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Extend Linear Expenditure System model is a collection of multiple linear models, and modeling is a clearly tedious process. The innovation of this paper is trying to find a simple way of ELES modeling, which means, in order to omit the modeling process one by one, we try to use Excel functionality to create a model workplace. As long as you replace the original sample data in the workspace, you can get the results you want.

ELES model is based on the assumption that people’s demand for various goods or services in a certain period is determined by people’s incomes and prices of various commodities, and the needs of people for a variety of goods are divided into two parts: people’s basic needs and additional requirements exceeding the basic needs. At the same time, the model assumes that people’s basic needs are irrelevant to the level of income, and people will allocate the residual income in accordance with the marginal propensity to consume to additional demands after people’s basic needs are met. However, it is mainly used to calculate the marginal propensity to consume, basic consumption, income elasticity and price elasticity, which are reference quantities to study on structure change.

There are structure researches on ELES model in paper [

ELES model is a collection of multiple linear models, and in the case of many types of data, ELES modeling is a clearly tedious process and huge workload. If you need to test it, the equations are as much as stars in the sky, the quantity will make you mad. The innovation of this paper is trying to make the modeling method more simple, which means, in order to omit the modeling process one by one, we try to use Excel functionality to create a model workplace. As long as you replace the original sample data in the workspace, you can get the results you want.

The basic expression of extended linear expenditure system (the abbreviation is ELES) is

or service i,

Transform the basic expression to get

assume

then,

Correspondingly, the econometric model is

Therefore, the sample data of consumption and income can be used to estimate model’s parameters.

At the same time, via summation on either side of the equal sign of formula ①, calculation formula of total

expenditure on basic consumption can be obtained as

get

In addition, you can also derive the income elasticity of commodity or service i

the cross price elasticity of commodity or service i

the own-price elasticity of commodity or service i

Suppose residents from n different income levels have m consumer items. Sample is used to estimate parameters of m regression equations by ordinary least squares estimate (its abbreviations is OLS), process is as follows:

Assume one of the m equations is

its residual sum is

According to the principle of OLS, equations are

Transform them into

then,

In this section, data of the annual per capita consumption expenditure of urban households in 2011, divided into n levels on the basis of their income, is selected as samples to illustrate the established ELES model workspace in Excel, steps as follows:

1) Excel book is divided into four workspaces, original data area (A1:J10), the preparation area for calculation (B11:J20), ElES workspace (A21:J23) and elasticity zone (A24:J33).

2) Income group labels are established in the A1:A10, each consumption item label is established in the B2:J10, input sample data in the B2:J10, wherein, data in B10:J10 is the average level of consumption expenditures of urban residents and from NBS (National Bureau of Statistics of China).

3) Input averaging formula AVERAGE (B2:B9) in B11 to figure out the average of B2:B9, and copy the formula into C11:J11.

4) Input the formula B2*B2 in B12, and copy the formula into the rectangular region B12:J20.

5) Input the formula (C20 − 8*B11*C11)/(B20-8*B11^2) in C22, and copy the formula into D22:J22. The formula calculates the slope of ELES model, which is the marginal propensity to consume.

6) Input the formula C11 − C22*B11 in C21, and copy the formula into D21:J21.

7) Input the formula C21 + C22*B25 in C23, and copy the formula into D23:J23, also, it calculates the basic expenditure for purchasing each commodity or service.

8) Input the formula C22*B10/C10 in C24, and copy the formula into D24:J24, also, it calculates the income elasticity of each commodity or service.

9) Input the formula SUM(C21:J21)/(1 − SUM(C22:J22)) in C25, which calculates total basic expenditure of urban households.

10) Input the formula-D22*C23/D10 in C27, and copy the formula into the rectangular region C26:J33. What is more, the formula calculates the cross price elasticity of every commodity or service.

11) Input the formula-C22*(C23 + B10 − B25)/C10 in C26, and copy the formula into cells D27, E28, F29. G30, H31, I32, J33. As well as, the formula calculates the own price elasticity of every commodity or service.

