In the simulation study, the sample size is selected to be 150 through this section.
Since the beta density function can have different shapes depending on the parameter values, we consider that rt are iid from the beta(a, b) distribution and (3.1) can be written as
Zt= rtZt−1+ εt, (3.5)
where rt
iid
∼ Beta(a, b), εt
iid
∼ N(0, σε), µr = E(rt) and σr = V ar(rt). Then, for any a, b > 0, the stationary parameter is
η = µ2r+ σr2
= ( a
a + b)2+ ab
(a + b + 1)(a + b)2
= ( a
a + b)2(a + b + 1
a + b + 1) + ab
(a + b + 1)(a + b)2
= a3+ a2b + a2+ ab (a + b + 1)(a + b)2
= a3+ a2b + a2+ ab
a3+ 3a2b + 3ab2+ b3 + a2 + 2ab + b2
= (a3+ a2b + a2+ ab)
(a3+ a2b + a2+ ab) + 2a2b + 3ab2+ b3+ ab + b2.
Since 2a2b + 3ab2+ b3+ ab + b2 > 0, the stationary condition (η < 1) is always satisfied.
In this simulation, we set the pairs (a, b) as (0.5,2), (2,2) and (2,0.5). The corre-sponding µr is 0.2, 0.5 and 0.8, respectively. Table 3.1 shows the rejection rates such that the type I error in our testing result is at level 0.05. The simulation replicants is 1000, and the rejection rates are very close 0.05 in difference parmaters setting.
Table 3.1: The EL method for testing the equality of two RCA(1) models at level 0.05
µr 0.2 0.5 0.8
(a, b) (0.5, 2) (2, 2) (2, 0.5)
σ2ε= 1 0.049 0.051 0.057
σ2ε= 2 0.052 0.045 0.061
σ2ε= 5 0.046 0.067 0.068
4 Application
In this chapter, we illustrate our testing method by a real data example. The data sets we use are the monthly sales of FamilyMart, President Chain Store and Poya in Taiwan. These data were obtained from Taiwan Economic Journal (TEJ), http://www.finasia.biz/ensite/.
First, we consider the data consists of 145 records for the monthly sales of Fami-lyMart and President Chain Store ranging from March 2000 to March 2012 presented in Figure 4.1, which show that the two time series are nonstationary. We take first differneces for both series in natural log scale. The sample autocorrelation and partial autocorrelation functions are also ploted in Figure 4.2 . Since the output in Figure 4.2 shows that the first differnece of the natural logarithms dies down very slowly at the seasonal level, we also take seasonal differneces with lag 12 for both series and denote them as Z1t and Z2t , t = 1, ..., n, respectively.
The ACF and PACF of the two series Z1t and Z2t are shown in Figure 4.3 and 4.4 which are used to identify a suitable model for the two time series. We determine the ARMA order of Z1t first. At seasonal level, the ACF and PACF of Z1t suggest that we may consider first-order seasonal MA model with the yearly seasonal period MA(1)12 or second-order seasonal AR model with the yearly seasonal period AR(2)12
to fit seasonal part. Since the coefficient of MA(1)12 we estimated is very close to 1, we prefer to ues AR(2)12 model tentatively. At nonseasonal level the PACF cuts off
at lag 3 and the ACF dies down. We may fit an AR(3) model. Although the partial autocorelation at lag 9 is significant, it is hard to explain why the sales depend on the past ninth month.
We combine the seasonal model and nonseasonal model above. This gives the overall model ARMA(3, 0)(2, 0)12for Z1t. Since the ACF and PACF of Z2t have similar pattern for Z1t, we directly use the same model to fit Z2t. We can see that both residuls of fited models look like white noise and their ACF and PACF in Figure 4.5 and 4.6 have no spikes in any lag. Hence, we conclude that our models is adequate and the coefficients we estimated are given in Table 4.5.
We performe our methods to test equality of two models. The testing statistic is
χ2 = ( ˆβx− ˆβy)′V∗ −1( ˆβx− ˆβy)
0.0149 0.0050 0.0011 −0.0014 −0.0024
0.0050 0.0160 0.0044 −0.0017 −0.0015
0.0011 0.0044 0.0143 0.0000 −0.0003
−0.0014 −0.0017 0.0000 0.0140 0.0085
−0.0024 −0.0015 −0.0003 0.0085 0.0134
where V∗ =
0.0071 0.0019 0.0003 −0.0005 −0.0010
0.0019 0.0073 0.0016 −0.0004 −0.0005
0.0003 0.0016 0.0069 0.0002 0.0000
−0.0005 −0.0004 0.0002 0.0074 0.0044
−0.0010 −0.0005 0.0000 0.0044 0.0070
0.0078 0.0031 0.0008 −0.0009 −0.0014
0.0031 0.0087 0.0028 −0.0013 −0.0010
0.0008 0.0028 0.0074 −0.0002 −0.0003
−0.0009 −0.0013 −0.0002 0.0066 0.0041
−0.0014 −0.0010 −0.0003 0.0041 0.0064
Since χ2 = 2.602746 <11.0705 = χ20.05(5), we did not reject the equality of two series under our model assumption.
