Empirical Research on Wheat Future Price by Samuelson Hypothesis

Empirical Research on Wheat Future Price by Samuelson Hypothesis
ZHANG Qi-wen, XING yuan-yuan
School of Economics & Management, Northeast Agricultural University, Heilongjiang, China
[email protected] or [email protected]
Abstract: The paper empirically analyzes the wheat future price in the Zhengzhou Commodity
Exchange by the generalized Autoregressive Conditional Heteroskedasticity model and establishes the
integrity ARMA-GARCH model. The conclusions show that the wheat future volatility have the high
durative. By adding the dummy variable, the paper proves that the most wheat future contracts certainly
exist the Samuelson hypothesis.
Key word: Wheat Future Price, volatility, ARCH mode, Samuelson hypothesis
1. Introduction
The future price should ascend with the coming of the maturity day. That hypothesis is the
Samuelson hypothesis or the (maturity effect) about the future marker. Since the Samuelson have found
the hypothesis that the maturity time have the mutuality with the future market’s price volatility in 1965,
many scholars in the foreign country studied that question. Rutledge 1976 Dusak-Miller(1979)
Castelino and Francis(1982) Milonas(1986) Galloway and Kolb(1996) Allen and Cruickshank(2000)
have found the Samuelson hypothesis. Though the most research though that many commodity future
exists the maturity effect, some scholars have the opposition view. For example Grauer(1977) havn’t
found the evidence of the Samuelson hypothesis in the ten commodity future productions.
Johnson(1998)also don’t have found it in his paper.
In our country, scholars have studied it for many years. Hu wei have researched the maturity
effect of the future market’s volatility by the Garman & Klass’s most highest price and lowest price, and
concluded that some productions in the form products surely exist the maturity effect. But when he
choosed the data, he didn’t used the integrity contracts, he intercepted ten productions data from
July,1999 to November,1999.Song bo(2006) have studied the Samuelson hypothesis of the soybean
future, he thinks that the soybean future surely have the Samuelson hypothesis. At present the study for
the wheat future isn’t abundance. The paper is based on the integrity future contracts, and further
analyses the daily price volatility data, conclude that the wheat future market surely exists the
Samuelson hypothesis by adding the dummy variable.
、
（ ）、
、
、
2. General Theories of the Samuelson Hypothesis and ARCH
Samuelson (1965) based his hypothesis on the assumptions that spot prices are generated by a first order
autoregressive process and that futures price sare unbiased predictors of the expiration price. That is,
given that St = αSt-1 + ut and the conditions E ut =0 E ut2 =σ2 , if α<1 the variance of the change
in futures prices will rise as a contract approaches expiry.
There are an important character about the ARCH model, it can describe the Volatility Clustering.
Volatility Clustering is that the volatility of the finance market usually have an trand. after the large
income, there will expect the large income and after the small income, there will expect the small
income.as the volatility in the currently usually have the positive relativity with the before time. The
ARCH family of models(including generalised ARCH), being consistent with a leptokurtic distribution,
are therefore more appropriate than standard models.
（ ） ，（ ）
3.Data explain
704
If the wheat future market have the maturity effect? In order to validate it, the article used the wheat
future data of the Zhengzhou Commodity Exchange, because the wheat future volume and open interest
are little after May,2006 and the price volatility don’t reflect the character of the wheat future market, so
the paper choose the data from October 2002 to May 2006.Wheat future contracts are bargain in the odd
month, the paper chooses all the integrity contract in the time span. Every contract is about 230 data. the
measure of daily futures return volatility used in this study is the absolute value of the continuously
 FT, t 
 ×100 where FT,t is the
 FT , t 
compounded rate of return multiplied by 100. That is, VF = log 
settlement price on day t of a contract expiring at time T. Descriptive statistics for daily volatility are
shown in table 1.
