for observations with zero weights.
$Drop Zero Weights = NO$: All observations are included in the model and the effective number of observations is N, i.e., $n = N$ .

Constraints:

If $WEIGHT ='W'$ , $WT (i) \geq 0.0$ , for $i = 1, 2, \dots, N$ ;
The effective number of observations $\geq 2$ .

12: NTAU – INTEGERInput

On entry: the number of quantiles of interest.

Constraint:

NTAU \geq 1

13: TAU(NTAU) – REAL (KIND=nag_wp) arrayInput

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

Constraint:

\sqrt{ε} < TAU (j) < 1 - \sqrt{ε}

where

ε

is the machine precision returned by X02AJF, for

j = 1, 2, \dots, NTAU

14: DF – REAL (KIND=nag_wp)Output

On exit: the degrees of freedom given by

n - k

, where

n

is the effective number of observations and

k

is the rank of the cross-product matrix

X^{T} X

15: B(IP,NTAU) – REAL (KIND=nag_wp) arrayInput/Output

On entry: if

Calculate Initial Values = NO

B (i, l)

must hold an initial estimates for

{\hat{β}}_{i}

, for

i = 1, 2, \dots, IP

and

l = 1, 2, \dots, NTAU

. If

Calculate Initial Values = YES

, B need not be set.

On exit:

B (i, l)

, for

i = 1, 2, \dots, IP

, contains the estimates of the parameters of the regression model,

\hat{β}

, estimated for

τ = TAU (l)

INTCPT ='Y'

B (1, l)

will contain the estimate corresponding to the intercept and

B (i + 1, l)

will contain the coefficient of the

j

th variate contained in DAT, where

ISX (j)

is the

i

th nonzero value in the array ISX.

INTCPT ='N'

B (i, l)

will contain the coefficient of the

j

th variate contained in DAT, where

ISX (j)

is the

i

th nonzero value in the array ISX.

16: BL(IP, $*$ ) – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array BL must be at least

NTAU

Interval Method \neq NONE

On exit: if

Interval Method \neq NONE

BL (i, l)

contains the lower limit of an

(100 \times α) %

confidence interval for

B (i, l)

, for

i = 1, 2, \dots, IP

and

l = 1, 2, \dots, NTAU

Interval Method = NONE

, BL is not referenced.

The method used for calculating the interval is controlled by the optional parameters Interval Method and Bootstrap Interval Method. The size of the interval,

α

, is controlled by the optional parameter Significance Level.

17: BU(IP, $*$ ) – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array BU must be at least

NTAU

Interval Method \neq NONE

On exit: if

Interval Method \neq NONE

BU (i, l)

contains the upper limit of an

(100 \times α) %

confidence interval for

B (i, l)

, for

i = 1, 2, \dots, IP

and

l = 1, 2, \dots, NTAU

Interval Method = NONE

, BU is not referenced.

The method used for calculating the interval is controlled by the optional parameters Interval Method and Bootstrap Interval Method. The size of the interval,

α

is controlled by the optional parameter Significance Level.

18: CH(IP,IP, $*$ ) – REAL (KIND=nag_wp) arrayOutput

Note: the last dimension of the array CH must be at least

NTAU

Interval Method \neq NONE

and

Matrix Returned = COVARIANCE

and at least

NTAU + 1

Interval Method \neq NONE

IID

BOOTSTRAP XY

and

Matrix Returned = H INVERSE

On exit: depending on the supplied optional parameters, CH will either not be referenced, hold an estimate of the upper triangular part of the covariance matrix,

Σ

, or an estimate of the upper triangular parts of

n J_{n}

and

n^{- 1} H_{n}^{- 1}

Interval Method = NONE

Matrix Returned = NONE

, CH is not referenced.

Interval Method = BOOTSTRAP XY

IID

and

Matrix Returned = H INVERSE

, CH is not referenced.

