Abstract
Time-varying coefficient regression is commonly used in the modeling of nonstationary stochastic processes. In this paper, we consider a time-varying coefficient convolution-type smoothed quantile regression (conquer). The covariates and errors are assumed to belong to a general class of locally stationary processes. We propose a local linear conquer estimator for the varying-coefficient function, and obtain the global Bahadur–Kiefer representation, which yields the asymptotic normality. Furthermore, statistical inference on simultaneous confidence bands is also studied. We investigate the finite-sample performance of the conquer estimator and confirm the validity of our asymptotic theory by conducting extensive simulation studies. We also consider financial volatility data as an example of a real-world application.
Keywords: Bahadur–Kiefer representation, convolution, time-varying coefficient model, quantile regression, locally stationary process
1. Introduction
Quantile regression (QR) is a useful technique for modeling the relationship between a response variable y and a multivariate predictor [13]. Compared with mean regression, QR allows the modeling of the entire conditional distribution of y given . Meanwhile, it is robust against outliers in the response and has few constraints on the response distribution, allowing us to perform QR for skewed or heavy-tailed response distribution, even in cases without correct specification of the likelihood. Many useful nonparametric quantile regression models have been investigated in Chapter 7 of [13], for example locally polynomial quantile regression, penalized methods, additive models and so on. However, there is a price to pay for QR. In the standard QR, the loss function is the so-called check function , which is not smooth and lacks second derivatives. Consequently, the asymptotic theory and statistical inference are not straightforward and include ancillary estimation of nuisance parameters. To tackle this issue, a wide range of technique has been explored, including the bootstrapping method [17], MCMC method [4], empirical likelihood [20], and strong approximation methods [22].
On the other hand, some scholars have attempted to smooth the check function in order to circumvent its non-differentiability. Horowitz [9] proposed to smooth the indicator part of to obtain
| (1) |
where is a smooth function ranging from 0 to 1. However, this smoothing technique comes at the cost of convexity. As is well-known, the local optimality is not necessarily global optimality in the non-convex optimization. In addition, there may be multiple local optimizations. These properties will undoubtedly increase the difficulty of solving. Recently, Fernandes et al. [6] introduced a convolution-type smoothed QR, referred to as conquer, that provides a convex and twice-differentiable loss function. This novel smoothing technique circumvents not only the non-differentiability of QR, but also the non-convexity of Horowitz-type smoothing function. Both convexity and differentiability make the optimization procedure become quite easy and standard. A couple of studies have applied conquer to the analysis of high-dimensional data: [8] studied the non-asymptotic properties of the conquer estimator; [23] proposed a conquer estimator with an iteratively reweighted -regularization. As far as we know, all of the existing literature about conquer is focused on linear QR. Linearity specification is a stringent restriction that may yield a non-negligible modeling bias and limit its applications in practical fields. There is an abundant literature on nonparametric QR: [3] for locally polynomial QR; [14] for partially linear QR; [11] for varying coefficients QR and [15] for additive QR.
Nonstationary time series is common and important in the fields of econometrics, finance, climatology, and environmental sciences, especially when looking at long time spans and/or political and economic shocks. A frequently used nonstationary series is called locally stationary process (LSP), which is introduced by [5]. The LSP is popular in that it can dynamically capture changes in data structure through the corresponding second-order moments' properties. There exists a great deal of research on time-varying coefficient model of LSP – see, for example, [26] and therein. This paper aims to develop a smoothed time-varying coefficient QR for the LSP, which will further enrich the theoretical understanding and applications of conquer. We propose a local linear conquer estimator for the varying-coefficient function, denoted as con-LLQR, and show that the bias term of con-LLQR is asymptotically negligible, which guarantees the consistency of the proposed estimator. We also present the global Bahadur–Kiefer representation of con-LLQR estimator, and obtain an asymptotic normality that yields simultaneous confidence bands (SCB) for the varying-coefficient function. Moreover, we construct the SCB based on bootstrap sampling methodology, and compare it with the Gaussian-based-SCB. Extensive simulation studies are conducted to validate the resulting theoretical findings. Comparison with existing methodology is also made.
Although our analysis procedure is parallel to [6], there exists great differences in the technique to yield asymptotic distribution. [6] considered i.i.d data, and used the functional exponential inequality [19] to yield Bahadur-Kiefer representation. However, in the context of LSP, their approach fails due to the autocorrelation of time series, and we resort to the exponential inequality of martingale difference [7]. The proof idea mainly comes from [26], yet we are involved in many technical details due to the complexity of convolution-type smoothed loss function. Specially, Lemma 9 given in the supplementary material provides important links between the smoothed QR loss and standard QR loss. It is found that the proposed smoothing technique has no effect on the asymptotic distribution except for bringing asymptotically negligible smoothing bias.
In the real-life data application, we consider a volatility series that is measured by the high–low range of the daily S&P 500 index during the period from 02/01/2020 to 31/12/2020, including 253 observations. We adopt 2 covariates (see Example 5.2 in Section 5), and our analysis shows a more flexible time-varying coefficient model is preferred. Moreover, the obtained results also provide a powerful evidence that the con-LLQR estimator can effectively alleviate the undersmooth phenomenon of local linear QR (LLQR) estimator, which increases the interpretability of dynamic effects.
