# Quantile Regression for General Spatial Panel Data Models with Fixed Effects

## Abstract

This paper considers the quantile regression model with both individual fixed effect and
time period effect for general spatial panel data. Instrumental variable quantile regression estimators will be proposed. Asymptotic properties of the proposed estimators will be developed. Simulations are conducted to study the performance of the proposed method. We will illustrate our methodologies using a cigarettes demand data set.

Keywords: Fixed effects, Instrumental variables, Quantile regression, Space-time panel models, Spatial autoregressive.

## 1 Introduction

Spatial econometric models have been widely used in many areas (e.g., economics, political science and public health) to deal with spatial interaction effects among geographical units (e.g., jurisdictions, regions, and states). Recently, the spatial econometrics literature has exhibited a growing interest in the specification and estimation of econometric relationships based on spatial panels, which typically refer to data containing time series observations of a number of spatial units. For instance, Kapoor et al. (2007) developed a generalized moments (GM) estimator for a space-time model with error components that are both spatially and time-wise correlated. Lee and Yu (2010) proposed the maximum likelihood (ML) estimator for the spatial autoregressive (SAR) panel model with both spatial lag and spatial disturbances. All these works were developed based on (conditional) mean regression methods. Compared with mean regression methods, the quantile regression (QR) method is more robust and can be adopted to deal with data characterized by different error distributions.

Recently, there has been a growing literature on estimating and testing of QR panel data models. Koenker (2004) introduced a novel approach for the estimation of a QR model for longitudinal data. Galvao et al. (2011) studied the quantile regression dynamic panel model with fixed effects. Galvao et al. (2013) investigated the estimation of censored QR models with fixed effects. Galvao et al. (2015) developed a new minimum distance quantile regression (MD-QR) estimator for panel data models with fixed effects. However, quantile regression estimation for spatial econometric panel models has not been studied in existing literature.

This paper focuses on the QR estimations in the general SAR panel data model with both individual fixed effects and time period specific effects (see, e.g., Lee and Yu, 2010). We employ the instrumental variable quantile regression (IVQR) method to estimate the parameters. The asymptotic properties of the IVQR estimator are also developed. The rest of the paper is organized as follows. Section 2 introduces the SAC panel data model with both individual fixed effects and time period fixed effects, and proposes the instrumental variable quantile regression (IVQR) estimation procedure. The asymptotic properties of the IVQR estimators are also discussed. Proofs of the theorems in Sections 2 are given in the Appendix. Section 3 reports a simulation study for assessing the finite sample performance of the proposed estimators. An empirical illustration is considered in Section 4. Section 5 concludes the paper.

## 2 General spatial autoregressive panel data quantile regression model with both individual and time effects

Lee and Yu (2010, Eq. (19)) considered the following general spatial autoregressive panel data model with both individual and time effects

(2.1) |

where is the dependent variable for subject at time , is a vector of nonstochastic time varying explanatory variables, is the th element of the spatial weight matrix reflecting spatial dependence on among cross sectional units, and is independent and identically distributed across and . Similarly, is the th element of the spatial weight matrix for the disturbances. The parameters are fixed effects for the regions while the parameters are fixed time effects. Interaction effects are reflected in the spatial-temporal lag variable (and associated scalar parameter ). In practice, may or may not be .

The model (2.1) can also be written in an alternative form as

(2.2) |

where , , , , , , is an matrix, is an matrix, is the vector with all the elements being 1, , , , , is an indicator variable for the individual effect , is an vector with the th element equal to 1 and the rest equal to 0, , is an indicator variable for the time effect , and is an vector with the th element equal to 1 and the rest equal to 0, .

Matrix form of model (2.2) is

(2.3) |

where is an vector with , is an vector with , is an matrix, , , , and are both spatial weight matrices, , , , and . Here we denote .

