A Non-parametric Regression Model for Consumption of Urban Residents and Foreign Exchange Reserve

The consumption of urban residents and the foreign exchange reserves are important indicators of economic development and between which there existed a stable relationship. Local polynomial regression estimation method of the non-parametric regression model is discussed and considering the historical data of China’s consumption of urban residents and foreign exchange reserves from 1981 to 2003, which was researched using the local polynomial regression estimation method of the non-parametric regression model, we built its non-parametric regression model and compare it with the polynomial fitting model. The results showed that non-parametric regression model is superior to polynomial fitting model.


nonparametric regression model
Definition 1.1:With given n pairs of observed value ) , ( , the relationship between response variable Y and covariate variables x is defined by following equation: where r is regression function.Variable x is also known as characteristic.The estimation of ) (x r is represented by ) ( ˆx r n .

Local polynomial estimation
Under the Nadaraya-Watson kernel estimation condition, consider an estimator

Define weight function
, and chose ) ( ˆx r a n ≡ to minimize the following weighted quadratic sum which is kernel regression estimation.
To improve the estimation, a p-order local polynomial is used.x is a fixed value which is used to estimate ) (x r .For value u that belongs to the neighborhood of x, define polynomial which minimizes the following weighted quadratic sum: (1.6) Estimater a ˆdepends on target value x .While .
(1.6) could be written as (1.8) By minimizing (1.8), a weighted least square estimation is obtained (1.9) and Y , then there is: Theorem 1.1: estimation for the local polynomial regression is: , and x X and x W were defined in (1.7).
The mean value and covariance value of this estimation are: (1.14) Theorem 1.2:If n rˆ is a linear smoothing value, then the leave-one-out cross-validation score ) ( ˆh R can be written as is the ith diagonal elements of the smooth matrix. The bandwidth h can be obtained by minimizing cross-validation formulas from theorem 1.2 (Larry Wasserman,2008,p.57-62).

A non-parametric regression model of the consumption of urban residents and the foreign exchange reserve
Define Y as foreign exchange reserves, X as consumption of urban residents.

Local polynomial regression estimation model
Build non-parameter local polynomial regression estimation model of consumption of urban criticizes and foreign exchange reserves according to theorem 1.1 and theorem 1.2, where p=3, h=250.The results are shown in Figure 1, Figure 2, Table 1, and Table 2.The fitting results of the foreign exchange reserves by polynomial fitting model and non-parameter local polynomial regression estimation model are shown in Figure 1, in which the dot-line represents the fitting curve of polynomial fitting model and the solid line represents the fitting curve of non-parameter local polynomial regression estimation model.Figure 2 shows the estimate value of the two models at x i , in which □ represents the fitting value y i at point x i based on polynomial fitting model, while ○ represents the estimate value at point x i based on non-parameter local polynomial regression estimation model.Symbol * represents the actual value in both Figure 1 and Figure 2.
As shown in Figure 1, the fitting curve in non-parameter local polynomial regression estimation model tallies with the actual situation better than that of polynomial fitting model.In Figure 2, fitting values in both models tallies with the actual values, but the former is superior to the latter.This can be drawn from table 1 and table 2. As can be seen from the above chart, at the beginning of China's reform and opening up, the relationship between the consumption of urban residents and foreign exchange reserves was not stable and the volatility couldn't be neglected.However, with the continuous deepening of China's reform and opening up, expending trade with other countries, improvement of the market economic system and perfection of national policy, the relationship between the consumption of urban residents and foreign exchange reserves in more and more stable, which creates a favorable condition for our studies Using polynomial fitting model and non-parameter local polynomial regression estimation model respectively, to predict the value of China's foreign exchange reserve in 2004, the difference between the former result and the actual result is large (error is larger than 1000) while the later is close to (error is less than 50).

Conclusion
In actual economic life, due to the unpredicted reasons, it is difficult to make specific assumption.With the improvement of the market economic system and increasingly close connection between China and world economy, the correlation between consumption of urban residents and foreign exchange reserve is more and more stable.Non-parametric local polynomial regression estimation model has the regression function form of arbitrariness, lower requirement than polynomial fitting model, and no need to consider the distribution of the sample, making non-parametric local polynomial regression estimation the best reflection of correlation between variables, and the errors are smaller than the polynomial fitting model.

Figure 1 .
Figure 1.Fitting curve in two models

Figure 2 .
Figure 2. Fitting value in two models The research sample is the historical data of China's foreign exchange reserves and consumption of urban residents date from 1981 to 2003.Data sources is from

Table 1 .
Comparison of MSE in two models

Table 2 .
Comparison of fitting values in two models