Optimal Combined Prediction of Industry Structure Base on GM(1, 1)

Optimal Combined Prediction of Industry Structure Base on GM(1, 1)
and ARMA(p, q) Model
ZHANG Yuchun, SHI Xiaolei
School of Economics and Management, Lanzhou University of Technology, China, 730050
[email protected]
Abstract: The optimal combined prediction model of industry structure base on ARMA(p, q) and
GM(1,1) model is proposed, it bases minimal residual square sum, power weightiness is calculated by
linear programming. The combined prediction model absorbs two model’s excellence, it has more
prediction precision, the difficult problem of industry structure prediction is availability settled.
Jiangsu’s industry structure is predicted by combined prediction model, that provide scientific gist for
adjust industry structure.
Keyword: Industry structure, ARMA (p,q) model, GM(1,1) model, combined prediction model
1. Introduction
Adjust the industrial structure still is focus to promote economic coordinated development. In order to
realize the strategic adjustment of industrial structure, we must grasp the changes of industrial structure,
which requires to accurately predicting the future development trends of the industrial structure. There
are many factors that impact changes of industrial structure; it is difficult using classical econometric
accurate prediction. Time series model is no longer a different causal relationship between variables, but
rather look for time-series changes of their own, only the past behavior of the prediction to predict the
future [1]. Gray prediction requires small sample data, convenient operation, and high precision
short-term prediction [2]. In this paper, to predict the industrial structure of ARMA (p, q) and GM (1, 1)
portfolio model, in accordance with the thinking of smallest residual sum of squares, the use of linear
programming optimization calculate the weight, so that the model absorb the merits of the two models,
with prediction accuracy, an effective solution to the problem of the industrial structure difficult
prediction. And predict industrial structure in Jiangsu.
2. ARMA (p, q) model
Time series model has three kinds of types: autoregressive model AR, moving average model MA and
autoregressive moving average model ARMA [3]. ARMA (p, q) model is commonly used for a class of
stochastic time series model, be able to understand better the nature of time series of structural
characteristics, to achieve minimum variance within the meaning of the optimal prediction.
2.1. Autoregressive Model AR (p)
X t for its pre-value and a linear function of random items, can be expressed as
X t = ϕ1 X t −1 + ϕ 2 X t − 2 + L + ϕ p X t − p + u t
(1)
Time-series
Where: p autoregressive model for the order;
error.
ϕi
(i = 1,2, ..., p) for the model coefficients; u t for
2.2. Moving Average model MA (q)
If u t is not a hundred noise, which for the current and pre-and random error of the linear function. Can
be expressed as
u t = ε t − θ 1ε t −1 − L − θ q ε t − q
(2)
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Where: q as a model of order;
θj
(j = 1,2, ..., q) for the model coefficients;
ε t for hundred noise.
2.3. Autoregressive moving average model ARMA (p, q)
ARMA (p, q) are auto-regressive model and the combination of moving average models, time series
X t for the current and pre-error and random items, as well as its early stage of the value of a linear
function. Can be expressed as
X t = ϕ1 X t −1 + ϕ 2 X t − 2 + L + ϕ p X t − p + ε t − θ1ε t −1 − L − θ q ε t − q
(3)
The introduction of lag operator B, the model can be expressed as[4]
（4）
ϕ ( B ) X t = θ ( B )u t
ϕ ( B) = (1 − ϕ1 B − ϕ 2 B 2 − L − ϕ p B P )
θ ( B) = (1 − θ1 B − θ 2 B − L − θ q B q
ARMA (p, q) process conditions are stable characteristic equation ϕ ( B ) = 0 , and reversible condition
θ ( B) = 0 that all roots outside the B plane unit circle. If the original time series non-stationary time
series, the steady after the difference, and then ARMA modeling, is referred to as ARMA (p, d, q)
model, in which the number of d for the difference time.
3. Identify and estimate ARMA ((p, q) model
3.1. Model Identification
3.1.1 AR (p) model. k period variance for lag
（5）
γ k = E ( X t − k X t ) = ϕ1γ k −1 + ϕ 2γ k − 2 + L + ϕ p γ k − p
autocorrelation function
ρk =
γk
= ϕ1 ρ k −1 + ϕ 2 ρ k − 2 + L + ϕ p ρ k − p
γ0
（6）
Autocorrelation function ACF give the Xt and Xt-1 overall relevance, but the overall correlation between
variables may be covered up completely different from the implied relations. In contrast, Xt and Xt-k
between the partial autocorrelation function PACF is eliminated between variables Xt-1, ..., Xt-k+1
brought after indirectly related to the direct relevance, it values are known sequence Xt-1, ..., Xt-k +1
conditions, Xt and Xt-k inter-relations measure.
3.1.2 MA (q) model. k period lag variance for
σ ε2 (1 + θ12 + θ 22 + L + θ q2 )
k =0
 2
rk = E ( X t X t −1 ) = σ ε (−θ k + θ1θ k +1 + L + θ q − k ϑq ) 1 ≤ k ≤ q
0
k>q

