Prediction and Analysis of the Gap of Per Capita Annual Income between Rural and City Households based on ARMA model
Abstract This paper applies ARMA model into the analysis of the gap of per capita annual income between rural and city households from 1978 to 2011. Firstly, it builds up several ARMA models based on 1978-2009 data; and then predicts the income gaps of 2010 and 2011. Compared to the real income gaps of 2010 and 2011, this paper shows that ARMA model works well in this time series prediction. Keyword B-J model, ARMA model, per capita annual income of rural and city households

1 Introduction
With the rapid development of the China economy, people’s life levels are rising year after year. However, there’re many social problems in the booming economy, such as the income gap between rural and city households. The increasing gap of per capita annual income between rural and city households has been a heat topic in the society, since it may result in the inequality in the social welfare distributions. Also, it will discourage the rural households’ working efficiency and reduce their happiness. In order to deal with this social problem in a proper way, many researchers have been thinking about it and offering advice to the policy makers. LU (2004) says that the phenomenon of urbanization has an important effect on the income gap between rural and city households [1]. YAO (2005) analyzes the relationship between the unbalanced financial development and the income gap of rural and city households [2]. WANG (2009) points out the negative effect of the increasing income gap on the China economy [3]. This paper aims at analyzing the time series of the income gap and offering the future predictions. Since the ARMA model, sometimes called Box-Jenkins model, is a very useful tool for understanding and predicting future values in the time series [4, 5], we apply it in this paper.

2 Data
2.1 Data source
We get the 1978-2011 per capita annual income of rural and city households
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from http://www.stats.gov.cn/ and http://www.gov.cn/. Table 1 1978-2011 per capita income of rural and city households Year City Rural gap Year City Rural gap 1978 343.4 133.6 209.8 1995 4283 1577.7 2705.3 1979 387 160.2 226.8 1996 4838.9 1926.1 2912.8 1980 477.6 191.3 286.3 1997 5160.3 2090.1 3070.2 1981 491.9 223.4 268.5 1998 5425.1 2162 3263.1 1982 526.6 270.1 256.5 1999 5854 2210.3 3643.7 1983 564 309.8 254.2 2000 6280 2253.4 4026.6 1984 651.2 355.3 295.9 2001 6859.6 2366.4 4493.2 1985 739.1 397.6 341.5 2002 7702.8 2475.6 5227.2 1986 899.6 423.8 475.8 2003 8472.2 2622.2 5850 1987 1002.2 462.6 539.6 2004 9421.6 2936.4 6485.2 1988 1181.4 544.9 636.5 2005 10493 3254.9 7238.1 1989 1373.9 601.5 772.4 2006 11759.5 3587 8172.5 1990 1510.2 686.3 823.9 2007 13786 4140 9646 1991 1700.6 708.6 992 2008 15780.8 4760.6 11020.2 1992 2026.6 784 1242.6 2009 17175 5153 12022 1993 2577.4 921.6 1655.8 2010 19109 5919 13190 1994 3496.2 1221 2275.2 2011 21810 6977 14833

2.2 Summary statistics
We use the income sample from 1978 to 2009 and leave 2010 and 2011 for prediction validation. Per capita annual income of rural households:

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12

10

Series: RURAL Sample 1978 2009 Observations 32 Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque-Bera Probability
0 1000 2000 3000 4000 5000

8

6

4

1622.228 1071.300 5153.000 133.6000 1432.353 0.881086 2.824325 4.181485 0.123595

2

0

Figure 1 Per capita annual income of rural households Per capita annual income of city households:

16 14 12 10 8 6 4 2 0 0 2500 5000 7500 10000 12500 15000 17500

Series: CITY Sample 1978 2009 Observations 32 Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque-Bera Probability 4788.772 3036.800 17175.00 343.4000 4828.103 1.091293 3.219928 6.416063 0.040436

Figure 2 Per capita annual income of city households Gap of per capita annual income between rural and city households:

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16 14 12 10 8 6 4 2 0 0 2000 4000 6000 8000 10000 12000

Series: GAP Sample 1978 2009 Observations 32 Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque-Bera Probability 3166.544 1965.500 12022.00 209.8000 3406.459 1.172749 3.369609 7.517296 0.023315

Figure 3 Gap of per capita annual income between rural and city households Time series plots of rural, city and gap:

20,000

16,000

12,000

8,000

4,000

0 1980 1985 1990 RURAL 1995 CITY 2000 GAP 2005

Figure 4 Time series plots of rural, city and gap Ratio between rural and city:

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9 8 7 6 5 4 3 2 1 0 0.30 0.35 0.40 0.45 0.50 0.55

Series: RATIO_R_C Sample 1978 2009 Observations 32 Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque-Bera Probability 0.397222 0.393548 0.549291 0.300029 0.075178 0.479546 2.328200 1.828231 0.400871

Figure 5 Ratio between rural and city (1)

RATIO_R_C
.56 .52 .48 .44 .40 .36 .32 .28 1980 1985 1990 1995 2000 2005

Figure 6 Ratio between rural and city (2) In the 32 years from 1978 to 2009: (1) The per capita annual income of rural households keeps increasing from 133.6 in 1978 to 5153 in 2009 with the mean of 1622.228. (2) The per capita annual income of city households keeps increasing from 343.4 in 1978 to 17175 in 2009 with the mean of 4788.772. (3) However, the gap of per capita annual income between rural households and city
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households also keeps increasing from 209.8 in 1978 to 12022 in 2009. (4) The ratio between rural and city, though reaching its peak around the mid-1980s, has the general trend of decreasing in the last 20 years.