We consider the following conditional -quantile of response variable:

(2.4) |

where is a quantile in the interval . We define the objection function by

(2.5) |

where is the check function and is the indicator function (see, e.g., Koenker, 2005). The estimator can then be obtained by

(2.6) |

### 2.1 Instrumental Variable Quantile Regression Estimator (IVQR)

In this section, we employ the instrumental variable quantile regression (IVQR) method for estimation. Let denote a scalar endogenous variable, which is related to a vector of instruments . The instruments are independent of . Consider the objection function for the conditional instrumental quantile relationship:

(2.7) |

Following Chernozhukov and Hansen (2006, 2008) and Galvao (2011), and assuming the availability of instrumental variables , we can derive the IVQR estimator via the following three steps:

Step 1: For a given quantile , define a suitable set of values . One then minimizes the objective function for to obtain the ordinary QR estimators of :

(2.8) |

Step 2: Choose among which makes a weighted distance function defined on closest to zero:

(2.9) |

where is a positive definite matrix.

Step 3: The estimation of can be obtained, which is respectively , , .

### 2.2 Asymptotic theory

In this section, we investigate the asymptotic properties of the IVQR estimator in Model (2.1). We impose the following regularity conditions:

A1 is independent and identically distributed (i.i.d.) for each fixed with conditional distribution function and differentiable conditional densities, , with bounded derivatives for and .

A2 For all , is in the interior of the set , and is compact and convex.

A3 Let ,

(2.10) |

and

(2.11) |

where and . The Jacobian matrices and are continuous and have full rank uniformly over . The parameter space is a connected set and the image of under the map is simply connected.

A4 Denote , where . Let . Then, the following matrices are positive definite:

(2.12) | ||||

(2.13) | ||||

(2.14) |

where and , . Let be a conformable partition of and . Hence, is invertible and is also invertible.

A5 , and .

###### Lemma 2.1

Denote , and let be a parameter vector in . Let

(2.15) |

Under conditions A1-A5, we have

###### Theorem 2.1 (Consistency)

Under conditions A1-A5, uniquely solves the equation over and is consistently estimable. Therefore, the parameters are also consistently estimable.

###### Theorem 2.2 (Asymptotic distribution)

Under conditions A1-A5 and Lemma 2.1, for a given , converges to a Gaussian distribution:

(2.16) |

where , , , , , , , , , , , , , and is a conformable partition of .

## 3 Monte Carlo simulations

In this section, we report the results of a Monte Carlo study in which we assess the finite sample performance of the IVQR estimators proposed in Section 2. For comparison purpose, we generate the samples being considered in the design of Lee anf Yu (2010):

where , and . Here, , , are drawn independently from and both the spatial weights matrices and are the same rook matrices. We use some combinations of , and . For the disturbance errors, we consider the standard normal (i.e., N(0, 1)) and Cauchy (i.e., ) distributions.

For each set of generated sample observations, we calculate the IVQR estimators. This step is repeated for 2000 times. We consider the bias and root mean squared error (RMSE) for the MLE, QMLE, OLS and IVQR. The quantile regression based estimators are calculated for quantiles . For the IVQR estimator, we employed as instrument. The results are summarized in Table 1-4.

Table 1-4 show that the IVQR estimator performs better than the other estimators in settings. In general, we find that the biases and RMSEs associated with are slightly smaller for the IVQR estimator. Under normal disturbance errors, the IVQR estimators for performs better than the other estimators while those for and have similar biases and RMSEs as the OLS estimators, but a bit larger than the MLE and QMLE estimators. For Cauchy disturbance errors, our proposed IVQR estimators outperform the other estimators as we do not impose any finite moment assumption on the distrubance errors. Therefore, we conclude that the proposed IVQR is more robust in practice.

