面板数据线性回归模型分析——SAS, Stata, LIMDEP, SPSS实战

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"这篇PDF文档是关于使用SAS, Stata, LIMDEP, 和 SPSS进行面板数据线性回归模型分析的指南，由Hun Myoung Park博士撰写，2005年至2009年间由Indiana University的University Information Technology Services (UITS)的Center for Statistical and Mathematical Computing发布。" 文档主要介绍了面板数据分析中的线性回归模型，这是一种统计方法，用于分析具有时间序列特性的面板数据集。面板数据集结合了多个个体在多个时间点上的观测值，这使得研究者能够考虑个体间的异质性和时间效应。 1. 面板数据基础：面板数据（Panel Data）是经济学和统计学中的一个重要概念，它将横截面数据（不同个体在特定时间点的数据）和时间序列数据（同一个体在多个时间点的数据）结合起来，提供了更丰富的信息和更高的估计效率。面板数据有助于识别并控制个体固定效应和时间趋势，减少随机误差项的共线性问题。 2. 线性回归模型：在面板数据设置下，线性回归模型可以分为固定效应模型和随机效应模型。固定效应模型假设个体间存在不可观测的异质性，这些异质性通过包括个体虚拟变量来处理。随机效应模型则认为个体异质性是随机的，并且可以被模型中的解释变量捕获。 3. SAS, Stata, LIMDEP, SPSS的使用： - SAS：这是一个强大的统计分析软件，提供了诸如PROC PANEL、PROC MIXED等命令来处理面板数据的线性回归。 - Stata：Stata的xtreg命令支持固定效应和随机效应模型，且提供了方便的选项进行交互式分析。 - LIMDEP：这个软件主要用于经济建模，其面板数据功能允许用户估计固定效应和随机效应模型，以及更复杂的动态面板数据模型。 - SPSS：虽然SPSS在面板数据处理上相对较少，但仍然可以通过扩展模块进行线性混合模型的估计。 4. 非线性模型：尽管文档主要关注线性模型，但面板数据也可以用于估计非线性模型，如逻辑斯蒂回归（logit）和 probit模型，这些模型通常用于处理二元响应变量的情况。该文档是针对研究者和学生的一个实用工具，他们希望通过统计软件来探索面板数据的复杂性，并进行有效的模型估计。通过了解如何在不同软件中实现这些模型，研究者可以更灵活地选择适合其研究需求的方法。

http://www.indiana.edu/~statmath

Table 2.2 compares how SAS, Stata, LIMDEP, and SPSS estimate LSDVs. SPSS is not able to

fit the LSDV3. In LIMDEP,

ONE indicates the intercept to be included. Cls: b(2)+b(3)=0 fits

the model under the condition that the sum of parameter estimates of

d1 (second parameter)

and

d2 (third parameter) is zero. In SPSS, pay attention to the /ORIGIN option for LSDV2.

http://www.indiana.edu/~statmath

3. Panel Data Models

Panel data models examine group (individual-specific) effects, time effects, or both. These

effects are either fixed effect or random effect. A fixed effect model examines if intercepts vary

across groups or time periods, whereas a random effect model explores differences in error

variances. A one-way model includes only one set of dummy variables (e.g., firm), while a two-

way model considers two sets of dummy variables (e.g., firm and year). Model 2 in Chapter 2,

in fact, is a one-way fixed group effect panel data model.

3.1 Functional Forms and Notation

The parameter estimate of a dummy variable is a part of the intercept in a fixed effect model

and a component of error in the random effect model. Slopes remain the same across groups or

time periods. The functional forms of one-way panel data models are as follows.

Fixed group effect model:

ititiit

vXuy 



)(

, where

),0(~

vit

IIDv



Random group effect model: )(

itiitit

vuXy 



, where ),0(~

vit

IIDv



Note that

u is a fixed or random effect and errors are independent identically distributed,

),0(~

vit

IIDv



Notations used in this document include,



i

y : dependent variable (DV) mean of group i.



: dependent variable (DV) mean at time t.



i

x : means of independent variables (IVs) of group i.



: means of independent variables (IVs) at time t.





y : overall means of the DV.





x : overall means of the IVs.



n: the number of groups or firms



T : the number of time periods



N=nT : total number of observations



k : the number of regressors excluding dummy variables



K=k+1 (including the intercept)

3.2 Fixed Effect Models

There are several strategies for estimating fixed effect models. The least squares dummy

variable model (LSDV) uses dummy variables, whereas the within effect model does not. These

strategies, of course, produce the identical parameter estimates of non-dummy independent

variables. The between effect model fits the model using group and/or time means of dependent

and independent variables without dummies. Table 3.1 summarizes pros and cons of these

models.

http://www.indiana.edu/~statmath

3.2.1 Estimations: LSDV, Within Effect, and Between Effect Models

As discussed in Chapter 2, LSDV is widely used because it is relatively easy to estimate and

interpret substantively. This LSDV, however, becomes problematic when there are many

groups or subjects in panel data. If T is fixed and



nT , only coefficients of regressors are

consistent. The coefficients of dummy variables,





, are not consistent since the number of

these parameters increases as nT increases (Baltagi 2001). This is the so called incidental

parameter problem. Under this circumstance, LSDV is useless and thus calls for another

strategy, the within effect model.

