行政院國家科學委員會專題研究計畫 期末報告
動態追蹤資料分量迴歸
計 畫 類 別 ： 個別型 計 畫 編 號 ： NSC 101-2410-H-004-011- 執 行 期 間 ： 101 年 08 月 01 日至 102 年 07 月 31 日 執 行 單 位 ： 國立政治大學經濟學系 計 畫 主 持 人 ： 林馨怡 計畫參與人員： 學士級-專任助理人員：蕭文鈺 碩士班研究生-兼任助理人員：高千涵 碩士班研究生-兼任助理人員：張珣 大專生-兼任助理人員：張伊婷 公 開 資 訊 ： 本計畫涉及專利或其他智慧財產權，2 年後可公開查詢中 華 民 國 102 年 10 月 31 日
中 文 摘 要 ： 無 中文關鍵詞： 無
英 文 摘 要 ： This paper develops a two-stage estimation of a dynamic panel quantile regression (DPQR) model with individual fixed effects. The regressors in the model include a lagged endogenous dependent variable and other explanatory variables, which are correlated with
the fixed effects. The estimation uses the fitted value of the endogenous variable from the first stage, and applies a penalized quantile regression method for panel data in the second stage. The Monte Carlo simulation shows that the proposed DPQR
estimation
effectively reduces the dynamic bias and performs better than other estimators in finite samples. The proposed approach is easy to implement and effective in several practical applications.
英文關鍵詞： Quantile regression, dynamic panel data, penalized method
1
Introduction
Quantile regression (QR) for a panel data model has received wide attention in the-oretical and empirical studies. Its advantages are that the QR reveals heterogeneity effects of regressors on the dependent variable, and the panel data can control for unobserved individual heterogeneity by including the individual effect. One issue associated with the model is that demeaning or differencing techniques for the inci-dental parameter problem cannot be used in the conditional quantile function. To solve the incidental parameter problem of QR for panel data, Koenker (2004) first proposes the penalized approach to estimate the QR for the fixed effect panel data model where the penalty of the estimation serves to shrink a vector of individual effects toward a common value. Lamarche (2010) further discusses the degree of this shrinkage and shows that a suitable selected tuning parameter can reduce the variability of the estimator. Recent papers on the QR for a panel data model include Geraci and Bottai (2007), Abrevaya and Dahl (2008), Wang and Fygenson (2009), Gamper-Rabindran et al. (2010), Canay (2011), and Kato et al. (2012).
The dynamic relationship of the panel data model is of great interest in empir-ical applications. The correlation between the lagged dependent variable and the fixed effect produces a dynamic bias in the estimation. Anderson and Hsiao (1982), Holtz-Eakin et al. (1988), and Arellano and Bond (1991) present that the two-stage least square estimation or the dynamic generalized method of moment (DGMM) in the first-differencing model can be used to eliminate the dynamic bias and produce consistent estimators. Such a first-differencing procedure is not feasible when ap-plying the QR to a dynamic panel data model, and the model should be estimated directly. It follows that the dynamic bias arises when applying QR to a dynamic panel data model. Galvao and Montes-Rojas (2010) use the instrumental variable quantile regression (IVQR) method of Chernozhukov and Hansen (2005, 2006, 2008) to estimate the penalized QR for a dynamic panel data. Galvao (2011) also con-siders the IVQR method for the dynamic panel model, without using the penalized method. Other papers related to the QR for a panel data model with endogenous variables are Arias et al. (2001), and Harding and Lamarche (2009).
This paper adopts a “fitted value” approach to eliminate the dynamic bias and develops a two-stage estimation procedure for the dynamic panel quantile regression (DPQR) model. The first stage consists of estimating a fitted value for the lagged
endogenous dependent variable. Under the assumption of independence between the instrumental variable and the disturbance term of the DPQR model, the dynamic bias can be eliminated by replacing the endogenous variable with its fitted value and adding a constant term in the regression. Moreover, the fixed effect in this paper is a pure location shift and does not depend on the quantiles. Specifying a dummy variable that identifies individuals for the fixed effect is not available in this setting. The fixed effects should be estimated directly, and this paper applies the penalized QR of Koenker (2004) to improve the estimation of common model parameters by controlling the variability introduced by the fixed effects. Therefore, the second step is to replace the endogenous variable in the DPQR model by its fitted value and to run a penalized QR for the panel data model to obtain the estimators.
The fitted value approach for the fixed effect DPQR model, by simply running two-stage regressions, is appealing in that it is generally applicable and easy to implement. The proposed estimator is extremely simple to compute and can be implemented in standard econometrics packages. The estimators introduced in this paper offers some computation advantages and should be viewed as a complement to those in Galvao and Montes-Rojas (2010). This paper shows that the proposed DPQR estimator is asymptotically normal with zero mean when both N and T are large. In addition, we compare the bias and root mean squared error (RMSE) of several estimators for the DPQR model under different scenarios. Using the Monte Carlo simulations, in a finite sample the dynamic bias is effectively reduced under the two-stage estimation. Comparing with the estimators of Koenker (2004), Gal-vao and Montes-Rojas (2010), and GalGal-vao (2011), the proposed estimator performs better than the other estimators regarding the bias and RMSE. Thus, our estimator competes efficiently with those methodologies applying to the DPQR model.
The remainder of this paper is organized as follows. Section 2 introduces the econometrics method, proposes the DPQR model with a two-stage estimating pro-cedure, and provides the large sample properties of the proposed estimator. Section 3 shows a Monte Carlo simulation.
2
Dynamic Panel Quantile Regression
2.1
Fitted Value Approach
Consider a dynamic panel data model with individual fixed effects:
yit = αyit−1+ x0itβ + ηi+ uit, ∀i = 1, · · · , N, t = 1, · · · , T, (1)
where yit is a real-valued dependent variable, yit−1 is the lagged dependent variable,
xit is a (dX× 1) vector of real-valued, continuously distributed, exogenous
explana-tory variables, ηi is the parameter that represents the individual fixed effects, α
and β are unknown parameters, and uit is the error term. The fixed effects ηi in
(1) capture some source of variability, or “unobserved heterogeneity,” that is not adequately controlled by other regressors in the model. When N is a large number, the estimation of N + k + 1 parameters is complicated and suffers from an incidental parameter problem. In addition, by construction, yit−1 is a function of the
unob-served individual effect ηi and is correlated with the error term. An endogeneity
problem thus arises in the dynamic panel data model. To eliminate the dynamic bias, Anderson and Hsiao (1982) suggest a two-stage least squares estimation by using further lags of the dependent variable as instruments for first-differenced lag dependent variable. Moreover, Holtz-Eakin et al. (1988) and Arellano and Bond (1991) suggest using the DGMM estimator, which is based on moment equations constructed from the first-differenced error term and lags of regressors.