Distribution | IVQR | MLE | QMLE | OLS | ||||
---|---|---|---|---|---|---|---|---|

Bias | 0.1278 | 0.1256 | 0.1278 | 0.0271 | 0.0121 | 0.1164 | ||

RMSE | 0.2293 | 0.2176 | 0.2286 | 0.1243 | 0.1408 | 0.1935 | ||

Bias | 0.0085 | 0.0056 | 0.0075 | -0.0904 | -0.0300 | 0.1877 | ||

RMSE | 0.0408 | 0.0406 | 0.0410 | 0.1618 | 0.1558 | 0.2382 | ||

Bias | 0.0096 | 0.0037 | 0.0049 | 0.0021 | -0.0020 | -0.0276 | ||

RMSE | 0.0975 | 0.0872 | 0.0976 | 0.0749 | 0.0764 | 0.0786 | ||

Bias | 0.2382 | 0.2381 | 0.2472 | -0.0382 | -0.0167 | 0.2264 | ||

RMSE | 0.2844 | 0.2749 | 0.2887 | 0.1129 | 0.1238 | 0.2599 | ||

Bias | -0.0001 | 0.0001 | 0.0001 | 0.0183 | 0.0017 | -0.0237 | ||

RMSE | 0.0041 | 0.0040 | 0.0041 | 0.1455 | 0.1658 | 0.2440 | ||

Bias | -0.0117 | -0.0057 | -0.0105 | -0.0017 | -0.0042 | -0.0299 | ||

RMSE | 0.1063 | 0.0926 | 0.1065 | 0.0733 | 0.0738 | 0.0817 | ||

Bias | 0.0199 | 0.0136 | 0.0183 | 0.0512 | 0.0528 | 0.2183 | ||

RMSE | 0.0579 | 0.0402 | 0.0598 | 0.1815 | 0.1775 | 0.2760 | ||

Bias | 0.0029 | 0.0022 | 0.0043 | -0.1057 | -0.1095 | 0.2114 | ||

RMSE | 0.0410 | 0.0404 | 0.0412 | 0.1937 | 0.1917 | 0.2608 | ||

Bias | 0.0081 | 0.0033 | -0.0007 | -1.3403 | -5.6184 | -0.5514 | ||

RMSE | 0.2253 | 0.1513 | 0.2243 | 57.2094 | 153.2562 | 47.4065 | ||

Bias | 0.0454 | 0.0311 | 0.0484 | -0.1077 | -0.1031 | 0.4463 | ||

RMSE | 0.0728 | 0.0502 | 0.0759 | 0.1841 | 0.1823 | 0.4645 | ||

Bias | 0.0031 | 0.0021 | 0.0030 | 0.0546 | 0.0551 | -0.0416 | ||

RMSE | 0.0041 | 0.0041 | 0.0041 | 0.1763 | 0.1740 | 0.1947 | ||

Bias | 0.0098 | 0.0023 | 0.0183 | -0.2703 | 0.8199 | -0.2556 | ||

RMSE | 0.2127 | 0.1521 | 0.2087 | 21.7709 | 17.4982 | 14.4871 |

Distribution | IVQR | MLE | QMLE | OLS | ||||
---|---|---|---|---|---|---|---|---|

