nag_regsn_quant_linear (g02qgc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_regsn_quant_linear (g02qgc)

Note: this function uses optional arguments 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 arguments, 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 arguments and to Section 12 for a detailed description of the monitoring information produced by the function.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_regsn_quant_linear (g02qgc) performs a multiple linear quantile regression. Parameter estimates and, if required, confidence limits, covariance matrices and residuals are calculated. nag_regsn_quant_linear (g02qgc) may be used to perform a weighted quantile regression. A simplified interface for nag_regsn_quant_linear (g02qgc) is provided by nag_regsn_quant_linear_iid (g02qfc).

2  Specification

#include <nag.h>
#include <nagg02.h>
void  nag_regsn_quant_linear (Nag_OrderType order, Nag_IncludeIntercept intcpt, Integer n, Integer m, const double dat[], Integer pddat, const Integer isx[], Integer ip, const double y[], const double wt[], Integer ntau, const double tau[], double *df, double b[], double bl[], double bu[], double ch[], double res[], const Integer iopts[], const double opts[], Integer state[], Integer info[], NagError *fail)

3  Description

Given a vector of n observed values, y = y i : i = 1, 2, , n , an n×p design matrix X, a column vector, x, of length p holding the ith row of X and a quantile τ 0 , 1 , nag_regsn_quant_linear (g02qgc) estimates the p-element vector β as the solution to
minimize β p i=1 n ρ τ y i - xiT 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 if z<0 and 0 otherwise. Weights can be incorporated by replacing X and y with WX and Wy respectively, where W is an n×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.
nag_regsn_quant_linear (g02qgc) uses the interior point algorithm of Portnoy and Koenker (1997), described briefly in Section 10, to obtain the parameter estimates β^, for a given value of τ.
Under the assumption of Normally distributed errors, Koenker (2005) shows that the limiting covariance matrix of β^-β has the form
Σ = τ 1 - τ n H n -1 J n H n -1
where Jn = n-1 i=1 n x i xiT  and Hn is a function of τ, as described below. Given an estimate of the covariance matrix, Σ^, lower (β^L) and upper (β^U) limits for an 100×α% confidence interval can be calculated for each of the p parameters, via
β^ Li = β^ i - t n-p , 1 + α / 2 Σ^ ii , β^ Ui = β^ i + t n-p , 1 + α / 2 Σ^ ii
where tn-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 XTX.
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 = τ 1 - τ n s τ 2 XT X -1
where s is the sparsity function. nag_regsn_quant_linear (g02qgc) estimates sτ from the residuals, ri = yi - xiT β^  and a bandwidth hn.
(ii) Powell Sandwich
Powell (1991) suggested estimating the matrix Hn by a kernel estimator of the form
H^ n = n cn -1 i=1 n K r i c n xi xiT
where K is a kernel function and cn satisfies lim n cn 0  and lim n n cn . When the Powell method is chosen, nag_regsn_quant_linear (g02qgc) uses a Gaussian kernel (i.e., K=ϕ) and sets
cn = minσr, qr3 - qr1 / 1.34 × Φ-1 τ + hn - Φ-1 τ - hn
where hn is a bandwidth, σr,qr1 and qr3 are, respectively, the standard deviation and the 25% and 75% quantiles for the residuals, ri.
(iii) Hendricks–Koenker Sandwich
Koenker (2005) suggested estimating the matrix Hn using
H^ n = n-1 i=1 n 2 hn xiT β^ τ + hn - β^ τ - hn xi xiT
where hn is a bandwidth and β^τ+hn denotes the parameter estimates obtained from a quantile regression using the τ+hnth quantile. Similarly with β^τ-hn.
(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 yi,xi (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, β^L and 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, hn. Two alternative methods for determining this are provided:
(i) Sheather–Hall
hn = 1.5 Φ-1αb ϕ Φ-1τ 2 n 2 Φ-1τ + 1 13
for a user-supplied value αb,
(ii) Bofinger
hn = 4.5 ϕ Φ-1τ 4 n 2 Φ-1τ + 1 2 15
nag_regsn_quant_linear (g02qgc) allows optional arguments to be supplied via the iopts and opts arrays (see Section 11 for details of the available options). If the default values for these optional arguments are sufficient then iopts and opts can be set to NULL, otherwise prior to calling nag_regsn_quant_linear (g02qgc) the optional argument arrays, must be initialized by calling nag_g02_opt_set (g02zkc) with optstr set to Initialize=g02qgc. If bootstrap confidence limits are required (Interval Method=BOOTSTRAP XY) then one of the random number initialization functions nag_rand_init_repeatable (g05kfc) (for a repeatable analysis) or nag_rand_init_nonrepeatable (g05kgc) (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  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     intcptNag_IncludeInterceptInput