The rest of this article is organized as follows. Section 2 describes the model setup and proposes the con-LLQR estimator. The asymptotic theory of the proposed estimation methodology is presented in Section 3. Section 4 addresses the practical problems of con-LLQR estimator, including the selection of bandwidth and implementation of simultaneous confidence bands. Simulation studies and a real-life data application are illustrated in Section 5. The requirements for validity of the asymptotic theories are presented in Appendix A, main proofs and complementary numerical studies are relegated to the supplementary materials.
2. Time-varying coefficient smoothed QR
Let be a sample of length T, where is a univariate response, is a -dimension covariate with , and ′ means the transposition of a vector or matrix. For a pre-specified quantile τ, time-varying coefficient QR model specifies the conditional τ-th quantile of given as
| (2) |
for . Let , model (2) also can be written as
| (3) |
where the error satisfies .
We define filtration and , where and are independent. Both and are supposed to be LSP [5,29] as below:
| (4) |
where and . Both and are measurable functions such that and are well defined for each . Note that and are both functions of , (4) considers a flexible cross correlation between covariates and random error . A special case of our model setup is the independent exogenous model that assumes .
Commonly, there exists two popular methods to fit a smooth univariate function, i.e. local polynomial approximation and spline approximation, which both have their pros and cons. On the one hand, spline fitting is a global and cost-saving method, but the asymptotic distribution is generally missing or can be derived when the bias term is assumed to be asymptotically negligible relative to the asymptotic variance, see, for example, [16,28]. On the other hand, at the cost of computing cost, local polynomial fitting can yield asymptotic distribution that provides specific form of bias term. Therefore, we can make statistical inference based on the obtained asymptotic distribution.
In this paper, we will adopt local linear fitting for the considered semiparametric QR model. Specifically, the LLQR estimator of regression model (3) is given by
| (5) |
where is the check function of τ-quantile, denotes the indicator on a set A, and for a given bandwidth h>0.
Motivated by the idea of conquer, we now propose a convolution-type smoothed QR local linear estimator, i.e. con-LLQR for the model (3), which is decided by the following optimization problem:
| (6) |
where , , and * denotes the convolution operator.
Note that the local smoothing is made for both check function and varying-coefficient functions. The objective function in (6) allows to choose different kernel functions and bandwidths in their respective local smoothing procedure.
We give some necessary notations. For a vector , define ; for a matrix A, denote ; and for a random vector or matrix , write if , and denote .
To describe the temporal dependence structures of the and , we adopt the following time series dependence measures, refer to [26].
Definition 2.1
Let be the filtration generated by , where are i.i.d. Let be an i.i.d. copy of . Assume that for all , , q>0. Define the physical dependence measure for the stochastic system in norm as
where .
Remark 2.1
By definition, measures the functional dependence of output on the input . [29] presents some examples that directly compute the dependence measure of a large class of locally stationary linear or non-linear precesses. We call a LSP is (i) long range dependent (in norm) if ; (ii) short range dependent if ; (iii) geometrically decaying if for some constant . In this paper, we suppose the covariates and errors are geometrically decaying processes, see Assumptions A2 and A3 given in Appendix A.
3. Asymptotic results
In this section, we will explore the asymptotic property of con-LLQR estimator, including smoothing bias, Bahadur–Kiefer representation and asymptotic normality.
Let , , , , denote
| (7) |
and . For any given , we represent and as the population parameters of convolution-type smoothed QR (6), which can be given by
| (8) |
with , . Similarly, let
and , then the population parameters and of standard QR (5) are decided by
where , .
Due to the convolution-type smoothing procedure made for the check function, the estimator that minimizes (7) actually approximates instead of . This means the con-LLQR estimator brings about a bias term . Let be the conditional density function of , be the partial derivative of f about j-th variable for j = 1, 2, and the partial derivative of about l-th variable for l = 1, 2. Furthermore, define
and , where for a given kernel function . Moreover, represent
with .
Theorem 3.1 presents the bias term caused by conquer's smoothing technique.
Theorem 3.1
Suppose assumptions A1 ∼ A4 and A8 hold. In addition, and as , then the population parameter of conquer is uniquely existing for a pre-specified quantile level . Moreover, we can express the smoothing bias term as
Theorem 3.1 shows that the bias term is dominated by , which implies that the smoothing bias shrinks to zero as the sample size T tends to infinity. Note that for cross-section data, Theorem 3.1 of [6] presents the asymptotically negligible smoothing bias for convolution-type linear QR. We generalize their result to the local linear estimator of semiparametric model. However, the generalization is not straightforward due to the complex data (LSP) and estimation methods (local linear fitting). Lemmas 1–4 in the supplementary material can be viewed as an analogy of Lemma 1 in [6], which is key to prove Theorem 3.1.
