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The Prediction of Engel's Coefficient and Education Expenditure Based on the Linear Regression Model for Heilongjiang and Ontario

It is meaningful to study trends in food and education expenditure as proportions of total household expenditure. In this study, based on year 2006 to 2017 data from Heilongjiang province in China and Ontario province in Canada, a linear regression model is used to forecast the Engel’s coefficients (proportion spent on food) and the education proportion from year 2018 to 2027 for those two regions. The results suggest that in both regions the Engel’s coefficients show a decreasing trend, while the education expenditure proportions show an increasing trend. The ratios of education expenditure to food expenditure in both places show an increasing trend.


Introduction
In general, household expenditures include food, housing, education, recreation, and so many other expenditures. Among these various types of expenditures, the fundamental one is food. As an economy develops and its residents' incomes increase, the proportion of food expenditure (FE) in the total of household expenditure tends to decrease since people's basic demand for food is met. As early as the nineteenth century, German statistician Ernst Engel (1821-1896) proposed Engel's coefficient to refer to the proportion of residents' income spent on food [1] . This coefficient is usually used to measure a population's standard of living. However, some other types of expenditure may be increasing. The proportion of education expenditure (EE) is one of those showing the increasing trend.
According to Maslow's hierarchy of needs, proposed by American psychologist Abraham Maslow, people pursue esteem needs, such as education needs [2] , in addition to the physiological needs, such as food needs. Dai and Zhou discussed the needs of happiness by the middle class in Guangdong province and concluded that education is one of the key contributing factors to happiness [3] . In a modern competitive society, education is a key prerequisite for winning the competition. So people are spending more and more money on education. Wang analyzed the advantages and disadvantages of measuring Shanghai citizens' living standard by using the Engel coefficient and concluded that the structure of food expenditure was gradually increasing with the improvement of people's living standard [4] . Gu's work, the Engel's coefficient in China was decreasing and the living standard of Chinese residents was improving [5] . Wang studied the influence of the price of commodities on Engel's coefficient and compared the similarities and differences of FE between urban and rural residents based on his econometric model [6] . Wang and Woo used Engel's coefficient to discuss the real level of the household income in China, from 2005 to 2008, the degree of corruption grew 91%. So this country must give institution reforms [7] . Tang used GM(1,1) [8] to predict the EE and school enrolment rates in the future. Also, to verify GM(1,1) can predict the same accurate number as NCES (National Center for Education Statistics). In the future, the GM(1,1) model can provide a short-term prediction on education [9] . It is clear that research on Engel's coefficient is abundant, but research on education as a proportion of household expenditure is limited. It is the first attempt to compare the changing trend of the proportion of EE and FE in the total household expenditure in two regions of a developing country and a developed country.
In this study, Heilongjiang (HLJ) province in China and Ontario province in Canada were chosen as comparable research regions. The reason is because HLJ is representative of a province in a developing country while Ontario is a special province in a developed country. HLJ is a landlocked province with relatively sluggish economic development. Compared with China's overall Gross Domestic Product (GDP) per capita of 59,210 CNY (about 7,786 USD) in 2017, GDP per capita of HLJ was only 41,916 CNY (about 6,208 USD) [10] . In addition, in 2017, the GDP growth rate of HLJ province was 6.3% [11] , much lower than the national annual GDP growth rate of 6.9% [12] . As the highest GDP province, Ontario's GDP reached 857,384M CAD (about 608,158M USD), contributing to 38.55% of Canadian total GDP (2,223,856M CAD, about 1,577,421M USD) in 2018 [13] .
In this study, a linear regression model is used to predict trends of both FE and EE proportion to the total household expenditure in HLJ and Ontario.

