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@ IJTSRD | Unique Paper ID – IJTSRD47716 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1759
The 2S-IVQR estimator can be obtained via the
following two stages.
Stage 1：For each individual and each quantile !,
estimates !th quantile regression of on and 6
using the data {( , , 6 ): = 1,2, ⋯ , ; =
1,2, ⋯ , }by the QR estimator:
( 7(!), 6
7(!)) = arg 5
;,<$
= = >-( − (!)
@
A
B
A
− C 6 (!) − ⋯ − C 6B(!)) (4)
where
6 (!)= (!) + (!), 6
7(!) = (6
7 (!), ⋯ , 6
7B(!))′, C =
F
1 GC H5I H 5J K JLM
0 C OC
P,>Q(L) = L(! − R(L < 0))
is the check function.
Stage 2: Estimate a 2SLS regression of 6
7 (!) on
using T as an instrument to get an estimator)(!)
of (!):
)(!)
= (U′VWU)X
(Z′VW6
7(!)), (5)
where U = ( , ⋯ , B) , [ = (T , ⋯ , TB)′, and
VW = [([′[)X
[′.
As we can see, for fixed !, the first stage only
needcalculate one time by introducing the dummy
variable. In addition, the second stage employs the
two-stage least squares method to get)]X 1234(!)
and)]X 1234(!)is a weighted combination of and
T .
3. Asymptotic theory
Now we briefly discuss the asymptotic properties of
the)]X 1234(!). To establish the asymptotic
properties of the )]X 1234(!), we impose the
following regularity assumptions.
Assumption 1: Observations are independent across
individuals, and for all = 1, ⋯ , , ( , ) are i.i.d.
across = 1, ⋯ , .
Assumption 2: The set of quantile indices _ is a
compact set included in (0,1).
Assumption3:(i) For all ! ∈ _ and = 1, ⋯ , ,
abcT (!)de = 0.(ii) As → ∞, X ∑ ai T j
B
A →
"kl and X ∑ aiT T j
B
A → "ll where "kl and
"ll are matrices with singular values bounded in
absolute value from above by mnand from below by
on. (iii) For all = 1, ⋯ , and = 1, ⋯ , , is
independent ofT conditional on( , ). (iv) For all
= 1, ⋯ , , ai||T ||pqrs j ≤ mn.
Assumption 4: As → ∞, /v
(log )/ → 0.
Assumption5: For all = 1, ⋯ , , satisfy the
moment conditions|| || ≤ mn, || || ≤ mn.
Assumption 6: For ! , ! ∈ _ and = 1, ⋯ , ,
∥ (!) − (! ) ∥≤ mz ∥ ! − ! ∥.
Assumption 7:(i) For all = 1, ⋯ , ,
a {OL|
Q∈}
~ (!)|pqrs• ≤ mn. (ii)As → ∞, for ! , ! ∈ _
X ∑ ai (!) (! )T T′ j
B
A → €(! , !). (iii) For
! , ! ∈ _, ∥ (!) − (! ) ∥≤ mz ∥ ! − ! ∥.
Assumption 8:(i) Let • (⋅) denote the conditional
density function of L = − − − given
( , ),for all ! ∈ _ and = 1, ⋯ , ,• (⋅) is
continuously differentiable, • (⋅) ≤ mƒ and • (0) ≥
oƒ. (ii) For = 1, ⋯ , ,the derivative • (⋅) satisfy
|• (L)| ≤ mƒ.
Theorem 1 Under Assumptions 1-8,
√ ()]X 1234(∙) - (∙)) → ‡(∙),
where ‡(∙) is a zero-mean Gaussian process with
uniformly continuous sample paths and covariance
function ˆ(! , !) = ‰€(! , !)‰′, €(! , !)is defined in
Assumption 7, ‰ = ("kl"ll
X
"′kl)X
"kl"ll
X
where
"kland "llare defined in Assumption 3.
Remark 1: Under Assumptions 1-8,
√ ()]X 1234(!) - (!)) → (0, Š),
where Š = ‰€(!, !)‰′.
4. Monte Carlo
The samples are generated from the following
model:
= ‹ + + Œ + K , = •T + Ž + L , Œ
= 5Ž − ,
whereK , Ž ∼ _(0,1), , T , L ∼ C |(0.25 ∗
(0,1)), ‹ = = 2, • = 5 = 1, = 0.5. Notice that
a(Œ ) = a(Œ |T ) = 5a(Ž ) − = 0.5 − 0.5 = 0. For
the sake of comparing the performance and
efficiency between different methods, we compare
the Bias and RMSE of the following estimators:
fixed effects quantile regression (QR) and grouped
IV quantile regression (Grouped IVQR) as in
Chetverikov, Larsen, and Palmer(2016).Here we
consider two different forms of generation of .