Remark 3.1
The assumptions imposed on Theorem 3.1 are rather weak and standard. Condition A1 specifies the degree of smoothness of the varying-coefficient function, which is often used in non/semi-parametric regression. Conditions A2 and A3 suppose that the covariates and error processes are stochastic Lipschitz continuous with geometrically decaying dependence (see Definition 2.1). These conditions are standard in the literature of LSP, see [26,29]. The conditional density of the error conditioning on the information sets of covariates is supposed to be Lipschitz continuity in A4, which is also a mild and common constraint. Similar to the assumption K1 of [6], condition A8 is imposed on kernel function, where specification (ii) is key in the proof of Theorem 3.1. It is easy to check it holds for widely used kernel functions, for example Gaussian kernel and Epanechnikov kernel etc.
Bahadur–Kiefer representation is a useful and popular tool for the asymptotic analysis of quantile estimators by approximating the estimators using mathematically tractable linear forms. Numerous studies have been done in the last few decades, say for example, Chapter 4.7 of [10] constructs the asymptotic representation of standard regression quantiles for i.i.d. dataset; [12] considers more general conditions allowing for misspecification of conditional quantile function; [21] investigates linear model in the framework of nonstationary and weak dependence; [26] and [29] develop the asymptotic representation for nonstationary semiparametric model.
Following their spirit, we will construct a Bahadur–Kiefer representation for the proposed con-LLQR estimator. Let
where and are specified in (8). Represent
and define
where for a given bandwidth h and . Furthermore, we denote and .
Theorem 3.2 shows the global Bahadur–Kiefer representation of con-LLQR estimator.
Theorem 3.2
Let . Under Assumptions A1 ∼ A6 and A8, if , and as , then we have
According to Theorem 3.2 we can uniformly approximate as
in the interval . Since is also a LSP with geometrically decaying, it can be uniformly approximated by a certain Gaussian process, see [26].
Remark 3.2
Some additional conditions are needed to derive Theorem 3.2. Condition A5 assumes the time-varying design matrix is non-degenerate on . It is a quite mild condition and can be found in [26]. Condition A6 supposes the dependence measure of the derivatives of the errors' conditional distributions and densities are also geometrically decaying. It is not difficult to verify A6 holds for a large class of LSP, can refer to [29] for specific examples. In addition, the conditions about bandwidths are also mild, for example, we may take and to satisfy the constraints given in Theorem 3.2.
Remark 3.3
Our Bahadur representation is quite different from [6], which is tailored for the convolution-type smoothed linear QR model of cross-section data. Theorem 3.1 of [26] establishes a global Bahadur representation of LLQR estimator for time-varying coefficient model of LSP. However, we can not directly use their representation due to the convolution-type smoothing technique. The proof of Theorem 3.2 shows that the major difference from [26] is that the check function in their Bahadur representation is replaced by due to the smoothing procedure. Following the idea of [26], we finish the proof after a series of routine but complicated computations. It should be pointed out that Lemma 9 given in the supplementary material is key to finish the proof because it shows that the difference of Bahadur representation between LLQR estimator and con-LLQR estimator is asymptotically negligible.
Now, we define and
| (9) |
Theorem 3.3 presents the asymptotic distribution of con-LLQR estimator pointwisely and simultaneously.
Theorem 3.3
Let . Suppose assumptions A1 ∼ A8 hold, if , and as , then for any fixed ,
Furthermore,
(10) where , , , and
with and .
Remark 3.4
To yield Theorem 3.3, we add condition A7, which means the time-varying long-run covariance matrices of the gradient vectors are non-degenerate on . This condition is also quite mild and used to derive the asymptotic distribution of LLQR estimator in the framework of LSP, see Assumption A5 of [26]. Combining the Bahadur representation obtained in Theorem 3.2, it is a standard practice to the derive the asymptotic distribution of con-LLQR estimator about parameter .
Notice that Theorems 3.1 and 3.3 ensure the consistency of con-LLQR estimator, i.e.
| (11) |
Compared with the asymptotic result of [26], the resultant bias adds a term , which is caused by convolution-type smoothed procedure according to Theorem 3.1. On the other hand, we also note that the conquer technique has no effect on the asymptotic variance, which is same with LLQR estimator considered in [26]. Therefore, we can adopt their method to get a consistent estimator of , which is presented in the supplementary material S2. Then, (10) will help us to establish a ( )% SCB via the maximal absolute deviation. The details are addressed in the next section.
He et al. [8] pointed out the smoothed high-dimensional QR can greatly save computing cost for large sample size. The computational expense is also investigated in our simulation study. It is found that con-LLQR has less elapsed time than LLQR even in flexible semiparametric model, and more cost is saved as the sample size increases. In addition, con-LLQR makes the estimated curve of component function smoother than LLQR, which will increase the interpretability of practical problem.
4. Implementations
Now we will address some practical issues that may arise with the con-LLQR estimation methodology.
Computational Method for con-LLQR
Following [8], we can solve con-LLQR by means of gradient descent with a Barzilai–Borwein update step, denoted as GD-BB. A good initial estimator is obtained by applying the GD-BB algorithm to the asymmetric Huber loss function. Specifically, we can directly use Algorithms 1 and 2 of [8] via replacing with , and as for any given .