Otherwise, for

i, j = 1, 2, \dots, IP, j \geq i

and

l = 1, 2, \dots, NTAU

If $Matrix Returned = COVARIANCE$ , $CH (i, j, l)$ holds an estimate of the covariance between $B (i, l)$ and $B (j, l)$ .
If $Matrix Returned = H INVERSE$ , $CH (i, j, 1)$ holds an estimate of the $(i, j)$ th element of $n J_{n}$ and $CH (i, j, l + 1)$ holds an estimate of the $(i, j)$ th element of $n^{- 1} H_{n}^{- 1}$ , for $τ = TAU (l)$ .

The method used for calculating

Σ

and

H_{n}^{- 1}

is controlled by the optional parameter Interval Method.

19: RES(N, $*$ ) – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array RES must be at least

NTAU

Return Residuals = YES

On exit: if

Return Residuals = YES

RES (i, l)

holds the (weighted) residuals,

r_{i}

, for

τ = TAU (l)

, for

i = 1, 2, \dots, N

and

l = 1, 2, \dots, NTAU

WEIGHT ='W'

and

Drop Zero Weights = YES

, the value of RES will be set to zero for observations with zero weights.

Return Residuals = NO

, RES is not referenced.

20: IOPTS( $*$ ) – INTEGER arrayCommunication Array

Note: the contents of IOPTS must not have been altered between calls to G02ZKF, G02ZLF, G02QGF and the selected problem solving routine.

On entry: optional parameter array, as initialized by a call to G02ZKF.

21: OPTS( $*$ ) – REAL (KIND=nag_wp) arrayCommunication Array

Note: the contents of OPTS must not have been altered between calls to G02ZKF, G02ZLF, G02QGF and the selected problem solving routine.

On entry: optional parameter array, as initialized by a call to G02ZKF.

22: STATE( $*$ ) – INTEGER arrayCommunication Array

Note: the actual argument supplied must be the array STATE supplied to the initialization routines G05KFF or G05KGF.

The actual argument supplied must be the array STATE supplied to the initialization routines G05KFF or G05KGF.

Interval Method = BOOTSTRAP XY

, STATE contains information about the selected random number generator. Otherwise STATE is not referenced.

23: INFO( $NTAU$ ) – INTEGER arrayOutput

On exit:

INFO (i)

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

τ = TAU (i)

Code	Warning
$0$	Model fitted and confidence limits (if requested) calculated successfully
$1$	The routine did not converge. The returned values are based on the estimate at the last iteration. Try increasing Iteration Limit whilst calculating the parameter estimates or relaxing the definition of convergence by increasing Tolerance.
$2$	A singular matrix was encountered during the optimization. The model was not fitted for this value of $τ$ .
$4$	Some truncation occurred whilst calculating the confidence limits for this value of $τ$ . See Section 10 for details. The returned upper and lower limits may be narrower than specified.
$8$	The routine did not converge whilst calculating the confidence limits. The returned limits are based on the estimate at the last iteration. Try increasing Iteration Limit.
$16$	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 as defined by the optional parameter Big.

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.

24: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 11$: On entry, $SORDER \neq 1$ or $2$ .

$IFAIL = 21$: On entry, $INTCPT \neq'Y'$ or $'N'$ .

$IFAIL = 31$: On entry, $WEIGHT \neq'U'$ or $'W'$ .

$IFAIL = 41$: On entry, $N < 2$ .

$IFAIL = 51$: On entry, $M < 0$ .

$IFAIL = 71$: On entry, $SORDER = 1$ , $LDDAT < N$ .

$IFAIL = 72$: On entry, $SORDER = 2$ , $LDDAT < M$ .

$IFAIL = 81$: On entry, $ISX (j) \neq 0$ or $1$ .

$IFAIL = 91$: On entry, $IP < 1$ or $IP \geq N$ .

$IFAIL = 92$: On entry, IP is not consistent with ISX and INTCPT.