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=Nag_Intercept
An intercept will be included in the model.
intcpt=Nag_NoIntercept
An intercept will not be included in the model.
Constraint: intcpt=Nag_NoIntercept or Nag_Intercept.
3:     nIntegerInput
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: n2.
4:     mIntegerInput
On entry: m, the total number of variates in the dataset.
Constraint: m0.
5:     dat[dim]const doubleInput
Note: the dimension, dim, of the array dat must be at least
  • pddat×m when order=Nag_ColMajor;
  • pddat×n when order=Nag_RowMajor.
Where DATi,j appears in this document, it refers to the array element
  • dat[j-1×pddat+i-1] when order=Nag_ColMajor;
  • dat[i-1×pddat+j-1] when order=Nag_RowMajor.
On entry: the ith value for the jth variate, for i=1,2,,n and j=1,2,,m, must be supplied in DATi,j 
The design matrix X is constructed from dat, isx and intcpt.
6:     pddatIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array dat.
Constraints:
  • if order=Nag_ColMajor, pddatn;
  • otherwise pddatm.
7:     isx[m]const IntegerInput
On entry: indicates which independent variables are to be included in the model.
isx[j-1]=0
The jth variate, supplied in dat, is not included in the regression model.
isx[j-1]=1
The jth variate, supplied in dat, is included in the regression model.
Constraints:
  • isx[j-1]=0 or 1, for j=1,2,,m;
  • if intcpt=Nag_Intercept, exactly ip-1 values of isx must be set to 1;
  • if intcpt=Nag_NoIntercept, exactly ip values of isx must be set to 1.
8:     ipIntegerInput
On entry: p, the number of independent variables in the model, including the intercept, see intcpt, if present.
Constraints:
  • 1ip<n;
  • if intcpt=Nag_Intercept, 1ipm+1;
  • if intcpt=Nag_NoIntercept, 1ipm.
9:     y[n]const doubleInput
On entry: y, observations on the dependent variable.
10:   wt[n]const doubleInput
On entry: optionally, the diagonal elements of the weight matrix W.
If weights are not provided then wt must be set to NULL.
When
Drop Zero Weights=YES
If wt[i-1]=0.0, the ith 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:
  • the effective number of observations 2;
  • wt[i]=0.0, for all i.
11:   ntauIntegerInput
On entry: the number of quantiles of interest.
Constraint: ntau1.
12:   tau[ntau]const doubleInput
On entry: the vector of quantiles of interest. A separate model is fitted to each quantile.
Constraint: ε<tau[j-1]<1-ε where ε is the machine precision returned by nag_machine_precision (X02AJC), for j=1,2,,ntau.
13:   dfdouble *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 XTX.
14:   b[ip×ntau]doubleInput/Output
Note: where Bi,l appears in this document, it refers to the array element b[l-1×ip+i-1].
On entry: If Calculate Initial Values=NO, Bi,l must hold an initial estimates for β^i, for i=1,2,,ip and l=1,2,,ntau. If Calculate Initial Values=YES, b need not be set.
On exit: Bi,l, for i=1,2,,ip, contains the estimates of the parameters of the regression model, β^, estimated for τ=tau[l-1].
If intcpt=Nag_Intercept, B1,l will contain the estimate corresponding to the intercept and Bi+1,l will contain the coefficient of the jth variate contained in dat, where isx[j-1] is the ith nonzero value in the array isx.
If intcpt=Nag_NoIntercept, Bi,l will contain the coefficient of the jth variate contained in dat, where isx[j-1] is the ith nonzero value in the array isx.
15:   bl[dim]doubleOutput
Note: the dimension, dim, of the array bl must be at least
  • if Interval MethodNONE, ip×ntau.
Where BLi,l appears in this document, it refers to the array element bl[l-1×ip+i-1].
On exit: if Interval MethodNONE, BLi,l contains the lower limit of an 100×α% confidence interval for Bi,l, for i=1,2,,ip and l=1,2,,ntau.
If Interval Method=NONE, bl is not referenced and can be set to NULL.
The method used for calculating the interval is controlled by the optional arguments Interval Method and Bootstrap Interval Method. The size of the interval, α, is controlled by the optional argument Significance Level.
16:   bu[dim]doubleOutput
Note: the dimension, dim, of the array bu must be at least
  • if Interval MethodNONE, ip×ntau.
Where BUi,l appears in this document, it refers to the array element bu[l-1×ip+i-1].
On exit: if Interval MethodNONE, BUi,l contains the upper limit of an 100×α% confidence interval for Bi,l, for i=1,2,,ip and l=1,2,,ntau.
If Interval Method=NONE, bu is not referenced and can be set to NULL.