The first case is being endogenous (correlated
with Œ through Ž ). In the second case, is
exogenous, where we set = T .In the simulations,
we report results
considering
{( , )|(25,25), (25,50), (100,25), (100,50)},
andquantiles! = {0.1,0.25,0.5,0.75,0.9}. We set the
number of replications to 1000.](https://image.slidesharecdn.com/263atwo-stageestimatorofinstrumentalvariablequantileregressionforpaneldatawithtime-invarianteffects-220308113111/85/A-Two-Stage-Estimator-of-Instrumental-Variable-Quantile-Regression-for-Panel-Data-with-Time-Invariant-Effects-3-320.jpg)
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@ IJTSRD | Unique Paper ID – IJTSRD47716 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1761
= 100, ˜ = 50
QR
Bias -0.0032 0.0079 -0.0039 0.0036 -0.0123 0.0062
RMSE 0.0087 0.0089 0.0086 0.0094 0.0094 0.0090
Grouped IVQR
Bias -0.0127 0.0042 -0.0053 0.0006 0.0085 0.0062
RMSE 0.0081 0.0089 0.0100 0.0097 0.0089 0.0091
2S-IVQR
Bias -0.0080 0.0052 -0.0025 0.0068 0.0029 0.0051
RMSE 0.0078 0.0076 0.0084 0.0083 0.0082 0.0081
The Bias and RMSE of the estimators when isendogenous are shown in Table 2. The results are similar to
those of Table 1. The results of Grouped IVQR and 2S-IVQR estimators are better than those of QR
estimators and 2S-IVQR estimator performs better than Grouped IVQR in terms of Avg.abs.bias and Avg.
RMSE. In addition, the Bias and RMSE decreases as and increases for 2S-IVQR estimators.
5. Conclusion
We explore a two-stage approach to instrumental
variable quantile regression for panel data with
time-invariant effects. In the first stage, the dummy
variables are introduced to represent the time-
invariant effects. By employing the dummy
variables, the number of estimation parameters can
be reduced. Then in the second stage, we apply
instrument variables approach and 2SLS method.
The proposed estimatoris a weighted combination
of and T .Moreover, the asymptotic properties of
the proposed estimators are studied in Section 3.
Monte Carlo simulation in various parameters sets
proves the validity of the proposed approach.
Monte Carlo simulation presents that with
increasing sample size, the Bias and RMSE of our
estimator are decreased. Besides, our estimator has
lower Bias and RMSE than those of the QR
estimator and the Grouped IVQR estimator.
6. Reference
[1] Koenker, R., 2004. Quantile regression for
longitudinal data. Journal of Multivariate
Analysis, 91(1):74-89.
[2] Lamarche, C., 2010. Robust penalized
quantile regression estimation for panel data.
Journal of Econometrics, 157(2):396-408.
[3] Galvao, A. F., Montes-Rojas, G., 2010.
Penalized quantile regression for dynamic
panel Data. Journal of Statistical Planning and
Inference, 140(11):3476-3497.
[4] Galvao, A. F., 2011. Quantile regression for
dynamic panel data with fixed effects. Journal
of Econometrics, 164(1):142-157.
[5] Harding, M., Lamarche, C., 2014. Hausman-
Taylor instrumental variable approach to the
penalized estimation of quantile panel
models. Economics Letters, 124:176-179.
[6] Lamarche, C., 2014. Penalized quantile
regression estimation for a model with
endogenous fixed effects.
[7] Tao, L., Zhang, Y. J., Tian, M. Z., 2019.
Quantile regression for dynamic panel data
using Hausman-Taylor instrumental
variables. Computational Economics,
53(3):1033-1069.
[8] Hausman, J.A., Taylor, W.E., 1981. Panel
data and unobservable individual effects.
Econometrica, 49(6):1377-1398.
[9] Canay, I. A., 2011. A simple approach to
quantile regression for panel data.
Econometrics Journal, 14(3):368-386.
[10] Galvao, A. F., Wang, L., 2015. Efficient
minimum distance estimator for quantile
regression fixed effects panel data. Journal of
Multivariate Analysis, 133:1-26.
[11] Galvao, A. F., Gu, J., Volgushev, S., 2020.On
the unbiased asymptotic normality of quantile
regression with fixed effects[J]. Journal of
Econometrics, 218(1):178-215.
[12] Chetverikov, D., Larsen, B., Palmer, C.,
2016. IV quantile regression for group-level
treatments, with an application to the
distributional effects of trade. Econometrica,
84(2):809-833.
[13] Gu, J., Volgushev, S., 2019. Panel data
quantile regression with grouped fixed
effects. Journal of Econometrics, 213(1):68-
91.
[14] Zhang, Y., Jiang, J., Feng, Y., 2021.
Penalized quantile regression for spatial panel
data with fixed effects. Communication in
Statistics-Theory and Methods,1-13.