$IFAIL = 111$: On entry, $WEIGHT ='W'$ and $WT (i) < 0.0$ for at least one $i$ .

$IFAIL = 112$: On entry, the effective number of observations is less than two.

$IFAIL = 121$: On entry, $NTAU < 1$ .

$IFAIL = 131$: On entry, TAU is invalid.

$IFAIL = 201$: On entry, one or more of the optional parameter arrays IOPTS and OPTS have not been initialized or have been corrupted.

$IFAIL = 221$: On entry, $Interval Method = BOOTSTRAP XY$ and STATE was not initialized or has been corrupted.

$IFAIL = 231$: On exit, problems were encountered whilst fitting at least one model. Additional information has been returned in INFO.

7 Accuracy

Not applicable.

8 Further Comments

G02QGF allocates internally approximately the following elements of real storage:

13 n + n p + 3 p^{2} + 6 p + 3 (p + 1) \times NTAU

. If

Interval Method = BOOTSTRAP XY

then a further

n p

elements are required, and this increases by

p \times NTAU \times Bootstrap Iterations

Bootstrap Interval Method = QUANTILE

. Where possible, any user-supplied output arrays are used as workspace and so the amount actually allocated may be less. If

SORDER = 2

WEIGHT ='U'

INTCPT ='N'

and

IP = M

an internal copy of the input data is avoided and the amount of locally allocated memory is reduced by

n p

9 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 two explanatory variables, a column of ones and household income. The model is fit for five different values of

τ

and the covariance matrix is estimated assuming Normal IID errors. Both the covariance matrix and the residuals are returned.

10 Algorithmic Details

By the addition of slack variables the minimization (1) can be reformulated into the linear programming problem

\underset{(u, v, β) \in ℝ_{+}^{n} \times ℝ_{+}^{n} \times ℝ^{p}}{minimize} τ e^{T} u + (1 - τ) e^{T} v ​   subject to y = X β + u - v

(2)

and its associated dual

\underset{d}{maximize} y^{T} d ​   subject to X^{T} d = 0, d \in {[τ - 1, τ]}^{n}

(3)

where

e

is a vector of

n

1

s. Setting

a = d + (1 - τ) e

gives the equivalent formulation

\underset{a}{maximize} y^{T} a ​   subject to X^{T} a = (1 - τ) X^{T} e, a \in {[0, 1]}^{n} .

(4)

The algorithm introduced by Portnoy and Koenker (1997) and used by G02QGF, uses the primal-dual formulation expressed in equations (2) and (4) along with a logarithmic barrier function to obtain estimates for

β

. The algorithm is based on the predictor-corrector algorithm of Mehrotra (1992) and further details can be obtained from Portnoy and Koenker (1997) and Koenker (2005). A good description of linear programming, interior point algorithms, barrier functions and Mehrotra's predictor-corrector algorithm can be found in Nocedal and Wright (1999).

10.1 Interior Point Algorithm

In this section a brief description of the interior point algorithm used to estimate the model parameters is presented. It should be noted that there are some differences in the equations given here – particularly (7) and (9) – compared to those given in Koenker (2005) and Portnoy and Koenker (1997).

10.1.1 Central path

Rather than optimize (4) directly, an additional slack variable

s

is added and the constraint

a \in {[0, 1]}^{n}

is replaced with

a + s = e, a_{i} \geq 0, s_{i} \geq 0

, for

i = 1, 2, \dots, n

The positivity constraint on

a

and

s

is handled using the logarithmic barrier function

B (a, s, μ) = y^{T} a + μ \sum_{i = 1}^{n} (\log a_{i} + \log s_{i}) .

