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American Journal of Mathematics and Statistics 2013, 3(6): 362-374 DOI: 10.5923/j.ajms.20130306.10
Study on the Error Component of Multiplicative Time Series Model Under Inverse Square Transformation
G. C. Ibeh1,*, C. R. Nwosu2
1Department of Maths/Statistics, School of Industrial and Applied Sciences, Federal Polytechnic Nekede, Owerri, Nigeria 2Department of Statistics, Faculty of Physical Sciences, NnamdiAzikiwe University, Awka, Nigeria
Abstract Thiswork examines the conditions for non-violation of the basic assumptions on the error component of a multiplicative time series model when inverse square transformation is applied to the error term. To achieve this, the curve shapes of the probability density functions (pdfs) of πππ‘π‘ and πππ‘π‘β² = 1
πππ‘π‘2 , ππβ(π₯π₯) ππππππ ( )h y were compared for some values of
ππ within the interval[0.01,0.25], and it was found that the distribution of the transformed variable loses its symmetry when ππ > 0.08. Rolleβs Theorem was also used to find the region where β(π¦π¦) satisfies the bell-shaped condition; and this is met whenππ β€ 0.094. Use of the simulated error terms shows that the transformed variable is normal for ππ < 0.08. Finally, from the functional expressions for πΈπΈ(πππ‘π‘β²) and ππ(πππ‘π‘β²), it was observed that the mean of πππ‘π‘β² is one and the increased variance is approximately 4 times the variance of πππ‘π‘ for ππ β€ 0.070. Therefore, the condition for successful inverse square transformation with respect to the error component of the multiplicative time series model is ππ β€ 0.070 . Keywords Error Component, Multiplicative Time Series Model, Left Truncated Normal Distribution, Inverse Square Transformation, Moments
1. Introduction According to[1], the general time series model is always
considered as a mixture of four components, namely the trend, seasonal movements, cyclical movements and irregular or random component. Hence, classifications of the time series model are
Multiplicative Model: πππ‘π‘ = πππ‘π‘πππ‘π‘πΆπΆπ‘π‘πππ‘π‘ (1)
Additive Model: πππ‘π‘ = πππ‘π‘+πππ‘π‘+πΆπΆπ‘π‘+πππ‘π‘ (2)
Mixed Model: πππ‘π‘ = πππ‘π‘πππ‘π‘πΆπΆπ‘π‘+πππ‘π‘ (3)
In short term series, the trend and cyclical components are merged to give the trend cycle component[2], hence
Equations (1) through (3) can be rewritten as πππ‘π‘ = πππ‘π‘πππ‘π‘πππ‘π‘ (4) πππ‘π‘ = πππ‘π‘+πππ‘π‘+πππ‘π‘ (5) πππ‘π‘ = πππ‘π‘πππ‘π‘ + πππ‘π‘ (6)
respectively, where πππ‘π‘ is the trend cycle component. Consider a random variable, ππ which is normally
distributed with a probability density function,
* Corresponding author: gabmicchuks@yahoo.com (G. C. Ibeh) Published online at http://journal.sapub.org/ajms Copyright Β© 2013 Scientific & Academic Publishing. All Rights Reserved
ππ(π₯π₯) = οΏ½1
ππβ2πππππ₯π₯ππ οΏ½β (π₯π₯β1)2
2ππ2 οΏ½ ,ββ < π₯π₯ < β,ππ2 > 00, πππ‘π‘βππππππππππππ
οΏ½ (7)
Most often, the random variable ππ, which is normally distributed with mean, 1 and ππ2 < β do not admit values less than or equal to zero. This usually leads to the truncation of all values of ππ β€ 0 to take care of the admissible region of ππ > 0.