Bias | 0.1267 | 0.1248 | 0.1274 | 0.0241 | 0.0056 | 0.1070 | ||

RMSE | 0.1800 | 0.1753 | 0.1789 | 0.0889 | 0.0988 | 0.1462 | ||

Bias | 0.0007 | 0.0009 | 0.0010 | -0.0779 | -0.0137 | 0.2109 | ||

RMSE | 0.0406 | 0.0410 | 0.0403 | 0.1198 | 0.1040 | 0.2312 | ||

Bias | 0.0012 | 0.0001 | 0.0013 | 0.0038 | -0.0001 | -0.0323 | ||

RMSE | 0.0660 | 0.0616 | 0.0666 | 0.0489 | 0.0500 | 0.0598 | ||

Bias | 0.2326 | 0.2246 | 0.2300 | -0.0305 | -0.0064 | 0.2328 | ||

RMSE | 0.2576 | 0.2489 | 0.2549 | 0.0794 | 0.0839 | 0.2491 | ||

Bias | 0.0002 | 0.0000 | 0.0001 | -0.0178 | -0.0005 | 0.0116 | ||

RMSE | 0.0041 | 0.0040 | 0.0041 | 0.0996 | 0.1126 | 0.1471 | ||

Bias | -0.0301 | -0.0268 | -0.0295 | 0.0001 | -0.0013 | -0.0293 | ||

RMSE | 0.0717 | 0.0689 | 0.0736 | 0.0471 | 0.0471 | 0.0577 | ||

Bias | 0.0131 | 0.0064 | 0.0105 | 0.0571 | 0.0613 | 0.2211 | ||

RMSE | 0.0397 | 0.0180 | 0.0344 | 0.1639 | 0.1615 | 0.2533 | ||

Bias | 0.0016 | 0.0003 | 0.0030 | -0.1065 | -0.1093 | 0.2330 | ||

RMSE | 0.0406 | 0.0408 | 0.0409 | 0.1781 | 0.1775 | 0.2565 | ||

Bias | -0.0034 | 0.0001 | -0.0008 | 4.5772 | 2.9534 | -2.0496 | ||

RMSE | 0.1365 | 0.0942 | 0.1316 | 151.1233 | 132.0682 | 48.2493 | ||

Bias | 0.0205 | 0.0132 | 0.0211 | -0.1010 | -0.1017 | 0.4500 | ||

RMSE | 0.0373 | 0.0242 | 0.0398 | 0.1696 | 0.1723 | 0.4606 | ||

Bias | 0.0001 | 0.0001 | -0.0001 | 0.0533 | 0.0527 | -0.0145 | ||

RMSE | 0.0040 | 0.0040 | 0.0041 | 0.1617 | 0.1609 | 0.1331 | ||

Bias | -0.0053 | -0.0048 | -0.0066 | -0.4902 | -2.9283 | 0.4147 | ||

RMSE | 0.1322 | 0.0947 | 0.1308 | 15.7727 | 82.8778 | 15.9012 |

Distribution | IVQR | MLE(QMLE) | OLS | ||||
---|---|---|---|---|---|---|---|

Bias | 0.0969 | 0.0935 | 0.0961 | 0.0096 | 0.0932 | ||

RMSE | 0.2088 | 0.1989 | 0.2127 | 0.1380 | 0.1756 | ||

Bias | -0.0028 | -0.0010 | 0.0010 | -0.0279 | 0.2372 | ||

RMSE | 0.0412 | 0.0409 | 0.0409 | 0.1485 | 0.2629 | ||

Bias | -0.0092 | -0.0076 | -0.0104 | -0.0027 | -0.0372 | ||

RMSE | 0.1004 | 0.0878 | 0.0973 | 0.0766 | 0.0804 | ||

Bias | 0.2121 | 0.2063 | 0.2176 | -0.0173 | 0.2125 | ||

RMSE | 0.2694 | 0.2552 | 0.2737 | 0.1163 | 0.2480 | ||

Bias | 0.0001 | 0.0000 | 0.0000 | 0.0021 | 0.0510 | ||

RMSE | 0.0041 | 0.0041 | 0.0041 | 0.1590 | 0.2172 | ||

Bias | -0.0366 | -0.0306 | -0.0370 | -0.0039 | -0.0340 | ||

RMSE | 0.1078 | 0.0942 | 0.1077 | 0.0737 | 0.0820 | ||

Bias | 0.0164 | 0.0088 | 0.0146 | -0.4973 | 0.1838 | ||

RMSE | 0.0555 | 0.0362 | 0.0523 | 0.5068 | 0.2449 | ||

Bias | 0.0018 | -0.0005 | -0.0002 | 0.4453 | 0.2798 | ||

RMSE | 0.0410 | 0.0408 | 0.0410 | 0.4489 | 0.3053 | ||

Bias | 0.0005 | 0.0019 | -0.0090 | -0.2086 | 0.2149 | ||

RMSE | 0.2122 | 0.1555 | 0.2172 | 6.7370 | 20.5035 | ||