Therefore,

Replace the sample data in Original data area with data of the annual per capita consumption expenditure of ur- ban households in 2012, to verify the above workspace in excel. The results are in

A | B | C | D | E | F | G | H | I | J | |
---|---|---|---|---|---|---|---|---|---|---|

1 | Income percentile | Income per capita | Food | Clothing | Housing | Household appliance | Tra&com | Culure, Edu&ent | Medical care | Others |

2 | Lowest income | 6876.1 | 2948.9 | 608 | 749 | 335.3 | 501.1 | 642.7 | 483.6 | 163.2 |

3 | Poor | 5398.2 | 2618.9 | 489.5 | 675.7 | 261.5 | 397.6 | 541.5 | 449.9 | 141.1 |

4 | Low income | 10672 | 3715.9 | 913.4 | 874.8 | 489.7 | 841.4 | 876.7 | 579.2 | 218.3 |

5 | Lower middle income | 14,498.3 | 4535.8 | 1250.9 | 1023 | 666.5 | 1150.2 | 1163.1 | 759.8 | 323.6 |

6 | Middle income | 19,544.9 | 5467.1 | 1628.6 | 1232.7 | 923.4 | 1762.3 | 1637.1 | 911 | 466.1 |

7 | Upper income | 26,420 | 6515 | 2045.8 | 1627.6 | 1277.1 | 2647.9 | 2238.1 | 1136.9 | 672.4 |

8 | High income | 35,579.2 | 7789.9 | 2598.3 | 2116.8 | 1736.4 | 3963 | 3155.7 | 1512.4 | 1033.8 |

9 | Highest income | 58,841.9 | 9681.7 | 3699.1 | 3272.7 | 2625.9 | 6912.7 | 5060.6 | 1959.8 | 1971.2 |

10 | Average | 21,809.8 | 5506.3 | 1674.7 | 1405 | 1023.2 | 2149.7 | 1851.7 | 969 | 581.3 |

11 | The preparation area for calculation | 22,228.8 | 5409.15 | 1654.2 | 1446.54 | 1039.48 | 2272.025 | 1914.44 | 974.075 | 623.7125 |

12 | 4.7E+07 | 20,276,931 | 4,180,669 | 5,150,199 | 2,305,556 | 3,445,614 | 4,419,269 | 3,325,282 | 1,122,180 | |

13 | 2.9E+07 | 14,137,346 | 2,642,419 | 3,647,564 | 1,411,629 | 2,146,324 | 2,923,125 | 2,428,650 | 761,686 | |

14 | 1.1E+08 | 39,656,085 | 9,747,805 | 9,335,866 | 5,226,078 | 8,979,421 | 9,356,142 | 6,181,222 | 2,329,698 | |

15 | 2.1E+08 | 65,761,389 | 1.8E+07 | 1.5E+07 | 9,663,117 | 16,675,945 | 1.7E+07 | 1.1E+07 | 4,691,650 | |

16 | 3.8E+08 | 1.07E+08 | 3.2E+07 | 2.4E+07 | 1.8E+07 | 34,443,977 | 3.2E+07 | 1.8E+07 | 9,109,878 | |

17 | 7E+08 | 1.72E+08 | 5.4E+07 | 4.3E+07 | 3.4E+07 | 69,957,518 | 5.9E+07 | 3E+07 | 17,764,808 | |

18 | 1.3E+09 | 2.77E+08 | 9.2E+07 | 7.5E+07 | 6.2E+07 | 1.41E+08 | 1.1E+08 | 5.4E+07 | 36,781,777 | |

19 | 3.5E+09 | 5.7E+08 | 2.2E+08 | 1.9E+08 | 1.5E+08 | 4.07E+08 | 3E+08 | 1.2E+08 | 1.16E+08 | |

20 | 6.2E+09 | 1.27E+09 | 4.3E+08 | 3.7E+08 | 2.9E+08 | 6.83E+08 | 5.3E+08 | 2.4E+08 | 1.89E+08 | |

21 | Intercept | 2416.02 | 308.85 | 355.63 | 35.97 | −480.89 | −0.16 | 316.80 | −141.31 | |

22 | Marginal propensity | 0.13 | 0.06 | 0.05 | 0.05 | 0.12 | 0.09 | 0.03 | 0.03 | |

23 | Basic expenditure | 3282.84 | 698.47 | 671.56 | 326.59 | 316.36 | 554.31 | 507.15 | 80.24 | |