Selection of Optimal Bandwidth
To perform con-LLQR estimation methodology, we need choose appropriate bandwidths and , as well as kernel functions and . As is well-known, the choice of the kernel function does not have a significant influence on the performance of local smoothing methodology. However, con-LLQR is sensitive to the selection of bandwidth. So, we preselect kernel functions and , and then find the optimal bandwidth based on modified multifold cross-validation (MM-CV) criterion introduced by [2]. The details about MM-CV are deferred to S1 of the supplementary material. In our numerical studies, the bandwidth grid ranges from 0.06 to 0.60 with increments of 0.03.
Jacknife Bias-corrected SCB
Note that most automatic bandwidth selectors pick the bandwidth for some c>0, which makes the bias term in the asymptotic distribution (10) is non-negligible. In order to avoid estimating the bias term, a common practice is to use Jacknife bias-corrected estimator of as below
| (12) |
where . Obviously, the bias-corrected estimator is equivalent to use the second order kernel .
Since Theorems 3.2 and 3.3 hold both for and , we conclude that on a possibly richer probability space, there exists i.i.d. random vectors , such that
| (13) |
Then the bias-corrected version of % SCB based on (10) can be given by
| (14) |
where is the k-th diagonal element of and .
Bootstrap-based SCB
Considering the convergence rate of (10) may be slow, we adopt a bootstrap sampling method to improve the finite-sample performance, see [26] and therein. According to (13),
whose distribution can be bootstrapped by generating a mass of i.i.d copies of
where are i.i.d. . The bootstrap procedure to generate a % SCB is summarized as follows.
Find an appropriate bandwidth according to MM-CV criterion to estimate for the quantile τ. Let and calculate according to (12).
Calculate based on the bandwidth .
- Generate i.i.d Gaussian vectors , and find the quantity
where is the second element of . - For a given confidence level , repeat step (iii) for B times, to obtain the estimated quantile , and construct the % SCB of as
(15)
5. Numerical studies
In this section, simulation experiments are investigated to explore the finite-sample performance of the proposed con-LLQR estimator. A real-life dataset is also considered to illustrate the usefulness of our method in practice.
Example 5.1
Let , , , and , where , , and with . Here , , are i.i.d standard normals. Define , and . We will consider three cases as follows.
- Case I:
The relationship between the response and covariates are given bywhere is the τ-th quantile of . This model considers a simple scenario in which the covariates and errors are independent.
(16) - Case II:
Replacing the last term in (16) with , which allows for the dependence between errors and covariates.
- Case III:
Let and in Case II. This setting aims to investigate the ability using SCB to identify the constancy of varying-coefficient component function.
To appraise the performance of the con-LLQR estimator, we introduce the root-mean-square error (RMSE), i.e.
and the weighted-average-mean-square error (WAMSE)
where denotes the range of a given function g, and we suppress the subscript on quantile level τ for a neat presentation.
Analysis for Case I
First, we investigate the sensitivity of the proposed con-LLQR estimator to bandwidth. To do this, we fix the sample size T = 200, and use same kernel function – Epanechnikov kernel in the smoothing procedure, that is . The quantile level is taken as and 0.9, and three kinds of bandwidths are considered: (i) the optimal bandwidth minimizing MM-CV criterion (S1.1) given in the supplementary material S1; (ii) a smaller bandwidth corresponding to the case of underfitting; and (iii) a larger bandwidth implying overfitting.
Based upon 200 Monte Carlo replications, Figure 1 presents the boxplot of WAMSEs for three quantile levels (1–3 denote and 0.9, respectively). The symbols ‘Optimal’, ‘Under’ and ‘Over’ represent the optimal case with bandwidth , underfitting with smaller bandwidth and overfitting with larger bandwidth, respectively. Moveover, Figure 1 in the supplementary material exhibits the boxplot of RMSEs for the varying-coefficient function at three quantile levels. It can be found from these figures that the bandwidth has insignificant influence on the finite-sample performance of con-LLQR estimator, and in most instances the estimation with optimal bandwidth outperforms over that with smaller or larger bandwidth.
Figure 1.

Boxplots of WAMSEs under different bandwidths based on 200 Monte Carlo replications: 1–3 correspond to quantile level and 0.9, respectively.
Next, we compare our methodology with three competitive alternatives: the first one is the standard QR [13] that assumes a linear model; the second one is the LLQR introduced by [27] for cross-section data, and later it is developed by [26] to fit a time-varying coefficient model in the analysis of nonstationary time series; the last one is Horowitz-type smoothed LLQR, which is determined by
| (17) |
where is the Horowitz's smoothed loss function given in (1).
We take sample size T = 100, 200, 300 and quantile level and 0.9. Based on 200 Monte Carlo replications, Table 1 compares the average RMSE (ARMSE) among QR estimator, LLQR estimator, con-LLQR estimator and Horowitz-type smoothed LLQR estimator (denoted as H-LLQR). The corresponding standard deviation is also given in parenthesis.
Table 1.