The resulting distribution after truncation was given by[3] as
ππβ(π₯π₯) = οΏ½ππ
ππβ2πππππ₯π₯ππ οΏ½β (π₯π₯β1)2
2ππ2 οΏ½ , 0 < π₯π₯ < β0 , ββ < π₯π₯ β€ 0
οΏ½ (8)
where ππ is the normalizing quantity. [4]obtained the value of k to be
ππ = 11βπποΏ½β1
πποΏ½ (9)
( ) ( )2
2
*
0, 0
11 exp ,0212 1
x
xx xfΟΟ Ο Ο
Ο
β β < β€
β = β < < β β β
β΄
(10) ππβ(π₯π₯) was shown to be a proper probability density
function[4]with mean and variance
πΈπΈβ(ππ) = 1 + ππππβ 1
2ππ2
β2πποΏ½1βπποΏ½β1πποΏ½οΏ½
(11)
American Journal of Mathematics and Statistics 2013, 3(6): 362-374 363
Table 1. Bartlettβs transformation for some values of π½π½
π½π½ 0 12
1 32
2 3 β1
Transformation No transformation οΏ½πππ‘π‘ πππππππππππ‘π‘ 1
οΏ½πππ‘π‘ 1
πππ‘π‘
1πππ‘π‘2
πππ‘π‘2
ππππππβ(ππ) =ππ2
2 οΏ½1 β πποΏ½β1πποΏ½οΏ½
οΏ½1 + ππππ οΏ½ππ(1)2 < 1
ππ2οΏ½οΏ½
β ππππβ 1
2ππ2
β2πποΏ½1βπποΏ½β1πποΏ½οΏ½
β οΏ½ ππππβ 1
2ππ2
β2πποΏ½1βπποΏ½β1πποΏ½οΏ½οΏ½
2
(12)
1.1. Data Transformation and Classification Data transformation is the application of a non-linear
function such as ππππππππ(πππ‘π‘),οΏ½πππ‘π‘ ,1πππ‘π‘
,πππ‘π‘2, 1πππ‘π‘2 to the original
data. According to[5], if the experimenter knows the theoretical
distribution of the observations, he may utilize this information in choosing an appropriate transformation, for example, if the observations follow the Poisson distribution, then the square root transformation ππ = βππ would be used. If the data follow log-normal distribution, then the logarithmic transformation ππ = ππππππππππ is appropriate. For binomial data expressed as fractions the arcsine transformation ππ = ππππππππππππππ is useful. When there is no obvious transformation, the experimenter usually empirically seeks a transformation that equalizes the variance regardless of the value of the mean.
[6] showed that the appropriate transformation is determined by the value of the slope (π½π½) in the linear relationship between the natural log of the periodic standard deviations (ππππππππππππ) and natural log of the periodic means (ππππππππππππ)given as
ππππππππππππ =β +π½π½ππππππππππππ (13) For the inverse square transformation, Ξ² should be
approximately equal to 3 [6] see Table 1.
1.2. Background of the Study [4] investigated the effect of logarithmic transformation
on the error component (πππ‘π‘) of a multiplicative time series model where πππ‘π‘~ππ(1,ππ2) and discovered that the logarithmic transformation πππ‘π‘ = πππππππππππ‘π‘ isnormally distributed with mean zero and the same variance, ππ2 if ππ < 0.1
[7]studied the effect of inverse transformation on the error component of the same multiplicative model and established that the inverse transformation πππ‘π‘ = 1
πππ‘π‘ is normally
distributed with mean, one and same variance provided ππ β€ 0.07, however, a more extensive study by the same authors using the functional expressions of the mean and variance, extended the region of successful transformation to Ο< 0.1.
[8] carried out a study on the effect of square root
transformation on the error component of the same model and concluded that the square root transformation πππ‘π‘ = οΏ½πππ‘π‘ is normally distributed with unit mean and variance, 1
4ππ2 for
ππ β€ 0.30 where ππ2 is the variance of the original error component before transformation.
[9] Investigated the effect of square transformation on the error component of the multiplicative time series model and observed that square transformation πππ‘π‘ = πππ‘π‘2 can be assumed to be normally distributed with unit mean and same variance for ππ β€ 0.027 they observed that ππππππ(πππ‘π‘2) >ππππππ(πππ‘π‘) ππππππ ππππππ ππ and that πππ‘π‘2~ππ(1.0,1.0),ππππππ ππ β€0.027.
Obviously, the overall aim of these studies is to establish conditions for successful transformation, hence provide better choice of right transformation. According to [10] choosing a good transformation improves analyses in three ways, namely (i) increase in visual clarity (ii) reduction or elimination of outliers (iii) increase in statistical clarity.
1.3. Need for the study The value of π½π½ in (13) classifies all the time seriesdata
into non-overlapping groups in the sense that any time series data requiring transformation belongs exclusively to one and only one group, hence can only be appropriately transformed by applying one of the six common transformations as shown in Table 1. Thus, despite the fact that Iwueze (2007), Chinwe et al (2010),Otuonye et al (2011) and Ohakwe et al (2013) have all carried out similar studies with respect to logarithmic, inverse, square root and square transformations respectively, this study on inverse square transformation is still very necessary since the results obtained for the above listed four transformations cannot be applied in the analysis of time series data requiring inverse square transformation.
1.4. Inverse Square Transformation
For π½π½ = 3, we adopt inverse square transformation on the multiplicative time series model given in equation (4) to obtain
πππ‘π‘ =1πππ‘π‘2
=1πππ‘π‘
2 .1πππ‘π‘2
.1πππ‘π‘2
= πππ‘π‘β²πππ‘π‘β²πππ‘π‘β² (14)
where πππ‘π‘β² = 1
πππ‘π‘2 , πππ‘π‘β² = 1
πππ‘π‘2 and πππ‘π‘β² = 1
πππ‘π‘2
Thus, it will be of interest to find the distribution of πππ‘π‘β² and the relationship between its variance and the variance of the original data.