The primal-dual form of the problem is used giving the Lagrangian

L (a, s, β, u, μ) = B (a, s, μ) - β^{T} (X^{T} a - (1 - τ) X^{T} e) - u^{T} (a + s - e)

whose central path is described by the following first order conditions

\begin{matrix} X^{T} a & = & (1 - τ) X^{T} e \\ a + s & = & e \\ X β + u - v & = & y \\ S U e & = & μ e \\ A V e & = & μ e \end{matrix}

(5)

where

A

denotes the diagonal matrix with diagonal elements given by

a

, similarly with

S, U

and

V

. By enforcing the inequalities on

s

and

a

strictly, i.e.,

a_{i} > 0

and

s_{i} > 0

for all

i

we ensure that

A

and

S

are positive definite diagonal matrices and hence

A^{- 1}

and

S^{- 1}

exist.

Rather than applying Newton's method to the system of equations given in (5) to obtain the step directions

δ_{β}, δ_{a}, δ_{s}, δ_{u}

and

δ_{v}

, Mehrotra substituted the steps directly into (5) giving the augmented system of equations

\begin{matrix} X^{T} (a + δ_{a}) & = & (1 - τ) X^{T} e \\ (a + δ_{a}) + (s + δ_{s}) & = & e \\ X (β + δ_{β}) + (u + δ_{u}) - (v + δ_{v}) & = & y \\ (S + Δ_{s}) (U + Δ_{u}) e & = & μ e \\ (A + Δ_{a}) (V + Δ_{v}) e & = & μ e \end{matrix}

(6)

where

Δ_{a}, Δ_{s}, Δ_{u}

and

Δ_{v}

denote the diagonal matrices with diagonal elements given by

δ_{a}, δ_{s}, δ_{u}

and

δ_{v}

respectively.

10.1.2 Affine scaling step

The affine scaling step is constructed by setting

μ = 0

in (5) and applying Newton's method to obtain an intermediate set of step directions

\begin{matrix} (X^{T} W X) δ_{β} & = & X^{T} W (y - X β) + (τ - 1) X^{T} e + X^{T} a \\ δ_{a} & = & W (y - X β - X δ_{β}) \\ δ_{s} & = & - δ_{a} \\ δ_{u} & = & S^{- 1} U δ_{a} - U e \\ δ_{v} & = & A^{- 1} V δ_{s} - V e \end{matrix}

(7)

where

W = {(S^{- 1} U + A^{- 1} V)}^{- 1}

Initial step sizes for the primal (

{\hat{γ}}_{P}

) and dual (

{\hat{γ}}_{D}

) parameters are constructed as

\begin{matrix} {\hat{γ}}_{P} = σ \min \{\min_{i, δ_{a_{i}} < 0} \{a_{i} / δ_{a_{i}}\}, \min_{i, δ_{s_{i}} < 0} \{s_{i} / δ_{s_{i}}\}\} \\ {\hat{γ}}_{D} = σ \min \{\min_{i, δ_{u_{i}} < 0} \{u_{i} / δ_{u_{i}}\}, \min_{i, δ_{v_{i}} < 0} \{v_{i} / δ_{v_{i}}\}\} \end{matrix}

(8)

where

σ

is a user-supplied scaling factor. If

{\hat{γ}}_{P} \times {\hat{γ}}_{D} \geq 1

then the nonlinearity adjustment, described in Section 10.1.3, is not made and the model parameters are updated using the current step size and directions.

10.1.3 Nonlinearity Adjustment

In the nonlinearity adjustment step a new estimate of

μ

is obtained by letting

\hat{g} ({\hat{γ}}_{P}, {\hat{γ}}_{D}) = {(s + {\hat{γ}}_{P} δ_{s})}^{T} (u + {\hat{γ}}_{D} δ_{u}) + {(a + {\hat{γ}}_{P} δ_{a})}^{T} (v + {\hat{γ}}_{D} δ_{v})

and estimating

μ

μ = {(\frac{\hat{g} ({\hat{γ}}_{P}, {\hat{γ}}_{D})}{\hat{g} (0, 0)})}^{3} \frac{\hat{g} (0, 0)}{2 n} .