Average RMSE and standard deviation of different estimators in Case I.
| T | Method | ||||||
|---|---|---|---|---|---|---|---|
| 100 | con-LLQR | 0.0947 | 0.0989 | 0.1160 | 0.1041 | 0.1097 | 0.1094 |
| (0.0244) | (0.0257) | (0.0328) | (0.0285) | (0.0285) | (0.0304) | ||
| LLQR | 0.1412 | 0.1279 | 0.1459 | 0.1149 | 0.1270 | 0.1318 | |
| (0.0269) | (0.0292) | (0.0370) | (0.0303) | (0.0300) | (0.0359) | ||
| H-LLQR | 0.3993 | 0.1864 | 0.2226 | 0.5882 | 0.3477 | 0.2969 | |
| (0.0397) | (0.0308) | (0.0334) | (0.1339) | (0.0607) | (0.0794) | ||
| QR | 1.0925 | 0.6188 | 1.3330 | 3.0243 | 0.6239 | 1.7514 | |
| (0.0905) | (0.0247) | (0.0075) | (0.1026) | (0.0284) | (0.0620) | ||
| 200 | con-LLQR | 0.0814 | 0.0743 | 0.0806 | 0.0845 | 0.0780 | 0.0918 |
| (0.0176) | (0.0205) | (0.0412) | (0.0534) | (0.0188) | (0.0428) | ||
| LLQR | 0.0871 | 0.0916 | 0.0950 | 0.0861 | 0.0908 | 0.1015 | |
| (0.0193) | (0.0190) | (0.0229) | (0.0225) | (0.0195) | (0.0262) | ||
| H-LLQR | 0.6619 | 0.3065 | 0.4272 | 0.6766 | 0.3340 | 0.3665 | |
| (0.0355) | (0.0302) | (0.0354) | (0.0527) | (0.0462) | (0.0467) | ||
| QR | 1.5852 | 0.6312 | 1.3893 | 2.2714 | 0.6423 | 1.5079 | |
| (0.0751) | (0.0266) | (0.0266) | (0.0823) | (0.0350) | (0.0429) | ||
| 300 | con-LLQR | 0.0617 | 0.0629 | 0.0680 | 0.0686 | 0.0625 | 0.0664 |
| (0.0342) | (0.0176) | (0.0212) | (0.0143) | (0.0165) | (0.0167) | ||
| LLQR | 0.0709 | 0.0754 | 0.0802 | 0.0715 | 0.0758 | 0.0793 | |
| (0.0188) | (0.0163) | (0.0210) | (0.0159) | (0.0172) | (0.0181) | ||
| H-LLQR | 0.7747 | 0.2944 | 0.4097 | 1.1021 | 0.4245 | 0.4537 | |
| (0.0377) | (0.0243) | (0.0366) | (0.0571) | (0.0417) | (0.0437) | ||
| QR | 1.5322 | 0.6351 | 1.3765 | 2.3877 | 0.6017 | 1.5767 | |
| (0.0778) | (0.0221) | (0.0209) | (0.0929) | (0.0052) | (0.0350) | ||
From Table 1 we can conclude that (i) con-LLQR estimator has the best performance whatever sample sizes and quantile levels, and its ARMSE will decrease as the sample size grows; (ii) LLQR estimator is next to con-LLQR estimator, and their similar performance shows that the convolution-type smoothing technique do not deteriorate the ability to estimate true functions; (iii) H-LLQR estimator behaves badly under different combinations of sample size and quantile level, implying that it is not a good choice for the considered semiparametric QR; (iv) standard QR estimator has the worst performance due to model misspecification, that is to say the resulting systematic bias can not be ignored in our setting.
Furthermore, based on 200 Monte Carlo replications, Table 1 in the supplementary material lists the average empirical relative efficiency about LLQR and H-LLQR. The obtained result implies that con-LLQR estimator indeed improve LLQR and H-LLQR estimator.
Again based on 200 Monte Carlo replications, Table 2 presents the average empirical coverage percentage (AECP) and average empirical length (AEL) for the 95% SCB of the varying-coefficient function. Here, we construct SCB in two ways: one is the bias-corrected version (14) based on asymptotic normality (denoted as ‘Gauss’); the other is given by (15) based upon bootstrap sampling method (written as ‘Boot’). The bootstrap sampling is repeated 500 times for each given sample data to derive the estimated quantile. Evaluations are done for sample size and quantile level and 0.9.
Table 2.
AECP and AEL (in parentheses) of 95% SCB in Case I.
| T | Method | ||||||
|---|---|---|---|---|---|---|---|
| 100 | Gauss | 0.8841 | 0.9131 | 0.9080 | 0.9218 | 0.8784 | 0.8816 |
| (0.5260) | (0.5169) | (0.5461) | (1.2945) | (0.6472) | (0.7604) | ||
| Boot | 0.8696 | 0.9039 | 0.8983 | 0.9162 | 0.8722 | 0.8746 | |
| (0.5056) | (0.4960) | (0.5224) | (1.2562) | (0.6283) | (0.7353) | ||
| 200 | Gauss | 0.9260 | 0.9255 | 0.9570 | 0.9301 | 0.9355 | 0.9117 |
| (0.4312) | (0.3973) | (0.4011) | (0.9638) | (0.4730) | (0.5170) | ||
| Boot | 0.9085 | 0.9136 | 0.9480 | 0.9195 | 0.9261 | 0.9027 | |
| (0.4083) | (0.3747) | (0.3782) | (0.9217) | (0.5528) | (0.5980) | ||
| 300 | Gauss | 0.9378 | 0.9482 | 0.9540 | 0.9574 | 0.9492 | 0.9482 |
| (0.3856) | (0.3759) | (0.3781) | (0.9208) | (0.6219) | (0.5274) | ||
| Boot | 0.9320 | 0.9462 | 0.9512 | 0.9472 | 0.9410 | 0.9408 | |
| (0.3594) | (0.3542) | (0.3522) | (0.8975) | (0.5884) | (0.5007) | ||
We notice that the AECPs of 95% SCB have similar performance for the two inference methodologies. Specifically, the SCB based on asymptotic distribution has slightly larger empirical coverage than the bootstrap method, however, the latter has smaller empirical interval length. Moreover, as the sample size grows, we also note that both AECPs are close to the given confidence level and the AEL decreases significantly. The result also indicates the obtained asymptotic distribution is reliable to construct SCB even for medium sample size.