Methods
Regression analysis generates a 'best-fit' mathematical equation that can be used to predict the values of the dependent variable as a function of the independent variable. In this study, a simple linear regression model is adopted, which is an effective predicting method used in many fields, such as finance, medicine, and economy.
This method mainly studies the relationship between the dependent variable and the independent variable. The equation of the linear regression is written as follows: (1) Where x is the independent variable, y is the dependent variable, a and b are regression parameters estimated from historical data, and Ɛ is an error random variable that has expected value 0. The ordinary least square (OLS) is used to estimate the parameters a and b.
Intuitively, OLS is fitting a line through the sample points such that the sum of squared residuals is as small as possible [14] .
The estimated regression line is written as: In Equation (2), The significance test of the linear regression involves F-test, t-test, R 2 and confidence interval.
The t-test can be used to test whether the mean of a random variable is equal to any particular number, even when the variance of the random variable is unknown. The t-test is written as follows: (5) t distribution critical value with significance level and (n-2) degrees of freedom, where n is the number of samples.
The F-test tests the significance of the equation and verifies the interpretation ability of x to y. The F-test is as follows: Where SSE is the sum of squared residuals, SSR is the sum of squares for regression and F statistic with (k-1) and (n-k) degrees of freedom. R 2 is goodness of fit, or the volatility of y relative to its mean. The larger R 2 is, the better the regression fitting is. The expression of the R 2 is as follows: The sample means are y and x. A confidence interval is used to measure the degree of uncertainty for predicting. In this study, the confidence interval refers to the probability that the predicted parameter will fall between two set values with 95% confidence level. The form of the confidence interval is written as follows.

Data
The linear regression model incorporated 2006 to 2017 data on Engel's coefficients, EE, and total household expenditure in HLJ and Ontario. The data for HLJ are obtained from the annual social development report [15][16] , shown in Table 1. The data for Ontario are obtained from Statistics Canada [17] , shown in Table 2.   The FE proportion curve for HLJ and Ontario can be seen from Figure 1, the FE proportion curve for HLJ shows a declining trend with fluctuation while the FE proportion curve for Ontario shows a more stable trend. Figure 2 shows that the EE in HLJ and Ontario have been overall increasing from 2006 to 2017. HLJ's EE proportion increased at an average of 0.08% per year. Ontario's EE proportion increased at an average of 0.04% per year. HLJ's EE proportion increased at a twice faster speed than Ontario's did.

HLJ's Engel's coefficient regression model
From the data in Table 1, OLS can be used to calculate the linear regression model for HLJ's Engel's coefficient as follows: y 1 = -0.7101x + 1461.6547 In this model, the slope is -0.7101, representing the decrease in HLJ's Engel's coefficient. The y-intercept is 1461.6547. The Engel's coefficient will gradually decrease with the increase of time x. The goodness of fit R 2 is 0.5127. HLJ's Engel's coefficient fitted value and original data are shown in Table 3 and Figure 4.   Figure 3, the coefficient of determination R 2 is 0.5127. F-test is 10.5224, t-test is 3.3192, -3.2438, so HLJ's regression model passes the F-test and t-test.

Ontario's Engel's coefficient regression model
From the data in  Table 4 and Figure 6.

HLJ's EE regression model
From the data in Table 1, OLS can be used to calculate the linear regression model for HLJ's EE as follows： z1 = 0.1207x -236.7083 In this model, the slope is 0.1207, representing the increase of the EE proportion in HLJ. The y-intercept is -236.7083. The EE proportion will gradually increase with the increase of time x. The goodness of fit R 2 is 0.3207. The EE proportion in HLJ fitted value and original data are shown in Table 5 and Figure 8.

Ontario's EE regression model
From the data in Table 2 Table 6 and Figure 10. In Figure 9, the coefficient of determination R 2 equals 0.6519. F-test is 18.7234, t-test is 4.3271, -4.2477. Ontario's regression model can pass the F-test and t-test.    Figure 11 shows the ratio curves of EE to FE for HLJ and Ontario from 2006 to 2017. These two curves show an increasing trend with fluctuation. The EE to FE proportion for HLJ decreased from 19.04 to 14.42 from 2006 to 2012 and increased from 20.27 to 24.82 for the period of 2012-2017. From 2008 to 2012, EE as a fraction of total household expenditure in HLJ is higher than that in Ontario. The FE proportion decreasing and the EE proportion increasing caused this increasing trend. The EE proportion in HLJ is higher than Ontario, but because the FE proportion in total expenditure is much higher, from 2008 to 2012, the ratio of EE to FE in HLJ is lower than Ontario. With the diminishing of the FE proportion in total expenditure, from 2013 to 2017, the ratio of EE to FE in HLJ is gradually higher than Ontario.

