ASYMPTOTICS AND INFERENCE 47

0.05.1.15.2

bandwidth

0 .2 .4 .6 .8 1

tau

Figure 2.10: Bandwidth parameter h_n. The blue solid line shows the Bofinger band-width forn = 500depending on the value ofτ. The red (α = 0.01) and green (α = 0.1) long-dashed lines present the Hall and Sheather bandwidth, the yellow (α = 0.01) and grey (α= 0.1) short-dashed lines indicate the Chamberlain bandwidth.

Hall and Sheather (1988) questioned Bofinger’s rule and suggested instead

h_n=n^−1/3z_α^2/3

µ1.5s(τ) s⁰⁰(τ)

¶_1/3

(2.59)

wherez_α= Φ⁻¹(1− ^α₂). The parameterαdenotes the desired size of the test. Since the Hall and Sheather rule is explicitly designed for confidence interval construction, rather than simply optimizing MSE-performance for the sparsity estimate itself, Koenker (1994) argued that it seems to be more reasonable for inferential purposes. If we again plug in the Gaussian distribution, we obtain

h_n=n^−1/3z^2/3_α

µ1.5(φ(Φ⁻¹(τ)))² 2(Φ⁻¹(τ))²+ 1

¶_1/3

(2.60)

Figure 2.9 also contains three examples of the Hall and Sheather bandwidth, each cal-culated forα = 0.05. It can be seen that the values are bigger than the Bofinger sand-wich for small sample sizes but (considerably) smaller for medium to big data sets.

A third and rather simple alternative to the bandwidth estimation was proposed by

Buchinsky (1991) and is derived from the asymptotics of binomially distributed quan-tiles (see also Chen (2001)). It is named Chamberlain bandwidth and also contains a significance level parameterα:

h_n=z_α

rτ(1−τ)

n (2.61)

Figure 2.10 pictures the different bandwidth choice rules for the whole range ofτ with n= 500. We imposed two different confidence levels for the Hall and Sheater bandwidth and the Chamberlain bandwidth, respectively.

Having presented three possibilities to determine the bandwidth parameterh_n, we now consider the question of how to calculate the estimated empirical quantile function Q(τb ) = Fb⁻¹(τ)in equation (2.55). A simple approach is to just take the residuals from a quantile regression fit at an arbitrarily chosen quantile (denoted withτ^∗):²⁶

b

u_i=y_i−x⁰_iβb_τ^∗ i= 1, . . . , n (2.62)

Sorting the resulting values, we obtain the corresponding order statistics bu_(i) : i = 1, . . . , nand can estimate an empirical quantile function by

Fb⁻¹(τ) =ub_(i) τ ∈[ⁱ⁻¹_n ,_nⁱ) (2.63)

Of course, now the estimated empirical quantile function refers to u_τ^∗ instead of the regressand. But since the sparsity is defined as a function of the difference of twoFb⁻¹’s and we have assumed i.i.d. errors, both approaches are valid. If one prefers a piecewise linear version of (2.63), one can use

Fe⁻¹(τ) =

26Unfortunately, Koenker (1994) is unclear at this point. Bassett and Koenker (1982) are more specific, they use not only the median but also (alternatively) a least squares preliminary estimation to obtain the residuals.

2.3. ASYMPTOTICS AND INFERENCE 49 withλ=τ n−j+ 1/2. The fact that theK residuals equal to zero may be problematic ifK/nis large relative toh_ncan easily be circumvented by ignoring the zero residuals.

This procedure can be compared to the usual degrees of freedom correction in least squares regression.

A perhaps more appealing approach to obtain the estimated empirical quantile function was proposed by Bassett and Koenker (1982). As we have already seen in the last subsection, the sample path of the conditional quantiles ofy is non-decreasing in τ at the mean ofx, so we can just use (see also Bassett and Koenker (1986))

Fb⁻¹(τ) = ¯x⁰_iβb_τ (2.65)

To give an impression of some typical sparsity values, figure 2.11 shows a very simple example. We drew 500 sample observations from a standard normal distribution, deter-mined the empirical quantile function (e.g. by just settingx= ιin (2.65)), and applied the three bandwidth choice rules. It can be seen that the Bofinger bandwidth yields the smoothest result (our plot shows the estimates forτ ∈[0.08,0.92]).

Other techniques for the sparsity estimation are also possible: Welsh (1988) used a ker-nel approach and gave greater weight to values with narrower bandwidth. In Koenker and Bassett (1982a), the sparsity function is estimated by twice differentiating a smoo-thed version ofR(τb ), which denotes the minimum value achieved by the objective func-tion at each regression quantile. Chamberlain (1994) obtained the sparsity from an estimated confidence band constructed from appropriate order statistics.

2.3.2.2 Non-i.i.d. errors

In equation (2.51), we have seen that in a non-i.i.d. setting, the asymptotic variance of the estimated quantile regression coefficients has the slightly unpleasant property of

24681012sparsity

0 .2 .4 .6 .8 1

tau

Figure 2.11: Sparsity estimation. The figure shows the estimated sparsity for a random sample of 500 standard normally distributed values with the Bofinger (solid blue line), the Hall and Sheather (long-dashed red line) and the Chamberlain (short-dashed green line) bandwidth.

So, in this case we somehow have to estimate the entire matrixH_τ. The first proposal (see Hendricks and Koenker (1992)) is to estimate the conditional densityf_i(Q_τ(y_i|x_i)) at each observation by

fb_i(Q_τ(y_i|x_i)) = 2h_n

x⁰_i(βb_τ+hn−βb_τ−hn) (2.67) with the same bandwidth parameterh_n(e.g. Bofinger, Hall and Sheather, or Chamber-lain bandwidth) as already discussed. Subsequently, the obtained values are substi-tuted intoH_n, and we have an estimator for the non-i.i.d. model. A possible drawback of this procedure is the fact that the denominator of (2.67) is not necessarily always pos-itive. As we have already seen, conditional quantile planes can cross, although usually only at the periphery of the design space. To correct for this “inconvenience”, we simply

2.3. ASYMPTOTICS AND INFERENCE 51

Another approach to estimateH_τ was proposed by Powell (1986, 1991) and is given as (see also Buchinsky (1998b)) function. So, we again have to choose an appropriate kernel and an optimal bandwidth, where the latter selection is more important. Buchinsky (1998b) suggests to use a cross-validation method for the choice ofh_n.

2.3.3 Bootstrapping

As we have seen in the last subsection, the estimation of the (asymptotic) covariance matrix can be a bit burdensome, in particular if one assumes the more realistic case of non-i.i.d. errors. So it comes to no surprise that several authors tried to circumvent the laborious estimation of the sparsity and proposed alternatives to obtain inferential information. A large part of these proposals is based on resampling methods known as bootstrapping(see Efron (1979) as an important initial contribution and e.g. Efron and Tibshirani (1993) for a comprehensive introduction).²⁷

As in other applications, there are several possible implementations of the bootstrap procedure. The first one is called residual bootstrap and was suggested by Efron (1982)

27Under a generous perspective, some parallels can be found between quantile regression and boot-strapping: Both approaches were “invented” in the late 70’s, both provided new insights unattainable by conventional methods, and both received a “career boost” in the 90’s due to improved availability of computational power.