To capture the heterogeneous covariate effects of the dependent variable, this paper applies the QR for the dynamic panel data with fixed effects. The DPQR model is able to capture the dynamic relationship of variables of interest, control for unobserved individual heterogeneity with ηi, and reveal heterogeneity effects of
regressors on the dependent variable. Since the first-differencing procedure is not feasible in the conditional quantile function, the QR for dynamic panel data model should be estimated directly, but the correlation between the lagged dependent variable and the fixed effect in (1) produces a dynamic bias in the estimation. Several studies propose to solve this endogeneity problem in the DPQR model. For example, Arias et al. (2001), following the control function approach, suggest a two-stage estimation. Harding and Lamarche (2009), Galvao and Montes-Rojas (2010), and Galvao (2011) introduce the IVQR method for panel data model.
This paper proposes the fitted value approach to eliminate the dynamic bias in the DPQR model, and develops a two-stage estimation procedure. First, let the endogenous lagged dependent variable can be divided into two parts: one is a function of exogenous and instrument variables ˆyit−1, and the other is the residual
between yit−1 and ˆyit−1. Let zit be a (dZ× 1) vector of the instrumental variable,
then we have
yit−1 = ˆyit−1+ vit, (2)
where ˆyit−1 := ˆyit−1(xit, zit) is a function of instruments, and vit = yit− ˆyit−1. In
this setting, vit is a real-valued unobserved random variable and is independent
of both xit and zit. Here, further lags of the first-differenced dependent variable,
∆yit−j, j = 1, 2, · · · , T − 1, can be used as instrumental variables, since the fixed
effect is eliminated by construction. For example, ∆yit−1, ∆yit−2 can be valid
instru-ments for yit−1. In addition, the exogenous variable xit is allowed to be incorporated
into the first stage.
To identify the estimation procedure used, we need the following assumptions. First, we need the consistency of the fitted value: ˆyit−1 should be a consistent
estimator for yit−1. Second, we need an assumption for the independence between
uit and zit as well as vit and ηi to identify the model.
Assumption 1. W p → 1, the function ˆyit−1(x, z) p
→ yit−1 uniformly in (x, z)
Assumption 2. uit is independent of zit, and vit is independent of ηi.
Replacing yit−1 in (1) by the function (2) yields:
yit = α(ˆyit−1+ vit) + x0itβ + ηi+ uit
= αˆyit−1+ x0itβ + ηi+ (αvit+ uit).
By construction of model (1), uitis the above model is independent of the exogenous
explanatory variable xit and the fixed effect ηi. Also, vit is independent of xit and
zit in model (2). With the independence assumption (Assumption 2), we obtain
that the conditional quantile function of Qαvit+uit(τ |xit, zit, ηi) equals an
uncondi-tional quantile function Qαvit+uit(τ ). Let Qαvit+uit(τ ) be c(τ ). Therefore, the τ -th
conditional quantile function for the DPQR model is:
Qyit(τ |xit, zit, ηi) = α(τ )ˆyit−1+ x
0
where Qyit(τ |xit, zit, ηi) is the τ -th conditional quantile function of the dependent
variable, and α(τ ) and β(τ ) are parameters at the τ -th quantile. In this paper the fixed effect ηi is a pure location shift effect and does not depend on the quantile
τ . The penalized QR approach of Koenker (2004) can then be used to obtain consistent estimators of α(τ ) and β(τ ) in (3), where c(τ ) is viewed as the coefficient of a constant term. This suggests that the parameters of the DPQR model can be estimated by a two-stage procedure. The first stage of the two-stage procedure is to construct a regression of yit−1 on zit and xit and obtain the fitted value ˆyit−1. In
the second step of the two-stage procedure, the fitted value ˆyit−1 is inserted in place
of the endogenous dependent variable yit−1, and the penalized QR approach for the
panel data model is used for (3). Therefore, the two-stage estimation corrects for endogeneity of the DPQR model by replacing yit−1 by ˆyit−1 and can be viewed as a
variant of the fitted value approach. One may note that Galvao and Montes-Rojas (2010) and Galvao (2011) also study the estimation and inference for the DPQR model. They consider using the IVQR method for the endogenous problem, whereas this paper uses a two-stage fitted value approach for the endogenous problem and can be viewed as a complement to their papers.
2.2
Estimation and Asymptotics
The estimation procedure consists of two-stages wherein the first stage estimates the fitted value of yit−1. A practical formulation for ˆyit−1 is to use the least squares
projection of yit−1 on xit and zit and possibly their powers. Efficiency can be
im-proved by choosing ˆyit−1 appropriately. In the second stage, we suggest using the
penalized QR of Koenker (2004) to improve the properties of the estimation of (3). The penalized QR method for the conditional quantile function (3) is to estimate (3) for several quantiles simultaneously with a penalty term, as characterized by:
min α,β,c K X k=1 N X i=1 T X t=1 ωkρτk(yit− x 0 itβ(τk) − α(τk)ˆyit−1− c(τk) − ηi) − λ N X i=1 |ηi|, (4)
where ωk is the weight for the τk-quantile, k = 1, · · · , K, ρτ(u) = u · (τ − 1(u < 0)) is
the check function as in Koenker and Bassett (1978), with 1(·) an indicator function, and λPN
i=1|ηi| is an `1 penalty term with tuning parameter λ. The weights ωk
control the relative influence of the q quantiles, {τ1, · · · , τK}, on the estimation of
resulting estimator with λ ≥ 0, and the tuning parameter λ controls the degree of the shrinkage. As Tibshirani (1996), Donoho et al. (1998), and Lamarche (2010) point out, when N is large relative to T , the `1 shrinkage is advantageous in controlling
the variability introduced by the large number of estimated ηi parameters, since the
fixed effects are estimated directly in the penalized method. Note that since c(τ ) can be viewed as the parameter of intercept, we therefore have q, τ -specific, estimates of the intercept in the estimation.
In addition to Assumptions 1 and 2, to obtain the asymptotic properties of the proposed estimator, we need more assumptions as follows.
Assumption 3. The variables yit are independent across individuals, stationary
with conditional distribution Fit, and continuous conditional densities fit are
uni-formly bounded away from 0 and ∞, for i = 1, · · · , N , and t = 1, · · · , T .