（7）
autocorrelation function
k =0
1
rk 
ρ k = = (−θ k + θ 1θ k +1 + L + θ q −k θ q ) /(1 + θ 12 + L + θ q2 )
r0 
k>q
0
1≤ k ≤ q
（8）
MA (q) model identification rules: if the random sequence of auto-correlation function of cut-off tail,
that is, since the q after, ρ k = 0 (k> q); and its partial autocorrelation function are trailing, then this
sequence are moving average MA (q) sequence.
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3.1.3 ARMA (p, q) autocorrelation function can be regarded as the mixture of the autocorrelation
function of the MA (q) and the autocorrelation function of the AR (p).
When p = 0 it has censored nature; when q = 0 it has a trailing character; when p, q are not 0, it has a
trailing character. From the identification point of view, usually, ARMA (p, q) process partial
autocorrelation function (PACF) may be p lag before the tip has several distinct spikes, but p steps lag
of the start gradually moving to zero ; and its autocorrelation function (ACF) is at the q steps lag before
there are a few obvious tip column items from the q steps lag and begin to tend to zero.
3.2. The model parameters estimated
AR (p), MA (q), ARMA (p, q) select the order of the model further to calculate the unknown parameters
of model. Here to choose the least square method to estimate the parameter values, the parameters can
use Eviews4.0 to estimate[5].
4. GM (1, 1) modeling steps:
1) Acquisition of raw data out[6]
X (0) = {x ( 0) (1), x ( 0) (2),L , x (0) (n)} Where: n is the number of collected samples.
Where: n is the number of collected samples.
2) The original sequence to do I-AGO data generation
X (1) = {x (1) (1), x (1) (2),L , x (1) (n)}
k
Where: x
(1)
= ∑ x ( 0) (i ),
k = 1,2,L, n
i =1
Z (1) For X (1) the neighboring mean generation sequence[7]
Z (1) = {z (1) (2), z (1) (3),L , z (1) (n)}
Among them,
z (1) (k ) =
1 (1)
( x (k ) + x (1) (k − 1)) , k = 2,3,L , n
2
3) Set up differential albino equations
dx (1)
+ ax (1) = b , a, b Parameters are to be determined.
dt
4) The use of least-squares method to solve the parameters a, b
 x (0) (2)
 (0) 
x (3) 
Y =

M


(0)
 x (n)
− z (1) (2) 1

 (1)
a
− z (3) 1
T
−1 T

B=
b  = ( B B) B Y

M
 


(1)
− z (3) 1
( 0)
( 0)
The time response sequence of GM (1,1) model x ( k ) + az ( k ) = b for
b
b