[15] Koenker, R., Bassett, G., 1978. Regression
Quantile. Econometrica, 46(1):33--50.](https://image.slidesharecdn.com/263atwo-stageestimatorofinstrumentalvariablequantileregressionforpaneldatawithtime-invarianteffects-220308113111/85/A-Two-Stage-Estimator-of-Instrumental-Variable-Quantile-Regression-for-Panel-Data-with-Time-Invariant-Effects-5-320.jpg)
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@ IJTSRD | Unique Paper ID – IJTSRD47716 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1762
7. Appendix
Proof of Theorem 1Denote™(!) = (U′VWU)X
(Z′VW6(!)), "
škl =
› W
B
, "
šll =
W W
B
，then
√ ()]X 1234(!) - ™(!)) = ("
škl"
šll
X
"
š′kl)X
"
škl"
šll
X
([′(6
7(!) − 6(!))/√ ).(6)
By Assumptions 3(iv) and 5 and Chebyshev’s inequality, we have
B
∑ ( T − ai T j)
B
A
.
→0.
Observe that X ∑ ai T j
B
A → "klby Assumption 3, it suffices to prove
"
škl = X ∑ T
B
A
.
→ "kl.
Similarly, it can be obtained
"
šll = X
= T T
B
A
.
→ "ll
Thus,
‰
7 = ("
škl"
šll
X
"
š′kl)X
"
škl"
šll
X
.
→ ("kl"ll
X
" kl)X
"kl"ll
X
= ‰ (7)
According to Chetverikov, Larsen, and Palmer(2016),
‰(!) =
1
√
= •6
7 (!) − 6 (!)ž
B
A
T = H.(1)(8)
uniformly over ! ∈ _.
Observe that(7) and(8)are satisfied, by (6)we have
√ ()]X 1234(!) - ™(!)) = H.(1).(9)
Next, we show √ (™(∙) − (∙)) → ‡(∙). Note that
√ (™(∙) − (∙)) = ‰
7 ∙
√B
∑ T
B
A (!).
and ‰
7
.
→ ‰by (7). In addition, Lemma 3 of Chetverikov, Larsen, and Palmer(2016) implies that
1
√
= T
B
A
(!) → ‡ (∙)
where ‡ (∙) is a zero-mean Gaussian process with uniformly continuous sample paths and covariance
function €(! , !), €(! , !)is defined in Assumption 7. Therefore, by Slutsky’s theorem,
√ (™(∙) − (∙)) → ‡(∙),
where ‡(∙) is a zero-mean Gaussian process with uniformly continuous sample paths and covariance
function ˆ(! , !) = ‰€(! , !)‰′.Combining (9) gives the asserted claim and completes the proof of
Theorem 1.](https://image.slidesharecdn.com/263atwo-stageestimatorofinstrumentalvariablequantileregressionforpaneldatawithtime-invarianteffects-220308113111/85/A-Two-Stage-Estimator-of-Instrumental-Variable-Quantile-Regression-for-Panel-Data-with-Time-Invariant-Effects-6-320.jpg)
A Two Stage Estimator of Instrumental Variable Quantile Regression for Panel Data with Time Invariant Effects
- 1. International Journal of Trend Volume 5 Issue 6, September-October @ IJTSRD | Unique Paper ID – IJTSRD A Two-Stage Estimator of Regression for Panel Data with Time School of Information, Beijing Wuzi ABSTRACT This paper proposes a two-stage instrumental variable quantile regression (2S-IVQR) estimation to estimate the time effects in panel data model. In the first stage, we introduce the dummy variables to represent the time quantile regression to estimate effects of individual covariates. The advantage of the first stage is that it can reduce calculations and the number of estimation parameters. Then in the second stage, we adapt instrument variables approach and 2SLS method. In addition, we present a proof of 2S-IVQR estimator's large sample properties. Monte Carlo simulation study shows that with increasing sample size, the Bias and RMSE of our estimator are decreased. Besides, our estimator has lower Bias and RMSE than those of the other two estimators. KEYWORDS: Time-invariant effects; Panel data; regression; Instrument variables 1. INTRODUCTION Panel data not only reflects the individual heterogeneity of cross-section data, but also shows the dynamic information about the time series data. Quantile regression can describe the independent variable for the dependent variable accurately, and capture systematic influences of covariates on the location, scale and shape of the conditional distribution of the response. Additionally, there is no need for quantile regression model to make any assumptions on the overall distribution, compared with the ordinary least squares method. Quantile regression model for panel data, can fully describe the conditional distribution of the response variable as well as control the variability. There are three types of estimation of quantile regression for panel data with fixed effects: the penalty estimation, two-step estimation, minimum distance estimation. The penalty estimation mainly considers adding a penalty item in the objective function. Koenker(2004) proposed a general approach to Trend in Scientific