Assumption 4. Let ˜xit = [1, ˆyit−1, x0it]
0and ˜β(τ ) = [c(τ ), α(τ ), β(τ )] be a (d
X+2)×1
vector of parameters. There exist positive definite matrices J0(τ ) and S(τ ) such that:
J0(τ ) = lim N →∞,T →∞ 1 N T ω1X0MD(τ1)0Φ(τ1)MD(τ1)X · · · 0 .. . . .. ... 0 · · · ωKX0MD(τK)0Φ(τK)MD(τK)X , S(τ ) = lim N →∞,T →∞ 1 N T Ω11X0MD(τ1)0MD(τ1)X · · · Ω1KX0MD(τ1)0MD(τK)X .. . . .. ... ΩK1X0MD(τK)0MD(τ1)X · · · ΩKKX0MD(τK)0MD(τK)X , where X = [˜xit], MD(τk) = I − PD(τk), PD(τk) = D(D0Φ(τk)D)−1D0Φ(τk), Φ(τk) = diag(fyit(˜x 0
itβ(τ ) + η˜ i)) and Ωjl= ωj(τj∧ τl− τjτl)ωl, with D = [dit] as an incidence
matrix of dummy variable, and ditas a dummy variable that identifies the N distinct
individuals in the sample.
Assumption 5. maxit||xit|| = O( √ N T ), and maxit||zit|| = O( √ N T ). Assumption 6. λT/ √ T → λ0.
Assumption 3 is standard in the QR literature. Assumption 4 uses the defi-nition of the positive definite matrices for the central limit theorem to obtain the asymptotic normality. Assumption 5 imposes bounds on the variable xit and ˜xit.
This assumption ensures the finite dimensional convergence of the objective func-tion. In Assumption 6, the shrinkage of the fixed effects toward a common value can decrease the variability caused by the presence of unobserved individual het-erogeneity. Similar assumptions can also be found in Lamarche (2010), Galvao and Montes-Rojas (2010), and Galvao (2011). Note that Kato et al. (2012) also discuss the asymptotics for panel data QR model, and use asymptotics that do not invoke the Bahadur representation in Koenker (2004). However, Kato et al. (2012) should impose substantially stronger conditions on the rate at which T is supposed to grow to infinity relative to N , and we rely on results from Koenker (2004), Galvao and Montes-Rojas (2010), and Galvao (2011). Under the assumptions above, we can establish the asymptotic normality of the proposed estimator.
Theorem 1. Suppose that Assumptions 1-6 hold. When N → ∞, T → ∞, and Na/T → 0, for some a > 0, then √N T (β(τ ) − ˜ˆ˜ β(τ )) converges to a normal
distri-bution with mean zero and covariance matrix J0(τ )−1S(τ )J0(τ )−1.
An asymptotically valid standard error of the estimator for the second step re-quires a correction for the first step estimation, which makes the variance function of the estimator very complicated. Thus, the bootstrap method is considered to calculate the standard error of the proposed estimator. Let yi· = {yi1, · · · , yiT},
xi· = {xi1, · · · , xiT}, and zi· = {zi1, · · · , ziT} be T × 1 vectors that contain T
ob-servations. For the bootstrapping procedure, first, a random sample {y∗i·, x∗i·, zi·∗} is randomly drawn from the data {yi·, xi·, zi·, i = 1, · · · , N } to create a resampled data
set {yi·∗, x∗i·, zi·∗, i = 1, · · · , N }. The DPQR estimates, βˆ˜∗(τ ), can be computed from the resampled data. Second, the above procedure is repeated B times to obtain DPQR estimates, βˆ˜1∗(τ ),β˜ˆ2∗(τ ), · · · ,βˆ˜B∗(τ ). Third and finally, a consistent variance estimator is obtained by:
d Var(β(τ )) =ˆ˜ 1 B − 1 B X b=1 ˆ ˜ βb∗(τ ) −β¯ˆ˜b∗(τ ) ˆ ˜ βb∗(τ ) −β¯ˆ˜b∗(τ ) 0 , with β¯ˆ˜b∗(τ ) = B−1PB b=1 ˆ ˜ βb∗(τ ).
3
Monte Carlo Simulations
This section studies the Monte Carlo study to investigate the small sample prop-erties of estimators for the DPQR model. We compare the bias and RMSE of the following estimators: (1) the two-stage estimator for the DPQR model in this paper (DPQR); (2) the penalized QR for panel data estimator in Koenker (2004) (PQR); (3) the penalized QR for panel data estimator using the IVQR method in Galvao and Montes-Rojas (2010) (PQR-IVQR); and (4) the fixed effect QR estimator of Galvao (2011) (FEQR). The latter two estimators use the IVQR method to reduce the dynamic bias. Three DPQR models are considered in this section: (A) the pure location shift model,
yit = ηi+ αyit−1+ βxit+ uit;
(B) the location-scale shift model I,
yit = ηi+ αyit−1+ βxit+ (γ0xit)uit;
and (C) the location-scale shift model II,
yit = ηi+ αyit−1+ βxit+ (1 + γ1xit)uit.
In all cases, we follow Galvao and Montes-Rojas (2010), set yi,−50 = 0 and
gen-erate yit for t = −49, −48, · · · , T , and discard the first 50 observations, using the
observations t = 0 through T for estimation. The error term uit follows the normal
distribution N (0, σu2) with σu2 = 1, 3, 5, the heavy-tail t-distribution with 3 degree of freedom (t3 distribution), or the χ2-distribution with 3 degree of freedom (χ23
distribution).
The regressor xit is generated according to xit = µi+ ξit, where the fixed effect:
µi = e1i+ 1 T T X t=1 xit, e1i∼ N (0, σ2e1),
and ξit follows the same distribution as uit. The fixed effects, ηi are generated as:
ηi = e2i+ 1 T T X t=1 it, e2i ∼ N (0, σ2e2).
From the above specification of the fixed effect, there is correlation between the individual effects and the explanatory variables, ensuring that the random effects are inconsistent. In the simulation, T = 10, N = 50, and the number of replications is 2000. We also compare different sample sizes, with T = {5, 15, 25}, and N = {50, 100, 150}. In addition, the parameters α = {0.3, 0.4, 0.5, 0.6, 0.7}, β = 1, and σe1 = σe2 = 1. For the location-scale shift models, we use γ0 = 0.5 and γ1 = 0.1.