The method used for calculating the interval is controlled by the optional arguments Interval Method and Bootstrap Interval Method. The size of the interval, α is controlled by the optional argument Significance Level.
17:   ch[dim]doubleOutput
Note: the dimension, dim, of the array ch must be at least
  • if Interval MethodNONE and Matrix Returned=COVARIANCE, ip×ip×ntau;
  • if Interval MethodNONE, IID or BOOTSTRAP XY and Matrix Returned=H INVERSE, ip×ip×(ntau+1).
The element CHi,j,l is stored in the array element ch[l-1×ip×ip+j-1×ip+i-1].
On exit: depending on the supplied optional arguments, 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 nJn and n-1Hn-1.
If Interval Method=NONE or Matrix Returned=NONE, ch is not referenced.
If Interval Method=BOOTSTRAP XY or IID and Matrix Returned=H INVERSE, ch is not referenced.
Otherwise, for i,j=1,2,,ip,ji and l=1,2,,ntau:
  • If Matrix Returned=COVARIANCE, CHi,j,l holds an estimate of the covariance between Bi,l and Bj,l.
  • If Matrix Returned=H INVERSE, CHi,j,1 holds an estimate of the i,jth element of nJn and CHi,j,l+1 holds an estimate of the i,jth element of n-1Hn-1, for τ=tau[l-1].
The method used for calculating Σ and Hn-1 is controlled by the optional argument Interval Method.
In cases where ch is not going to be referenced it can be set to NULL.
18:   res[n×ntau]doubleOutput
On exit: if Return Residuals=YES, res[l-1×n+i-1] holds the (weighted) residuals, ri, for τ=tau[l-1], for i=1,2,,n and l=1,2,,ntau.
If wt is not NULL and Drop Zero Weights=YES, the value of res will be set to zero for observations with zero weights.
If Return Residuals=NO, res is not referenced and can be set to NULL.
19:   iopts[dim]const IntegerCommunication Array
Note: the dimension, dim, of the array iopts must be at least liopts (see nag_g02_opt_set (g02zkc)).
On entry: If the default values of the optional arguments are sufficient, then iopts can be set to NULL, otherwise the optional argument array, as initialized by a call to nag_g02_opt_set (g02zkc) must be supplied. The contents of iopts MUST NOT be modified directly between calls to any of the functions nag_regsn_quant_linear (g02qgc), nag_g02_opt_set (g02zkc) and nag_g02_opt_get (g02zlc).
20:   opts[dim]const doubleCommunication Array
Note: the dimension, dim, of the array opts must be at least lopts (see nag_g02_opt_set (g02zkc)).
On entry: If the default values of the optional arguments are sufficient, then opts can be set to NULL, otherwise the optional argument array, as initialized by a call to nag_g02_opt_set (g02zkc) must be supplied. The contents of opts MUST NOT be modified directly between calls to any of the functions nag_regsn_quant_linear (g02qgc), nag_g02_opt_set (g02zkc) and nag_g02_opt_get (g02zlc).
21:   state[dim]IntegerCommunication Array
Note: the dimension, dim, of the array state must be at least lstate when Interval Method=BOOTSTRAP XY. See nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).
If Interval Method=BOOTSTRAP XY, state contains information about the selected random number generator. Otherwise state is not referenced and can be set to NULL.
22:   info[ntau]IntegerOutput
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 function 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 function 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 argument 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.
23:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_ARRAY_SIZE
On entry, pddat=value and m=value.
Constraint: pddatm.
On entry, pddat=value and n=value.
Constraint: pddatn.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INITIALIZATION
On entry, either the option arrays have not been initialized or they have been corrupted.
NE_INT
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n2.
On entry, ntau=value.
Constraint: ntau1.
NE_INT_2
On entry, ip=value and n=value.
Constraint: 1ip<n.
NE_INT_ARRAY
On entry, isx[value]=value.
Constraint: isx[i]=0 or 1 for all i.
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.
NE_INVALID_STATE
On entry, state vector has been corrupted or not initialized.
NE_IP_INCOMP_SX
On entry, ip is not consistent with isx or intcpt: ip=value, expected value=value.
NE_NEG_WEIGHT
On entry, wt[value]=value.
Constraint: wt[i]0.0 for all i.
NE_OBSERVATIONS
On entry, effective number of observations=value.
Constraint: effective number of observationsvalue.
NE_REAL_ARRAY
On entry, tau[value]=value is invalid.
NW_POTENTIAL_PROBLEM
A potential problem occurred whilst fitting the model(s).
Additional information has been returned in info.