Research and Development October 2021 Available Online: www.ijtsrd.com IJTSRD47716 | Volume – 5 | Issue – 6 | Sep-Oct Stage Estimator of Instrumental Variable Quantile Regression for Panel Data with Time-Invariant Effects Tao Li School of Information, Beijing Wuzi University, Beijing, China stage instrumental variable quantile estimation to estimate the time-invariant effects in panel data model. In the first stage, we introduce the dummy variables to represent the time-invariant effects, and use quantile regression to estimate effects of individual covariates. The the first stage is that it can reduce calculations and estimation parameters. Then in the second stage, we adapt instrument variables approach and 2SLS method. In IVQR estimator's large sample nte Carlo simulation study shows that with increasing sample size, the Bias and RMSE of our estimator are Besides, our estimator has lower Bias and RMSE than invariant effects; Panel data; Quantile How to cite Stage Estimator of Instrumental Variable Quantile Regression for Panel Data with Time-Invariant Effects" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456 Volume-5 | Issue October 2021, pp.1757-1762, URL: www.ijtsrd.com/papers/ijtsrd47716.pdf Copyright © International Scientific Research Journal. This distributed under terms of the Commons Attribution License (http://creativecommons.org/licenses/by/4.0 Panel data not only reflects the individual section data, but also shows the dynamic information about the time series data. Quantile regression can describe the independent for the dependent variable range accurately, and capture systematic influences of ion, scale and shape of the conditional distribution of the response. Additionally, there is no need for quantile regression model to make any assumptions on the overall distribution, compared with the ordinary el for panel data, can fully describe the conditional distribution of the response variable as well as control the variability. There are three types of estimation of quantile regression for panel data with fixed effects: the penalty stimation, minimum distance The penalty estimation mainly considers adding a penalty item in the objective function. Koenker(2004) proposed a general approach to estimate quantile regression models for longitudinal data employing Lamarche(2010) proved, in theory, that there existing optimal penalty parameter of penalized quantile regression for panel data with fixed effect. Galvao and Montes penalized quantile regression for dynamic panel data, using instrumental variables to resolve the problem of endogeneity. Galvao(2011) also adopted instrumental variables approach to study a quantile regression dynamic panel model with fixed effects. Harding and Lamarche(2014) gave an quantile regression estimator which adapted the Hausman–Taylor instrumental variable approach. Lamarche(2014) proposed a penalized quantile regression estimator for panel data that explicitly considers individual heterogeneity associated with the covariates. Tao, Zha proposed two new panel data instrumental variable estimators that combine the Huasman instrumental variables of Huasman and Taylor(1981) and shrinkage of Koenker(2004) to resolve biased parameter estimation problem Development (IJTSRD) www.ijtsrd.com e-ISSN: 2456 – 6470 Oct 2021 Page 1757 Instrumental Variable Quantile Invariant Effects University, Beijing, China this paper: Tao Li "A Two- Stage Estimator of Instrumental Variable Quantile Regression for Panel Data with Invariant Effects" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, 5 | Issue-6, October 2021, 1762, URL: www.ijtsrd.com/papers/ijtsrd47716.pdf © 2021 by author (s) and International Journal of Trend in Research and Development This is an Open Access article under the the Creative Attribution (CC BY 4.0) //creativecommons.org/licenses/by/4.0) estimate quantile regression models for longitudinal regularization methods. proved, in theory, that there existing optimal penalty parameter of penalized quantile regression for panel data with fixed effect. Galvao and Montes-Rojas(2010) proposed penalized quantile regression for dynamic panel using instrumental variables to resolve the Galvao(2011) also adopted instrumental variables approach to study a quantile regression dynamic panel model with fixed effects. Harding and Lamarche(2014) gave an ℓ1 penalized gression estimator which adapted the Taylor instrumental variable approach. Lamarche(2014) proposed a penalized quantile regression estimator for panel data that explicitly considers individual heterogeneity associated with the covariates. Tao, Zhang and Tian(2017) proposed two new panel data instrumental variable estimators that combine the Huasman-Taylor instrumental variables of Huasman and shrinkage of Koenker(2004) to resolve biased parameter estimation problem IJTSRD47716