2. Aim and Objectives of the Study
364 G. C. Ibeh et al.: Study on the Error Component of Multiplicative Time Series Model Under Inverse Square Transformation
The aim of this study is to obtain the distribution of the inverse square transformed error component of the multiplicative time series model and the objectives are:
i) To examine the nature of the distribution ii) To verify the satisfaction of the assumption on the
mean of the error terms; ππ = 1 iii) To establish the relationship between the variance
of the original series and the transformed series and identify the conditions for such relationship to exist.
2.1. Derivation of the Probability Density Function (pdf) of the Inverse Square Transformation of the Error Component of the Multiplicative Time Series Model
Let et =X and, πππ‘π‘β² = ππ = 1ππ2, then
ππ = οΏ½1πποΏ½
12 = ππβ
12 (15)
Adapting equations (10 & 15) and using the transformation of variable technique,
h(y)=ππβ(π₯π₯) οΏ½πππ₯π₯πππ¦π¦οΏ½ [11] (16)
hence 21
2 112
32
y ,012 2 1
0 , 0
y
eh yy
y
Ο
Ο Ο Ο Οβ
βββ
= < < ββ β
β < β€
οΌ οΌ (17)
Observe that h(y) is a proper probability density function (pdf).
2.2. Plot of the Probability Density Functions ππβ(ππ) ππππππ ππ(ππ)
Using the probability density functions of the two variables as given in Equations, (10) and (17), the functions ππβ(π₯π₯) ππππππ β(π¦π¦) were plotted for some values of ππ (see Figures (1) to (5))
Figure 1. Curves Shapes for ππ =0.02
Figure 2. Curves Shapes for ππ =0.04
American Journal of Mathematics and Statistics 2013, 3(6): 362-374 365
Figure 3. Curves Shapes for ππ =0.06
Figure 4. Curves Shapes for ππ =0.08
Figure 5. Curves Shapes for ππ =0.10
2.3. Region of Normality for ππ(ππ)
Since h(y) has one maximum point π¦π¦πππππ₯π₯ (mode) hence one maximum value β(π¦π¦πππππ₯π₯ ) for all values of Ο.
Rolleβs Theorem was used to find the values of Ο that satisfy the bell-shaped and symmetric condition of modeβ1βmean.
Let ππ = 1
2ππβ2πποΏ½1βππ(β1ππ)οΏ½
in (17)
then, β(π¦π¦) = πππ¦π¦β32ππ
β 12ππ2οΏ½π¦π¦
β12 β1οΏ½
2
Using the product rule,
366 G. C. Ibeh et al.: Study on the Error Component of Multiplicative Time Series Model Under Inverse Square Transformation
ββ²(π¦π¦ ) =
ππ οΏ½π¦π¦β32 οΏ½ 1
2ππ2 οΏ½π¦π¦β1
21οΏ½ π¦π¦β32ππβ
οΏ½π¦π¦β12β1οΏ½
2
2ππ2 οΏ½ οΏ½+ππβ 1
2ππ2οΏ½π¦π¦β1
2β1οΏ½2
οΏ½β 32π¦π¦β
52οΏ½οΏ½οΏ½(18)
Equating ββ²(π¦π¦) = 0 gives 1π¦π¦ππ2 οΏ½1 β π¦π¦
12οΏ½ β 3 = 0
3ππ2π¦π¦ + π¦π¦12 β 1 = 0 (19)
substituting π¦π¦ = π₯π₯2ππππππππ ( 19), ππππππππππ 3ππ2π₯π₯2 + π₯π₯ β 1 = 0 (20)
π₯π₯ = β1Β±οΏ½1+12ππ2
6ππ2 , Since π¦π¦πππππ₯π₯ is positive then
π₯π₯ =β1+οΏ½1+12ππ2
6ππ2
But π¦π¦ = π₯π₯2, π¦π¦πππππ₯π₯ = οΏ½β1+οΏ½1+12ππ2
6ππ2 οΏ½2
The bell-shaped condition would imply π¦π¦πππππ₯π₯ β 1, see Table 2 for the numerical computation of π¦π¦πππππ₯π₯ =
οΏ½β1+οΏ½1+12ππ2
6ππ2 οΏ½2
and Table 3 for the summary values of π¦π¦πππππ₯π₯
Table 2. Numerical Computation of
22
max 21 1 12
6y Ο
Ο
β + + =
ππ π¦π¦πππππ₯π₯ = οΏ½β1 + β1 + 12ππ2
