Conditional Variance Estimation in Heteroscedastic Regression Models

Conditional Variance Estimation in Heteroscedastic Regression Models

Lu-Hung Chen¹, Ming-Yen Cheng² and Liang Peng³

Abstract

First, we propose a new method for estimating the conditional variance in het- eroscedasticity regression models. For heavy tailed innovations, this method is in general more efficient than either of the local linear and local likelihood estimators.

Secondly, we apply a variance reduction technique to improve the inference for the conditional variance. The proposed methods are investigated through their asymp- totic distributions and numerical performances.

Keywords. Conditional variance, local likelihood estimation, local linear estimation, log-transformation, variance reduction, volatility.

Short title. Conditional Variance.

1Department of Mathematics, National Taiwan University, Taipei 106, Taiwan. Email:

r91090@csie.ntu.edu.tw

2Department of Mathematics, National Taiwan University, Taipei 106, Taiwan. Email:

cheng@math.ntu.edu.tw

3School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332-0160, USA. Email:

peng@math.gatech.edu

1 Introduction

Let {(Y_i, X_i)} be a two-dimensional strictly stationary process having the same marginal distribution as (Y, X). Let m(x) = E(Y |X = x) and σ²(x) = V ar(Y |X = x) be re- spectively the regression function and the conditional variance, and σ²(x) > 0. Write Y_i = m(X_i) + σ(X_i) ε_i. (1.1) Thus E(ε_i|X_i) = 0 and V ar(ε_i|X_i) = 1. When X_i = Y_i−1, model (1.1) includes ARCH(1) time series (Engle, 1982) and AR (1) processes with ARCH(1) errors as special cases. See Borkovec (2001), Borkovec and Kl¨uppelberg (2001) for probabilistic properties and Ling (2004) and Chan and Peng (2005) for statistical properties of such AR(1) processes with ARCH(1) errors.

There exists an extensive study on estimating the nonparametric regression func- tion m(x); see for example Fan and Gijbels (1996). Here we are interested in estimat- ing the volatility function σ²(x). Some methods have been proposed in the literature;

see Fan and Yao (1998) and references cited therein. More specifically, Fan and Yao (1998) proposed to first estimate m(x) by the local linear technique, i.e.,m(x) =b ba if

(ba, bb) = argmin_(a,b)

n

X

i=1

Yi− a − b(X_i− x) 2

KX_i− x h₂



, (1.2)

where K is a density function and h2 > 0 is a bandwidth, and then estimate σ²(x) byσb₁²(x) =αb₁ if

(αb1, bβ1) = argmin_(α1,β1)

n

X

i=1



rbi− α1− β1(Xi− x) 2

W

X_i− x h₁



, (1.3)

where br_i = {Y_i−m(Xb _i)}², W is a density function and h₁ > 0 is a bandwidth. The main drawback of this conditional variance estimator is that it is not always positive.

Recently, Yu and Jones (2004) studied the local Normal likelihood estimation, men- tioned in Fan and Yao (1998), with expansion for log σ²(x) instead of σ²(x). The main advantage of doing this is to ensure that the resulting conditional variance estimator

is always positive. More specifically, the estimator proposed by Yu and Jones (2004) is bσ²₂(x) = exp{αb₂}, where

(αb₂, bβ₂) = argmin_(α2,β2)

n

X

i=1

h

br_iexp−α₂−β₂(X_i−x) +α₂+β₂(X_i−x)i

WXi− x h₁

 . (1.4) Yu and Jones (2004) derived the asymptotic limit ofbσ₂²(x) under very strict conditions which require independence of (X_i, Y_i)⁰s and ε_i ∼ N (0, 1).

Motivated by empirical evidences that financial data may have heavy tails, see for example Mittnik and Rachev (2000), we propose the following new estimator for σ²(x). Rewrite (1.1) as

log r_i = ν(X_i) + log(ε²_i/d) (1.5) where r_i = {Y_i − m(X_i)}², ν(x) = log(dσ²(x)) and d satisfies E{log(ε²_i/d)} = 0.

Based on the above equation, first we estimate ν(x) by bν(x) =αb₃ where (αb₃, bβ₃) = argmin_(α3,β3)

n

X

i=1

 log(br_i+ n⁻¹) − α₃− β₃(X_i− x) 2

WX_i− x h₁



. (1.6) Note that we employ log(br_i+ n⁻¹) instead of logbr_i to avoid log(0). Next, by noting that E(ε²_i|Xi) = 1 and ri = exp{ν(Xi)} ε²_i/d, we estimate d by

d =b h1 n

n

X

i=1

br_iexp{−ν(Xb _i)}i−1

. Therefore our new estimator for σ²(x) is defined as

bσ₃²(x) = exp{bν(x)}/ bd.