On the other hand, we are also interested in the variation of FamilyMart’s and Poya’s monthly sales in the same periods that we analyzed above. We directly take log and differneces in seasonal and nonseasonal lag of Poya’s monthly sales and denote it by Z3t. The model ARMA(3, 0)(2, 0)12 is considered as well. We performe bonferroni method and chi-square method for testing the equality of paramaters which we esti-mated to FamilyMart’s and Poya’s monthly sales. The chi-square statistic 11.60805 is larger than χ20.05(5). This means that there are significantly difference between Fam-ilyMart’s and Poya’s relationships between past sales and future sales. However, the critical value by bonferroni approach is z1−0.05/(2∗5) = 2.575829 and z-value for the five coefficirnt are 2.16, 0.97, 1.49, 1.63, 0.64. There are not any significantly difference between the parameters in two ARMA(3, 0)(2, 0)12 models by bonferroni approach.
Table 4.1: The 145 records for the monthly sales of FamilyMart
1176297 1203511 1298980 1341173 1448275 1412358 1377681 1420655 1324594 1370169 1424020 1257277 1393381 1401237 1493782 1561164 1700699 1742855 1557325 1606533 1496431 1630714 1603934 1625136 1699289 1743039 1897031 1958004 2022666 2000199 1857083 1883538 1736939 1817184 1823917 1757139 1842472 1889945 2034507 2115684 2290404 2231388 2170777 2152775 2047819 2052770 2192003 1972375 2126223 2143391 2357776 2346279 2525170 2551546 2356080 2331998 2213939 2248976 2186831 2172723 2263059 2364689 2434188 2273040 2627003 2703091 2646670 2635123 2736831 2640543 2738463 2338039 2270752 2493470 2694421 2783042 2740841 2692549 2542157 2670539 2408413 2414213 2416360 2502406 2555435 2551918 2728505 2698956 3029872 3344389 3081076 2937270 2753587 2818224 3006540 3026450 2921727 3081081 3274186 3159104 3456286 3470185 3228930 3221119 2952750 3033530 3228163 2899779 3036564 2956038 3258665 3212950 3632916 3679114 3434640 3541790 3171044 3154109 3184423 3260491 3325341 3268256 3717275 3632261 4003762 3960021 3704590 3783590 3547716 3566106 3495592 3308275 3507977 3796799 3927056 3892256 4186706 4203710 4444292 4411963 4145609 4107737 4523114 3898680 4269153
Table 4.2: The 145 records for the monthly sales of President Chain Store
4491115 4757277 4917705 5317334 5151417 4944261 5114518 4688220 4851881 5245460 4551829 5006136 4955858 5402798 5796654 6071025 6134253 5310996 5542568 5071628 5662084 5440804 5478789 5664451 5795443 6204563 6358800 6711169 6613039 6059449 6141493 5703570 5831072 5902463 5909544 5948880 6123298 6549613 6634948 7244164 7106254 6868259 6737832 6369917 6270463 7005126 6193627 6329060 6378422 6926771 6892067 7314727 7363984 6762306 6739042 6445986 6585437 6427478 6693389 6636050 6874298 8426333 9389209 10189078 8048829 7797608 7920930 7751074 7486858 7957398 6906009 7772104 8182944 8531847 8088966 9657812 9241675 8694701 8637923 8097559 8187449 7775909 8573222 8272678 8087873 9447797 9192908 9296334 8561123 8253002 8471859 7997619 8403231 8286532 8240892 7955810 7937409 8451530 8620256 9401185 9279031 8521024 8904654 8407819 8185085 8678734 7479726 7982511 8173621 9085255 8521446 8795788 8813036 8322949 8602572 8440369 8860415 8761825 8861694 9267065 8895036 9686463 9495227 10289154 10189891 10040908 10105227 9341217 9730864 9336890 9540881 9406303 9534967 10243258 10393956 10806588 10657507 10811993 10941694 10379517 10659172 11675088 9969214 10604789
Table 4.3: The 145 records for the monthly sales of Poya
134572 139882 151795 142373 169890 158217 156978 144326 131484 146331 159908 154044 167596 164856 159300 160481 175428 174350 178467 171466 159049 170817 153637 154914 158808 159861 169925 159645 166328 190791 228079 203493 191163 203540 217475 209811 227167 202533 226091 213989 241005 239018 245058 239038 217248 260849 266234 237624 220658 225923 245033 231372 239919 280470 289460 279621 240378 284378 271904 270610 255527 256112 265460 262181 278458 275864 272014 281022 270523 319695 295752 269613 263292 264167 276487 272641 288919 298156 331839 296026 307825 352557 309917 338401 319048 312168 338692 317888 359587 366206 402671 363113 347163 395244 380332 416281 374001 376488 382949 382992 391630 477309 436935 443974 423617 448455 524183 415991 405746 401773 451212 412836 474039 511685 508552 494973 460700 556776 523646 508097 490038 471239 499398 468760 534932 550774 525182 516074 465039 538118 546607 513171 457284 465630 469685 513494 564373 563342 566643 538218 486194 571026 619879 511171 486322