- 2. International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47716 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1758 caused by lagged response variables and random error. The main idea of the two-step estimation is to eliminate the fixed effect in the first step, use simple quantile regression for transformed data in the second step. Canay(2011) introduced a two-step estimator for panel data quantile regression models. It is noted that the two-step estimation method eliminates the fixed effect in the first step, which can greatly reduce the estimated parameters in quantile regression. Notethatdue to the first step, the two-step estimation cannot adapt the time- invariant independent variables in the panel data model. Galvao and Wang (2015) developed a new minimum distance quantile regression (MD-QR) estimator for panel data models with fixed effects. The MD-QR is defined as the weighted average of the individual quantile regression slope estimators. However, from the definition of MD-QR we can know, the model cannot adapt the time-invariant independent variables. Besides, MD-QR estimator ignores the endogenous problems. Galvao, Gu and Volgushev(2018) provided new insights on the asymptotic properties of the MD-QR estimator under the two different assumption, the assumption that data within individuals are independent and the assumption that data are dependence across time while maintain independence across individuals. There have been other growing studies on quantile regression for panel data, see e.g., Galvao(2011), Chetverikov, Larsen, and Palmer(2016), Gu and Volgushev(2019), Zhang, Jiang and Feng(2021). As the two-step estimation and minimum distance estimation fail when there are time-invariant independent variables in the model, we propose a two-stage instrumental variable quantile regression(2S-IVQR) estimation to estimate the time-invariant effects in panel data model. In the first stage, we introduce the dummy variables and perform quantile regression with all the data to estimate effects of individual covariates. Then in the second stage, we adapt instrument variables approach and 2SLS method. Moreover, we study the asymptotic properties of the proposed estimators. Monte Carlo simulation in various parameters sets proves the validity of the proposed approaches. We compare the bias and root mean squared error (RMSE) of the proposed estimators with the QR estimator's of Koenker and Bassett (1978) and the Grouped IVQR estimator's of Chetverikov, Larsen, and Palmer (2016). The rest of the paper is organized as follows. Section 2 presents two-stage instrumental variable quantile regression (2S-IVQR) estimator for panel data with time-invariant effects. Section 3 is devoted to the asymptotic behavior of the proposed estimators. Section 4 describes the Monte Carlo experiment. Finally, section 5 concludes the paper. 2. Model and methods 2.1. Basic Model Consider the panel data model that contains time- varying as well as time-invariant regressors: = + + + , = 1,2, ⋯ , ; = 1,2, ⋯ , (1) where = + , (2) is a × 1 vector of time-varying variables, and is an × 1 vector of observed individual-specific variables that only vary over the cross-section units . In addition to , the outcomes, , is also governed by unobserved individual specific effects, . We consider the following model for the !th conditional quantile functions of the response of the th observation on the th individual , "#$% (!| , , ) = (!) + (!) + (!), (3) There has been a lot of research on how to estimate (!), see Koenker(2004), Canay(2011), Galvao and Wang (2015), etc. Besides, the estimation method proposed by Canay(2011), Galvao and Wang (2015) can not identify (!). Therefore, we are primarily interested in estimation (!) inthispaper,that is,the focus of the following analysis is on estimation and inference involving the elements of (!). 