*3.3. Dynamic Expansion of AIDS Model*

Equation (6) is a static model, but the change in the income distribution is a dynamic process, so the combination of the two will also be a dynamic system. Based on model (6), the expenditure share *μ<sup>i</sup>* of the commodity *i* can be regarded as a function of the income *m* and the price vector *p*, which is *μi*(*m*, *p*); if *p* is kept constant, *μi*(*m*, *p*) is similar to the Engel equation of commodity *i*. Taking the two periods as an example, the evolution of the consumption structure caused by the income changes can be decomposed as follows:

$$\begin{cases} \mu\_{i1}(y\_1, p\_1) \stackrel{
eqref{\text{left}}}{\rightarrow} \eta\_{i1} = \mu\_i(y\_1, p\_2) \\ \eta\_{i1} = \mu\_i(y\_1, p\_2) \stackrel{
eqref{\text{right}}}{\rightarrow} \eta\_{i2} = \mu\_i(\xi\_1, p\_2) \stackrel{
eqref{\text{right}}}{\rightarrow} \eta\_{i3} = \mu\_i(\xi\_2, p\_2) \stackrel{
eqref{\text{right}}}{\rightarrow} \mu\_{i2}(y\_2, p\_2) \end{cases},\tag{11}$$

The above process is clearly also a counterfactual decomposition process and its first step is to eliminate the impact of the price change. The mean, variance, and residual effects of the evolution of the consumption structure caused by income distribution change are then denoted by Δ1, Δ2, and Δ3, respectively. So,

$$\begin{aligned} \Delta\_1 &= \eta\_{i2} - \eta\_{i1} = \alpha\_i \ln(\not\zeta\_1 / \not\chi\_1) \\ \Delta\_2 &= \eta\_{i3} - \eta\_{i2} = \beta\_i \ln(\not\zeta\_2 / \not\chi\_1) \\ \Delta\_3 &= \mu\_{i2} - \eta\_{i3} = \theta\_i \ln(\not\chi\_2 / \not\chi\_2) \end{aligned} \tag{12}$$

Then, the dynamic expansion model of AIDS can be obtained as follows:

$$\begin{aligned} \mu\_{i2} &= \eta\_{i1} + \Delta\_1 + \Delta\_2 + \Delta\_3 \\ &= a\_i^\* + \sum\_{j=1}^n r\_{ij}^\* \ln(p\_{j2}) + b\_i \ln(y\_1/p\_2) + a\_i \ln(\mathfrak{J}\_1/y\_1) + \beta\_i \ln(\mathfrak{J}\_2/\mathfrak{J}\_1) + \theta\_i \ln(y\_2/\mathfrak{J}\_2) \end{aligned} \tag{13}$$

Equation (13) is the dynamic expansion of the AIDS model, which includes the factors of the income distribution change. The dependent variable is the consumption share of each commodity in the second period, and the independent variable contains two kinds of price factors and income distribution factors: ∑*<sup>n</sup> <sup>j</sup>*=<sup>1</sup> *r*<sup>∗</sup> *ij* ln(*pj*2) denotes the impact of its own price and the interactive price, and ln(*y*1/*p*2) represents the change in real income due to the price changes. These factors measure the effect of the price on various expenditures together, as the nominal income remains constant. Obviously, the three items ln(*ξ*1/*y*1), ln(*ξ*2/*ξ*1), and ln(*y*2/*ξ*2) indicate the impact of the changes in the income distribution, which is the focus of this study.

#### **4. Data Preparation and Regression Equation Setting**

#### *4.1. Data Preparation*

The focus of this study is to examine the issue of China's domestic demand, not the dynamic evolution trend of the consumption structure. As can be seen from the consumption rate curve in Figure 1 (source: [1]), the significant decline in consumption rate occurred between 2000 and 2010 is what we want to explain from the structural level. In addition, as China's economy enters a new normal, the statistical caliber of urban residents' income has also changed since 2012. After 2010, only two years of data are available. For the robustness of the results, the empirical discussion focuses on the provincial panel data from 2000 to 2010.