Bias | 0.0349 | 0.0231 | 0.0367 | -0.7019 | 0.4122 | ||

RMSE | 0.0654 | 0.0450 | 0.0777 | 0.7774 | 0.4327 | ||

Bias | 0.0001 | 0.0001 | -0.0001 | 0.6524 | -0.0024 | ||

RMSE | 0.0042 | 0.0041 | 0.0041 | 0.7311 | 0.1918 | ||

Bias | -0.0050 | -0.0056 | -0.0021 | -0.8846 | -0.1423 | ||

RMSE | 0.2011 | 0.1484 | 0.2052 | 17.5186 | 11.0912 |

Distribution | IVQR | MLE(QMLE) | OLS | ||||
---|---|---|---|---|---|---|---|

Bias | 0.0955 | 0.0920 | 0.0960 | 0.0040 | 0.0946 | ||

RMSE | 0.1570 | 0.1523 | 0.1586 | 0.0949 | 0.1358 | ||

Bias | -0.0014 | 0.0002 | 0.0004 | -0.0110 | 0.2602 | ||

RMSE | 0.0409 | 0.0407 | 0.0414 | 0.0945 | 0.2698 | ||

Bias | -0.0047 | -0.0055 | -0.0022 | -0.0005 | -0.0364 | ||

RMSE | 0.0647 | 0.0588 | 0.0630 | 0.0492 | 0.0602 | ||

Bias | 0.2123 | 0.2130 | 0.2115 | -0.0066 | 0.2118 | ||

RMSE | 0.2379 | 0.2370 | 0.2361 | 0.0762 | 0.2283 | ||

Bias | 0.0000 | 0.0001 | 0.0000 | 0.0007 | 0.0755 | ||

RMSE | 0.0041 | 0.0041 | 0.0041 | 0.1053 | 0.1636 | ||

Bias | -0.0335 | -0.0316 | -0.0338 | -0.0011 | -0.0362 | ||

RMSE | 0.0713 | 0.0718 | 0.0716 | 0.0466 | 0.0621 | ||

Bias | 0.0078 | 0.0050 | 0.0082 | -0.3896 | 0.1907 | ||

RMSE | 0.0253 | 0.0164 | 0.0249 | 0.4027 | 0.2215 | ||

Bias | 0.0005 | 0.0015 | -0.0001 | 0.3629 | 0.2919 | ||

RMSE | 0.0411 | 0.0410 | 0.0405 | 0.3696 | 0.3064 | ||

Bias | 0.0008 | 0.0003 | -0.0007 | -1.4438 | -2.3930 | ||

RMSE | 0.1358 | 0.0952 | 0.1349 | 22.4584 | 34.5131 | ||

Bias | 0.0202 | 0.0105 | 0.0194 | -0.0082 | 0.4118 | ||

RMSE | 0.0363 | 0.0204 | 0.0351 | 0.4977 | 0.4207 | ||

Bias | -0.0001 | 0.0001 | -0.0001 | -0.0192 | 0.0218 | ||

RMSE | 0.0041 | 0.0040 | 0.0041 | 0.5188 | 0.1456 | ||

Bias | -0.0066 | -0.0030 | -0.0067 | 0.5884 | -1.0812 | ||

RMSE | 0.1263 | 0.0904 | 0.1274 | 25.4738 | 47.3648 |

Remark: Lee and Yu (2010) demonstrated the transformation approach (QMLE) and the direct approach (MLE) yield the same estimate of when considering the individual effects only.

## 4 Illustration

In this section, we use the cigarette demand data set to illustrate our methodologies. The data set is based on a panel of 46 states over 30 time periods (1963-1992), which has been analyzed by many authors (see, Baltagi and Levin, 1992; Baltagi, 2001; Baltagi, Griffin, and Xiong, 2000; Yang, 2006; Elhorst, 2005; Kelejian and Piras, 2013). The QR model with both individual and time-period effects is given by:

(4.1) |

where is real per capita sales of cigarettes by persons of smoking age (14 years and older), , , , is the average retail price of a pack of cigarettes measured in real terms, and is real per capita disposable income. Here, we choose as instruments.