6ππ2 οΏ½2
1 β π¦π¦πππππ₯π₯ ππ π¦π¦πππππ₯π₯ = οΏ½β1 + β1 + 12ππ2
6ππ2 οΏ½2
1 β π¦π¦πππππ₯π₯
0.010 0.999400 0.0005996 0.051 0.984692 0.0153081 0.011 0.999275 0.0007253 0.052 0.984098 0.0159023 0.012 0.999137 0.0008631 0.053 0.983493 0.0165071 0.013 0.998987 0.0010127 0.054 0.982878 0.0171225 0.014 0.998826 0.0011743 0.055 0.982252 0.0177484 0.015 0.998652 0.0013477 0.056 0.981615 0.0183848 0.016 0.998467 0.0015331 0.057 0.980968 0.0190316 0.017 0.998270 0.0017303 0.058 0.980311 0.0196887 0.018 0.998061 0.0019393 0.059 0.979644 0.0203562 0.019 0.997840 0.0021602 0.060 0.978966 0.0210339 0.020 0.997607 0.0023928 0.061 0.978278 0.0217218 0.021 0.997363 0.0026373 0.062 0.977580 0.0224198 0.022 0.997106 0.0028935 0.063 0.976872 0.0231279 0.023 0.996839 0.0031615 0.064 0.976154 0.0238461 0.024 0.996559 0.0034411 0.065 0.975426 0.0245742 0.025 0.996267 0.0037325 0.066 0.974688 0.0253122 0.026 0.995964 0.0040356 0.067 0.973940 0.0260601 0.027 0.995650 0.0043502 0.068 0.973182 0.0268177 0.028 0.995323 0.0046765 0.069 0.972415 0.0275851 0.029 0.994986 0.0050144 0.070 0.971638 0.0283621 0.030 0.994636 0.0053638 0.071 0.970851 0.0291488 0.031 0.994275 0.0057248 0.087 0.957011 0.0429894 0.032 0.993903 0.0060972 0.088 0.956070 0.0439295 0.033 0.993519 0.0064811 0.089 0.955122 0.0448780 0.034 0.993124 0.0068764 0.090 0.954165 0.0458348 0.035 0.992717 0.0072832 0.091 0.953200 0.0467999 0.036 0.992299 0.0077012 0.092 0.952227 0.0477733 0.037 0.991869 0.0081306 0.093 0.951245 0.0487547 0.038 0.991429 0.0085713 0.094 0.950256 0.0497443 0.039 0.990977 0.0090232 0.095 0.949258 0.0507418 0.040 0.990514 0.0094863 0.096 0.948253 0.0517472 0.041 0.990039 0.0099606 0.097 0.947239 0.0527605 0.042 0.989554 0.0104460 0.098 0.946218 0.0537816 0.043 0.989057 0.0109425 0.099 0.945190 0.0548105 0.044 0.988550 0.0114500 0.100 0.944153 0.0558469 0.045 0.988031 0.0119686 0.046 0.987502 0.0124980 0.047 0.986962 0.0130384 0.048 0.986410 0.0135897 0.049 0.986410 0.0141517 0.050 0.985848 0.0147245
American Journal of Mathematics and Statistics 2013, 3(6): 362-374 367
Table 3. Conditions for Mode βMean β1, Where ππππππππ β ππ
Decimal Places Mode β mean β1,
2 0<ππβ€0.028
1 0<ππβ€0.094
Thus β(π¦π¦) is symmetrical about one with Mode β
1 β
mean correct to two decimal places when 0 < ππ β€ 0.028
correct to one decimal place when 0 < ππ β€ 0.094
2.4. Use of Simulated Error Terms To find the region where the bell-shaped conditions are
satisfied, artificial data were generated from ππ(1,ππ2) for πππ‘π‘ , subsequently transformed to obtain πππ‘π‘β² = 1
πππ‘π‘2 for 0.01 β€ ππ β€
0.10. Values of the required statistics were obtained for each of the variables, πππ‘π‘ ππππππ πππ‘π‘β² as shown in Tables 4 to 8. For each configuration of (ππ = 100,0.01 β€ π₯π₯ β€ 0.10), 1000 replications were performed for values of ππ in steps of 0.01. For want of space the result of the first 25 replications are shown for the configurations, ( ππ = 100,ππ = 0.05)
and .