**Figure 1.** Consumption Rate of Chinese Urban Residents during 1995–2014.

The data used are mainly from the National Statistical Yearbook 2000-2010 data in respect of the urban residents in the provinces. The indicators used include the expenditure data for eight categories of goods, the disposable income data, the classified consumer price index, and the annual nominal interest rate data. The eight categories of goods are food, clothing, residence, household facilities, medical care, transportation and communications, entertainment and education, and other miscellaneous expenses and services. The nominal interest rate used is a 1-year time deposit rate, which is actually the average annual interest rate. Because the government only announces the corresponding interest rate of the adjustment day, if there are multiple interest rates in a given year, this study will average the different interest rates by the number of days. In addition, we still need to construct some new indicators based on the available data.

For the income indexes, which are the last three income items in model (13), the structure of the two potential income variables *ξ*<sup>1</sup> and *ξ*<sup>2</sup> are relative to the income of the previous year, so the change in the income distribution is measured between the adjacent two years. The standard deviation will be used when we calculate *ξ*2; the income data of the yearbook are derived from the same household survey data, so the standard deviation is calculated by using the data of seven income levels in each year, and the proportion of the population in each group is 0.1, 0.1, 0.2, 0.2, 0.2, 0.1, and 0.1, respectively (see Table 1).


**Table 1.** Disposable Income of Chinese Residents by Group.

Note: The per capita income data are not calculated and are the original data in the China Statistical Yearbook. All data were converted using 2000 as the base year.

For the price indicators we use the 2000–2010 consumer price indices for the various commodities based on 2000. We take the discount rate 1/(1 + *r*) as the price of the savings, where the real interest rate *r* is obtained by using the annual nominal interest rate minus the total consumer price index. In order to maintain the order of magnitude consistent with the other commodity price indices, the savings price is also converted at the 2000 base price. The total price index also needs to be reconstructed assuming that the share of savings is *s*, then the "total price index = (1 − *s*)× total expenditure price index +*s*× savings price index."

#### *4.2. Regression Equation Setting*

With the rapid economic growth, residents' living standards have undergone great changes, so the consumer spending structure is bound to change significantly as a consequence of this. Therefore, the problem of structural mutation must be considered when we estimate the model. In order to avoid the estimated error caused by the artificial set of abrupt points, this study chooses to use Hansen's threshold method to find the abrupt points [50–52], which is completely determined by the data, so we introduce dummy variables in model (13) as Hansen did. The equation for setting a single mutation point is as follows:

$$\begin{split} \mu\_{il} &= \sum\_{j=1}^{n} r\_{ij}^{\*} \ln(p\_{l^{\bar{l}}}) + b\_{l} \ln(y\_{l-1}/p\_{l}) + \left[ a\_{l1}^{\*} + a\_{l1} \ln(\xi\_{1l}/y\_{l-1}) + \beta\_{l1} \ln(\xi\_{2l}/\xi\_{1l}) + \theta\_{l1} \ln(y\_{l}/\xi\_{2l}) \right] \cdot h\_{1}(t$$

where *T* is the structural break point, *h*1(*t* < *T*) and *h*2(*t* ≥ *T*) are both indicator functions. If *t* < *T*, then *h*<sup>1</sup> = 1, and *h*<sup>2</sup> has a similar definition. This study only adds the dummy variable to the intercept term and the income change factor; the price variable is not the focus, so we take it as the control variable. Because of the use of the panel data, the equation intercept also introduces province dummy variables to remove the impact of the individual effects, which is not reflected in the equation.

When the structural mutation is not considered, the degree of freedom of the system is 92, so in order to ensure the robustness of the estimation results, the time interval span set in this study is not less than three years. The sample point for each year is 31, which ensures that there are no less than 93 data observations per interval. Therefore, the potential mutation points considered in this study are in the 2003–2007 range. In addition, Hansen chooses the residual squared sum as the criterion of fitting and then determines the mutation point, and this study chooses the logarithmic maximum likelihood statistic in the regression result. The larger the likelihood value, the more likely that the corresponding potential mutation year is real.

From the above results (Table 2), the corresponding logarithmic likelihood is largest in 2004, and the results of the likelihood ratio test show that the original hypothesis of structural change is rejected at the significance level of 1%, so the structural mutation point *T* is 2004. So far, the final regression equation used in this study has been obtained.



Note: (a) Tests of potential mutation years, (b) Nonlinear test with 2004 as the mutation year. \* indicates that the value is the largest. The "df" is the degree of freedom and the likelihood-ratio test is based on the results of Table 2(a).