This estimate, along with the nonlinear terms (

Δ u

Δ s

Δ a

and

Δ v

) from (6) are calculated using the values of

δ_{a}, δ_{s}, δ_{u}

and

δ_{v}

obtained from the affine scaling step.

Given an updated estimate for

μ

and the nonlinear terms the system of equations

\begin{matrix} (X^{T} W X) δ_{β} & = & X^{T} W (y - X β + μ (S^{- 1} - A^{- 1}) e + S^{- 1} Δ_{s} Δ_{u} e - A^{- 1} Δ_{a} Δ_{v} e) + (τ - 1) X^{T} e + X^{T} a \\ δ_{a} & = & W (y - X β - X δ_{β} + μ (S^{- 1} - A^{- 1})) \\ δ_{s} & = & - δ_{a} \\ δ_{u} & = & μ S^{- 1} e + S^{- 1} U δ_{a} - U e - S^{- 1} Δ_{s} Δ_{u} e \\ δ_{v} & = & μ A^{- 1} e + A^{- 1} V δ_{s} - V e - A^{- 1} Δ_{a} Δ_{v} e \end{matrix}

(9)

are solved and updated values for

δ_{β}, δ_{a}, δ_{s}, δ_{u}, δ_{v}, {\hat{γ}}_{P}

and

{\hat{γ}}_{D}

calculated.

10.1.4 Update and convergence

At each iteration the model parameters

(β, a, s, u, v)

are updated using step directions,

(δ_{β}, δ_{a}, δ_{s}, δ_{u}, δ_{v})

and step lengths

({\hat{γ}}_{P}, {\hat{γ}}_{D})

Convergence is assessed using the duality gap, that is, the differences between the objective function in the primal and dual formulations. For any feasible point

(u, v, s, a)

the duality gap can be calculated from equations (2) and (3) as

\begin{matrix} τ e^{T} u + (1 - τ) e^{T} v - d^{T} y & = & τ e^{T} u + (1 - τ) e^{T} v - {(a - (1 - τ) e)}^{T} y \\ = & s^{T} u + a^{T} v \\ = & e^{T} u - a^{T} y + (1 - τ) e^{T} X β \end{matrix}

and the optimization terminates if the duality gap is smaller than the tolerance supplied in the optional parameter Tolerance.

10.1.5 Additional information

Initial values are required for the parameters

a, s, u, v

and

β

. If not supplied by the user, initial values for

β

are calculated from a least squares regression of

y

X

. This regression is carried out by first constructing the cross-product matrix

X^{T} X

and then using a pivoted

Q R

decomposition as performed by F08BFF (DGEQP3). In addition, if the cross-product matrix is not of full rank, a rank reduction is carried out and, rather than using the full design matrix,

X

, a matrix formed from the first

p

-rank columns of

X P

is used instead, where

P

is the pivot matrix used during the

Q R

decomposition. Parameter estimates, confidence intervals and the rows and columns of the matrices returned in the parameter CH (if any) are set to zero for variables dropped during the rank-reduction. The rank reduction step is performed irrespective of whether initial values are supplied by the user.

Once initial values have been obtained for

β

, the initial values for

u

and

v

are calculated from the residuals. If

|r_{i}| < ε_{u}

then a value of

\pm ε_{u}

is used instead, where

ε_{u}

is supplied in the optional parameter Epsilon. The initial values for the

a

and

s

are always set to

1 - τ

and

τ

respectively.

The solution for

δ_{β}

in both (7) and (9) is obtained using a Bunch–Kaufman decomposition, as implemented in F07MDF (DSYTRF).

10.2 Calculation of Covariance Matrix

G02QGF supplies four methods to calculate the covariance matrices associated with the parameter estimates for

β

. This section gives some additional detail on three of the algorithms, the fourth, (which uses bootstrapping), is described in Section 3.