7  Accuracy

Not applicable.

8  Further Comments

nag_regsn_quant_linear (g02qgc) allocates internally approximately the following elements of double storage: 13n+ np+ 3p2+ 6p+ 3p+1×ntau. If Interval Method=BOOTSTRAP XY then a further np elements are required, and this increases by p×ntau×Bootstrap Iterations if 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 order=Nag_RowMajor, wt is NULL, intcpt=Nag_NoIntercept and ip=m an internal copy of the input data is avoided and the amount of locally allocated memory is reduced by np.

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.

9.1  Program Text

Program Text (g02qgce.c)

9.2  Program Data

Program Data (g02qgce.d)

9.3  Program Results

Program Results (g02qgce.r)

10  Algorithmic Details

By the addition of slack variables the minimization (1) can be reformulated into the linear programming problem
minimize u, v, β + n × + n × p τ eT u + 1 - τ eT v ​   subject to   y = Xβ + u - v (2)
and its associated dual
maximize d yT d ​   subject to   XTd=0, d τ-1,τ n (3)
where e is a vector of n 1s. Setting a=d+1-τe gives the equivalent formulation
maximize a yT a ​   subject to   XT a = 1 - τ XT e , a 0,1 n . (4)
The algorithm introduced by Portnoy and Koenker (1997) and used by nag_regsn_quant_linear (g02qgc), 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 0,1 n  is replaced with a+s=e , a i 0 , s i 0 , for i= 1,2,,n .
The positivity constraint on a and s is handled using the logarithmic barrier function
B a,s,μ = yT a + μ i=1 n logai + logsi .
The primal-dual form of the problem is used giving the Lagrangian
L a,s,β,u,μ = B a,s,μ - βT XT a - 1 - τ XT e - uT a + s - e
whose central path is described by the following first order conditions
XTa = 1-τ XTe a+s = e Xβ+u-v = y SUe = μe AVe = μe (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., ai>0 and si>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 δβ,δa,δs,δu  and δv , Mehrotra substituted the steps directly into (5) giving the augmented system of equations
XT a + δa = 1-τXTe a + δa + s + δs = e X β + δβ + u + δu - v + δv = y S + Δs U + Δu e = μe A + Δa V + Δv e = μe (6)
where Δa,Δs,Δu  and Δv  denote the diagonal matrices with diagonal elements given by δa,δs,δu  and δv  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
XT W X δ β = XT W y - X β + τ - 1 XT e + XT a δ a = W y-Xβ - 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 (7)
where W = S-1 U + A-1 V -1 .
Initial step sizes for the primal (γ^P) and dual (γ^D) parameters are constructed as
γ^P = σ × min min i , δ a i < 0 ai / δ a i , min i , δ s i < 0 si / δ s i γ^D = σ × min min i , δ u i < 0 ui / δ u i , min i , δ v i < 0 vi / δ v i (8)
where σ is a user-supplied scaling factor. If γ^P × γ^D 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
g^γ^P,γ^D = s + γ^P δs T u + γ^D δu + a + γ^P δa T v + γ^D δv
and estimating μ as
μ = g^γ^P,γ^D g^0,0 3 g^0,0 2n .
This estimate, along with the nonlinear terms (Δu, Δs, Δa and Δv) from (6) are calculated using the values of δa,δs,δu  and δv  obtained from the affine scaling step.
Given an updated estimate for μ and the nonlinear terms the system of equations
XT W X δ β = XT 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 XT e + XT a δ a = W y-Xβ - 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 Δs Δu e δ v = μ A-1 e + A-1 V δ s - V e - A-1 Δa Δv e (9)
are solved and updated values for δβ,δa,δs,δu,δv,γ^P  and 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, δβ,δa,δs,δu,δv  and step lengths γ^P,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
