Step 1: Analyze the stationarity of the data (unit root test)
According to the formal procedure, the panel data model needs to check the stationarity of the data before regression. Li Zinai once pointed out that some non-stationary economic time series often show a common trend of change, and these series do not necessarily have a direct correlation. The result is of no practical significance. This situation is called spurious regression or spurious regression. He believes that the true meaning of stationary is: after a time series has eliminated the invariant mean (which can be regarded as the intercept) and the time trend, the remaining series has zero mean and homoscedasticity, that is, white noise. Therefore, there are three test modes in the unit root test: both trend and intercept, only intercept, and none of the above.
Therefore , in order to avoid spurious regression and ensure the validity of the estimated results, we must test the stationarity of each panel series. The most commonly used method to test the stationarity of data is the unit root test. First of all, we can draw a time series diagram for the panel series, and roughly observe whether the polyline representing the variable contains a trend item and/or an intercept item from each observation value in the time series diagram, so as to make a further test mode for the unit root test. Prepare.
Literature Review of Unit Root Test Methods: In the asymptotic process of non-stationary panel data, Levin and Lin (1993) found very early that the limiting distribution of these estimators is Gaussian, and these results were also applied to panel data with heteroscedasticity , and established an earlier version of the test for panel unit roots. Later, improved by Levin et al. (2002), LLC method for testing panel unit root was proposed. Levin et al. (2002) pointed out that this method allows for different intercepts and temporal trends, heteroscedasticity and high-order series correlation, and is suitable for medium dimensions (time series between 25 and 250, and the number of sections between 10 and 250). ) panel unit root test. Im et al. (1997) also proposed the IPS method to test the panel unit root, but Breitung (2000) found that the IPS method is extremely sensitive to the setting of the limiting trend, and proposed the Breitung method of the panel unit root test. Maddala and Wu (1999) proposed ADF-Fisher and PP-Fisher panel unit root test methods.
It can be seen from the above review that five methods of LLC, IPS, Breintung, ADF-Fisher and PP-Fisher can be used for panel unit root test.
Where LLC-T, BR-T, IPS-W, ADF-FCS, PP-FCS, HZ refer to Levin, Lin & Chu t* statistics, Breitung t statistics, lm Pesaran & Shin W statistics, ADF-Fisher The null hypothesis for the Chi-square statistic, PP-Fisher Chi-square statistic, Hadri Z statistic, and Levin, Lin & Chu t* statistic, Breitung t statistic is the existence of an ordinary unit root process, lm Pesaran & Shin The null hypothesis of W statistic, ADF-Fisher Chi-square statistic, and PP-Fisher Chi-square statistic is that there is an effective unit root process, and the null hypothesis of the Hadri Z statistic test is that there is no ordinary unit root process.
Sometimes, for convenience, only two panel data unit root test methods are used, namely the same root unit root test LLC (Levin-Lin-Chu) test and the different root unit root test Fisher-ADF test (Note: For ordinary series (non-panel) The unit root test method of the sequence) is usually the ADF test). If the original hypothesis of the existence of the unit root is rejected in both tests, we say that the sequence is stationary, otherwise it is not stationary .
If we use T(trend) to represent the trend item in the series, I(intercept) to represent the series to contain the intercept item, T&I to represent both, and N(none) to represent neither of the two, then we can use the previous time series diagram. The conclusion drawn, select the corresponding test mode in the unit root test.
But the conclusion based on the timing diagram is rough after all. Strictly speaking, those inspection structures need to be inspected one by one. For specific operations, please refer to Li Zinai’s statement: ADF test is done through three models. First, the model with intercept and trend term is tested, then the model with only intercept term is tested, and finally the model without both is tested. . And we believe that the time series is non-stationary only when the test results of the three models cannot reject the null hypothesis, and as long as the test results of one of the models reject the null hypothesis, the time series can be considered stationary.
In addition, the unit root test generally starts from the level sequence. If there is a unit root, the first-order difference is performed on the sequence and then the test is continued. If there is still a unit root, the second-order or even higher-order difference is performed. test until the series is stationary. We record I(0) as the zero-order single integer, I(1) as the first-order single integer, and so on, I(N) as the N-order single integer.
Step 2: Cointegration Test or Model Revision
Case 1: If the variables are found to be single integral of the same order based on the results of the unit root test, then we can conduct a cointegration test. Cointegration test is a method to examine the long-term equilibrium relationship between variables. The so-called cointegration means that if two or more non-stationary variable series, the series after a certain linear combination is stationary. At this point, we say that there is a cointegration relationship between these variable sequences. Therefore , the requirement or premise of cointegration is single integration of the same order.