Moreover, we visualize the con-LLQR estimator for T = 200 and . Figure 2 compares the con-LLQR estimator (‘SQR’, dashed line) with the true function (‘True’, solid line) at the quantile . In addition, we also figure the 95% SCB (14) based on Gaussian distribution (‘SCB’, longdash lines) and bootstrap method (15) (‘Boot-SCB’, dotdash lines). The figures at and 0.5 are presented in Figure 2 of the supplementary material.
Figure 2.
Estimation at in Case I. (a)–(c) visualize varying-coefficient functions : true function (True, solid line), con-LLQR estimator (SQR, dashed line), Gaussian-based SCB (14) (SCB, longdash lines) and bootstrap-based SCB (15) (Boot-SCB, dotdash lines).
The resulting figures show the newly-proposed con-LLQR estimator is close to the true component function, and the 95% SCB based upon asymptotic distribution performs almost as good as the bootstrap-based methodology. In summary, it provides a strong evidence that the proposed estimation methodology and inference procedure have good performance in finite sample.
Finally, we assess the computational efficiency via elapsed time it takes to predict values of varying-coefficient function on rescaled time , where T is the sample size. Figure 3 (a)–(c) in the supplementary material compares elapsed time with LLQR estimator for different sample sizes at and 0.9, respectively. We note that con-LLQR estimator saves computing cost and time difference becomes larger as the sample size grows for the considered quantile levels and 0.9.
Analysis for Case II
Figure 3.
Estimation at in Case II. (a)–(c) visualize : true function (True, solid line), con-LLQR estimator (SQR, dashed line), Gaussian-based SCB (14) (SCB, longdash lines) and bootstrap-based SCB (15) (Boot-SCB, dotdash lines).
We still take sample size and quantile level and 0.9. Based on 200 Monte Carlo replications, we list the average RMSE (ARMSE) and standard deviation (SD) of con-LLQR estimator in Table 3, which exhibits a good performance even in a more complex model setup. Although the ARMSE becomes slightly larger than that in Case I, it remains within an acceptable scope, which is not larger than 0.19 even for T = 100. Moreover, for a given sample size, the median con-LLQR estimator performs best, whilst the estimator at is better in most cases than the one at . On the other hand, when the sample size grows, both ARMSE and SD decrease significantly whatever quantile levels. In addition, all the standard deviation is not larger than 0.06, which manifests stability of the proposed estimator.
Table 3.
Average RMSE and standard deviation (SD) in Case II.
| T | quantile level | ARMSE | SD | ARMSE | SD | ARMSE | SD |
|---|---|---|---|---|---|---|---|
| 100 | 0.1297 | 0.0332 | 0.1814 | 0.0491 | 0.1747 | 0.0533 | |
| 0.0945 | 0.0246 | 0.1510 | 0.0398 | 0.1542 | 0.0457 | ||
| 0.1530 | 0.0368 | 0.1684 | 0.0499 | 0.1793 | 0.0473 | ||
| 200 | 0.0984 | 0.0288 | 0.1235 | 0.0379 | 0.1218 | 0.0419 | |
| 0.0913 | 0.0364 | 0.1065 | 0.0303 | 0.1080 | 0.0278 | ||
| 0.1429 | 0.0326 | 0.1246 | 0.0326 | 0.1339 | 0.0396 | ||
| 300 | 0.0877 | 0.0201 | 0.1084 | 0.0293 | 0.1100 | 0.0348 | |
| 0.0701 | 0.0322 | 0.0839 | 0.0225 | 0.0861 | 0.0244 | ||
| 0.1187 | 0.0279 | 0.1061 | 0.0309 | 0.1117 | 0.0352 | ||
Similar to Table 2, we also compare 95% SCB of the varying-coefficient function using (14) and (15). Again based upon 200 Monte Carlo replications, the AECP and AEL are presented in Table 4, which displays a similar property to Case I, indicating the statistical inference on simultaneous confidence band is still reasonable in this scenario.
Table 4.