τ eT u + 1-τ eT v - dT y = τ eT u + 1-τ eT v - a - 1 - τ e T y = sT u + aT v = eT u - aT y + 1 - τ eT X β
and the optimization terminates if the duality gap is smaller than the tolerance supplied in the optional argument 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 on X. This regression is carried out by first constructing the cross-product matrix XTX and then using a pivoted QR decomposition as performed by nag_dgeqp3 (f08bfc). 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 XP is used instead, where P is the pivot matrix used during the QR decomposition. Parameter estimates, confidence intervals and the rows and columns of the matrices returned in the argument 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 ri<εu then a value of ±εu is used instead, where εu is supplied in the optional argument Epsilon. The initial values for the a and s are always set to 1-τ and τ respectively.
The solution for δβ in both (7) and (9) is obtained using a Bunch–Kaufman decomposition, as implemented in nag_dsytrf (f07mdc).

10.2  Calculation of Covariance Matrix

nag_regsn_quant_linear (g02qgc) 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, sτ, which nag_regsn_quant_linear (g02qgc) estimates as follows:
(a) Let ri denote the residuals from the original quantile regression, that is ri = yi - xiT β^ .
(b) Drop any residual where ri is less than εu, supplied in the optional argument Epsilon.
(c) Sort and relabel the remaining residuals in ascending order, by absolute value, so that εu < r1 < r2 < .
(d) Select the first l values where l=hnn, for some bandwidth hn.
(e) Sort and relabel these l residuals again, so that r1 < r2 < < rl  and regress them against a design matrix with two columns (p=2) and rows given by xi = 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 Hn, the quantity
cn = minσr, qr3 - qr1 / 1.34 × Φ-1 τ + hn - Φ-1 τ - hn
is calculated. Dependent on the value of τ and the method used to calculate the bandwidth (hn), it is possible for the quantities τ±hn to be too large or small, compared to machine precision (ε). More specifically, when τ-hnε, or τ+hn1-ε, a warning flag is raised in info, the value is truncated to ε or 1-ε respectively and the covariance matrix calculated as usual.
(iii) Hendricks–Koenker Sandwich
The Hendricks–Koenker Sandwich requires the calculation of the quantity di= xiT β^ τ + hn - β^ τ - hn . As with the Powell Sandwich, in cases where τ-hnε, or τ+hn1-ε, a warning flag is raised in info, the value truncated to ε or 1-ε respectively and the covariance matrix calculated as usual.
In addition, it is required that di>0, in this method. Hence, instead of using 2 hn / di  in the calculation of Hn, max 2 hn / di + εu ,0  is used instead, where εu is supplied in the optional argument Epsilon.

11  Optional Arguments

Several optional arguments in nag_regsn_quant_linear (g02qgc) 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 nag_regsn_quant_linear (g02qgc) by first calling nag_g02_opt_set (g02zkc) with optstr set to Initialize=g02qgc.
Each optional argument has an associated default value; to set any of them to a nondefault value, use nag_g02_opt_set (g02zkc). The current value of an optional argument can be queried using nag_g02_opt_get (g02zlc).
The remainder of this section can be skipped if you wish to use the default values for all optional arguments. The following is a list of the optional arguments available and a full description of each optional argument is provided in Section 11.1.