2.5. Derivation of the Mean and Variance of
By definition,
πΈπΈ(ππ) = (21)
= πππ¦π¦
= 1
2ππβ2πποΏ½1βπποΏ½β1πποΏ½οΏ½
ππβ1
2οΏ½π¦π¦β
12β1ππ οΏ½
2
πππ¦π¦ (22)
Let u = π¦π¦β12 β1ππ
, π‘π‘βππππ π¦π¦ = (ππππ + 1)β2 and πππ¦π¦ = β2ππ(ππππ + 1)β3ππππ for β< u<
β΄ πΈπΈ(ππ) =1
2ππβ2ππ οΏ½1 β πποΏ½β1πποΏ½οΏ½
ππβππ22 (β2ππ(ππππ + 1)β3ππππ)
= 1
β2πποΏ½1βπποΏ½β1πποΏ½οΏ½
(23)
Using the binomial expansion[12]
(1 + π₯π₯)ππ = 1 + ππ1!π₯π₯ + ππ(ππβ1)
2!π₯π₯2 + ππ(ππβ1)(ππβ2)
3!π₯π₯3 + β― (24)
β΄ (1 + ππππ)β2 = 1 β 2(ππππ) + 3(ππππ)2 β 4(ππππ)3 + β― and
πΈπΈ(ππ) = 1
β2πποΏ½1βπποΏ½β1πποΏ½οΏ½
E(Y) = 1
οΏ½1βπποΏ½β1πποΏ½οΏ½
β£β’β’β’β‘
β¦β₯β₯β₯β€
(25)
β΄ πΈπΈ(ππ) =1
οΏ½1 β ππ(β1πποΏ½οΏ½ β
2ππππβ1
2ππ2
β2ππβ
3ππππβ1
2ππ2
β2ππ+
3ππ2
2οΏ½1 + Pr οΏ½ < 1
ππ2οΏ½οΏ½ β4ππππβ
12ππ2
β2ππβ
8ππ3ππβ1
2ππ2
β2ππ+ β―οΏ½
= 1 β 9ππππβ 1
2ππ2
β2πποΏ½1βππ(β1πποΏ½
+3ππ2οΏ½1+PrοΏ½ππ2
(1)< 1 ππ2οΏ½οΏ½
2οΏ½1βππ(β1πποΏ½
β 8ππ3ππβ 1
2ππ2
β2πποΏ½1βππ(β1πποΏ½
+β¦ (26)
To find the variance, we first obtain the second moment;
( )100, 0.08n Ο= = ( )100, 0.10n Ο= =
( )h y
( )0
yh y dyβ
β«21
21 12
30 212 2 1
y
eyy
Ο
Ο Ο ΟΟ
β β
β β
β β
β«
12
0
yβ
β
β«
1Ο
β
( )( )1
2 2
1
1uΟ
Οββ
β
β+β«
( )2
2 2
11
u
u e du
Ο
Οββ
β
β
+β«
( ) ( ) ( )2
2 3 2
1(1 2 3 4 ) du
u
u u u e
Ο
Ο Ο Οβ
β
β
β + β +β¦β«2
2 2 22 32
2 32 2 2
1 1 1 1
2 3 4
2 2 2 2
uu u ue
du ue du u e du u e du
Ο Ο Ο Ο
Ο Ο Ο
Ο Ο Ο Ο
ββ β β
β β β β
β β β β
β + β + β¦β« β« β« β«
( )11 Ο
Οβ β
( )21Ο
368 G. C. Ibeh et al.: Study on the Error Component of Multiplicative Time Series Model Under Inverse Square Transformation
πΈπΈ(ππ2) =
= 12ππβ2πποΏ½1βππ(β1
πποΏ½ (27)
Let u = π¦π¦β12 β1ππ
, y = (Οu + 1)β2and πππ¦π¦ = β2πππ¦π¦32ππππfor β< u <
then
E( )=1
2ππβ2πποΏ½1βππ(β1πποΏ½
=1
β2πποΏ½1βππ(β1πποΏ½
(28)
πππππ‘π‘ (ππππ + 1)β4 = 1 β 4(ππππ) + 10(ππππ)2 β 20(ππππ)3 + β―
β΄E( )=1
β2πποΏ½1βππ(β1πποΏ½
= 1οΏ½1βππ(β1
πποΏ½οΏ½β« ππβ
ππ22
β2ππββ1ππ
ππππ β β« 4ππππππβππ22
β2ππππππ + β« 10(ππππ )2ππβ
ππ22
β2ππββ1ππ
ββ1ππ
ππππ β β« 20(ππππ )3ππβππ22
β2ππππππβ
β1ππ
+ β―οΏ½ (29)
πΈπΈ(ππ2) =1
οΏ½1 β ππ(β1πποΏ½οΏ½οΏ½1 β ππ(β1
πποΏ½ β4ππππβ
12ππ2
β2ππβ
10ππππβ1
2ππ2
β2ππ+ 5ππ2 Pr οΏ½ππ2
(1) < 1ππ2οΏ½ + 5ππ2 β
20ππππβ1
2ππ2
β2ππβ
40ππ3ππβ1
2ππ2
β2ππ
+ β―οΏ½
= 1 β 34ππππβ 1
2ππ2
β2πποΏ½1βππ(β1πποΏ½
+5ππ2οΏ½1+PrοΏ½ππ2
(1)< 1ππ2οΏ½οΏ½
οΏ½1βππ(β1πποΏ½
β 40ππ3ππβ 1
2ππ2
β2πποΏ½1βππ(β1πποΏ½
+ β― (30)
Subsequent terms in series 26 and 30 for E(Y) and E ( 2Y ) respectively are all zeros as they have the factor 21
2e Οβ which
is zero for Ο β€ 0.23 Thus,
(31)
( )2
0
y h y dyβ
β«21
21 11 22
0
y
y e dyΟ
β β
β β β«
β 1Ο
β
2Y2
11 32 2 22
u
y e y duΟ
Οβ
β
β
β
β«
( )2
4 2
11
u
u e du
Ο
Ο β ββ
β
+β«
2Y ( ) ( ) ( )2
2 3 2
1(1 4 10 20 )
u
u u u e du
Ο
Ο Ο Οβ
β
β
β + β +β¦β«
( )( ) ( )
2 2 2 21 12 2
21 15 1 Pr 3 1 Pr
1 11 11 ( 2 1 (
V YΟ Ο Ο Ο
Ο Ο
Ο ΟΟ Ο
+ < + < = + β +
β β β β
( )( ) ( )
22 2 2 2
1 12 2
1 12 1 Pr 3 1 Pr
1 11 ( 2 1 (V Y
Ο Ο Ο ΟΟ Ο
Ο ΟΟ Ο
+ < + < = β
β β β β
Ο<0.24
A
mer
ican
Jour
nal o
f Mat
hem
atic
s and
Sta
tistic
s 201
3, 3
(6):
362-
374
1
Tabl
e 4.