Intuitively, the log-transformation in (1.5) makes data less skewed, and thus the new estimator may be more efficient in dealing with heavy tailed errors than other estimators such as bσ²₁(x) and bσ²₂(x). Peng and Yao (2003) investigated this effect in least absolute estimation of the parameters in ARCH and GARCH models. Note that this new estimatorσb₃²(x) is always positive.

We organize this paper as follows. In Section 2, the asymptotic distribution of this new estimator is given and some theoretical comparisons withbσ²₁(x) andbσ₂²(x) are

addressed as well. In Section 3, we apply the variance reduction technique in Cheng, et al. (2007) to the conditional variance estimators σb₁²(x) and bσ²₃(x) and provide the limiting distributions. Bandwidth selection for the proposed estimators is discussed in Section 4. A simulation study and a real application are presented in Section 5.

All proofs are given in Section 6.

2 Asymptotic Normality

To derive the asymptotic limit of our new estimator, we impose the following regu- larity conditions. Denote by p(·) the marginal density function of X.

(C1) For a given point x, the functions E(Y³|X = z), E(Y⁴|X = z) and p(z) are continuous at the point x, and ¨m(z) = _dz^d22m(z) and ¨σ²(z) = _dz^d22σ²(z) are uniformly continuous on an open set containing the point x. Further, assume p(x) > 0;

(C2) E(Y^4(1+δ)) < ∞ for some δ ∈ (0, 1);

(C3) The kernel functions W and K are symmetric density functions each with a bounded support in (−∞, ∞). Further, there exists M > 0 such that |W (x₁) − W (x₂)| ≤ M |x₁−x₂| for all x₁and x₂ in the support of W and |K(x₁)−K(x₂)| ≤ M |x₁− x₂| for all x₁ and x₂ in the support of K;

(C4) The strictly stationary process {(Y_i, X_i)} is absolutely regular, i.e., β(j) := sup

j≥1

E sup

A∈F_i+j^∞

|P (A|F₁ⁱ) − P (A)| → 0 as j → ∞,

where F_i^j is the σ-field generated by {(Y_k, X_k) : k = i, · · · , j}, j ≥ i. Further, for the same δ as in (C2),

∞

X

j=1

j²β^δ/(1+δ)(j) < ∞;

(C5) As n → ∞, h_i → 0 and lim inf

n→∞ nh⁴_i > 0 for i = 1, 2.

Our main result is as follows.

Theorem 1. Under the regularity conditions (C1)–(C5), we have pnh₁

σb₃²(x) − σ²(x) − θ_n3 d

→ N

0, p(x)⁻¹σ⁴(x)λ²(x)R(W ) , where λ²(x) = E{(log(ε²/d))²|X = x), R(W ) =R W²(t) dt and

θ_n3 = 1

2h²₁σ²(x)¨ν(x) Z

t²W (t) dt

−1

2h²₁σ²(x)E ¨ν(X₁) Z

t²W (t) dt + o h²₁+ h²₂
.

Next we compareσb₃²(x) withσb₁²(x) and bσ²₂(x) in terms of their asymptotic biases and variances. It follows from Fan and Yao (1998) and Yu and Jones (2004) that, for i = 1, 2,

pnh₁

σb_i²(x) − σ²(x) − θ_ni d

→ N

0, p⁻¹(x)σ⁴(x)¯λ²(x)R(W ) , where ¯λ²(x) = E{(ε²− 1)²|X = x},

θ_n1 = 1

2h²₁σ¨²(x) Z

t²W (t) dt + o(h²₁+ h²₂) and θ_n2 = ¹₂h²₁σ²(x)_dx^d22{log σ²(x)}R t²W (t) dt + o h²₁+ h²₂
.

Remark 1. If ˆν(X_i) in ˆd is replaced by another local linear estimate with either a smaller order of bandwidth than h₁ or a higher order of kernel than W , then the asymptotic squared bias of bσ₃²(x) is the same as that of bσ₂²(x), which may be larger or smaller than the asymptotic squared bias of σb₁²(x); see Yu and Jones (2004) for detailed discussions.

