The panel data contains T cross sections. If we use the distance between the cross sections to measure the similarity, then we obtain a similarity matrix, and it is a symmetrical matrix. The similarity matrix is as follows:

is a dissimilarity degree measure between the t-th cross section and the s-th cross section, which also is a measure of the distance. When the two time sections are very similar, its value is close to zero.

Here are several kinds of commonly used method for measuring distance between cross sections. As shown below:

1) Euclidean Distance:.

2) Squared Euclidean Distance:.

3) Minkowski Distance:. When p = 2, Minkowski distance is the Euclidean distance.

4) Manhattan Distance:. Manhattan distance is a special case of the Minkowski distance when p = 1.

5) Chebyshev Distance:.

The clustering analysis of panel data can divide time sections into several divisions. Building model on one of the division can ignore unobservable time effect, which has important significance on the application. Zhu and Chen [13] studied the clustering analysis of panel data and its application, and focused on the cluster in cross section.

The basic principle of clustering analysis is: for the panel data, first of all, we divide each cross section into a class, then we have a total of T classes; secondly, according to the above distance calculation, we obtain a similarity matrix of panel data, then we merge the nearest two time sections into a class, so we have classes; again, according to the similarity matrix, we merge the nearest two time sections into a class, so we have classes; by analogy, we eventually merge all T time series into a class.

To build model on panel data, we must first determine the form of the model. General panel data model is as follows:

Among them, is a vector, and is a vector, and K is the number of explanatory variables. is the intercept item, and its value is related to the individual, and it is regarded as the fixed parameter to estimate here. is a random error term, and it is not associated with explanatory variables, and its mean is zero, and its variance is, and it is independent and identically distributed.

The common situation of model (1) is as follows:

1) when, model (1) is called the basic model or mixed regression model;

2) when, model (1) is called the variable intercept model;

3) when, model (1) is called the variable coefficient model.

The common test for determining the model forms is the analysis of covariance, also is called F test. The test contains two main hypotheses:

Hypothesis 1: The slopes are the same, but the intercepts are not the same. The model is:

Hypothesis 2: The intercepts and slopes are the same in different cross sections and different time series. The model is:

According to the method in parameter constraint test, we can construct test statistics for the above two hypotheses¹. Test statistics for hypothesis 1 and hypothesis 2 respectively are:

Among them, respectively are the sums of squared residuals for model (1), (2) and (3) under ordinary least square method.

When hypothesis 1 is correct,. When hypothesis 2 is correct, . Obviously, if we accept hypothesis 2, we don’t need to test hypothesis 1, and we should build model (3). If we reject hypothesis 2, we should test hypothesis 1. If we accept hypothesis 1, we should build model (2). If we reject hypothesis 1, we should build model (1).

For fixed effect variable coefficient model (1), it can be rewritten as:

Among them,

The matrix form is:

Among them,.

Fixed effect variable coefficient model is also called seeming unrelated regression model. The model considers that coefficients don’t change with time for each individual. It is put forward by Zellnerin 1962. The selection of parameter estimation method depends on the random disturbance term². If and, model (4) can be estimated by ordinary least square method, which is the classic method in single equation econometric model. Namely we take each time series as sample, and use ordinary least squares method to estimate respectively, or adopt the generalized least square method to estimate at the same time. The two kinds of estimation results are consistent. If, we can use the generalized least square method to estimate. We write, then the covariance matrix of is:

So the generalized least square estimation of the parameters is:.

The modeling data is the per capita disposable income and the per capita cash expenditure of Chinese urban residents’ eight income levels from 2002 to 2012³. In order to eliminate the rising factor of price⁴, we regarded cpi of 2002 as 100, and recalculated cpi from 2002 to 2012. Then dividing the original data by recalculated cpi, and multiplying it by 100, finally we obtained the per capita disposable income panel data and the per capita cash expenditure panel data eliminated the rising factor of price. Using SPSS 19.0, we carried out clustering analysis of the panel data respectively. The following is the comparison of the cluster tree.

From Figure 1, we can classify the per capita disposable income from 2007 to 2012 into the same cluster. From Figure 2, we can classify the per capita cash expenditure from 2007 to 2012 into the same cluster. Therefore, we can build panel data model on the per capita disposable income and the per capita cash expenditure of Chinese urban residents’ eight income levels from 2007 to 2012.

Table 1 and Table 2 are two original panel data from 2007 to 2012. Because China’s cpi is calculated based the previous year as the base period 100, not based a certain date as the base period, we needed to recount cpi since 2007. The calculation results are shown in Table 3. We needed to eliminate the rising factor of the panel data in Table 1 and Table 2. Then it could be put into the model. Namely dividing the original data by recalculated cpi in Table 3 respectively, and multiplying it by 100.

Due to the structure of consumption and income from 2007 to 2012 belongs to the same type, so we can set the model parameters as unaffected by time. The form is:

Among them, is the per capita cash expenditure of the i-th income group in the t-th year. is the per capita disposable income of the i-th income group in the t-th year. The two panel data have been eliminated the

rising factor of price based cpi of 2007 as 100. In addition, due to the model studied each income group’s own data, so the parameters can be regarded as fixed parameters to estimate. Namely the model is the fixed effect model.

By the parameter estimation results, the following conclusions can be drawn:

1) When income levels are different, there are obvious differences in marginal consumption propensity. And the marginal consumption propensity is decreasing with the improvement of income level. It shows that income disparity exactly is the decisive factor in influencing consumption, and the higher the income is, the weaker the marginal consumption desire is. That is consistent with the saying “diminishing marginal returns” in economics.

2) The intercept item is increasing with the improvement of income level. It shows that the absolute consumption level of urban residents is increasing by increased income.

3) In general, the marginal consumption propensity of different income levels is over 50%. It shows that no matter what the income levels of residents are, their consumption desire is very high. But different income levels may pursue different consumption direction.