We estimate the parameters using the IVQR, MLE, and OLS methods. The results are presented in Table 5. The first three columns are the IVQR estimates for , and the last two columns correspond to the MLE and OLS estimates respectively. The top half of table presents the estimations with both the individual and time-period effects while the bottom half of table shows the estimations with individual effects only. We can see that the IVQR estimates at different quantiles (i.e., ) are quite different from the MLE and OLS estimates. In particular, the signs of the estimates for and are different among IVQR, MLE and OLS methods. Besides, the sign of the MLE estimates for and change when the individual effect is omitted from the analysis.

Parameter | IVQR | MLE | OLS | ||
---|---|---|---|---|---|

With both individual and time-period effects | |||||

-0.5249 | -0.7183 | -0.3087 | 0.3547 | -0.1747 | |

0.1900 | 0.4800 | 0.0400 | -0.3289 | -0.1747 | |

log average cigarettes retail price | -0.4108 | -0.4284 | -0.4552 | -1.0886 | -0.6115 |

log disposable income | 0.2652 | 0.3308 | 0.3320 | 0.4706 | 0.3955 |

With individual effects only | |||||

-0.4617 | -0.9111 | -1.1878 | 0.3938 | -0.1191 | |

0.1800 | 0.7200 | 0.9900 | 0.6168 | -0.1191 | |

log average cigarettes retail price | -0.3699 | -0.3699 | -0.4173 | -3.2356 | -0.5464 |

log disposable income | 0.3073 | 0.3446 | 0.3713 | 0.5310 | 0.4370 |

Figure 1 presents a complete analysis, which considers other quantiles of the conditional cigarettes demand distribution. Similarly, the top panel presents the estimations which are with both individual and time-period effects and the bottom panel shows the estimations which are with individual effects only. The -axis presents the quantiles and -axis presents the estimations of parameters (red lines) and their corresponding confidence intervals (blue lines). We find that the cigarettes retail price has negative effect to the capita sales of cigarettes and disposable income has positive effect to the capita sales of cigarettes. In the presence of both individual and time-period effects, the estimates of capita sales of cigarettes are larger at extreme quantiles than those at other quantiles. The estimates of disposable income become larger along with the higher quantiles. However, in the presence of individual effect only, the estimates of parameters are larger at the middle quantiles.

## 5 Conclusion

In this paper, we investigate the instrumental variable quantile (IVQR) estimation of general spatial autoregressive panel data model with fixed effects. The model with both individual and time-period effects is considered. The asymptotic properties are studied. Monte Carlo results are provided to show that the proposed methodology is robust to error distributions with un-defined moments.

## 6 Acknowledgements

The work was partially supported by National Natural Science Foundation of China (No.11271368), Project supported by the Major Program of Beijing Philosophy and Social Science Foundation of China (No. 15ZDA17), Project of Ministry of Education supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20130004110007), The Key Program of National Philosophy and Social Science Foundation Grant (No. 13AZD064), The Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (No. 15XNL008), and The Project of Flying Apsaras Scholar of Lanzhou University of Finance & Economics.

## Appendix: Proofs

Proof of Lemma 2.1 is similar to that of Lemma 2 in Galvao (2011) and is hence omitted here.

### 1 Proof of Theorem 2.1

Proof. Firstly, following Chernozhukov and Hansen (2006), uniquely solves the problem for each .

To prove the consistency of the parameter, we need to show that under conditions A1-A5, . Let

and is continuous. Under condition Lemma 2.1, we have that for , which implies that . By Corollary 3.2.3 in van der Vaart and Wellner (1996), we have . Therefore, , , , and . Hence, and the theorem follows.

### 2 Proof of Theorem 2.2

For any , we can write the objective function defined in equation (2.7) as

where , , and

For fixed , we can consider the behavior of . Let and

Let

and

Expanding , we obtain