Prediction and Comparison
The linear regression model is then used to forecast Engel's coefficient and EE from 2018 to 2027, as shown in Table 8 Table 8 and Figure 8. The predictions of EE are shown in Table 9 and Figure 9.  Figure 12 the trend of Engel's coefficient in HLJ and Ontario. Engel's coefficients in both provinces are forecasted to gradually decrease in the period of 2018 to 2027. Ontario's Engel's coefficient to 12.76%, while HLJ's Engel's coefficient to 20.89%. Engel's coefficient in HLJ is expected to decrease at a greater rate than in Ontario. As time goes on, the difference between the Engel's coefficient of the two provinces will likely become smaller and smaller.  Figure 13. The EE proportion in HLJ and Ontario from 2018 to 2027 Figure 13 the trend of the EE proportion in HLJ and Ontario. In the next decade, the EE in the two provinces are gradually increasing. The EE in HLJ increased faster than in Ontario. As time goes on, the differences between the EE of the two provinces could become smaller and smaller. But in 2027, the EE proportion in HLJ is 7.93%, and the EE proportion in Ontario is 4.21%, so the EE in HLJ is still higher than Ontario's.

Historical data comparison: from 2006 to 2017
(1) Engel's coefficient in HLJ province decreased from 33.3% in 2006 to 28.7% in 2017 with an average annual decrease rate of 4.6%.
(2) It indicates that the standard of living in HLJ's residents is still improving. (4) The two provinces both show a decreasing trend in their Engel's coefficients. The decreasing rate in HLJ's is higher than Ontario's, but Ontario's Engel's coefficient is lower than HLJ's Engel's coefficient. The standard of living of Ontario's residents is still higher than HLJ's residents' living level.
(5) The EE proportion in HLJ has increased from 6.34% in 2006 to 7.12% in 2017, and Ontario's EE proportion increased from 2.66% in 2006 to 3.34% in 2017. The increasing trend of the two provinces indicates that residents spend more and more money on education. From 2006 to 2017, the EE proportion in HLJ increased by an average of 0.08% per year. The EE proportion in Ontario increased by an average of 0.04% per year. Although both provinces have shown an increasing trend, HLJ's EE proportion is still higher than Ontario's.

Conclusion
Using data from 2006 to 2017, a linear regression model is used to analyze trends in Engel's coefficient and the EE proportion in HLJ and Ontario, as well as to forecast 2018 to 2027. Key insights from the forecast include: (1)HLJ's Engel's coefficient shows a decreasing trend, with average annual decline rate of 0.38% during the period while the change of Ontario's Engel's coefficient during the research period is relatively steady, with an average annual decline rate of only 0.14%. Ontario's Engel's coefficient is lower than HLJ's Engel's coefficient in each year.
(2)HLJ's EE shows an increasing trend, with average annual increase rate of 0.08% during the period while the change of Ontario's EE during the research period is relatively steady, with an average annual increase rate of only 0.04%.
(3)The ratio of EE to FE in HLJ shows an increasing trend, with average annual increase rate of 0.53% during the period; the ratio of EE to FE in Ontario shows an increasing trend, with average annual increase rate of 0.58% during the period.
(4)The predicted results and comparison of Engel's coefficient and EE show that HLJ's standard of living is relatively low. In the future, HLJ's Engel's coefficient will reach 20.89% in 2027, while Ontario's Engel's coefficient will reach 12.76% in 2027. The living level of residents in the two provinces is rising, and the differences are getting smaller and smaller.