The estimators are analyzed for three quantiles τ = (0.25, 0.5, 0.75). For the DPQR, PQR-IVQR and FEQR estimators, we consider instruments yit−2 and xit
for the lagged dependent variable. Tables 1 and 2 report the bias and RMSE results for estimates of the autoregressive parameter values for the location shift and the location-scale shift models, respectively. Both Tables 1 and 2 show that, under the normal error distribution, the autoregressive coefficient is biased upward for the PQR and PQR-IVQR estimations, and is slightly biased for the DPQR estimation. Tables 1 and 2 present that the DPQR estimator has lower bias and RMSE of ˆα than those of the PQR and PQR-IVQR estimators. Thus, the DPQR estimator performs best among the estimations for the three DPQR models. Tables 1 and 2 also shows that ˆβ is biased upward for all three estimators, except for some values of the DPQR estimators. Regarding the bias and the RMSE of β, in Table 1 and the upper panel of Table 2, the DPQR estimator also performs better than the other two estimators. In the lower panel of Table 2, regarding the bias and RMSE, the DPQR estimator performs better than the other two estimators for the lower to middle quartiles (τ = 0.25, 0.5) and the PQR-IVQR estimator performs better than the other two estimators for the higher quartile (τ = 0.75).
Tables 3 and 4 report the bias and RMSE results under the N (0, 3), N (0, 5), t3,
and χ23 distributions, for the location and location-scale shift models, respectively. In Table 3, the autoregressive estimates of PQR and PQR-IVQR are biased upward, and those of DPQR are biased downward. Regarding the bias and RMSE, Table 3 shows that for the normal distributions N (0, 3) and N (0, 5), and t3 distributions, the
PQR estimator performs best among the three estimators; for the χ23 distribution, the DPQR estimator performs best among the three estimators. Table 4 presents that the PQR estimator has the lowest bias and RMSE for different error distribu-tions except the χ23 distribution. The DPQR estimator performs better than the other two estimators for the χ23 distribution for τ = 0.5, 0.75. Moreover, regarding the bias and RMSE of ˆβ, both Tables 3 and 4 show that, in all three DPQR models,
the DPQR estimator of β has lower bias and RMSE than the other two estimators for all error distributions and across all quartiles.
Table 5 reports the bias and RMSE results with different panel sizes. The au-toregressive parameter and β estimates are biased upward for all three estimators. Both the bias and RMSE are larger for small T and decrease as T increases, but the bias and RMSE do not depend on n. In addition, it is seen that the PQR estimator performs better than the other two estimators for the autoregressive parameter; the DPQR estimator performs better than the other two estimators for β in the loca-tion shift model. Similar results can be found in the localoca-tion-scale shift models in the lower two panels of Table 5. In the final part of the simulation, we consider different values of tuning parameter λ. It is noted that the special case λ = 0 of the PQR-IVQR estimator is the FEQR estimator of Galvao (2011).
Table 6 shows the bias and RMSE results for different lambda values. For λ = 0, the PQR, DPQR, and FEQR estimators of α and β are biased downward; for λ = 0.5 and 1, the DPQR estimator of α is biased downward; for λ = 0.5, the DPQR estimator of β is biased downward. For other values of λ, the three estimators of α and β are biased upward. In the location shift model the FEQR estimator performs best among the three estimator when λ = 0, and for other values of λ the DPQR estimator has a lower bias and RMSE values. In the location-scale shift models the PQR estimator performs best for λ = 0, 0.5, while the DPQR estimator performs best for λ = 1, 1.5, 2 regarding the bias and RMSE of α and β. The results suggest that the DPQR estimator performs well in large values of λ for all three dynamic panel models.
4
Conclusions
This paper has proposed a two-stage estimator for the DPQR model. In this paper, the two-stage estimation method, adjusting the dynamic bias, depends crucially on the assumption of independence between the instrumental variable and the distur-bance term of the DPQR model. Thus, it would be useful to extend this assumption. In addition, future research could also benefit by investigating the issue of efficiency of the estimation, and the selection of λ.
References
Abrevaya, J., Dahl, C.M. (2008). The effects of birth inputs on birthweight: Evidence from quantile estimation on panel data, Journal of Business and Economic Statistics, 26, 370–397.’
Anderson, T.W., Hsiao, C. (1982). Formulation and estimation of dynamic models using panel data, Journal of Econometrics, 18, 47–82.
Arellano, M., Bond, S. (1991). Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations, Review of Economic Studies, 58, 277–297.
Arias, O., Hallock, K.F., Sosa-Escudero, W. (2001). Individual heterogeneity in the returns to schooling: Instrumental variables quantile regression using twins data, Empirical Economics, 26, 7-40.
Canay, I.A. (2011). A simple approach to quantile regression for panel data, The Econometrics Journal, 14(3), 368–386.
Cat˜ao, L., Terrones, M. (2005). Fiscal deficits and inflation, Journal of Monetary Economics, 52, 529–554.
Chernozhukov, V., Hansen, C. (2005). Notes and comments an IV model of quantile treatment effects, Econometrica, 73, 245–261.
Chernozhukov, V., Hansen, C. (2006). Instrumental quantile regression inference for structural and treatment effect models, Journal of Econometrics, 132, 491–525.
Chernozhukov, V., Hansen, C. (2008). Instrumental variable quantile regression: A robust infer-ence approach, Journal of Econometrics, 142, 379–398.
Donoho, D., Chen, S., Saunders, M. (1998). Atomic decomposition by basis pursuit, SIAM Journal of Scientific Computing, 20, 33–61.
Galvao, A.F. (2011). Quantile regression for dynamic panel data with fixed effects, Journal of Econometrics, 164(1), 142–157.
Galvao, A.F., Montes-Rojas, G.V. (2010). Penalized quantile regression for dynamic panel data, Journal of Statistical Planning and Inference, 140, 3476–3497.
Gamper-Rabindran, S., Khan, S., Timmins, C. (2010). The impact of piped water provision on infant mortality in Brazil: A quantile panel data approach, Journal of Development Economics, 92, 188–200.
Geraci, M., Bottai, M. (2007). Quantile regression for longitudinal data using the asymmetric Laplace distribution, Biostatistics, 8, 140–154.
Harding, M., Lamarche, C. (2009). A quantile regression approach for estimating panel data models using instrumental variables, Economics Letters, 104, 133–135.
Holtz-Eakin, D., Newey, W., Rosen, H. (1988). Estimating vector antoregressions with panel data, Econometrica, 56, 1371–1395.
Kato, K., Galvao, A.F., Montes-Rojas, G.V. (2012). Asymptotics for panel quantile regression models with individual effects, Journal of Econometrics, 170(1), 76–91.