Remark 2. Suppose that given X = x, ε has a t-distribution with degrees of freedom m. Then the ratios of λ²(x) to ¯λ²(x) are 0.269, 0.497, 0.674, 0.848, 1.001 for m = 5, 6, 7, 8, 9, respectively. That is, bσ²₃(x) may have a smaller variance than both bσ₁²(x) andσb₂²(x) when ε has a heavy tailed distribution.

Remark 3. Note that the regularity conditions (C1)–(C5) were employed by Fan and Yao (1998) as well. However, it follows from the proof in Section 6 that we

could replace condition (C2) by E{| log(ε²)|^2+δ} < ∞ for some δ > 0 in deriving the asymptotic normality of ν(x). Condition (C2) is only employed to ensure thatb d − d = Ob _p(n^−1/2). As a matter of fact we only need bd − d = o_p(nh₁)^−1/2

to derive the asymptotic normality of bσ²₃(x). In other words, the asymptotic normality ofσb₃²(x) may still hold even when E(Y⁴) = ∞. This is different from other conditional variance estimators such as bσ₁²(x) and σb₂²(x), which require at least E(Y⁴) < ∞ to ensure asymptotic normality.

3 Variance Reduced Estimation

Here we apply the variance reduction techniques proposed by Cheng, et al. (2007), which concerns nonparametric estimation of m(x), to the conditional variance esti- mators σb₁²(x) and bσ₃²(x). The reason why we do not consider bσ₂²(x) here is that the asymptotic normality was derived by Yu and Jones (2004) under much more stringent conditions than those required by the other two estimators. The idea of our variance reduction strategy is to construct a linear combination of either σb₁²(z) or σb₃²(z) at three points around x such that the asymptotic bias is unchanged. The details are given below.

For any given point x, let β_x,0, β_x,1, β_x,2 be a grid of equally spaced points, with bin width γh₁ = β_x,1− β_x,0, such that x = β_x,1+ lγh₁ for some l ∈ [−1, 1]. Then, like Cheng et al. (2007), our variance reduction estimators for σ²(x) are defined as

eσ²_j(x) = l(l − 1)

2 bσ_j²(β_x,0) + (1 − l²)bσ²_j(β_x,1) + l(l + 1)

2 σb_j²(β_x,2), (3.1) for j = 1 and 3. Suppose that Supp(σ) is bounded, Supp(σ) = [0, 1] say, since β_x,0 < x < β_x,2, β_x,0 and β_x,2 would be outside Supp(σ) if x is close to the end- points. Therefore we take γ(x) = minγ, x/(1 + l)h₁, (1 − x)/(1 − l)h₁

so that

β_x,0, β_x,1, β_x,2 ∈ Supp(σ) = [0, 1] all the time.

The following theorem gives the asymptotic limits of the variance reduced esti-

mators.

Theorem 2. Under the conditions of Theorem 1, for interior point x we have pnh₁

eσ₁²(x) − σ²(x) − θ_n1 d

→ N

0,R(W ) − l²(1 − l²) C(γ) p(x)⁻¹σ⁴(x)¯λ²(x) and

pnh₁

eσ₃²(x) − σ²(x) − θ_n3 d

→ N

0,R(W ) − l²(1 − l²) C(γ) p(x)⁻¹σ⁴(x)λ²(x) , where C(s, t) =R W (u − st)W (u + st) du and C(s) = ³₂C(0, s) − 2C(¹₂, s) +¹₂C(1, s).

Hence eσ²_j(x) has the same asymptotic bias as σb_j²(x) for j = 1, 3. Note that 0 ≤ l²(1 − l²) ≤ 1/4 for all l ∈ [−1, 1] and it attains the maximum at l = ±2^−1/2. Moreover, for symmetric kernel W the quantity C(γ) is nonnegative for all γ ≥ 0;

0 ≤ C(γ) ≤ (3/2)R(W ) and C(γ) is increasing in γ if W is symmetric and concave; see Cheng et al. (2007). So, the variance reduction estimators have smaller asymptotic variances and asymptotic mean squared errors.

By choosing l = ±2^−1/2, we achieve the most variance reduction regardless what h1, γ and W are and the resulting estimators are

eσ_j,(1)² (x) = 1

4(1 − 2^1/2)σb_j² x − (1 + 2^−1/2)γh₁
+1

2bσ_j² x − 2^−1/2γh₁
+1

4(1 + 2^1/2)bσ_j² x − (2^−1/2− 1)γh₁

(3.2) and

σe_j,(2)² (x) = 1

4(1 + 2^1/2)σb_j² x + (2^−1/2− 1)γh₁
+1

2bσ_j² x + 2^−1/2γh₁
+1

4(1 − 2^1/2)bσ²_j x + (2^−1/2+ 1)γh₁

(3.3) for j = 1 and 3.