24 | Income elasticity | 0.533 | 0.788 | 0.762 | 0.962 | 1.256 | 1.014 | 0.666 | 1.291 | |

25 | Total basic expenditure | 6437.52 | ||||||||

26 | Price elasticity | Food | −0.456 | −0.017 | −0.016 | −0.008 | −0.008 | −0.014 | −0.012 | −0.002 |

27 | Clothing | −0.119 | −0.581 | −0.024 | −0.012 | −0.011 | −0.020 | −0.018 | −0.003 | |

28 | Housing | −0.115 | −0.024 | −0.560 | −0.011 | −0.011 | −0.019 | −0.018 | −0.003 | |

29 | Hou^{*} app^{*} | −0.145 | −0.031 | −0.030 | −0.693 | −0.014 | −0.024 | −0.022 | −0.004 | |

30 | Tra&com | −0.189 | −0.040 | −0.039 | −0.019 | −0.904 | −0.032 | −0.029 | −0.005 | |

31 | Cul, edu&ent | −0.153 | −0.032 | −0.031 | −0.015 | −0.015 | −0.741 | −0.024 | −0.004 | |

32 | Medical | −0.100 | −0.021 | −0.020 | −0.010 | −0.010 | −0.017 | −0.485 | −0.002 | |

33 | others | −0.194 | −0.041 | −0.040 | −0.019 | −0.019 | −0.033 | −0.030 | −0.915 |

we use eview 5.0 software to regress the equations, whose results are in

Therefore, comparing

A | B | C | D | E | F | G | H | I | J | |
---|---|---|---|---|---|---|---|---|---|---|

1 | Income percentile | Income per capita | Food | Clothing | Housing | Household appliance | Tra&com | Culure, edu&ent | Medical care | Others |

2 | Lowest income | 8215.1 | 3310.4 | 706.8 | 832.6 | 405.4 | 602.8 | 723 | 548.3 | 172.1 |

3 | Poor | 6520 | 2979.3 | 589.8 | 759.6 | 333.1 | 495.3 | 613.9 | 466.5 | 129.2 |

4 | Low income | 12,488.6 | 4147.4 | 1045.5 | 924.5 | 569.3 | 954.4 | 1034.9 | 669.6 | 265 |

5 | Lower middle income | 16,761.4 | 5028.6 | 1408.2 | 1160.4 | 760 | 1393 | 1326.6 | 832.9 | 371.1 |

6 | Middle income | 22,419.1 | 6061.4 | 1765.9 | 1384.3 | 1033.6 | 2063.3 | 1785.5 | 1096 | 529.9 |

7 | Upper income | 29,813.7 | 7102.4 | 2213.8 | 1708.7 | 1346.2 | 2960.6 | 2449.1 | 1248.9 | 800.4 |

8 | High income | 39,605.2 | 8561 | 2767.5 | 2154.3 | 1827.9 | 4304.1 | 3432.8 | 1580 | 1169.4 |

9 | Highest income | 63,824.2 | 10,323.1 | 3928.5 | 3123.3 | 2807.3 | 7971.1 | 5431.6 | 1951.1 | 2125.7 |

10 | Average | 24,564.7 | 6040.9 | 1823.4 | 1484.3 | 1116.1 | 2455.5 | 2033.5 | 1063.7 | 657.1 |

11 | The preparation area for calculation | 24,955.9 | 5939.2 | 1803.25 | 1505.96 | 1135.35 | 2593.075 | 2099.68 | 1049.16 | 695.35 |