Si
mul
atio
n R
esul
ts w
hen ππ
=0.
05
e tN
(1,ππ
2 ),ππ
=0.
05
ππ π‘π‘β²=
1 ππ π‘π‘2~
N(1
,ππ2 ),ππ
=0.
05
mea
n m
edia
n St
D
Var
πΎπΎ 1
πΎπΎ 2
A
D
P-V
alue
m
ean
med
ian
Std
Var
πΎπΎ 1
πΎπΎ 2
A
D
P-V
alue
πποΏ½πππ‘π‘β² οΏ½
ππ( πππ‘π‘)
1 1.
0008
0.
05
0.00
25
0.01
-0
.05
0.18
3 0.
908
1.00
75
0.99
84
0.10
19
0.01
04
0.42
0.
16
0.45
5 0.
264
4
1 1.
0002
0.
05
0.00
25
0.00
0.
20
0.19
5 0.
889
1.00
75
0.99
97
0.10
22
0.01
04
0.49
0.
63
0.41
9 0.
322
4
1 1.
0024
0.
05
0.00
25
0.00
0.
22
0.23
4 0.
790
1.00
75
0.99
52
0.10
22
0.01
04
0.49
0.
53
0.49
2 0.
213
4
1 1.
0031
0.
05
0.00
25
0.00
-0
.33
0.17
8 0.
918
1.00
75
0.99
39
0.10
20
0.01
04
0.43
0.
16
0.48
9 0.
217
4
1 1.
0037
0.
05
0.00
25
0.10
0.
05
0.43
5 0.
294
1.00
75
0.99
26
0.10
14
0.01
03
0.38
0.
50
0.41
5 0.
328
4
1 1.
0031
0.
05
0.00
25
0.00
-0
.03
0.17
8 0.
918
1.00
75
0.99
38
0.10
20
0.01
04
0.43
0.
16
0.49
0 0.
217
4
1 1.
0011
0.
05
0.00
25
0.07
-0
.04
0.13
7 0.
976
1.00
75
0.99
78
0.10
15
0.01
03
0.36
0.
07
0.35
2 0.
461
4
1 0.
9951
0.
05
0.00
25
0.05
0.
10
0.19
6 0.
888
1.00
75
1.00
99
0.10
16
0.01
03
0.39
0.
12
0.45
8 0.
258
4
1 1.
0003
0.
05
0.00
25
0.01
0.
06
0.20
0 0.
880
1.00
75
0.99
94
0.10
20
0.01
04
0.44
0.
25
0.48
6 0.
221
4
1 1.
0037
0.
05
0.00
25
0.10
0.
05
0.43
5 0.
294
1.00
75
0.99
26
0.10
14
0.01
03
0.38
0.
50
0.41
5 0.
328
4
1 0.
9992
0.
05
0.00
25
-0.0
1 -0
.05
0.18
3 0.
908
1.00
75
1.00
16
0.10
21
0.01
04
0.46
0.
40
0.32
1 0.
525
4
1 0.
9986
0.
05
0.00
25
0.10
0.
10
0.25
0 0.
739
1.00
75
1.00
29
0.10
14
0.01
03
0.39
0.
54
0.26
6 0.
684
4
1 1.
0008
0.
05
0.00
25
0.18
0.
05
0.20
9 0.
859
1.00
74
0.99
84
0.10
06
0.01
01
0.24
-0
.07
0.35
5 0.
454
4
1 1.
0023
0.
05
0.00
25
0.03
-0
.00
0.19
5 0.
889
1.00
75
0.99
53
0.10
18
0.01
04
0.42
0.
28
0.43
9 0.