Koenker, R. (2004). Quantile regression for longitudinal data, Journal of Multivariate Analysis, 91, 74–89.
Koenker, R., Bassett, G. (1978), Regression Quantiles, Econometrica, 46, 33-50.
Lamarche, C. (2010). Robust penalized quantile regression estimation for panel data, Journal of Econometrics, 157, 396–408.
Ruppert, D., Carroll, R. (1980). Trimmed least squares estimation in the linear model, Journal of the American Statistical Association, 75, 828–838.
Tibshirani, R. (1996). Regression shrinkage and selection via the lasso, Journal of the Royal Statistical Society, B, 58, 267–288.
Wang, H., Fygenson, M. (2009). Inference for censored quantile regression models in longitudinal studies, Annals of Statistics, 37, 756–781.
T able 1: Bias and RMSE results for d ifferen t au toregressiv e p arameter v alues (The lo cation shift mo del) τ = 0 .25 τ = 0 .5 τ = 0 .75 α 0.3 0.4 0.5 0.6 0.7 0 .3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7 PQR ˆα Bias 0.1120 0.1221 0.1 293 0.1289 0.1198 0.1118 0.1219 0.1 292 0.1292 0.1197 0.1116 0.1218 0.12 91 0.1294 0.1197 RMSE 0.1194 0 .1281 0.1339 0.1318 0.1212 0.1190 0 .1278 0.1336 0.1319 0.1211 0.1190 0 .1280 0.1336 0.1322 0.1211 ˆ β Bias 0.0956 0.0923 0.0906 0.0814 0.0633 0.0958 0.0920 0.0904 0.0816 0.0633 0.0959 0.0915 0.0906 0.0817 0.0635 RMSE 0.1112 0.1097 0.1078 0.0998 0.0864 0.1110 0.1086 0.1073 0.0993 0.0855 0.1116 0.1085 0.1082 0.1000 0.0864 DPQR ˆα Bias -0.0378 -0.0487 -0.0312 0.0151 0.0683 -0.0369 -0.0484 -0.0308 0.0153 0.0684 -0.0369 -0.0488 -0.02 95 0.0159 0.0681 RMSE 0.0699 0.0766 0.0702 0.0628 0.0791 0.0681 0.0755 0.0688 0.0622 0.0791 0.0694 0.0770 0.0686 0.0624 0.0791 ˆ β Bias 0.0265 0.0170 0.0201 0.0256 0.0222 0.0250 0.0181 0.0188 0.0242 0.0228 0.0246 0.0164 0.0148 0.0240 0.0236 RMSE 0.0711 0.0699 0.0750 0.0801 0.0818 0.0646 0.0657 0.0663 0.0700 0.0728 0.0708 0.0703 0.0723 0.0791 0.0809 PQR-IV QR ˆα Bias 0.2029 0.2199 0.2032 0.1721 0.1388 0.1812 0.2118 0.2021 0.1724 0.1388 0.2025 0.2201 0.2029 0.1727 0.1388 RMSE 0.2248 0.2296 0.2076 0.1744 0.1401 0.2118 0.2246 0.2066 0.1745 0.1400 0.2243 0.2299 0.2071 0.1749 0.1401 ˆ β Bias 0.0722 0.0660 0.0642 0.0577 0.0454 0.0794 0.0674 0.0644 0.0576 0.0454 0.0728 0.0645 0.0643 0.0578 0.0455 RMSE 0.0951 0.0902 0.0885 0.0828 0.0749 0.0993 0.0902 0.0879 0.0819 0.0738 0.0954 0.0891 0.0889 0.0829 0.0747
T able 2: Bias and RMSE results for diff eren t auto regressiv e par ameter v alues (The lo cation-scale shift mo dels) Mo del I τ = 0 .25 τ = 0 .5 τ = 0 .75 α 0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7 PQR ˆα Bias 0.0868 0.1002 0.1 122 0.1206 0.1176 0.0867 0.1002 0.1121 0.1207 0 .1177 0.0867 0.1002 0.1121 0.1208 0 .1177 RMSE 0.0917 0 .1047 0.1161 0.1231 0.1188 0.0915 0.1046 0. 1160 0.1231 0.1189 0.0915 0.1046 0. 1160 0.1232 0.1190 ˆ β Bias 0.0554 0.0573 0.0562 0.0549 0.0483 0.0551 0.0575 0.0563 0.0555 0.0481 0.0549 0.0574 0.0561 0.0560 0.0481 RMSE 0.0724 0.0738 0.0744 0.0715 0.0666 0.0700 0.0726 0.0724 0.0700 0.0639 0.0717 0.0743 0.0742 0.0721 0.0664 DPQR ˆα Bias -0.0528 -0.0492 -0.0143 0.0442 0 .0899 -0.0529 -0.0494 -0. 0156 0.0429 0.0882 -0.0526 -0.0489 -0.0147 0.0447 0.0900 RMSE 0.0711 0.0718 0.0610 0.0698 0.0953 0.0709 0.0717 0.0610 0.0690 0.0934 0.0711 0.0715 0.0603 0.0707 0.0952 ˆ β Bias 0.0014 -0.0016 0.0003 0.0090 0.0089 0.00 12 -0.0040 -0.0013 0.007 5 0.0078 0.0005 -0.0026 0.0003 0.0099 0.0082 RMSE 0.0579 0.0607 0.0646 0.0659 0.0716 0.0482 0.0507 0.0537 0.0558 0.0590 0.0584 0.0617 0.0664 0.0673 0.0724 PQR-IV QR ˆα Bias 0.1438 0.1978 0.2053 0.1777 0.1430 0.1107 0.1796 0.2010 0.1777 0.1430 0.1439 0.1979 0.2056 0.1782 0.1431 RMSE 0.1648 0.2122 0.2099 0.1792 0.1438 0.1433 