AECP and AEL (in parentheses) of 95% SCB in Case II.
| T | Method | ||||||
|---|---|---|---|---|---|---|---|
| 100 | Gauss | 0.9296 | 0.9018 | 0.8864 | 0.9496 | 0.8908 | 0.8923 |
| (0.7916) | (0.9180) | (0.9609) | (0.8934) | (0.8519) | (0.8786) | ||
| Boot | 0.9156 | 0.8906 | 0.8738 | 0.9402 | 0.8795 | 0.8824 | |
| (0.7517) | (0.8747) | (0.9055) | (0.8563) | (0.8153) | (0.8374) | ||
| 200 | Gauss | 0.9456 | 0.9218 | 0.9415 | 0.9662 | 0.9202 | 0.9350 |
| (0.6345) | (0.7135) | (0.6997) | (0.8794) | (0.7925) | (0.7660) | ||
| Boot | 0.9360 | 0.9112 | 0.9349 | 0.9634 | 0.9122 | 0.9262 | |
| (0.6088) | (0.6819) | (0.6707) | (0.8265) | (0.7590) | (0.7339) | ||
| 300 | Gauss | 0.9512 | 0.9474 | 0.9556 | 0.9635 | 0.9510 | 0.9377 |
| (0.5438) | (0.5746) | (0.5903) | (0.7265) | (0.7387) | (0.6281) | ||
| Boot | 0.9478 | 0.9382 | 0.9458 | 0.9584 | 0.9492 | 0.9316 | |
| (0.5176) | (0.5452) | (0.5631) | (0.6839) | (0.7075) | (0.5989) | ||
Furthermore, for the sample size T = 200 and quantile , Figure 3 plots con-LLQR estimator and 95% SCB for the varying-coefficient function. Compared with Case I, the width of 95% SCB becomes a little bigger due to the heteroscedasticity. However, the proposed estimation methodology still provides a good approximation for the true function, and the resulting 95% SCB remains valuable for reliable statistical inference. In the supplementary material, Figure 4 presents the visualization at the quantile and 0.5. In addition, the computational efficiency is investigated in Figure 3 (d)–(f) of the supplementary material.
Figure 4.
Visualization of con-LLQR estimator at in Case III. (a)–(c) figure the varying-coefficient function: true function (True, solid line), con-LLQR estimator (SQR, dashed line), and 95% SCB based on bootstrap method (15) (Boot-SCB, dotdash lines).
Analysis for Case III
In this setting, we aims to investigate the performance of the proposed statistical inference methodology when it appears some constant coefficient functions. For the sample size T = 200 and , Figure 4 presents the con-LLQR estimator (‘SQR’, dashed line) and 95% SCB based on bootstrap method (‘Boot-SCB’, dotdash lines), Figure 5 in the supplementary material gives analogues for and 0.5.
Figure 5.
The high–low range of the S&P 500 index from 02/01/2020 to 31/12/2020: (a) figures the high–low range series ; (b) plots the histogram of .
On the one hand, we note that the con-LLQR estimator is close to true function whatever the varying-coefficient function is time-varying or constant. On the other hand, for the constant coefficient functions and , their respective 95% SCB completely covers the true constant (solid horizontal line), implying the graph of SCB can help us intuitively judge whether the varying-coefficient function is really time-varying. It will provide a useful insight into the model selection.
It should be pointed out that our methodology can also handle heavy tails with infinite variance. Example 3 in the supplementary material assumes the error follows distribution and investigates the finite-sample performance similar to Example 5.1.
Example 5.2
In this example, we will illustrate our methodology through the analysis of financial volatility time series. A commonly used daily volatility measure is the so-called high-low range, which is defined as the difference between the highest and lowest logarithmic prices of a day–see for example, [1,18,25]. We consider the high–low range series (denoted as at day t) of S&P 500 index from 02/01/2020 to 31/12/2020. The historical data of S&P 500 index is open, and can be loaded from the website https://www.nasdaq.com/zh/market-activity/index/spx/historical. Figure 5(a) plots the time series , and (b) presents the corresponding histogram, which indicates the distribution of is skewed and roughly concentrates on the interval .
Asymmetry and non-normality of motivate us to adopt quantile regression instead of classical mean regression. Following [24], we adopt two covariates: one is the lag-one high–low range (that is ), and the other is the lag-one daily return (denoted as ). Formally, a general time-varying coefficient QR model is considered as below
(18) for .
We consider three competitive methods: (i) standard QR estimator of linear model corresponding to constant coefficient functions in the model (18); (ii) LLQR estimator and (iii) con-LLQR estimator. We evaluate different models using BIC information criterion, prediction RMSE(PRMSE) that is defined by and run time measuring the computational efficiency. As an illustration, Table 5 lists the evaluation index for median regression, i.e. .
Table 5.
Evaluation index for median regression.
| Index | con-LLQR | LLQR | QR |
|---|---|---|---|
| BIC | −0.5013 | −0.4953 | −0.3599 |
| PRMSE | 0.0066 | 0.0069 | 0.0076 |
| run time | 0.16s | 0.49s | 0.01s |
The BIC criterion shows that standard QR is not a preferable choice for the S&P 500 index data. Meanwhile, con-LLQR estimator has the smallest PRMSE, implying that it is comparable to LLQR estimator although we approximate the check function with convolution-type smoothed loss function. Above all, con-LLQR estimator can greatly save computing cost than LLQR estimator, about twice as much time being saved for this dataset. In addition, Figure 6 visualizes con-LLQR estimator of varying-coefficient function, whilst Figure 6 in the supplementary material plots median LLQR estimator. From these figures, we find the curve of con-LLQR estimator are smoother than LLQR estimator due to the smoothing technique of loss function. Therefore, con-LLQR estimator can provide more informative insights into real data.