(i)

Independent, identically distributed (IID) errors

When assuming IID errors, the covariance matrices depend on the sparsity,

s (τ)

, which G02QGF estimates as follows:

(a)	Let $r_{i}$ denote the residuals from the original quantile regression, that is $r_{i} = y_{i} - x_{i}^{T} \hat{β}$ .
(b)	Drop any residual where $\|r_{i}\|$ is less than $ε_{u}$ , supplied in the optional parameter Epsilon.
(c)	Sort and relabel the remaining residuals in ascending order, by absolute value, so that $ε_{u} < \|r_{1}\| < \|r_{2}\| < \dots$ .
(d)	Select the first $l$ values where $l = h_{n} n$ , for some bandwidth $h_{n}$ .
(e)	Sort and relabel these $l$ residuals again, so that $r_{1} < r_{2} < \dots < r_{l}$ and regress them against a design matrix with two columns ( $p = 2$ ) and rows given by $x_{i} = \{1, i / (n - p)\}$ using quantile regression with $τ = 0.5$ .
(f)	Use the resulting estimate of the slope as an estimate of the sparsity.

(ii)

Powell Sandwich

When using the Powell Sandwich to estimate the matrix

H_{n}

, the quantity

c_{n} = \min (σ_{r}, (q_{r 3} - q_{r 1}) / 1.34) \times (Φ^{- 1} (τ + h_{n}) - Φ^{- 1} (τ - h_{n}))

is calculated. Dependent on the value of

τ

and the method used to calculate the bandwidth (

h_{n}

), it is possible for the quantities

τ \pm h_{n}

to be too large or small, compared to machine precision (

ε

). More specifically, when

τ - h_{n} \leq \sqrt{ε}

, or

τ + h_{n} \geq 1 - \sqrt{ε}

, a warning flag is raised in INFO, the value is truncated to

\sqrt{ε}

1 - \sqrt{ε}

respectively and the covariance matrix calculated as usual.

(iii)

Hendricks–Koenker Sandwich

The Hendricks–Koenker Sandwich requires the calculation of the quantity

d_{i} = x_{i}^{T} (\hat{β} (τ + h_{n}) - \hat{β} (τ - h_{n}))

. As with the Powell Sandwich, in cases where

τ - h_{n} \leq \sqrt{ε}

, or

τ + h_{n} \geq 1 - \sqrt{ε}

, a warning flag is raised in INFO, the value truncated to

\sqrt{ε}

1 - \sqrt{ε}

respectively and the covariance matrix calculated as usual.

In addition, it is required that

d_{i} > 0

, in this method. Hence, instead of using

2 h_{n} / d_{i}

in the calculation of

H_{n}

\max (2 h_{n} / (d_{i} + ε_{u}), 0)

is used instead, where

ε_{u}

is supplied in the optional parameter Epsilon.

11 Optional Parameters

Several optional parameters in G02QGF control aspects of the optimization algorithm, methodology used, logic or output. Their values are contained in the arrays IOPTS and OPTS; these must be initialized before calling G02QGF by first calling G02ZKF with OPTSTR set to

Initialize = G02QGF

Each optional parameter has an associated default value; to set any of them to a nondefault value, use G02ZKF. The current value of an optional parameter can be queried using G02ZLF.

The remainder of this section can be skipped if you wish to use the default values for all optional parameters.

The following is a list of the optional parameters available. A full description of each optional parameter is provided in Section 11.1.

11.1 Description of the Optional Parameters

For each option, we give a summary line, a description of the optional parameter and details of constraints.

The summary line contains:

the keywords, where the minimum abbreviation of each keyword is underlined (if no characters of an optional qualifier are underlined, the qualifier may be omitted);
a parameter value, where the letters $a$ , $i and r$ denote options that take character, integer and real values respectively;
the default value, where the symbol $ε$ is a generic notation for machine precision (see X02AJF).

Keywords and character values are case and white space insensitive.