0.2020 0.2073 0.1792 0.1438 0.1651 0.2122 0.2101 0.1797 0.1439 ˆ β Bias 0.0524 0.0503 0.0464 0.0441 0.0378 0.0598 0.0543 0.0470 0.0444 0.0375 0.0515 0.0507 0.0465 0.0448 0.0375 RMSE 0.0723 0.0702 0.0684 0.0643 0.0599 0.0747 0.0715 0.0664 0.0621 0.0567 0.0711 0.0719 0.0682 0.0646 0.0596 Mo del II τ = 0 .25 τ = 0 .5 τ = 0 .75 α 0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7 PQR ˆα Bias 0.1121 0.1207 0.1 266 0.1293 0.1183 0.1122 0.1208 0.1267 0.1294 0 .1186 0.1125 0.1208 0.1268 0.1295 0 .1189 RMSE 0.1198 0 .1268 0.1318 0.1323 0.1198 0.1196 0.1266 0. 1316 0.1321 0.1199 0.1201 0.1268 0. 1318 0.1323 0.1203 ˆ β Bias 0.0950 0.0924 0.0865 0.0787 0.0620 0.0959 0.0932 0.0867 0.0794 0.0627 0.0967 0.0940 0.0871 0.0802 0.0632 RMSE 0.1101 0.1084 0.1036 0.0969 0.0848 0.1104 0.1089 0.1032 0.0969 0.0847 0.1117 0.1101 0.1039 0.0983 0.0857 DPQR ˆα Bias -0.0403 -0.0509 -0.0339 0.0127 0 .0642 -0.0382 -0.0486 -0. 0321 0.0153 0.0666 -0.0362 -0.0476 -0.0294 0.0177 0.0693 RMSE 0.0699 0.0775 0.0706 0.0609 0.0760 0.0680 0.0752 0.0691 0.0616 0.0777 0.0685 0.0755 0.0690 0.0632 0.0805 ˆ β Bias -0.0301 -0.0347 -0.0323 -0.0249 -0.0239 0.0250 0.0178 0.0177 0.0234 0.0242 0.0796 0.0736 0.0660 0.0705 0.0686 RMSE 0.0720 0.0773 0.0754 0.0780 0.0810 0.0640 0.0638 0.0648 0.0691 0.0730 0.1042 0.0996 0.0959 0.1019 0.1030 PQR-IV QR ˆα Bias 0.2034 0.2190 0.2018 0.1724 0.1387 0.1825 0.2124 0.2005 0.1723 0.1389 0.2035 0.2193 0.2023 0.1725 0.1393 RMSE 0.2248 0.2297 0.2066 0.1747 0.1399 0.2124 0.2257 0.2056 0.1745 0.1400 0.2251 0.2299 0.2069 0.1747 0.1405 ˆ β Bias 0.0723 0.0665 0.0600 0.0561 0.0442 0.0803 0.0690 0.0605 0.0569 0.0448 0.0747 0.0680 0.0602 0.0580 0.0455 RMSE 0.0938 0.0896 0.0842 0.0815 0.0742 0.0992 0.0906 0.0838 0.0808 0.0735 0.0957 0.0906 0.0843 0.0823 0.0748
T able 3: Bias and RMSE results for differen t error distributions (The lo cation shift mo del) τ = 0 .25 τ = 0 .5 τ = 0 .75 u N (0 , 3) N (0 , 5) t3 χ 2 3 N (0 , 3) N (0 , 5) t3 χ 2 3 N (0 , 3) N (0 , 5) t3 χ 2 3 PQR ˆα Bias 0.0569 0.0207 0.0616 0.2343 0.0568 0.0206 0.0618 0.2796 0.0568 0.0207 0.0 620 0.3429 RMSE 0.0735 0.0519 0.0689 0.2346 0.0725 0.0505 0.0688 0.2800 0.0732 0.0520 0.0693 0.3436 ˆ β Bias 0.1380 0.1626 0.0581 0.1750 0.1386 0.1623 0.0581 0.1813 0.1390 0.1620 0.0581 0.1971 RMSE 0.1692 0.2023 0.07 21 0.1805 0.1688 0.1999 0.0719 0.1893 0.1703 0.2009 0.0722 0.2142 DPQR ˆα Bias -0.1191 -0.1582 -0.1336 -0.1615 -0.1211 -0.1602 -0.1347 -0. 1526 -0.1211 -0.1597 -0.1326 -0.146 2 RMSE 0.1389 0.1739 0.14 23 0.1692 0.1391 0.1749 0.1426 0.1611 0.1403 0.1763 0.1412 0.1583 ˆ β Bias 0.0493 0.0731 -0.0165 -0.0161 0.0524 0.0739 -0.0178 -0.0215 0.0495 0.0753 -0.0188 -0.0217 RMSE 0.1229 0.1617 0.06 25 0.0532 0.1174 0.1501 0.0526 0.0595 0.1260 0.1639 0.0614 0.0830 PQR-IV QR ˆα Bias 0.1161 0.0548 0.1277 0.2642 0.1069 0.0472 0.1134 0.3126 0.1154 0.0526 0.1278 0.3821 RMSE 0.1452 0.0978 0.14 62 0.2644 0.1391 0.0919 0.1376 0.3130 0.1436 0.0968 0.1461 0.3828 ˆ β Bias 0.1169 0.1484 0.0583 0.0956 0.1217 0.1532 0.0620 0.0924 0.1184 0.1501 0.0582 0.0896 RMSE 0.1553 0.1941 0.07 30 0.0956 0.1569 0.1935 0.0754 0.1069 0.1567 0.1939 0.0729 0.1217