Table 4.5: The estimation of the parimaters of ARM A(3, 0)(2, 0)12model FamilyMart
parameter φx1 φx2 φx3 φx12 φx24
estimation -0.2714 -0.1231 -0.3302 -0.8729 -0.4028 S.E. 0.0842 0.0854 0.0829 0.0858 0.0837 z-value 3.223278 1.4415 3.9831 10.1737 4.8124 variance of residuals estimated as 0.001545
President Chain stores
parameter φy1 φy2 φy3 φy12 φy24
estimation -0.3972 -0.1788 -0.2086 -0.8623 -0.4187 S.E. 0.0882 0.0931 0.0857 0.0814 0.0800 z-value 4.5034 1.9205 2.4341 10.5934 5.2338 varance of residuals estimated as 0.002339
Poya
parameter φz1 φz2 φz3 φz12 φz24
estimation -0.5362 -0.2480 -0.1504 -0.6702 -0.3224 S.E. 0.0889 0.0969 0.0874 0.0904 0.0927 z-value 6.0315 2.5593 1.7208 7.4137 3.4779 variance of residuals estimated as 0.004062
FamilyMart’s and President Chain Store’s Sales
Months
Sales (Thousand Dollars)
FamilyMart
President Chain Store
Mar−00 Mar−02 Mar−04 Mar−06 Mar−08 Mar−10 Mar−12
1e+064e+068e+061.2e+07
Poya’s Sales
Months
Sales (Thousand Dollars)
Mar−00 Mar−02 Mar−04 Mar−06 Mar−08 Mar−10 Mar−12
2e+053e+054e+055e+056e+05
Figure 4.1: Monthly Sales of FamilyMart, President Chain stores and Poya
0 5 10 15 20 25
−1.0−0.50.00.51.0
Lag
ACF
FamilyMart
0 5 10 15 20 25 30 35
−1.0−0.50.00.51.0
Lag
ACF
President Chain Store
0 5 10 15 20 25
−1.0−0.50.00.51.0
Lag
ACF
Poya
Figure 4.2: the ACF of the first differneces of the natural logarithms.
0 5 10 15 20 25 30 35
−1.0−0.50.00.51.0
Lag
ACF
FamilyMart
0 5 10 15 20 25 30 35
−1.0−0.50.00.51.0
Lag
ACF
President Chain Store
0 5 10 15 20 25 30 35
−1.0−0.50.00.51.0
Lag
ACF
Poya
Figure 4.3: the ACF of both series took log and two difference (lag 1 and lag 12)
0 5 10 15 20 25 30 35
−1.0−0.50.00.51.0
Lag
Partial ACF
FamilyMart
0 5 10 15 20 25 30 35
−1.0−0.50.00.51.0
Lag
Partial ACF
President Chain Store
0 5 10 15 20 25 30 35
−1.0−0.50.00.51.0
Lag
Partial ACF
Poya
Figure 4.4: the PACF of both series took log and two difference (lag 1 and lag 12)
0 5 10 15 20
−1.0−0.50.00.51.0
Lag
ACF
ACF of FamilyMart’s Residuals
5 10 15 20
−1.0−0.50.00.51.0
Lag
Partial ACF
PACF of FamilyMart’s Residuals
Figure 4.5: the ACF and PACF of residuals of the ARM A(3, 0)(2, 0)12model of Z1t
0 5 10 15 20 25 30 35
−1.0−0.50.00.51.0
Lag
ACF
ACF of President Chain Store’s Residuals
0 5 10 15 20 25 30 35
−1.0−0.50.00.51.0
Lag
Partial ACF
PACF of President Chain Store’s Residuals
Figure 4.6: the ACF and PACF of residuals of the ARM A(3, 0)(2, 0)12model of Z2t.
0 5 10 15 20 25 30 35
−1.0−0.50.00.51.0
Lag
ACF
ACF of Poya’s Residuals
0 5 10 15 20 25 30 35
−1.0−0.50.00.51.0
Lag
Partial ACF
PACF of Poya’s Residuals
Figure 4.7: the ACF and PACF of residuals of the ARM A(3, 0)(2, 0)12model fitting Z3t.
5 Conclusions
In this thesis, we study and review the literature on estimation and inference for ARMA models based on MLE method. For comparing time series, we proposed an approach to test the equality of the parameters estimated from two time series. We also presente the Bonferroni approach for multiple testing. In addition to the classical ARMA based methods to compare two time series, we considered the RCA models as well. We performe the empirical likelihood estimation for both RCA models, and then test equality of their means of random coefficients by the F distribution. We also considere beta distribution for the random coefficient of the RCA(1) model and show that the stationary condition is always satisfied. For testing for ARMA models or RCA models, our simulations verify the testing results can attain a desired level mostly.
Finally, we practice our methods for real data. The data consists of three companies’
monthly sales, namely, FamilyMart, President Chain Store and Poya. In our analysis, we conclude that there are significantly difference between FamilyMart’s and Poya’s sales behavior.
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