Table1. Descriptive statistics for daily volatility
contract
sample
Mean
Std Dev
Skewness
Kurtosis
Jarque-Bera
Prob
wt401
wt403
wt405
wt407
wt409
wt411
wt501
wt503
wt505
wt507
wt509
wt511
wt601
wt603
wt605
-0.367699
-0.306422
0.080741
-0.023973
0.104281
-0.236307
-0.273875
-0.341521
-0.195731
0.438151
0.131506
0.241068
0.993783
0.348217
-0.871960
5.714570
6.06.759
5.018288
4.432229
4.652848
6.186206
7.867162
6.862026
6.887691
9.516724
5.689759
8.950372
14.31836
7.845413
7.253100
91.62071
123.4218
50.39048
24.21511
26.82890
101.1589
239.8929
143.5631
1443987
372.9070
66.95313
328.1634
1166.491
176.7275
184.8886
0.000000
0.000000
0.000000
0.000006
0.000001
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
278
304
295
283
232
234
240
224
227
207
220
221
212
177
210
0.29277
0.042050
-0.006890
-0.004414
0.080620
-0.050883
-0.089465
-0.131465
-0.090819
-0.129628
-0.082336
-0.083853
-0.063020
-0.059684
-0.067911
1.134648
1.146600
1.114543
1.126530
1.133856
0.828111
0.756109
0.710366
0.702180
0.823152
0.700434
0.661103
0.747326
1.202168
0.655924
4. Empirical results
The paper found that the daily wheat price volatility series have the autocorrelation and partial
correlation with the autocorrelation analysis and stationary inspection for the zheng zhou wheat future
price. The ADF inspection shown that the future contracts are stationary at 1 probability level. So the
paper can build the ARMA model, The paper inspect the residuals of the model with the ARCH-LM, the
conclusion shown that the second-order and eight-order inspection results of 14 contracts are significant
at 1 probability level.wt505 are significant at 5 probability level as well. But the inspection results of
wt507 are non-significant. As a whole, the residuals of the daily volatility series of wheat future
contracts surely exists the ARCH effects, and especially the high-level ARCH effects exists.
In order to depict the daily price volatility of the wheat future contracts, the paper builds the
integrity ARMA 1 1 -GARCH 1 1 and ARMA 2 2 -GARCH 1 1 model. The results
was shown in table 5 and table 6.we can conclude that the model can depict the process of price
volatility of the wheat future very well. All of the parameter of the GARCH model about the 15
contracts are significant at 5 probability level, especially 12 contracts are significant at 1 probability
level. the contracts are 13 which content to the parameter restriction conditions, that is α+β 1.
％
％
％
（，）
（，）
（，）
（，）
％
（ ，1）-GARCH（1，1）and ARMA（2，2）-GARCH（1，1）
Table2 testing for the ARMA 1
contract
401
403
（）
AR 1
-0.98896
(0.0000)
-1.079032
％
〈
（）
AR 2
-0.926945
（）
MA 1
0.95931
(0.0000)
1.094404
705
（）
MA 2
0.986985
α
0.37578
(0.0001)
0.347242
β
0.576919
(0.0000)
0.590640
405
407
409
411
501
503
505
507
509
511
601
603
605
(0.0000)
-0.028534
(0.9667)
0.09961
(0.8781)
0.414078
(0.0385)
-0.480504
0.1872
0.973457
(0.0000)
0.083495
(0.0086)
0.545064
(0.0465)
0.454924
(0.0343)
0.261729
(0.4623)
0.420028
0.0087
-0.191935
0.5298
-0.016131
(0.9414)
0.426089
0.0190
（
）
(0.0000)
0.08034
(0.8490)
-0.748430
(0.0000)
0.774851
(0.0001)
(0.0000)
-0.013491
(0.9843)
-0.18734
(0.7683)
-0.52977
0.0088
0.441301
0.2470
-0.994517
(0.0000)
-0.199657
(0.8377)
-0.648276
(0.0043)
0.409193
(0.0409)
-0.377167
(0.2753)
-0.486834
0.2786
-0.078909
0.7938
-0.517375
(0.0027)
-0.637477
0.0001
（
（
）
）
（
（
）
）
（
（
）
）
（
）
（
）
（，）
(0.0000)
0.01581
(0.9972)
0.752128
(0.0000)
-0.396826
(0.0364)
(0.00001)
0.184790
(0.0042)
0.226692
(0.0147)
0.245272
(0.0001)
0.217819
(0.0001)
