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雷达卡
In recent years, estimation techniques that use time series cross-sectional (panel) data approaches have become widely used. The PANEL procedure in SAS/ETS software fits classes of linear models that arise when time series and cross-sectional data are combined. It is capable of fitting the following models:
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- • one-way and two-way models
- • random-effects and fixed-effects models
- • autoregressive and moving average models
- o Parks method
- o dynamic panel method (GMM)
- o Da Silva method
• one-way and two-way models• random-effects and fixed-effects models• autoregressive and moving average models o Parks method o dynamic panel method (GMM) o Da Silva methodThis paper uses simulated data to compare these techniques and outline their advantages and disadvantages. The paper starts with a brief theoretical overview of panel data methods. Several examples are given to demonstrate these techniques and their implementation in the PANEL procedure. The PANEL procedure is then compared with other SAS procedures.
PANEL MODELS
In this paper, the term panel refers to pooled data on time series cross-sectional bases. Typical examples of panel data include observations on households, countries, firms, trade, etc. For example, in the case of survey data on household income, the panel is created by repeatedly surveying the same households in different time periods
(years). The model is typically written in the following form:
demonstrate results from various models.
DATA GENERATING PROCESS
The following AR(1) panel data generating process (DGP) was adopted from Bond, Bowsher, and Windmeijer (2001). One hundred cross sections, each containing six time periods, were generated. The DGP can be described as follows:
- data two;
- delta = 0.4;
- array x[6];
- do i=1 to 100;/*i从1到100*/
- e_i = 4*rannor(1234);/*N(O,4)*/
- mu_i = 36*rannor(32444);/*N(0,36)*/
- do t=1 to 6;/*t从1到6*/
- do k = 1 to 6;
- x[k] = 3+4*(rannor(58785));/*系数β=3*/
- end;
- if t = 1 then
- y = mu_i/(1-delta) + x1 + 5*x2 +10*x3 + e_i;/*β=5*/
- else do;
- v_it = 4*rannor(34454);
- y = delta * y_t1 + mu_i + x1 + 5*x2 +10*x3 + v_it;
- end;
- output;
- y_t1 = y;
- end;
- end;
- run;
data two;delta = 0.4;array x[6];do i=1 to 100;/*i从1到100*/ e_i = 4*rannor(1234);/*N(O,4)*/ mu_i = 36*rannor(32444);/*N(0,36)*/ do t=1 to 6;/*t从1到6*/ do k = 1 to 6; x[k] = 3+4*(rannor(58785));/*系数β=3*/ end; if t = 1 then y = mu_i/(1-delta) + x1 + 5*x2 +10*x3 + e_i;/*β=5*/ else do; v_it = 4*rannor(34454); y = delta * y_t1 + mu_i + x1 + 5*x2 +10*x3 + v_it; end; output; y_t1 = y; end;end;run;
MODEL ESTIMATION
When introduced to a new method, users are likely to compare it with other estimation frameworks and techniques that are available. In this section, the generated data are used and several different models are estimated. We start with a simple OLS model that ignores the time series cross-sectional nature of the data by using the REG procedure.
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- ods graphics on;
- proc reg data = two plot(unpackpanel)=all;
- model y = x1 x2 x3 /noint;
- run;
- ods graphics off;
ods graphics on;proc reg data = two plot(unpackpanel)=all;model y = x1 x2 x3 /noint;run;ods graphics off;The model is estimated for three different cross-sectional error specifications, and the new PLOT option is used to obtain fit diagnostics for residuals (Figure 1).
Using the following statements, one-way fixed- and random-effects models are estimated using the PANEL procedure for the same three cross-sectional error specifications:
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- proc panel data = two plot=all;
- id i t;
- model y =x1 x2 x3 /fixone ranone noint;
- run;
- ods graphics off;
ods graphics on;proc panel data = two plot=all;id i t;model y =x1 x2 x3 /fixone ranone noint;run;ods graphics off;
The PLOT=ALL option is used to obtain two diagnostic panels to examine the fit of the model. The panels for the oneway random-effects model are presented in Figures 2 and 3. The first panel was created using all 100 cross sections; the second panel depicts only the first 10 cross sections.
Since the model specification in Equation (2) includes a lagged dependent variable,One-way to correct for the inefficiencies is by using GMM for panel models developed by Arellano and Bond (1991), as follows :
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- proc panel data=two plot(unpackpanel)=all;
- id i t;
- instrument depvar exogenous = (x4 x5 x6);
- model y = x1 x2 x3 /gmm twostep maxband=5 nolevels noint;
- run;
- ods graphics off;
ods graphics on;proc panel data=two plot(unpackpanel)=all;id i t;instrument depvar exogenous = (x4 x5 x6);model y = x1 x2 x3 /gmm twostep maxband=5 nolevels noint;run;ods graphics off;
It can be clearly seen from Equation (2) that the dependent variable y is not exogenous, since its values depend on its previous realizations. Arellano and Bond (1991) show that the dependent variable can still be used as one of the instruments if properly lagged. This is accomplished by using the DEPVAR option in the INSTRUMENT statement.
The INSTRUMENT statement can include other variables that are not correlated with the error term. In this model, x4, x5, and x6 are considered to be purely exogenous and uncorrelated with the error. In other models, it is possible that future values of available instruments are correlated with the error term but their past and current realizations are not. The fact that the past and present realizations are not correlated with the error enables us to use them as instruments with the PREDETERMINED option. The following INSTRUMENT statement is used to describe a model with two exogenous and one predetermined variables:
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