12 | 6.7E+07 | 27,195,267 | 5,806,433 | 6,839,892 | 3,330,402 | 4,952,062 | 5,939,517 | 4,504,339 | 1,413,819 | |

13 | 4.3E+07 | 19,425,036 | 3,845,496 | 4,952,592 | 2,171,812 | 3,229,356 | 4,002,628 | 3,041,580 | 842,384 | |

14 | 1.6E+08 | 51,795,220 | 1.3E+07 | 1.2E+07 | 7,109,760 | 11,919,120 | 1.3E+07 | 8,362,367 | 3,309,479 | |

15 | 2.8E+08 | 84,286,376 | 2.4E+07 | 1.9E+07 | 1.3E+07 | 23,348,630 | 2.2E+07 | 1.4E+07 | 6,220,156 | |

16 | 5E+08 | 1.36E+08 | 4E+07 | 3.1E+07 | 2.3E+07 | 46,257,329 | 4E+07 | 2.5E+07 | 11,879,881 | |

17 | 8.9E+08 | 2.12E+08 | 6.6E+07 | 5.1E+07 | 4E+07 | 88,266,440 | 7.3E+07 | 3.7E+07 | 23,862,885 | |

18 | 1.6E+09 | 3.39E+08 | 1.1E+08 | 8.5E+07 | 7.2E+07 | 1.7E+08 | 1.4E+08 | 6.3E+07 | 46,314,321 | |

19 | 4.1E+09 | 6.59E+08 | 2.5E+08 | 2E+08 | 1.8E+08 | 5.09E+08 | 3.5E+08 | 1.2E+08 | 1.36E+08 | |

20 | 7.6E+09 | 1.53E+09 | 5.1E+08 | 4.1E+08 | 3.4E+08 | 8.57E+08 | 6.4E+08 | 2.8E+08 | 2.3E+08 | |

21 | Intercept | 2649.14 | 341.01 | 461.20 | 44.59 | −667.84 | −28.68 | 383.35 | −175.76 | |

22 | Marginal propensity | 0.13 | 0.06 | 0.04 | 0.04 | 0.13 | 0.09 | 0.03 | 0.03 | |

23 | Basic expenditure | 3537.07 | 735.64 | 743.16 | 338.97 | 212.22 | 545.73 | 563.04 | 59.34 | |

24 | Income elasticity | 0.536 | 0.789 | 0.693 | 0.962 | 1.307 | 1.030 | 0.616 | 1.305 | |

25 | Total basic expenditure | 6735.17 | ||||||||

26 | Price elasticity | Food | −0.466 | −0.016 | −0.016 | −0.007 | −0.005 | −0.012 | −0.012 | −0.001 |

27 | Clothing | −0.114 | −0.597 | −0.024 | −0.011 | −0.007 | −0.018 | −0.018 | −0.002 | |

28 | Housing | −0.100 | −0.021 | −0.524 | −0.010 | −0.006 | −0.015 | −0.016 | −0.002 | |

29 | Hou^{*} app^{*} | −0.139 | −0.029 | −0.029 | −0.711 | −0.008 | −0.021 | −0.022 | −0.002 | |

30 | Tra&com | −0.188 | −0.039 | −0.040 | −0.018 | −0.960 | −0.029 | −0.030 | −0.003 | |

31 | Cul, edu&ent | −0.148 | −0.031 | −0.031 | −0.014 | −0.009 | −0.771 | −0.024 | −0.002 | |

32 | Medical | −0.089 | −0.018 | −0.019 | −0.009 | −0.005 | −0.014 | −0.461 | −0.001 | |

33 | Others | −0.188 | −0.039 | −0.039 | −0.018 | −0.011 | −0.029 | −0.030 | −0.950 |

C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | |
---|---|---|---|---|---|---|---|---|

Intercept | 2666.681 | 345.9387 | 460.5944 | 44.34132 | −677.737 | −32.43652 | 386.2129 | −178.5756 |

Slope | 0.131814 | 0.058587 | 0.041865 | 0.043708 | 0.130679 | 0.085289 | 0.026676 | 0.034909 |

t-Statistic | 12.45697 | 24.57792 | 69.801650 | 89.87068 | 27.84007 | 59.28272 | 13.61498 | 27.82504 |

P-t | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

p-white | 0.060171 | 0.098096 | 0.62869 | 0.280624 | 0.293578 | 0.584147 | 0.096307 | 0.283249 |

In the above discussion, perhaps the selected data happens to have just no heteroscedasticity. However, if there is heteroscedasticity data, weighted least squares estimate (its abbreviation is WLS) will be used to estimate the parameters, whose principle is adding a weight in front of residuals, and the derivation process is as follows:

Its residual sum is

according to the principle of WLS, equations are

and transform them into

then,

In the process of the establishment, put the

Ningning Song,Yiqing Liu, (2015) Convenient Way of Extend Linear Expenditure System Modeling without Regression. Open Journal of Statistics,05,519-524. doi: 10.4236/ojs.2015.56055