Band Width Alpha

r

Default

= 1.0

A multiplier used to construct the parameter

α_{b}

used when calculating the Sheather–Hall bandwidth (see Section 3), with

α_{b} = (1 - α) \times Band Width Alpha

. Here,

α

is the Significance Level.

Constraint:

Band Width Alpha > 0.0

Band Width Method

a

Default

='SHEATHER HALL'

The method used to calculate the bandwidth used in the calculation of the asymptotic covariance matrix

Σ

and

H^{- 1}

Interval Method = HKS

KERNEL

IID

(see Section 3).

Constraint:

Band Width Method = SHE ATHER HAL L

BOF INGER

Big

r

Default

= {10.0}^{20}

This parameter should be set to something larger than the biggest value supplied in DAT and Y.

Constraint:

Big > 0.0

Bootstrap Interval Method

a

Default

='QUANTILE'

Interval Method = BOOTSTRAP XY

, Bootstrap Interval Method controls how the confidence intervals are calculated from the bootstrap estimates.

$Bootstrap Interval Method = T$: $t$ intervals are calculated. That is, the covariance matrix, $Σ = \{σ_{i j} : i, j = 1, 2, \dots, p\}$ is calculated from the bootstrap estimates and the limits calculated as $β_{i} \pm t_{(n - p, (1 + α) / 2)} σ_{i i}$ where $t_{(n - p, (1 + α) / 2)}$ is the $(1 + α) / 2$ percentage point from a Student's $t$ distribution on $n - p$ degrees of freedom, $n$ is the effective number of observations and $α$ is given by the optional parameter Significance Level.
$Bootstrap Interval Method = QUANTILE$: Quantile intervals are calculated. That is, the upper and lower limits are taken as the $(1 + α) / 2$ and $(1 - α) / 2$ quantiles of the bootstrap estimates, as calculated using G01AMF.

Constraint:

Bootstrap Interval Method = T

QUA NTILE

Bootstrap Iterations

i

Default

= 100

The number of bootstrap samples used to calculate the confidence limits and covariance matrix (if requested) when

Interval Method = BOOTSTRAP XY

Constraint:

Bootstrap Iterations > 1

Bootstrap Monitoring

a

Default

='NO'

Bootstrap Monitoring = YES

and

Interval Method = BOOTSTRAP XY

, then the parameter estimates for each of the bootstrap samples are displayed. This information is sent to the unit number specified by Unit Number.

Constraint:

Bootstrap Monitoring = YES

NO

Calculate Initial Values

a

Default

='YES'

Calculate Initial Values = YES

then the initial values for the regression parameters,

β

, are calculated from the data. Otherwise they must be supplied in B.

Constraint:

Calculate Initial Values = YES

NO

Defaults

This special keyword is used to reset all optional parameters to their default values.

Drop Zero Weights

a

Default

='YES'

If a weighted regression is being performed and

Drop Zero Weights = YES

then observations with zero weight are dropped from the analysis. Otherwise such observations are included.

Constraint:

Drop Zero Weights = YES

NO

Epsilon

r

Default

= \sqrt{ε}

ε_{u}

, the tolerance used when calculating the covariance matrix and the initial values for

u

and

v

. For additional details see Section 10.2 and Section 10.1.5 respectively.

Constraint:

Epsilon \geq 0.0

Interval Method

a

Default

='IID'

The value of Interval Method controls whether confidence limits are returned in BL and BU and how these limits are calculated. This parameter also controls how the matrices returned in CH are calculated.

$Interval Method = NONE$: No limits are calculated and BL, BU and CH are not referenced.
$Interval Method = KERNEL$: The Powell Sandwich method with a Gaussian kernel is used.
$Interval Method = HKS$: The Hendricks–Koenker Sandwich is used.
$Interval Method = IID$: The errors are assumed to be identical, and independently distributed.
$Interval Method = BOOTSTRAP XY$: A bootstrap method is used, where sampling is done on the pair $(y_{i}, x_{i})$ . The number of bootstrap samples is controlled by the parameter Bootstrap Iterations and the type of interval constructed from the bootstrap samples is controlled by Bootstrap Interval Method.