Either of the variance reduction estimators eσ²_j,(1)(x) and eσ_j,(2)² (x) uses more in- formation from data points on one side of x than the other side; see (3.2) and (3.3).

One way to balance this finite sample bias effect is to take the average

σe_j,(3)² (x) = 1 2



eσ_j,(1)² (x) +eσ_j,(2)² (x)

(3.4)

for j = 1 and 3. When Supp(σ) = [0, 1], to keep the points β_x,0, β_x,1, β_x,2 with l = ±2^−1/2 all within the data range [0, 1] we let γ(x) = minγ, x/(1 + 2^−1/2)h₁, (1 − x)/(1 + 2^−1/2)h₁ for a positive constant γ, γ = 1 say.

Theorem 3. Under the conditions of Theorem 1, for interior point x we have pnh₁

eσ_1,(3)² (x)−σ²(x)−θ_n1 d

→ N

0,R(W )−C(γ)/4−D(γ)/2 p(x)⁻¹σ⁴(x)¯λ²(x) and

pnh₁

eσ_3,(3)² (x)−σ²(x)−θ_n3 d

→ N

0,R(W )−C(γ)/4−D(γ)/2 p(x)⁻¹σ⁴(x)λ²(x) , where

D(γ) = R(W ) − C(γ) 4 − 1

16 n

4 1 +√

2
C √

2 − 1, γ/2
+ 3 + 2√

2
C 2 −√

2, γ/2
+2C √

2, γ/2
+ 4 1 −√

2
C √

2 + 1, γ/2
+ 3 − 2√

2
C √

2 + 2, γ/2
o .

Remark 4. Note that, for any kernel W , 0 ≤ D(γ) ≤ (5/8)R(W ). Hence, eσ²_j,(3)(x) has a smaller asymptotic variance than both eσ_j,(1)² (x) andσe_j,(2)² (x) for j = 1 and 3.

Remark 5. In Cheng et al. (2007) the variance reduction techniques are applied to nonparametric estimation of the regression m(x). The results in Theorems 2 and 3 are nontrivial given the theory developed therein.

Remark 6. When estimating the conditional variance σ²(x), it does not provide any gain, in asymptotic terms, by replacing m(Xb _i) with the variance reduced regression estimator of Cheng, et al. (2007) in the squared residuals br_i = Yi − m(Xb _i) 2

, i = 1, · · · , n.

Remark 7. In (1.6), the term n⁻¹ is added to avoid log 0 and it can be replaced by n^−η, for any η > 0, without affecting the theoretical results in Theorems 1–

3. However, in finite sample cases, a too small value of η would increase the bias and a too large value of η would increase the variability. In the simulation study summarized in Section 5.1, we also experimented with η = 0.5 and 2. We found that

η = 0.5 is undesirable as the MADE boxplot is always above the others even though it is narrower. When η = 2, besides the MADE boxplot is wider than the others, the performance is not stable with the MADE boxplot lower than the others only in some settings, not all.

4 Bandwidth Selection

In the construction of our estimator bσ₃²(x), two bandwidths are needed: h₂ is the bandwidth in (1.2) to get the squared residualsbr₁, · · · ,br_nand h₁ is the bandwidth in (1.6) to estimate the conditional variance. Since both (1.2) and (1.6) are local linear fittings based on data (X_i, Y_i), i = 1, · · · , n and (X_i, log(rb_i + n⁻¹), i = 1, · · · , n respectively, we suggest to employ the same bandwidth procedure in the two steps.

Let ˆh(X₁, · · · , X_n; Y₁, · · · , Y_n) denote any data-driven bandwidth rule for the local linear fitting (1.2).

1. Take h2 = ˆh(X1, · · · , Xn; Y1, · · · , Yn) in the local linear regression (1.2) to obtain the regression estimates m(Xb _i), i = 1, · · · , n, and the squared residuals br_i =

Y_i−m(Xb _i) 2

, i = 1, · · · , n.

2. Use bandwidth h₁ = ˆh X₁, · · · , X_n; log(br₁+n⁻¹), · · · , log(rb_n+n⁻¹)
in the local linear fitting (1.6) to get bσ₃²(x).