11.1  Description of the Optional Arguments

For each option, we give a summary line, a description of the optional argument and details of constraints.
The summary line contains:
Keywords and character values are case and white space insensitive.
Band Width AlpharDefault =1.0
A multiplier used to construct the parameter αb used when calculating the Sheather–Hall bandwidth (see Section 3), with αb=1-α×Band Width Alpha. Here, α is the Significance Level.
Constraint: Band Width Alpha>0.0.
Band Width MethodaDefault ='SHEATHER HALL'
The method used to calculate the bandwidth used in the calculation of the asymptotic covariance matrix Σ and H-1 if Interval Method=HKS, KERNEL or IID (see Section 3).
Constraint: Band Width Method=SHEATHER HALL or BOFINGER.
BigrDefault =10.020
This argument should be set to something larger than the biggest value supplied in dat and y.
Constraint: Big>0.0.
Bootstrap Interval MethodaDefault ='QUANTILE'
If 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 ij : i,j = 1,2,,p  is calculated from the bootstrap estimates and the limits calculated as β i ± t n - p , 1 + α / 2 σ ii  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 argument 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 nag_double_quantiles (g01amc).
Constraint: Bootstrap Interval Method=T or QUANTILE.
Bootstrap IterationsiDefault =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 MonitoringaDefault ='NO'
If 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 or NO.
Calculate Initial ValuesaDefault ='YES'
If 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 or NO.
Defaults
This special keyword is used to reset all optional arguments to their default values.
Drop Zero WeightsaDefault ='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 or NO.
EpsilonrDefault =ε
εu, 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: Epsilon0.0.
Interval MethodaDefault ='IID'
The value of Interval Method controls whether confidence limits are returned in bl and bu and how these limits are calculated. This argument 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 yi,xi. The number of bootstrap samples is controlled by the argument Bootstrap Iterations and the type of interval constructed from the bootstrap samples is controlled by Bootstrap Interval Method.
Constraint: Interval Method=NONE, KERNEL, HKS, IID or BOOTSTRAP XY.
Iteration LimitiDefault =100
The maximum number of iterations to be performed by the interior point optimization algorithm.
Constraint: Iteration Limit>0.
Matrix ReturnedaDefault ='NONE'
The value of Matrix Returned controls the type of matrices returned in ch. If Interval Method=NONE, this argument 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=NONE, COVARIANCE or H INVERSE.
MonitoringaDefault ='NO'
If 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 or NO.
QR TolerancerDefault =ε0.9
The tolerance used to calculate the rank, k, of the p×p cross-product matrix, XTX. Letting Q be the orthogonal matrix obtained from a QR decomposition of XTX, then the rank is calculated by comparing Qii with Q11×QR Tolerance.
If the cross-product matrix is rank deficient, then the parameter estimates for the p-k columns with the smallest values of Qii 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 QR decomposition used can be found in nag_dgeqp3 (f08bfc).
Constraint: QR Tolerance>0.0.
Return ResidualsaDefault ='NO'
If Return Residuals=YES, the residuals are returned in res. Otherwise res is not referenced.
Constraint: Return Residuals=YES or NO.
SigmarDefault =0.99995
The scaling factor used when calculating the affine scaling step size (see equation (8)).
Constraint: 0.0<Sigma<1.0.
Significance LevelrDefault =0.95
α, the size of the confidence interval whose limits are returned in bl and bu.
Constraint: 0.0<Significance Level<1.0.
TolerancerDefault =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 NumberiOutput sent to stdout
The unit number to which any monitoring information is sent. See nag_open_file (x04acc) for details on how to assign a file to a unit number. If no unit number is specified then any monitoring information will be sent to standard output (stdout).
Constraint: Unit Number>1.

12  Description of Monitoring Information

See the description of the optional argument Monitoring.

nag_regsn_quant_linear (g02qgc) (PDF version)
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NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012