0.180259
(0.0000)
0.239137
(0.0001)
0.346755
0.0003
0.368861
(0.0032)
0.096044
(0.0146)
0.133233
(0.0004)
0.207339
(0.0000)
0.336677
(0.0083)
0.346216
(0.0000)
（
(0.0000)
0.750489
(0.0000)
0.593295
(0.0000)
0.789336
(0.0000)
0.764403
(0.0000)
0.774851
(0.0001)
0.760283
(0.0000)
0.506476
(0.0000)
0.419917
(0.0109)
0.868167
(0.0000)
0.852725
(0.0000)
0.845158
(0.0000)
0.626305
(0.0000)
0.600532
(0.0000)
）
（ ，1）and ARMA（2，2）-GARCH（1，1）
Table 3testing for the ARMA 1 1 -GARCH 1
contract
401
403
405
407
409
411
501
503
505
507
509
511
601
603
605
α+β
0.952691
0.937882
0.939779
0.819987
1.034608
0.982222
0.955110
0.99942
0.853231
0.788878
0.964211
0.985957
1.052497
0.962982
0.946748
T
-0.52784(0.0154)
0.756417(0.0080)
-0.55121(0.0027)
-0.288383 0.0344
0.183655 0.0060
-0.258102 0.0245
-0.177238 0.0722
-0.206330(0.6263)
-0.139023(0.1010)
0.246063(0.0319)
-0.228445(0.0357)
0.462498 0.1928
0.212983 0.0240
0.357532(0.0904)
-0.310019 0.0001
（
（
（
（
）
）
）
）
（
（
（
）
）
）
n
16
11
10
52
9
33
29
24
35
18
18
12
16
12
15
706
AIC
2.730322
2.751512
2.804034
2.972627
2.648529
2.222386
1.981051
1.771142
1.927631
2.249634
1.980081
1.886808
1.749640
2.821077
1.792799
SC
2.821904
2.862087
2.916241
3.063029
2.752845
2.326065
2.112356
1.878094
2.033577
2.394534
2.088407
1.994787
1.860470
2.947176
1.904744
The paper adopted the method which have been used by D.E.Allen and S.N.Cruichshank, the paper
add the dummy t, t was the distance to the maturity. If the distance to the expiry was less than n, the
value was 1, If the distance to the expiry was more than n, the value was 0. the n was enactment in
advance. If the maturity exist, the parameter of the t was more than o. the conclude was shown in table 6.
the n of the 10 contracts less than 20, so the maturity was exist about 20 days.
5. Conclusion
By analysis the volatility of the wheat future, the paper concluded that wheat future exist the
high-level ARCH effects. ARMA 1 1 -GARCH 1 1
ARMA 2 2 -GARCH 1 1
can depict the daily yield series of the wheat future very well. The volatility of the wheat future are
highly durative. If the yield will fluctuate abnormly when it is concussed, it is difficult to eliminate
it. The n of the most contracts less than 30, the maturity effect exist, but the paper can not find the
exist of the maturity effect when n more than 30.
（，）
（ ， ）和
（，）
（，）
References
[1]Allen D.E. & Cruickshank S.N.Empirical Testing of the Samuelson hypothesis: An Application to
Futures Markets in Australia, Singapore and the UK. School of Finance and Business Economics,
paper 2000 9-12 19-25
[2]Bollerslev T.Generalized autoregressive conditional heteroskedasticity Journal of Econometrics 1986
31 307-327.
[3][ ]Walter Enders. Applied econometric time series high education press Beijing 2006
[4] Huu N. Duong and Petko S. Kalev An Intraday Analysis of the Samuelson Hypothesis for
Commodity Futures Contracts, Department of Accounting and Finance Monash University paper 2006
8-12
[5]Song bo Empirical research on the wheat future price’s Samuelson hypothesis paper 2006 5-8
[6]Yi danhui data analysis and Eviews apply china Statistic pess Beijing 2002
英
ZHANG Qi-wen ,male, was borned in August,1967, Xiuyan city of Liaoning Province, research
direction was rural finance and insurance. Associate professor, management doctor, the postdoctoral of
Heilongjiang Province agriculture academy of science, the director and chief assistant in .Northeast
Agricultural college economy management institute
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