2.2. 2S-IVQR estimator Chetverikov, Larsen, and Palmer(2016) are interested in estimating (!), they propose the Grouped IVQR estimator. The Grouped IVQR estimator consists of two stages. First, estimate !th quantile regression of on using the data {( , ): = 1,2, … , }by the classical quantile regression(QR) estimator of Koenker and Bassett (1978). Second, estimate a 2SLS regression of intercept term coefficient on using instrument variablesto get an estimator )*+,-./0 1234(!) of (!). However, the Grouped IVQR estimator calculates each individual data separately, that is, it needs to calculate 5 times in the first stage. To simplify computation and improve the estimation accuracy, we propose a two-stage instrumental variable quantile regression estimator(2S-IVQR) for model (3).
- 3. International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47716 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1759 The 2S-IVQR estimator can be obtained via the following two stages. Stage 1：For each individual and each quantile !, estimates !th quantile regression of on and 6 using the data {( , , 6 ): = 1,2, ⋯ , ; = 1,2, ⋯ , }by the QR estimator: ( 7(!), 6 7(!)) = arg 5 ;,<$ = = >-( − (!) @ A B A − C 6 (!) − ⋯ − C 6B(!)) (4) where 6 (!)= (!) + (!), 6 7(!) = (6 7 (!), ⋯ , 6 7B(!))′, C = F 1 GC H5I H 5J K JLM 0 C OC P,>Q(L) = L(! − R(L < 0)) is the check function. Stage 2: Estimate a 2SLS regression of 6 7 (!) on using T as an instrument to get an estimator)(!) of (!): )(!) = (U′VWU)X (Z′VW6 7(!)), (5) where U = ( , ⋯ , B) , [ = (T , ⋯ , TB)′, and VW = [([′[)X [′. As we can see, for fixed !, the first stage only needcalculate one time by introducing the dummy variable. In addition, the second stage employs the two-stage least squares method to get)]X 1234(!) and)]X 1234(!)is a weighted combination of and T . 3. Asymptotic theory Now we briefly discuss the asymptotic properties of the)]X 1234(!). To establish the asymptotic properties of the )]X 1234(!), we impose the following regularity assumptions. Assumption 1: Observations are independent across individuals, and for all = 1, ⋯ , , ( , ) are i.i.d. across = 1, ⋯ , . Assumption 2: The set of quantile indices _ is a compact set included in (0,1). Assumption3:(i) For all ! ∈ _ and = 1, ⋯ , , abcT (!)de = 0.(ii) As → ∞, X ∑ ai T j B A → "kl and X ∑ aiT T j B A → "ll where "kl and "ll are matrices with singular values bounded in absolute value from above by mnand from below by on. (iii) For all = 1, ⋯ , and = 1, ⋯ , , is independent ofT conditional on( , ). (iv) For all = 1, ⋯ , , ai||T ||pqrs j ≤ mn. Assumption 4: As → ∞, /v (log )/ → 0. Assumption5: For all = 1, ⋯ , , satisfy the moment conditions|| || ≤ mn, || || ≤ mn. Assumption 6: For ! , ! ∈ _ and = 1, ⋯ , , ∥ (!) − (! ) ∥≤ mz ∥ ! − ! ∥. Assumption 7:(i) For all = 1, ⋯ , , a {OL| Q∈} ~ (!)|pqrs• ≤ mn. (ii)As → ∞, for ! , ! ∈ _ X ∑ ai (!) (! )T T′ j B A → €(! , !). (iii) For ! , ! ∈ _, ∥ (!) − (! ) ∥≤ mz ∥ ! − ! ∥. Assumption 8:(i) Let • (⋅) denote the conditional density function of L = − − − given ( , ),for all ! ∈ _ and = 1, ⋯ , ,• (⋅) is continuously differentiable, • (⋅) ≤ mƒ and • (0) ≥ oƒ. (ii) For = 1, ⋯ , ,the derivative • (⋅) satisfy |• (L)| ≤ mƒ. Theorem 1 Under Assumptions 1-8, √ ()]X 1234(∙) - (∙)) → ‡(∙), where ‡(∙) is a zero-mean Gaussian process with uniformly continuous sample paths and covariance function ˆ(! , !) = ‰€(! , !)‰′, €(! , !)is defined in Assumption 7, ‰ = ("kl"ll X "′kl)X "kl"ll X where "kland "llare defined in Assumption 3. Remark 1: Under Assumptions 1-8, √ ()]X 1234(!) - (!)) → (0, Š), where Š = ‰€(!, !)‰′. 4. Monte Carlo The samples are generated from the following model: = ‹ + + Œ + K , = •T + Ž + L , Œ = 5Ž − , whereK , Ž ∼ _(0,1), , T , L ∼ C |(0.25 ∗ (0,1)), ‹ = = 2, • = 5 = 1, = 0.5. Notice that a(Œ ) = a(Œ |T ) = 5a(Ž ) − = 0.5 − 0.5 = 0. For the sake of comparing the performance and efficiency between different methods, we compare the Bias and RMSE of the following estimators: fixed effects quantile regression (QR) and grouped IV quantile regression (Grouped IVQR) as in Chetverikov, Larsen, and Palmer(2016).Here we consider two different forms of generation of . The first case is being endogenous (correlated with Œ through Ž ). In the second case, is exogenous, where we set = T .In the simulations, we report results considering {( , )|(25,25), (25,50), (100,25), (100,50)}, andquantiles! = {0.1,0.25,0.5,0.75,0.9}. We set the number of replications to 1000.