但也有如下的宽限说法：如果变量个数多于两个，即解释变量个数多于一个，被解释变量的单整阶数不能高于任何一个解释变量的单整阶数。另当解释变量的单整阶数高于被解释变量的单整阶数时，则必须至少有两个解释变量的单整阶数高于被解释变量的单整阶数。如果只含有两个解释变量，则两个变量的单整阶数应该相同。
也就是说，单整阶数不同的两个或以上的非平稳序列如果一起进行协整检验，必然有某些低阶单整的，即波动相对高阶序列的波动甚微弱(有可能波动幅度也不同)的序列，对协整结果的影响不大，因此包不包含的重要性不大。而相对处于最高阶序列，由于其波动较大，对回归残差的平稳性带来极大的影响，所以如果协整是包含有某些高阶单整序列的话(但如果所有变量都是阶数相同的高阶，此时也被称作同阶单整，这样的话另当别论)，一定不能将其纳入协整检验。
协整检验方法的文献综述：(1)Kao(1999)、Kao and Chiang(2000)利用推广的DF和ADF检验提出了检验面板协整的方法,这种方法零假设是没有协整关系,并且利用静态面板回归的残差来构建统计量。(2)Pedron(1999)在零假设是在动态多元面板回归中没有协整关系的条件下给出了七种基于残差的面板协整检验方法。和Kao的方法不同的是,Pedroni的检验方法允许异质面板的存在。(3)Larsson et al(2001)发展了基于Johansen(1995)向量自回归的似然检验的面板协整检验方法，这种检验的方法是检验变量存在共同的协整的秩。
主要采用的是Pedroni、Kao、Johansen的方法。
通过了协整检验，说明变量之间存在着长期稳定的均衡关系，其方程回归残差是平稳的。因此可以在此基础上直接对原方程进行回归，此时的回归结果是较精确的。
这时，我们或许还想进一步对面板数据做格兰杰因果检验(因果检验的前提是变量协整)。但如果变量之间不是协整(即非同阶单整)的话，是不能进行格兰杰因果检验的，不过此时可以先对数据进行处理。引用张晓峒的原话，“如果y和x不同阶，不能做格兰杰因果检验，但可通过差分序列或其他处理得到同阶单整序列，并且要看它们此时有无经济意义。”
下面简要介绍一下因果检验的含义：这里的因果关系是从统计角度而言的，即是通过概率或者分布函数的角度体现出来的：在所有其它事件的发生情况固定不变的条件下，如果一个事件X的发生与不发生对于另一个事件Y的发生的概率(如果通过事件定义了随机变量那么也可以说分布函数)有影响，并且这两个事件在时间上又有先后顺序(A前B后)，那么我们便可以说X是Y的原因。考虑最简单的形式，Granger检验是运用F-统计量来检验X的滞后值是否显著影响Y(在统计的意义下，且已经综合考虑了Y的滞后值；如果影响不显著，那么称X不是Y的“Granger原因”(Granger cause)；如果影响显著，那么称X是Y的“Granger原因”。同样，这也可以用于检验Y是X的“原因”，检验Y的滞后值是否影响X(已经考虑了X的滞后对X自身的影响)。
Eviews好像没有在POOL窗口中提供Granger causality test，而只有unit root test和cointegration test。说明Eviews是无法对面板数据序列做格兰杰检验的，格兰杰检验只能针对序列组做。也就是说格兰杰因果检验在Eviews中是针对普通的序列对(pairwise)而言的。你如果想对面板数据中的某些合成序列做因果检验的话，不妨先导出相关序列到一个组中(POOL窗口中的Proc/Make Group)，再来试试。
情况二：如果如果基于单位根检验的结果发现变量之间是非同阶单整的，即面板数据中有些序列平稳而有些序列不平稳，此时不能进行协整检验与直接对原序列进行回归。但此时也不要着急，我们可以在保持变量经济意义的前提下，对我们前面提出的模型进行修正，以消除数据不平稳对回归造成的不利影响。如差分某些序列，将基于时间频度的绝对数据变成时间频度下的变动数据或增长率数据。此时的研究转向新的模型，但要保证模型具有经济意义。因此一般不要对原序列进行二阶差分，因为对变动数据或增长率数据再进行差分，我们不好对其冠以经济解释。难道你称其为变动率的变动率？
步骤三：面板模型的选择与回归
面板数据模型的选择通常有三种形式：
一种是混合估计模型(Pooled Regression Model)。如果从时间上看，不同个体之间不存在显著性差异；从截面上看，不同截面之间也不存在显著性差异，那么就可以直接把面板数据混合在一起用普通最小二乘法(OLS)估计参数。一种是固定效应模型(Fixed Effects Regression Model)。如果对于不同的截面或不同的时间序列，模型的截距不同，则可以采用在模型中添加虚拟变量的方法估计回归参数。一种是随机效应模型(Random Effects Regression Model)。如果固定效应模型中的截距项包括了截面随机误差项和时间随机误差项的平均效应，并且这两个随机误差项都服从正态分布，则固定效应模型就变成了随机效应模型。
在面板数据模型形式的选择方法上，我们经常采用F检验决定选用混合模型还是固定效应模型，然后用Hausman检验确定应该建立随机效应模型还是固定效应模型。
检验完毕后，我们也就知道该选用哪种模型了，然后我们就开始回归：
During regression, the weights can be weighted by cross-section (cross-section weights), especially for the case where the number of cross-sections is greater than the number of time series, which means that different cross-sections are allowed to have heteroscedasticity. The estimation method adopts the PCSE (Panel Corrected Standard Errors, panel corrected standard error) method. The PCSE estimation method introduced by Beck and Katz (1995) is an innovation of the panel data model estimation method, which can effectively deal with complex panel error structures, such as simultaneous correlation, heteroscedasticity, serial correlation, etc. It is especially useful when the sample size is not large enough .