xˆ (1) (k + 1) =  x (0) (1) − e −ak +
k = 1,2,L , n
a
a

；
（9）
Restore the value
b

xˆ ( 0) (k + 1) = (1 − e a ) x (0) (1) − e −ak
a

5. Combination forecasting model
； k = 1,2,L, n
（10）
Combination prediction is a prediction methods that weighted appropriate average of the prediction
outcome of each individual as the ultimate outcome, portfolio prediction theory suggests that: for a
159
problem of the prediction, since the combination prediction methods contained in all of single prediction
useful information, so that the number of prediction based on different assumptions of the linear
combination method can effectively improve the model fitting capabilities, enhance its ability to adapt
to future changes as well as the prediction accuracy. The key of prediction combination model that
combination weights are determined, the smallest residual sum of squares as the basis for calculating the
optimal combination weights of standards in the paper, use linear programming optimization to calculate
weights. The principle is: set up the actual value of an industry in a certain period for y it (i = 1,2,3; t =
1,2, ..., n), if the industry has m kinds of feasible prediction method, predictive value for y ijt (i = 1,2,3;
j = 1, ..., m; t = 1,2, ..., n), also located m kinds of prediction methods of weighting coefficients
normalized to satisfy constraint conditions:
m
∑W
ij
=1
(i = 1,2,3; j = 1,L m)
(11)
j =1
Combination forecasting model can be expressed as
m
yˆ it = ∑ Wij y ijt
(i = 1,2,3; j = 1,L m; t = 1,2,L n)
(12)
j =1
m
u it = y it − yˆ it = y it − ∑ Wij y ijt
(i = 1,2,3; j = 1,L m; t = 1,2,L n)
(13)
j =1
Weighting coefficients must satisfy non-negative constraint conditions so that the residual u it sum of
squares smaller as possible. The above model can be translated into the following linear programming
model
n
n
n
t =1
t =1
t =1
min U = ∑ u12t + ∑ u 22t + ∑ u 32t
(14)
m

−
y
 1t ∑ W1 j y1 jt = u1t
j =1

m

−
y
 2t ∑ W 2 j y 2 jt = u 2t
j =1

m

 y 3t − ∑ W3 j y 3 jt = u 3t

j =1
m
m
m
∑ W1 j yˆ1 jt + ∑ W 2 j y 2 jt + ∑ W3 j y 3 jt = 1
s.t. j =1
j =1
j =1
m
∑ W1 j = 1
 j =1
m
∑ W 2 j = 1
 j =1
m
 W =1
3j
∑
j =1