2.3 Data transformation
Since the gap of per capita annual income between rural households and city households has the exponential trend, we transform it into log(gap), which is almost linear.

LOGGAP
10

9

8

7

6

5 1980 1985 1990 1995 2000 2005

Figure 7 log(gap)

3 Model
3.1 Trend regression
Table 2 Trend regression

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Model: E[loggap] = -276.0548 + 0.142158 * year Adj-R = 0.982925 is very close to 1, but D-W statistic = 0.270908 < dL(32, 1) = 1.37. Thus there exists first-order autoregression in residuals.
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3.2 ARMA
3.2.1 Unit root test
Unit root test of loggap: Table 3 Unit root test of loggap

Since the t statistic is larger than the critical values at all three levels, the series is not stationary. Thus, we need to take the first difference to make the series stationary. First difference: dloggapt = (1 - L)loggapt

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DLOGGAP
.4

.3

.2

.1

.0

-.1 1980 1985 1990 1995 2000 2005

Figure 8 dloggap Compared to loggap, we cannot see a trend in dloggap. Besides, dloggap is almost positive. Unit root test of dloggap: Table 4 Unit root test of dloggap

Comparing the t statistic with the critical values at all three levels, we can see that the series is stationary at 10% and 5% level, but not stationary at 1% level. Thus, we take the second difference to make the series stationary. Second difference: ddloggapt = (1 - L)dloggapt

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DDLOGGAP
.2

.1

.0

-.1

-.2

-.3

-.4 1980 1985 1990 1995 2000 2005

Figure 9 ddloggap We can see that ddloggap also doesn’t have a trend and changes around the zero axis. Unit root test of ddloggap: Table 5 Unit root test of ddloggap

The t statistic is smaller than all the critical values at the three levels, so the series is stationary at all the three levels.

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3.2.2 ACF and PACF:
Table 6 ACF and PACF of ddloggap

From the ACF and the PACF of ddloggap, we can see that the series is stationary, since they are all around zero. Now we determine ARMA(p, q). Since the ACF is not significantly different from zero after the first lag, we set q = 1. Since the PACF is significantly different from zero at the first, fourth and fifth lag, we set p = 1, 4 and 5. Hence, we should compare models – ARMA(1, 1), ARMA(4, 1) and ARMA(5, 1). Before we test the models, we should test if the series of ddloggap has the mean of zero.

3.2.3 Zero mean test scalar m = @mean(ddloggap) m = 3.0315 * 10-4 scalar s = @stdev(ddloggap) * @sqrt((1 + 2 * (-0.308)) / (@obs(ddloggap))) s = 0.0115 Since m is in (-s, s), we can say this series has a zero mean.

3.2.4 ARMA(1, 1)
Equation estimation: d(log(gap), 1) ar(1) ma(1).

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Table 7 ARMA(1, 1)

Table 8 ARMA(1, 1) residual test

3.2.5 ARMA(4, 1)
Equation estimation: d(log(gap), 2) ar(1) ar(2) ar(3) ar(4) ma(1).

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Table 9 ARMA(4, 1)

Table 10 ARMA(4, 1) residual test

3.2.6 ARMA(5, 1)
Equation estimation: d(log(gap), 2) ar(1) ar(2) ar(3) ar(4) ar(5) ma(1).
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Table 11 ARMA(5, 1)

Table 12 ARMA(5, 1) residual test

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3.2.7 Comparison
Table 13 Comparison of ARMA(1, 1), ARMA(4, 1) and ARMA(5, 1) R2 AIC SC ARMA(1, 1) 0.407970 -2.204343 -2.110046 ARMA(4, 1) 0.447036 -2.309928 -2.067986 ARMA(5, 1) 0.702771 -2.944820 -2.652290 We can see that ARMA(5, 1) has the largest R2 and smallest AIC and SC. ARMA(5, 1) also has inverted AR and MA roots inside the unit circle. What’s more, ARMA(5, 1)’s residuals are all around zero. Thus, we choose ARMA(5, 1).

3.2.8 Prediction and comparison for 2010 and 2011
Table 14 Prediction and comparison for 2010 and 2011 gap\year 2010 predicted gap 13065.93 real gap 13190 predicted / real 99.1%

2011 13859.23 14833 93.4%

We see that the predicted gap of 2010 is very good, while that of 2011 is good.

4 Conclusion and Improvement
In this paper, we apply ARMA model in the analysis of the gap of per capita annual income between rural and city households from 1978 to 2011. We find that ARMA model makes a good predicting job in the time series of the gap. However, in order to get a more precise result, we need more data. We also can try further difference, since all the three ARMA models above have a root close to 1, which results in instability.

Reference
[1] LU Ming; Urbanization and income gap of rural and city; Economic Research, 2004. [2] YAO Yaojun; The empirical analysis of the relationship between the financial development and the income gap of rural and city; Financial Research, 2005. [3] WANG Shaoping; 中国城乡收入差距对实际经济增长的阈值效应; 中国社会科 学, 2009. [4] http://en.wikipedia.org/wiki/Autoregressive%E2%80%93moving-average_model [5] Walter Enders; Applied Econometric Time Series; John Wiley & Sons, 2010: P52.
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