A simple modification of the above bandwidth procedure can be used to imple- ment our variance reduction estimatorσe_3,(3)² (x). Comparing Theorem 1 and Theorem 3, the asymptotically optimal global (or local) bandwidths ofσe_3,(3)² (x) andbσ₃²(x) differ by the constant multiplier R(W ) − C(γ)/4 − D(γ)/2 ^1/5 which depend only on the known W and γ. Therefore, no matter whether ˆh is a global bandwidth or a local bandwidth, the modification proceeds as, in step 2 above, using

h₁ =R(W ) − C(γ)/4 − D(γ)/2 ^1/5ˆh X₁, · · · , X_n; log(br₁+ n⁻¹), · · · , log(br_n+ n⁻¹)
(4.1)

in (1.6) to obtain bσ²₃(z), z ∈ {β_x,0, β_x,1, β_x,2} with l = ±2^−1/2. Then one can form the linear combinations, specified in (3.2), (3.3) and (3.4), to get σe_3,(3)² (x).

For the conditional variance estimator bσ²₁(x), Fan and Yao (1998) recommended a bandwidth principle analogous to what we specify for bσ²₃(x) in the above. To modify the bandwidth rule of σb₁²(x) for use in eσ_1,(3)² (x), apply the same constant factor adjustment as in (4.1) when computing bσ²₁(z) for z ∈ {β_x,0, β_x,1, β_x,2} with l = ±2^−1/2.

5 Numerical Study

The five estimators σb₁²(x), σb₂²(x), bσ²₃(x), eσ²_1,(3)(x) and σe_3,(3)² (x) are compared based on their finite sample performances via a simulation study and an application to the motorcycle data set.

5.1 Simulation

Consider the regression model

Y_i = aX_i+ 2 exp −16X_i²
+ σ (X_i) ε_i, (5.1) where σ (x) = 0.4 exp (−2x²) + 0.2, a = 0.5, 1, 2, or 4, X_i ∼ Uniform[−2, 2] and ε_i is independent of Xi and follows either the N (0, 1) or the (1/√

3) t3 distribution.

For each of the settings, 1000 samples of size n = 200 were generated from model (5.1). The plug-in bandwidth of Ruppert et al. (1995) was employed as the band- width selector ˆh in Section 4. To implement bσ₂²(x), h₁ was taken as the data-driven bandwidth given in Yu and Jones (2004). Both K and W , kernels in the regression and conditional variance estimation stages, were taken as the Epanechnikov kernel K(u) = (3/4)(1 − u²)I(|u| < 1). The parameter γ in σb_1,(3)² (x) and bσ²_3,(3)(x) was set to 1. Performance of an estimator bσ(·) of σ(·) is measured by the mean absolute

0.010.020.030.040.050.06

normal error a=0.5

0.010.020.030.040.050.06

t error a=0.5

0.010.020.030.040.050.06

normal error a=1

0.010.020.030.040.050.06

t error a=1

0.010.020.030.040.050.06

normal error a=2

0.010.020.030.040.050.06

t error a=2

0.010.020.030.040.050.06

normal error a=4

0.010.020.030.040.050.06

t error a=4

Figure 1: MADE boxplots under model (5.1). In each panel, the MADE boxplots of bσ₁(·), σb₂(·), bσ₃(·), eσ_1,(3)(·) and eσ_3,(3)(·) are arranged from left to right. The left and right columns respectively give the results for the Normal and t errors. From the top, the rows correspond to a = 0.5, 1, 2 and 4.

deviation error or the mean squared deviation error, respectively defined by

M ADE (σ) =b 1 g

g

X

i=1



bσ (xi) − σ (xi)

, M SDE (bσ) = 1 g

g

X

i=1



bσ (xi) − σ (xi) 2

,

where {x_i, i = 1, · · · , g} is a grid on [-2,2] with g = 101. Here, we measure the performance by MADE or MSDE of estimating σ(·) instead of those of estimating σ²(·) since the latter would seriously down-weight errors in estimating small values of σ²(·).