- 4. International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47716 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1760 Table 1Bias and RMSE of estimators when •– is endogenous — 0.1 0.25 0.5 0.75 0.9 Avg.abs. = 25, = 25 QR Bias 0.3607 0.3909 0.3754 0.3929 0.4031 0.3846 RMSE 0.0241 0.0256 0.0247 0.0258 0.0267 0.0254 Grouped IVQR Bias -0.0636 -0.1002 -0.1022 -0.1254 -0.0464 0.0876 RMSE 0.0208 0.0399 0.0390 0.0557 0.0389 0.0388 2S-IVQR Bias -0.0662 -0.0720 -0.0711 -0.1186 0.0032 0.0662 RMSE 0.0208 0.0256 0.0254 0.0599 0.0668 0.0397 = 25, = 50 QR Bias 0.3570 0.3796 0.3864 0.4090 0.4172 0.3899 RMSE 0.0149 0.0158 0.0161 0.0169 0.0173 0.0162 Grouped IVQR Bias -0.0243 -0.0468 -0.0239 -0.0337 -0.0438 0.0345 RMSE 0.0111 0.0126 0.0130 0.0115 0.0111 0.0118 2S-IVQR Bias -0.0265 -0.0526 -0.0348 -0.0282 -0.0314 0.0347 RMSE 0.0078 0.0091 0.0090 0.0082 0.0080 0.0084 = 100, = 25 QR Bias 0.3686 0.3772 0.3787 0.4033 0.4157 0.3887 RMSE 0.0244 0.0247 0.0247 0.0262 0.0272 0.0254 Grouped IVQR Bias -0.0670 -0.0236 -0.0762 -0.0371 -0.0772 0.0562 RMSE 0.0307 0.0504 0.0246 0.0711 0.0372 0.0428 2S-IVQR Bias -0.0544 -0.0316 -0.0632 -0.0321 -0.0642 0.0491 RMSE 0.0268 0.0444 0.0199 0.0645 0.0333 0.0378 = 100, = 50 QR Bias 0.3548 0.3838 0.3832 0.4029 0.4138 0.3877 RMSE 0.0182 0.0195 0.0194 0.0204 0.0211 0.0197 Grouped IVQR Bias -0.0183 -0.0451 -0.0516 -0.0166 -0.0192 0.0301 RMSE 0.0105 0.0181 0.0149 0.0102 0.0100 0.0127 2S-IVQR Bias -0.0166 -0.0430 -0.0537 -0.0120 -0.0244 0.0299 RMSE 0.0101 0.0179 0.0141 0.0090 0.0093 0.0121 Table 1 provides the Bias and RMSE of the estimators in the case is endogenous. It is clear that the Bias and RMSE of Grouped IVQR and 2S-IVQR estimators are better than QR estimators. Moreover, the Bias and RMSE of 2S-IVQR estimator are smaller than those of Grouped IVQR estimator at most quantiles. Meanwhile, we also find that the Bias and RMSE decreases as and increases for 2S- IVQR estimators. Table 2 Bias and RMSE of estimators when •– is exogenous — 0.1 0.25 0.5 0.75 0.9 Avg.abs. = 25, ˜ = 25 QR Bias 0.0254 0.0292 0.0008 0.0131 -0.0188 0.0174 RMSE 0.0139 0.0145 0.0127 0.0139 0.0136 0.0137 Grouped IVQR Bias -0.0085 0.0083 0.0274 0.0145 0.0012 0.0120 RMSE 0.0167 0.0194 0.0212 0.0192 0.0165 0.0186 2S-IVQR Bias 0.0036 0.0069 -0.0047 0.0219 0.0027 0.0080 RMSE 0.0118 0.0136 0.0123 0.0124 0.0121 0.0124 = 25, ˜ = 50 QR Bias 0.0135 -0.0023 0.0027 -0.0078 -0.0085 0.0070 RMSE 0.0098 0.0091 0.0096 0.0101 0.0096 0.0975 Grouped IVQR Bias 0.0037 0.0052 0.0033 0.0003 -0.0072 0.0039 RMSE 0.0119 0.0137 0.0142 0.0120 0.0119 0.0127 2S-IVQR Bias 0.0069 -0.0040 0.0037 -0.0031 -0.0029 0.0041 RMSE 0.0078 0.0086 0.0089 0.0083 0.0086 0.0084 = 100, ˜ = 25 QR Bias 0.0308 -0.0140 -0.0035 -0.0161 0.0108 0.0150 RMSE 0.0141 0.0129 0.0128 0.0128 0.0137 0.0133 Grouped IVQR Bias 0.0139 -0.0286 0.0073 -0.0123 0.0237 0.0172 RMSE 0.0130 0.0137 0.0145 0.0135 0.0137 0.0137 2S-IVQR Bias 0.0049 -0.0166 -0.0013 -0.0052 0.0179 0.0092 RMSE 0.0121 0.0119 0.0127 0.0117 0.0125 0.0122
- 5. International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47716 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1761 = 100, ˜ = 50 QR Bias -0.0032 0.0079 -0.0039 0.0036 -0.0123 0.0062 RMSE 0.0087 0.0089 0.0086 0.0094 0.0094 0.0090 Grouped IVQR Bias -0.0127 0.0042 -0.0053 0.0006 0.0085 0.0062 RMSE 0.0081 0.0089 0.0100 0.0097 0.0089 0.0091 2S-IVQR Bias -0.0080 0.0052 -0.0025 0.0068 0.0029 0.0051 RMSE 0.0078 0.0076 0.0084 0.0083 0.0082 0.0081 The Bias and RMSE of the estimators when isendogenous are shown in Table 2. The results are similar to those of Table 1. The results of Grouped IVQR and 2S-IVQR estimators are better than those of QR estimators and 2S-IVQR estimator performs better than Grouped IVQR in terms of Avg.abs.bias and Avg. RMSE. In addition, the Bias and RMSE decreases as and increases for 2S-IVQR estimators. 