A within group effect model does not need dummy variables, but it uses deviations from group

means. Thus, this model is the OLS of

)()'()(













iitiitiit

xxyy





without an

intercept.

The incidental parameter problem is no longer an issue. The parameter estimates of

regressors in the within effect model are identical to those of LSDV. The within effect model in

turn has several disadvantages.

Since this model does not report dummy coefficients, you need to compute them using the

formula







iii

xyd

Since no dummy is used, the within effect model has larger degrees of

freedom for error, resulting in small MSE (mean square error) and incorrect (smaller) standard

errors of parameter estimates. Thus, you have to adjust the standard error using the formula

knnT

knT

sese

LSDV

error

Within

error







. Finally, R

of the within effect model is not correct

because the intercept is suppressed.

Table 3.1 Comparison of Fixed Effect Models

LSDV1 Within Effect Between Effect

Functional form

iiii

Xiy























iitiitiit

xxyy



iii











Dummy Yes No No

Dummy coefficient Presented Need to be computed N/A

Transformation No Deviation from the group means Group means

Intercept (estimation) Yes No Yes

Correct Incorrect

SSE Correct Correct

MSE Correct Smaller

Standard error of



Correct Incorrect (smaller)

error

nT-n-k nT-k (n larger) n-K

Observations nT nT n

The between group effect model, so called the group mean regression, uses group means of the

dependent and independent variables. This data aggregation reduces the number of

You need to follow three steps: 1) compute group means of the dependent and independent variables; 2)

transform variables to get deviations from the group means; 3) run OLS with the transformed variables without the

intercept.

http://www.indiana.edu/~statmath

observations down to n. Then, run OLS of

iii











. Table 3.1 contrasts LSDV, the

within effect model, and the between group models.

3.2.2 Testing Group Effects

In a regression of

ititiit











 ' , the null hypothesis is that all dummy parameters

except for one for the dropped are zero: 0...:

110



n



. This hypothesis is tested by the

F test, which is based on loss of goodness-of-fit. The robust model in the following formula is

LSDV (or within effect model) and the efficient model is the pooled regression.

),1(~

)()1(

)1()(

)()'(

)1()''(

knnTnF

knnTR

nRR

knnTee

neeee

Robust

EfficientRobust

Robust

RobustEfficient









If the null hypothesis is rejected, you may conclude that the fixed group effect model is better

than the pooled OLS model.

3.2.3 Fixed Time Effect and Two-way Fixed Effect Models

For the fixed time effects model, you need to switch

n and T, and i and t in the formulas.

 Model:

itittit











 '

 Within effect model: )()'()(

tittittit

xxyy

















 Dummy coefficients:



ttt

xyd





 Correct standard errors:

kTTn

kTn

sese

LSDV

error

Within

error







 Between effect model:

ttt











 0...:

110



T



 F-test: ),1(~

)()'(

)1()''(

kTTnTF

kTTnee

Teeee

Within

WithinPooled







The fixed group and time effect model uses slightly different formulas. The within effect model

of this two-way fixed model is estimated by five strategies (see Section 6.1).



Model:

itittiit













 .



Within effect Model:



 yyyyy

tiitit

and



 xxxxx

tiitit



Dummy coefficients:



)'()(



 xxyyd

iii

and



)'()(



 xxyyd

ttt

When comparing fixed effect and random effect models, the fixed effect estimates are considered as the robust

estimates and random effect estimates as the efficient estimates.

http://www.indiana.edu/~statmath

 Correct standard errors:







kTnnT

knT

sese

LSDV

error

Within

error

 0...:

110



n



and 0...



T



 F-test: )]1(),2[(~

)1()'(

)2()''(











kTnnTTnF

kTnnTee

Tneeee

Robust

RobustEfficient

3.3 Random Effect Models

The one-way random group effect model is formulated as

itiitit

vuXy 









itiit

vuw





where ),0(~

IIDu



and ),0(~

vit

IIDv



. The

u are assumed independent of

v and

X ,

which are also independent of each other for all

i and t. This assumption is not necessary in the

fixed effect model. The components of )(),(

jsitjsit

wwEwwCov



are



 if i=j and t=s and



if i=j and st  .

Thus, the  matrix or the variance structure of errors looks like,



















2222

...

............

...

vuuu

uvuu

uuvu



A random effect model is estimated by generalized least squares (GLS) when the variance

structure is known, and by feasible generalized least squares (FGLS) when the variance is

unknown. Compared to fixed effect models, random effect models are relatively difficult to

estimate. This document assumes panel data are balanced.

3.3.1 Generalized Least Squares (GLS)

When

 is known (given), GLS based on the true variance components is BLUE and all the

feasible GLS estimators considered are asymptotically efficient as either

n or T approaches

infinity (Baltagi 2001).

In GLS, you just need to compute



using the  matrix:











Then transform

variables as follows.





iitit

yyy







iitit

xxx



for all X





 1

This implies that

),(

jsit

wwCorr

is 1 if i=j and t=s, and )(

222

vuu



 if i=j and st  .

If 0



, run pooled OLS. If 1



and 0





, then run the within effect model.

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