Figure 1 presents the MADE boxplots of the five estimators σbj(·), j = 1, 2, 3, and eσ_j,(3)(·), j = 1, 3. Under all of the configurations, eσ_j,(3)(·) outperforms σb_j(·) for j = 1, 3 and our log-tranform based methods improve on the Fan and Yao (1998) estimator. When the error distribution is Normal, the Yu and Jones (2004) estimator bσ2(·) is somehow the best since its MADE median is the lowest. The reason for this optimality is that bσ₂(·) is derived from a local Normal likelihood model which now coincides with the true error model. However, its MADE boxplot is always much wider than those of the other four and this instability is intrinsic to local likelihood methods. Interestingly, our estimator eσ_3,(3)(·) is nearly optimal even under Normal errors: compared to bσ₂(·), the MADE median is roughly the same and the MADE

0.000.100.20

normal error npred=2

0.000.100.20

t error npred=2

0.00.20.4

normal error npred=3

0.00.40.81.2

t error npred=3

Figure 2: MADE boxplots of predictions under model (5.2). The layout is the same as in Figure 1, except that the top and bottom rows respectively represent the two- and three-step predictions here.

upper quartile is lower. Further,bσ₃(·) and eσ_3,(3)(·) become decisively better than any of the others under t errors. Therefore,σb₃(·) andeσ_3,(3)(·) are very robust against heavy tailed errors, and σb₂(·) is not robust against departure from Normality. Although eσ1,(3)(·) performs better thanσb1(·) all the time but its behaviors under different error distributions is predetermined by that of bσ₁(·). The MSDE boxplots are not given here, but they provide similar conclusions.

Another setting we considered is the following nonlinear time series model X_t+1 = 0.235X_t(16 − X_t) + ε_t (5.2) where ε_t ∼ N (0, 0.3²) is independent of X_t. From model (5.2), 500 samples of size n = 500 were simulated. The conditional variance of X_t+1 given the past data

Xt, X_t−1, · · · is a constant function. Hence we investigate estimation of the condi- tional variances in two-step and three-step prediction problems with Y_t = X_t+2 and Y_t = X_t+3 respectively. Figure 2 presents the MADE boxplots, which convey similar conclusions as in the previous example. In particular, eσ_3,(3)(·) is quite reliable and robust.

In the construction of bσ3(·) and eσ3,(3)(·), the term n⁻¹ is added in (1.6) to avoid log 0. We also experimented with the term n⁻¹ replaced by n^−0.5 or n⁻². To save space, the results are not given here. We found that with n^−0.5 the MADE boxplot is always above the others even though it is narrower. When using n⁻², the MADE boxplot is wider than the others, but the performance is not stable with the MADE boxplot lower than the others only in some settings.

5.2 An Application

The estimators bσ₁²(x),σb₂²(x),σb₃²(x),σe_1,(3)² (x) and eσ²_3,(3)(x) were employed to estimate the conditional variance for the motorcycle data given by Schmidt et al. (1981). The covariate X is the time (in milliseconds) after a simulated impact on motorcycles and the response variable Y is the head acceleration (in gram) of a test object. The

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Figure 3: Motorcycle data. Panel (a) depicts the motorcycle data and the local linear regression estimatem(·). The absolute residuals are plotted against the design pointsb in panels (b)–(f), which respectively show the estimates σb₁(·), bσ₂(·), bσ₃(·), eσ_1,(3)(·) and eσ_3,(3)(·) (solid lines) and 10%, 50% and 90% variability curves of their resampled versions (dashed lines).

sample size is 132. As before, the squared residuals rb₁, · · · ,br_n are used to estimate the conditional variance. We took ˆh in Section 4 as the bandwidth selector of Rupprt, et al. (1995). Then the bandwidth h₂ in m(·) was 4.0145, and the bandwidth hb ₁ in bσ₁²(·), bσ²₃(·), σe_1,(3)² (·) and eσ_3,(3)² (·) was respectively 6.1775, 4.5763, 5.1053 and 3.7821.

The bandwidth h₁ in bσ²₃(·) was selected by the method of Yu and Jones (2004) and was 13.4188. In Figure 3, panel (a) depicts the original data and the local linear regression estimate m(·), and the solid lines in panels (b)–(f) are respectively theb estimates bσ₁(·), bσ₂(·), σb₃(·), eσ_1,(3)(·) and eσ_3,(3)(·). In Figure 4, panel (a) plots the residuals, and panels (b)–(f) depicts the Normal Q-Q plots of the residuals divided by the estimates of the conditional standard deviations. Panels (b) and (e) suggest that the motorcycle data has a heavy left tail in the error distribution. Panels (d) and (f) show that our estimators σ(·) andb eσ3,(3)(·) effectively correct the heavy left tail. Panel (c) indicates a departure from normality whenbσ₂(·) is applied to this data