Figure 6.
The con-LLQR estimator for S&P 500 index data. (a)–(c) plot the varying-coefficient functions , k = 0, 1, 2 for : con-LLQR estimator (SQR, solid lines), SCB based on (14) (SCB, longdash lines) and (15) (Boot-SCB, dotdash lines); (d)–(f) for and (g)–(i) for . (a) , . (b) , . (c) , . (d) , . (e) , . (f) , . (g) , . (h) , and (i) , .
Figure 6 considers quantile level , and plots the con-LLQR estimator (solid line) of for k = 0, 1, 2, the 95% SCB using (14) (‘SCB’, longdash lines) and (15) (‘Boot-SCB’, dotdash lines). Each row corresponds the same quantile level and each column presents the same varying-coefficient function. As a comparison, we also add a horizontal line to mark the constant estimator obtained in linear QR.
From the resultant figures, we can draw the following conclusions:
The time-varying property of the varying-coefficient function may vary with different quantile levels. Specifically speaking, the trend function is time-invariant for and 0.9, and time-variant for ; keeps time-variant for the three quantile levels; while is time-variant for quantile and 0.9 and time-invariant for .
The median trend function flattens when rescaled time u lies between 0.4 and 0.6, then increases rapidly and reaches peak about u = 0.8, and decreases after that. For and 0.9, although a constant trend is obvious, there exists a noteworthy difference between con-LLQR and QR estimator, partly due to the neglect of time-varying property of some varying-coefficient functions.
For and 0.5, the coefficient function for has a similar pattern, i.e. ascending before u = 0.2, and then descending. However, for , it declines gradually until about u = 0.4 and then go up to the peak at about u = 0.6, followed by a continuous declination since then. The coefficient function for also has obvious differences under three quantile levels.
The analysis also provides a powerful evidence that a simple linear quantile regression is not enough to model the high-low range for all quantile levels and varying-coefficient functions. In other words, we introduce a more flexible and adaptive method which includes time-invariant or/and time-variant coefficient functions, and can intuitively judge the time-invariant property through visualizing the con-LLQR estimator and SCB.
In a word, con-LLQR estimator is superior to LLQR estimator from the viewpoints of interpretability, computing cost and prediction accuracy.
6. Concluding remarks
The time-varying coefficient model is a useful tool to fit locally stationary time series data. In this paper, we have proposed a novel local linear quantile estimator for varying-coefficient component function based upon convolution-type smoothed technique, named as con-LLQR. The global Bahadur–Kiefer representation and asymptotic normality have been constructed. We have also considered statistical inference on simultaneous confidence bands. Extensive simulation studies show that the newly-proposed methodology outperforms several competitive alternatives.
The obtained large sample property further enriches theory of conquer methodology and widens its application field in the real world. However, our work does not involve prediction of conditional quantile curve of response. In fact, it is interesting and challenging to predict conditional quantile curve of response that satisfies monotonicity constraint. It is also alluring to generalize the convolution-type smoothed quantile technique to a more flexible semiparametric regression model that fits complex data structure, for example functional data or repeatedly measured data. Our future work will go in this direction.
Supplementary Material
Acknowledgments
The authors are grateful to the editor, an associate editor and two referees for their constructive comments that have substantially improved the quality of this article.
Appendix A. Assumption sets
Let be the space of all continuous functions g defined on , and denote the space of all r-order smooth function defined on as , where is the r-order derivative of g. Define the class of Lipschitz-continuous functions for some fixed constant C>0 as . Let C be a finite generic constant that may vary from line to line.
The following assumptions are necessary to construct a series of asymptotic results.
-
(A1)
For every , and for and some constant C>0.
-
(A2)
for , and for some constant . For , for some v>1.
-
(A3)
There exists , such that . Furthermore, for some constant . For , it holds that .
-
(A4)
is Lipschitz continuous on , is Lipschitz continuous about x and .
-
(A5)
Denote as the smallest eigenvalue of . There exists some positive constant η such that .
-
(A6)Denote . For any integer s, define
where . Assume that for some positive constant , and are bounded for , . In addition, represent as the conditional density of . -
(A7)
The smallest eigenvalue of , the time-varying long-run covariance matrices defined in Equation (9), is bounded away from 0 on .
-
(A8)
The kernel functions and are probability density functions defined on , which are even, integrable and twice differentiable with bounded first and second derivatives such that (i) and and (ii) with .
Funding Statement
Dr Hu's research was supported by the National Social Science Foundation of China (SSFC) (No. 21BTJ039).
Supplementary Materials
The supplement provides detailed proofs of main theorems, additional numerical results and complementary implementation issue. (.pdf type)
Disclosure statement
No potential conflict of interest was reported by the author(s).
Supplemental Material
Supplemental data for this article can be accessed online at http://dx.doi.org/10.1080/02664763.2024.2440056.
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