T able 4: Bias and RMSE results for d ifferen t er ror distribution (The lo cation-scale shift mo dels) Mo del I τ = 0 .25 τ = 0 .5 τ = 0 .75 u N (0 , 3) N (0 , 5) t3 χ 2 3 N (0 , 3) N (0 , 5) t3 χ 2 3 N (0 , 3) N (0 , 5) t3 χ 2 3 PQR ˆα Bias 0.0752 0.0550 0.0418 0.0804 0.0752 0.0548 0.0417 0.0817 0.0752 0.0547 0.0 416 0.0780 RMSE 0.0811 0.0626 0.0474 0.0829 0.0809 0.0622 0.0472 0.0854 0.0812 0.0623 0.0472 0.0859 ˆ β Bias 0.0791 0.0928 0.0483 1.0687 0.0787 0.0923 0.0480 1.6140 0.0786 0.0917 0.0476 2.4501 RMSE 0.1121 0.1352 0.07 74 1.0771 0.1078 0.1294 0.0737 1.6235 0.1118 0.1345 0.0755 2.4638 DPQR ˆα Bias -0.0897 -0.1188 -0.1390 -0.1909 -0.0925 -0.1203 -0.1412 -0. 1787 -0.0916 -0.1183 -0.1391 -0.176 2 RMSE 0.1065 0.1332 0.14 53 0.1995 0.1074 0.1326 0.1468 0.1888 0.1076 0.1323 0.1453 0.1916 ˆ β Bias 0.0027 0.0153 -0.0183 0.7142 0.0037 0.0129 -0.0178 1.1981 0.0051 0.0118 -0.0201 1.9371 RMSE 0.0970 0.1257 0.08 78 0.7377 0.0806 0.0995 0.0722 1.2204 0.0983 0.1238 0.0887 1.9650 PQR-IV QR ˆα Bias 0.1508 0.1161 0.0884 0.1194 0.1349 0.1013 0.0730 0.1208 0.1517 0.1154 0.0883 0.1164 RMSE 0.1678 0.1356 0.10 46 0.1223 0.1580 0.1269 0.0951 0.1258 0.1682 0.1352 0.1046 0.1266 ˆ β Bias 0.0691 0.0814 0.0510 0.8951 0.0735 0.0872 0.0553 1.4414 0.0680 0.0815 0.0497 2.2893 RMSE 0.1075 0.1295 0.08 04 0.9092 0.1049 0.1256 0.0800 1.4574 0.1070 0.1292 0.0787 2.3114 Mo del II τ = 0 .25 τ = 0 .5 τ = 0 .75 u N (0 , 3) N (0 , 5) t3 χ 2 3 N (0 , 3) N (0 , 5) t3 χ 2 3 N (0 , 3) N (0 , 5) t3 χ 2 3 PQR ˆα Bias 0.0560 0.0200 0.0596 0.1828 0.0557 0.0202 0.0594 0.2206 0.0554 0.0202 0.0 594 0.2733 RMSE 0.0730 0.0525 0.0672 0.1836 0.0720 0.0511 0.0669 0.2216 0.0722 0.0519 0.0671 0.2751 ˆ β Bias 0.1348 0.1655 0.0558 0.4019 0.1359 0.1662 0.0562 0.5280 0.1368 0.1666 0.0567 0.7295 RMSE 0.1653 0.2047 0.06 96 0.4093 0.1652 0.2032 0.0697 0.5370 0.1669 0.2046 0.0706 0.7442 DPQR ˆα Bias -0.1209 -0,1611 -0.1392 -0.1851 -0.1190 -0.1608 -0.1356 -0. 1753 -0.1160 -0.1577 -0.1297 -0.170 4 RMSE 0.1394 0.1772 0.14 70 0.1933 0.1367 0.1758 0.1430 0.1852 0.1358 0.1745 0.1384 0.1844 ˆ β Bias -0.0510 -0.0531 -0.0710 0.1375 0.0482 0.0 781 -0.0191 0.2254 0.1443 0.2050 0.0313 0.3739 RMSE 0.1213 0.1551 0.09 25 0.1619 0.1120 0.1516 0.0534 0.2478 0.1840 0.2477 0.0654 0.4029 PQR-IV QR ˆα Bias 0.1133 0.0522 0.1246 0.2228 0.1019 0.0478 0.1103 0.2657 0.1123 0.0520 0.1237 0.3261 RMSE 0.1415 0,0985 0.14 40 0.2235 0.1342 0.0937 0.1354 0.2667 0.1398 0.0975 0.1431 0.3280 ˆ β Bias 0.1135 0.1537 0.0565 0.2735 0.1207 0.1573 0.0612 0.3801 0.1182 0.1561 0.0583 0.5548 RMSE 0.1510 0.1981 0.07 10 0.2848 0.1534 0.1963 0.0741 0.3934 0.1532 0.1977 0.0723 0.5765
Table 5: Bias and RMSE results for different panel sizes (τ = 0.5)
T = 5 T = 15 T = 25
N = 50 N = 100 N = 150 N = 50 N = 100 N = 150 N = 50 N = 100 N = 150
The location shift model PQR ˆ α Bias 0.2134 0.2147 0.2157 0.0751 0.0749 0.0747 0.0400 0.0398 0.0400 RMSE 0.2174 0.2165 0.2169 0.0822 0.0785 0.0771 0.0467 0.0432 0.0424 ˆ β Bias 0.1477 0.1472 0.1468 0.0587 0.0595 0.0595 0.0314 0.0324 0.0321 RMSE 0.1713 0.1594 0.1552 0.0758 0.0680 0.0657 0.0482 0.0411 0.0383 DPQR ˆ α Bias 0.1937 0.1978 0.1988 -0.1281 -0.1242 -0.1254 -0.1719 -0.1716 -0.1716 RMSE 0.2029 0.2023 0.2015 0.1349 0.1279 0.1279 0.1745 0.1729 0.1726 ˆ β Bias 0.0566 0.0584 0.0577 -0.0075 -0.0055 -0.0054 -0.0240 -0.0221 -0.0229 RMSE 0.1119 0.0899 0.0793 0.0490 0.0342 0.0275 0.0445 0.0345 0.0315 PQR-IVQR ˆ α Bias 0.2947 0.2981 0.2998 0.1430 0.1697 0.1820 0.0881 0.1021 0.1089 RMSE 0.2988 0.2998 0.3008 0.1635 0.1813 0.1891 0.1059 0.1122 0.1160 ˆ β Bias 0.0641 0.0654 0.0639 0.0558 0.0558 0.0548 0.0327 0.0336 0.0338 RMSE 0.1141 0.0934 0.0842 0.0739 0.0651 0.0618 0.0491 0.0423 0.0398
The location-scale shift model I PQR ˆ α Bias 0.2284 0.2260 0.2274 0.0491 0.0476 0.0477 0.0220 0.0218 0.0216 RMSE 0.2309 0.2272 0.2282 0.0521 0.0493 0.0487 0.0242 0.0228 0.0224 ˆ β Bias 0.1142 0.1185 0.1179 0.0281 0.0280 0.0272 0.0133 0.0126 0.0123 RMSE 0.1345 0.1286 0.1244 0.0440 0.0363 0.0329 0.0277 0.0213 0.0187 DPQR ˆ α Bias 0.2268 0.2274 0.2306 -0.1344 -0.1336 -0.1337 -0.1800 -0.1795 -0.1795 RMSE 0.2323 0.2303 0.2324 0.1383 0.1356 0.1351 0.1813 0.1801 0.1800 ˆ β Bias 0.0175 0.0202 0.0185 -0.0170 -0.0164 -0.0165 -0.0224 -0.0222 -0.0227 RMSE 0.0876 0.0631 0.0526 0.0431 0.0319 0.0279 0.0375 0.0305 0.0286 PQR-IVQR ˆ α Bias 0.3125 0.3142 0.3160 0.0945 0.1166 0.1371 0.0625 0.0715 0.0726 RMSE 0.3148 0.3151 0.3166 0.1127 0.1330 0.1505 0.0716 0.0766 0.0761 ˆ β Bias 0.0526 0.0558 0.0548 0.0335 0.0346 0.0344 0.0159 0.0156 0.0153 RMSE 0.0918 0.0775 0.0690 0.0483 0.0426 0.0396 0.0298 0.0237 0.0212