Constraint:

Interval Method = NON E

KER NEL

HKS

IID

BOO TSTRAP XY

Iteration Limit

i

Default

= 100

The maximum number of iterations to be performed by the interior point optimization algorithm.

Constraint:

Iteration Limit > 0

Matrix Returned

a

Default

='NONE'

The value of Matrix Returned controls the type of matrices returned in CH. If

Interval Method = NONE

, this parameter is ignored and CH is not referenced. Otherwise:

$Matrix Returned = NONE$: No matrices are returned and CH is not referenced.
$Matrix Returned = COVARIANCE$: The covariance matrices are returned.
$Matrix Returned = H INVERSE$: If $Interval Method = KERNEL$ or $HKS$ , the matrices $J$ and $H^{- 1}$ are returned. Otherwise no matrices are returned and CH is not referenced.

The matrices returned are calculated as described in Section 3, with the algorithm used specified by Interval Method. In the case of

Interval Method = BOOTSTRAP XY

the covariance matrix is calculated directly from the bootstrap estimates.

Constraint:

Matrix Returned = NON E

COV ARIANCE

H INV ERSE

Monitoring

a

Default

='NO'

Monitoring = YES

then the duality gap is displayed at each iteration of the interior point optimization algorithm. In addition, the final estimates for

β

are also displayed.

The monitoring information is sent to the unit number specified by Unit Number.

Constraint:

Monitoring = YES

NO

QR Tolerance

r

Default

= ε^{0.9}

The tolerance used to calculate the rank,

k

, of the

p \times p

cross-product matrix,

X^{T} X

. Letting

Q

be the orthogonal matrix obtained from a

Q R

decomposition of

X^{T} X

, then the rank is calculated by comparing

Q_{i i}

with

Q_{11} \times QR Tolerance

If the cross-product matrix is rank deficient, then the parameter estimates for the

p - k

columns with the smallest values of

Q_{i i}

are set to zero, along with the corresponding entries in BL, BU and CH, if returned. This is equivalent to dropping these variables from the model. Details on the

Q R

decomposition used can be found in F08BFF (DGEQP3).

Constraint:

QR Tolerance > 0.0

Return Residuals

a

Default

='NO'

Return Residuals = YES

, the residuals are returned in RES. Otherwise RES is not referenced.

Constraint:

Return Residuals = YES

NO

Sigma

r

Default

= 0.99995

The scaling factor used when calculating the affine scaling step size (see equation (8)).

Constraint:

0.0 < Sigma < 1.0

Significance Level

r

Default

= 0.95

α

, the size of the confidence interval whose limits are returned in BL and BU.

Constraint:

0.0 < Significance Level < 1.0

Tolerance

r

Default

= \sqrt{ε}

Convergence tolerance. The optimization is deemed to have converged if the duality gap is less than Tolerance (see Section 10.1.4).

Constraint:

Tolerance > 0.0

Unit Number

i

Default taken from X04ABF

The unit number to which any monitoring information is sent.

Constraint:

Unit Number > 1

12 Description of Monitoring Information

See the description of the optional argument Monitoring.

G02QGF (PDF version)

G02 Chapter Contents

G02 Chapter Introduction

NAG Library Manual

NAG Library Routine DocumentG02QGF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

10 Algorithmic Details

10.1 Interior Point Algorithm

10.1.1 Central path

10.1.2 Affine scaling step

10.1.3 Nonlinearity Adjustment

10.1.4 Update and convergence

10.1.5 Additional information

10.2 Calculation of Covariance Matrix

11 Optional Parameters

11.1 Description of the Optional Parameters

12 Description of Monitoring Information

NAG Library Routine Document

G02QGF