5. Conclusion We explore a two-stage approach to instrumental variable quantile regression for panel data with time-invariant effects. In the first stage, the dummy variables are introduced to represent the time- invariant effects. By employing the dummy variables, the number of estimation parameters can be reduced. Then in the second stage, we apply instrument variables approach and 2SLS method. The proposed estimatoris a weighted combination of and T .Moreover, the asymptotic properties of the proposed estimators are studied in Section 3. Monte Carlo simulation in various parameters sets proves the validity of the proposed approach. Monte Carlo simulation presents that with increasing sample size, the Bias and RMSE of our estimator are decreased. Besides, our estimator has lower Bias and RMSE than those of the QR estimator and the Grouped IVQR estimator. 6. Reference [1] Koenker, R., 2004. Quantile regression for longitudinal data. Journal of Multivariate Analysis, 91(1):74-89. [2] Lamarche, C., 2010. Robust penalized quantile regression estimation for panel data. Journal of Econometrics, 157(2):396-408. [3] Galvao, A. F., Montes-Rojas, G., 2010. Penalized quantile regression for dynamic panel Data. Journal of Statistical Planning and Inference, 140(11):3476-3497. [4] Galvao, A. F., 2011. Quantile regression for dynamic panel data with fixed effects. Journal of Econometrics, 164(1):142-157. [5] Harding, M., Lamarche, C., 2014. Hausman- Taylor instrumental variable approach to the penalized estimation of quantile panel models. Economics Letters, 124:176-179. [6] Lamarche, C., 2014. Penalized quantile regression estimation for a model with endogenous fixed effects. [7] Tao, L., Zhang, Y. J., Tian, M. Z., 2019. Quantile regression for dynamic panel data using Hausman-Taylor instrumental variables. Computational Economics, 53(3):1033-1069. [8] Hausman, J.A., Taylor, W.E., 1981. Panel data and unobservable individual effects. Econometrica, 49(6):1377-1398. [9] Canay, I. A., 2011. A simple approach to quantile regression for panel data. Econometrics Journal, 14(3):368-386. [10] Galvao, A. F., Wang, L., 2015. Efficient minimum distance estimator for quantile regression fixed effects panel data. Journal of Multivariate Analysis, 133:1-26. [11] Galvao, A. F., Gu, J., Volgushev, S., 2020.On the unbiased asymptotic normality of quantile regression with fixed effects[J]. Journal of Econometrics, 218(1):178-215. [12] Chetverikov, D., Larsen, B., Palmer, C., 2016. IV quantile regression for group-level treatments, with an application to the distributional effects of trade. Econometrica, 84(2):809-833. [13] Gu, J., Volgushev, S., 2019. Panel data quantile regression with grouped fixed effects. Journal of Econometrics, 213(1):68- 91. [14] Zhang, Y., Jiang, J., Feng, Y., 2021. Penalized quantile regression for spatial panel data with fixed effects. Communication in Statistics-Theory and Methods,1-13. [15] Koenker, R., Bassett, G., 1978. Regression Quantile. Econometrica, 46(1):33--50.
- 6. International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47716 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1762 7. Appendix Proof of Theorem 1Denote™(!) = (U′VWU)X (Z′VW6(!)), " škl = › W B , " šll = W W B ，then √ ()]X 1234(!) - ™(!)) = (" škl" šll X " š′kl)X " škl" šll X ([′(6 7(!) − 6(!))/√ ).(6) By Assumptions 3(iv) and 5 and Chebyshev’s inequality, we have B ∑ ( T − ai T j) B A . →0. Observe that X ∑ ai T j B A → "klby Assumption 3, it suffices to prove " škl = X ∑ T B A . → "kl. Similarly, it can be obtained " šll = X = T T B A . → "ll Thus, ‰ 7 = (" škl" šll X " š′kl)X " škl" šll X . → ("kl"ll X " kl)X "kl"ll X = ‰ (7) According to Chetverikov, Larsen, and Palmer(2016), ‰(!) = 1 √ = •6 7 (!) − 6 (!)ž B A T = H.(1)(8) uniformly over ! ∈ _. Observe that(7) and(8)are satisfied, by (6)we have √ ()]X 1234(!) - ™(!)) = H.(1).(9) Next, we show √ (™(∙) − (∙)) → ‡(∙). Note that √ (™(∙) − (∙)) = ‰ 7 ∙ √B ∑ T B A (!). and ‰ 7 . → ‰by (7). In addition, Lemma 3 of Chetverikov, Larsen, and Palmer(2016) implies that 1 √ = T B A (!) → ‡ (∙) where ‡ (∙) is a zero-mean Gaussian process with uniformly continuous sample paths and covariance function €(! , !), €(! , !)is defined in Assumption 7. Therefore, by Slutsky’s theorem, √ (™(∙) − (∙)) → ‡(∙), where ‡(∙) is a zero-mean Gaussian process with uniformly continuous sample paths and covariance function ˆ(! , !) = ‰€(! , !)‰′.Combining (9) gives the asserted claim and completes the proof of Theorem 1.