The location-scale shift model II PQR ˆ α Bias 0.2129 0.2131 0.2137 0.0737 0.0732 0.0737 0.0389 0.0391 0.0387 RMSE 0.2167 0.2151 0.2149 0.0806 0.0766 0.0761 0.0453 0.0426 0.0411 ˆ β Bias 0.1493 0.1469 0.1481 0.0589 0.0579 0.0583 0.0340 0.0326 0.0326 RMSE 0.1714 0.1588 0.1563 0.0752 0.0665 0.0643 0.0492 0.0417 0.0384 DPQR ˆ α Bias 0.1942 0.1946 0.1969 -0.1282 -0.1272 -0.1266 -0.1722 -0.1722 -0.1724 RMSE 0.2037 0.1992 0.1999 0.1352 0.1306 0.1290 0.1749 0.1735 0.1733 ˆ β Bias 0.0576 0.0578 0.0572 -0.0065 -0.0074 -0.0077 -0.0216 -0.0217 -0.0227 RMSE 0.1129 0.0885 0.0803 0.0488 0.0347 0.0291 0.0429 0.0343 0.0310 PQR-IVQR ˆ α Bias 0.2948 0.2996 0.2993 0.1421 0.1654 0.1803 0.0881 0.1021 0.1063 RMSE 0.2989 0.3012 0.3004 0.1618 0.1777 0.1878 0.1060 0.1120 0.1136 ˆ 17
Table 6: Bias and RMSE results for different tuning parameter values (τ = 0.5)
λ 0 0.5 1 1.5 2
The location shift model PQR ˆ α Bias -0.0755 0.0310 0.1234 0.1829 0.2173 RMSE 0.0860 0.0507 0.1293 0.1863 0.2198 ˆ β Bias -0.0098 0.0486 0.0943 0.1108 0.1078 RMSE 0.0625 0.0757 0.1113 0.1255 0.1243 DPQR ˆ α Bias -0.3640 -0.2088 -0.0481 0.1031 0.1986 RMSE 0.3666 0.2147 0.0781 0.1198 0.2049 ˆ β Bias -0.0652 -0.0227 0.0189 0.0443 0.0385 RMSE 0.0841 0.0647 0.0656 0.0782 0.0773 PQR-IVQR; FEQR ˆ α Bias -0.0719 0.0579 0.2127 0.2771 0.3003 RMSE 0.0932 0.0926 0.2260 0.2808 0.3024 ˆ β Bias -0.0172 0.0530 0.0687 0.0500 0.0307 RMSE 0.0646 0.0786 0.0925 0.0805 0.0723
The location-scale shift model I PQR ˆ α Bias -0.0234 0.0312 0.1003 0.1671 0.2110 RMSE 0.0298 0.0368 0.1048 0.1707 0.2135 ˆ β Bias -0.0046 0.0245 0.0568 0.0799 0.0851 RMSE 0.0424 0.0482 0.0712 0.0919 0.0983 DPQR ˆ α Bias -0.3442 -0.1982 -0.0478 0.1113 0.2138 RMSE 0.3467 0.2018 0.0706 0.1279 0.2194 ˆ β Bias -0.0661 -0.0347 -0.0018 0.0199 0.0161 RMSE 0.0818 0.0596 0.0498 0.0573 0.0597 PQR-IVQR; FEQR ˆ α Bias -0.0238 0.0540 0.1805 0.2787 0.3064 RMSE 0.0393 0.0717 0.2024 0.2841 0.3079 ˆ β Bias -0.0066 0.0312 0.0538 0.0420 0.0302 RMSE 0.0430 0.0529 0.0702 0.0642 0.0604
The location-scale shift model II PQR ˆ α Bias -0.0728 0.0291 0.1203 0.1820 0.2162 RMSE 0.0836 0.0490 0.1265 0.1855 0.2187 ˆ β Bias -0.0089 0.0491 0.0926 0.1085 0.1082 RMSE 0.0596 0.0746 0.1089 0.1235 0.1229 DPQR ˆ α Bias -0.3643 -0.2114 -0.0510 0.1001 0.1964 RMSE 0.3669 0.2172 0.0785 0.1178 0.2030 ˆ β Bias -0.0647 -0.0230 0.0190 0.0433 0.0380 RMSE 0.0832 0.0627 0.0635 0.0773 0.0747 PQR-IVQR; FEQR ˆ α Bias -0.0677 0.0558 0.2131 0.2770 0.3003 RMSE 0.0914 0.0909 0.2271 0.2807 0.3023 ˆ
國科會補助計畫衍生研發成果推廣資料表
日期:2013/10/31國科會補助計畫
計畫名稱: 動態追蹤資料分量迴歸 計畫主持人: 林馨怡 計畫編號: 101-2410-H-004-011- 學門領域: 數理與數量方法無研發成果推廣資料
101 年度專題研究計畫研究成果彙整表
計畫主持人：林馨怡 計畫編號： 101-2410-H-004-011-計畫名稱：動態追蹤資料分量迴歸 量化 成果項目 實際已達成 數（被接受 或已發表） 預期總達成 數(含實際已 達成數) 本計畫實 際貢獻百 分比 單位 備 註 （ 質 化 說 明：如 數 個 計 畫 共 同 成 果、成 果 列 為 該 期 刊 之 封 面 故 事 ... 等） 期刊論文 0 0 100% 研究報告/技術報告 0 0 100% 研討會論文 0 0 100% 篇 論文著作 專書 0 0 100% 申請中件數 0 0 100% 專利 已獲得件數 0 0 100% 件 件數 0 0 100% 件 技術移轉 權利金 0 0 100% 千元 碩士生 0 0 100% 博士生 0 0 100% 博士後研究員 0 0 100% 國內 參與計畫人力 （本國籍） 專任助理 0 0 100% 人次 期刊論文 0 0 100% 研究報告/技術報告 0 0 100% 研討會論文 0 0 100% 篇 論文著作 專書 0 0 100% 章/本 申請中件數 0 0 100% 專利 已獲得件數 0 0 100% 件 件數 0 0 100% 件 技術移轉 權利金 0 0 100% 千元 碩士生 0 0 100% 博士生 0 0 100% 博士後研究員 0 0 100% 國外 參與計畫人力 （外國籍） 專任助理 0 0 100% 人次其他成果