288
4
1 1.
0026
0.
05
0.00
25
0.05
-0
.12
0.14
1 0.
972
1.00
75
0.99
49
0.10
16
0.01
03
0.37
0.
05
0.34
2 0.
486
4
1 0.
9979
0.
05
0.00
25
0.27
0.
18
0.31
0 0.
552
1.00
74
1.00
42
0.10
01
0.01
00
0.19
0.
16
0.27
5 0.
654
4
1 1.
0005
0.
05
0.00
25
-0.1
4 -0
.47
0.26
2 0.
699
1.00
76
0.99
89
0.10
27
0.01
06
0.49
-0
.11
0.55
1 0.
152
4
1 0.
9986
0.
05
0.00
25
0.03
-0
.04
0.18
2 0.
911
1.00
75
1.00
29
0.10
17
0.01
04
0.39
0.
03
0.49
5 0.
210
4
1 0.
9965
0.
05
0.00
25
0.02
0.
27
0.15
0 0.
962
1.00
75
1.00
70
0.10
21
0.01
04
0.49
0.
68
0.35
9 0.
445
4
1 0.
9949
0.
05
0.00
25
0.25
0.
04
0.29
0 0.
606
1.00
74
1.01
03
0.10
02
0.01
00
0.19
0.
04
0.21
6 0.
843
4
1 0.
9942
0.
05
0.00
25
0.16
0.
04
0.45
0 0.
270
1.00
74
1.01
17
0.10
09
0.01
02
0.30
0.
36
0.35
9 0.
443
4
1 0.
9959
0.
05
0.00
25
0.09
-0
.10
0.30
6 0.
559
1.00
75
1.00
83
0.10
12
0.01
02
0.31
-0
.11
0.54
2 0.
160
4
1 0.
9989
0.
05
0.00
25
0.01
-0
.13
0.19
9 0.
882
1.00
75
1.00
21
0.10
19
0.01
04
0.40
-0
.00
0.49
0 0.
216
4
1 0.
9952
0.
05
0.00
25
0.19
-0
.14
0.21
6 0.
841
1.00
74
1.00
97
0.10
04
0.01
01
0.21
-0
.15
0.22
3 0.
823
4
1 0.
9954
0.
05
0.00
25
0.25
-0
.09
0.31
1 0.
546
1.00
74
1.00
93
0.10
00
0.01
00
0.14
0.
26
0.35
8 0.
446
4
NB
(i) 3
Ο2 = 3
(0.0
025)
= 0
.007
5
(ii) E
(πππ‘π‘β² )
β 1
+3Ο2
American Journal of Mathematics and Statistics 2013, 3(6): 362-376 369
2 G
. C. I
beh
et a
l.:
Stud
y on
the
Erro
r Com
pone
nt o
f Mul
tiplic
ativ
e Ti
me
Serie
s
Mod
el U
nder
Inve
rse
Squa
re T
rans
form
atio
n
Tabl
e 5.
Si
mul
atio
n R
esul
ts fo
r ππ
=0.
08
e tN
(1,ππ
2 ),ππ
=0.
08
ππ π‘π‘β²=
1 ππ π‘π‘2~
N(1
,ππ2 ),ππ
=0.
08
Mea
n St
D
Var
M
edia
n πΎπΎ 1
πΎπΎ 2
A
D
P-V
alue
M
ean
StD
V
ar
Med
ian
πΎπΎ 1
πΎπΎ 2
AD
P-
valu
e ππ(ππ π‘π‘β²
)ππ(ππ π‘π‘
)
1 0.
08
0.00
64
1.00
13
0.01
-0
.05
0.18
3 0.
908
1.01
96
0.16
81
0.02
83
0.99
75
0.69
0.
63
0.81
3 0.
034
4
1 0.
08
0.00
64
1.00
13
0 0.
2 0.
195
0.88
9 1.
0197
0.
1693
0.
0287
0.
9994
0.
82
1.35
0.
794
0.03
8 4
1 0.
08
0.00
64
1.00
39
0 0.
22
0.23
4 0.
79
1.01
97
0.16
93
0.02
87
0.99
23
0.81
1.
13
0.89
3 0.
022
4
1 0.
08
0.00
64
1.00
49
0 -0
.03
0.17
8 0.
918
1.01
96
0.16
83
0.02
83
0.99
02
0.7
0.59
0.
896
0.02
1 4
1 0.
08
0.00
64
1.00
59
0.1
0.05
0.
435
0.29
4 1.
0194
0.
1671
0.
0279
0.
9883
0.
71
1.15
0.
663
0.08
1 4
1 0.
08
0.00
64
1.00
49
0 -0
.03
0.17
8 0.
918
1.01
96
0.16
83
0.02
83
0.99
02
0.7
0.59
0.
896
0.02
1 4
1 0.
08
0.00
64
1.00
18
0.07
-0
.04
0.13
7 0.