Wij ≥ 0 i = 1,2,3; j = 1,L , m.
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(15)
Solute the linear programming problem, you can get the weighting coefficients Wij of optimal
combination based on the smallest residual sum of squares, when the new historical data come in
ever-increasing, to retake the recent historical data, from the formula to recalculated Wij so as to arrive
at optimal portfolio predictive value.
6. Study
According to Statistical Yearbook of Jiangsu acquire the 1978-2005 year observed values of the
proportion of primary industry, secondary industry, tertiary industry y1 , y 2 and y 3 (%) in Jiangsu ①.
ARMA (p, q) theory can only be used for stationary time series, therefore required to test the proportion
of three industries smooth nature in Jiangsu. The three industries respectively ADF unit root test and
found that the proportion of three industries are non-stationary time series. After carried out the
first-order differential unit root test, three industries are smooth time series, that is all I (1). Three
industries in accordance with observations by the smallest AIC criteria use Eviews4.0 to set up three
industries differential time series ARMA (p, q) model②.
∆y11t = −0.950 − 0.739∆y11t −1 − 0.215∆y11t − 2 + ε t − 1.212ε t −1 − 0.994ε t −2
(16)
2
(-2.246) (-3.456) (-1.376)
(16.317) (6.979) R =0.546 DW=1.966
∆y 21t = 0.177 − 0.959∆y 21t −1 − 0.423∆y 21t − 2 + ε t − 1.037ε t −1 − 0.994ε t − 2
(17)
2
(0.429)
(-4.628)
(-2.448)
(27.248) (12.732) R =0.429 DW=1.790
∆y 31t = 0.779 + 0.266∆y 31t −1 + 0.296∆y 31t − 2 − 0.274∆y 31t −3 + ε t − 0.209ε t −1
(1.378)
(1.034)
(1.159)
(-1.137)
− 0.196ε t − 2 − 0.931ε t −3
(-2.856)
(14.933)
(-2.738)
(18)
2
R =0.419
DW=2.164
∆y i1t (i = 1,2,3; t = 1,2, ..., n) express the first prediction method of ARMA (p, q) differential three
industries. y i1t (i = 1,2,3; t = 1,2, ..., n) express the three industries observations of the first prediction
method. Difference model of the above appropriate transform available three industries ARMA (p, q)
model, and predict the proportion of three industries predictive value in Jiangsu's, as shown in table 1.
y11t = −0.950 + 0.261 y11t −1 + 0.527 y11t −2 + 0.215 y11t −3 + ε t − 1.212ε t −1 − 0.994ε t − 2 (19)
y 21t = 0.177 + 0.041 y 21t −1 + 0.527 y 21t − 2 + 0.432 y 21t −3 + ε t − 1.037ε t −1 − 0.994ε t − 2 (20)
y 31t = 0.779 + 1.266 y 31t −1 + 0.030 y 31t − 2 − 0.057 y 31t −3 + 0.274 y31t − 4 + ε t
− 0.209ε t −1 − 0.196ε t − 2 − 0.931ε t −3
(21)
According to original data appropriate metabolism, and set up three industries gray GM (1,1) prediction
model is as follows in Jiangsu, y i 2t ( k + 1) (i = 1,2,3; t = 1,2, ..., n; k = 1,2 ... , n) that the second
prediction method of GM (1,1) three industries time series, and forecast the proportion of three
industries in Jiangsu, in table 1.
①
Source: Jiangsu Statistical Bureau, Jiangsu Statistical Yearbook 2005
②
The number in brackets below the parameters estimated in Regression model are t statistics of corresponding
parameters.
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Table 1 ARMA (p, q) and GM (1,1) model prediction of the proportion of three industries
ARMA(p, q) Model
Model
GM(1,1) Model
Primary
industry
Secondary
industry
Tertiary
industry
Primary
industry
Secondary
industry
Tertiary
industry
2007
5.07
55.26
38.73
7.42
56.27
38.71
2008
5.12
55.37
39.51
6.86
55.14
38
2009
4.87
55.4
39.73
6.62
55.23
38.15
2010
3.22
55.71
41.07
6.04
55.31
38.65
Years
y12t (k + 1) = −665.582 exp(−0.057 * k ) + 695.082
y 22t (k + 1) = 9912.845 exp(0.004 * k ) − 9865.245
y 32 t (k + 1) = 704.653 exp(0.026 * k ) − 684.853
(22)
(23)
(24)
The data into the equation (14) and (15), acquire a combination of the primary industry weight (0.529,
0.471), the combination of second industry weight (0.529, 0.471) and the combination of the tertiary
industry weight (0.125, 0.875). Therefore acquire three industries final optimal prediction model in
Jiangsu.
The combination prediction model of three industries in Jiangsu:
yˆ1t = 0.529 y11t + 0.471 y12t
(25)
yˆ 2t = 0.529 y 21t + 0.471 y 22t
(26)
yˆ 3t = 0.125 y 31t + 0.875 y 32 t
(27)
Where: yˆ 1t , yˆ 2 t , yˆ 3t for the respective three industries combination prediction; y i1t for the respective
three industries ARMA (p, q) model prediction; y i 2 t for the respective three industries GM (1,1) model
prediction.
Use combination forecasting models to predict the future proportion of the first industry, secondary
industry and tertiary industry in Jiangsu. in table 2.
Years
Table 2 portfolio model prediction of the proportion of three industries
Primary industry
Secondary industry
Tertiary industry
2007
6.10
56.00
37.90
2008
5.94
55.88
38.18
2009
5.69
55.96
38.35
2010
5.55
56.08
38.37
7. Conclusion
In this paper put forward the ARMA (p, q) and GM (1,1) model prediction model combination, in
accordance with the thinking of the smallest residual sum of squares, use linear programming calculate
the weight, so that the combination forecasting model absorb the advantages of ARMA (p , q) and GM
(1,1) model and have high accuracy characteristics of with the prediction. Analyze the industrial
structure prediction in Jiangsu. Effectively solve the difficult problem of the industrial structure
prediction, will provide a scientific basis for realize the strategic adjustment of industrial structure.
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About the author:
ZHANG Yu Chun Doctor, an associate professor of School of Economics and Management, which
belongs to Lanzhou University of Technology, The main research are directions and methods of
management decision-making theory, economic analysis and evaluation of industry
SHI Xiao Lei a student of School of Economics and Management, which belongs to Lanzhou University
of Technology.
Postal Code: 730050
Telephone number: 13993183818, 13679487620
E-mail: [email protected], [email protected]
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