976
1.01
95
0.16
7 0.
0279
0.
9964
0.
62
0.41
0.
703
0.06
4 4
1 0.
08
0.00
64
0.99
21
0.05
0.
1 0.
196
0.88
8 1.
0195
0.
1675
0.
028
1.01
59
0.65
0.
45
0.83
3 0.
031
4
1 0.
08
0.00
64
1.00
05
0.01
0.
06
0.2
0.88
1.
0196
0.
1684
0.
0284
0.
999
0.72
0.
66
0.90
5 0.
02
4
1 0.
08
0.00
64
1.00
59
0.1
0.05
0.
435
0.29
4 1.
0194
0.
1671
0.
0279
0.
9883
0.
71
1.15
0.
063
0.08
1 4
1 0.
08
0.00
64
0.99
87
-0.0
1 -0
.05
0.18
3 0.
908
1.01
96
0.16
89
0.02
85
1.00
25
0.77
1.
08
0.63
0.
098
4
1 0.
08
0.00
64
0.99
77
0.1
0.1
0.25
0.
739
1.01
94
0.16
72
0.02
8 1.
0046
0.
72
1.27
0.
509
0.19
4 4
1 0.
08
0.00
64
1.00
13
0.18
0.
05
0.20
9 0.
859
1.01
92
0.16
46
0.02
71
0.99
74
0.49
0.
11
0.66
7 0.
079
4
1 0.
08
0.00
64
1.00
38
0.03
0
0.19
5 0.
889
1.01
96
0.16
8 0.
0282
0.
9925
0.
71
0.83
0.
827
0.03
2 4
1 0.
08
0.00
64
1.00
41
0.05
-0
.12
0.14
1 0.
972
1.01
95
0.16
72
0.02
79
0.99
18
0.63
0.
43
0.68
3 0.
072
4
1 0.
08
0.00
64
0.99
67
0.27
0.
18
0.31
0.
55
1.01
91
0.16
34
0.02
67
1.00
67
0.47
0.
5 0.
481
0.22
7 4
1 0.
08
0.00
64
1.00
09
-0.1
4 -0
.47
0.26
2 0.
699
1.01
98
0.17
0.
0289
0.
9983
0.
72
0.32
0.
934
0.01
7 5
1 0.
08
0.00
64
0.99
77
0.03
-0
.04
0.18
2 0.
911
1.01
95
0.16
76
0.02
81
1.00
46
0.64
0.
35
0.89
5 0.
022
4
1 0.
08
0.00
64
0.99
45
0.02
0.
27
0.15
0.
962
1.01
96
0.16
92
0.02
86
1.01
12
0.83
1.
47
0.73
5 0.
054
4
1 0.
08
0.00
64
0.99
18
0.25
0.
04
0.29
0.
606
1.01
91
0.16
35
0.02
67
1.01
66
0.46
0.
28
0.42
3 0.
314
4
1 0.
08
0.00
64
0.99
07
0.16
0.
04
0.45
0.
27
1.01
93
0.16
56
0.02
74
1.01
88
0.62
0.
94
0.56
4 0.
141
4
1 0.
08
0.00
64
0.99
34
0.09
-0
.1
0.30
6 0.
559
1.01
94
0.16
61
0.02
76
1.01
34
0.55
0.
1 0.
92
0.01
9 4
1 0.
08
0.00
64
0.99
83
0.01
-0
.13
0.19
9 0.
882
1.01
96
0.16
79
0.02
82
1.00
34
0.65
0.
28
0.91
1 0.
02
4
1 0.
08
0.00
64
0.99
23
0.19
-0
.14
0.21
6 0.
841
1.01
92
0.16
4 0.
0269
1.
0155
0.
45
0.04
0.
453
0.26
6 4
1 0.
08
0.00
64
0.99
26
0.25
-0
.09
0.31
1 0.
546
1.01
91
0.16
28
0.02
65
1.01
49
0.37
-0
.17
0.59
4 0.
118
4
NB
(i) 3
Ο2 = 3
(0.0
025)
= 0
.007
5
(ii) E
(πππ‘π‘β² )
β 1
+3Ο2
370 G. C. Ibeh et al.: Study on the Error Component of Multiplicative Time Series Model Under Inverse Square Transformation
A
mer
ican
Jour
nal o
f Mat
hem
atic
s and
Sta
tistic
s 201
3, 3
(6):
362-
374
3
Tabl
e 6.
Si
mul
atio
n R
esul
ts fo
r ππ
=0.
10
e tN
(1,ππ
2 ),ππ
=0.
10
ππ π‘π‘β²=
1 ππ π‘π‘2~
N(1
,ππ2 ),ππ
=0.
10
Mea
n St
D
Var
M
edia
n πΎπΎ 1
πΎπΎ 2
A
D
P-V
alue
M
ean
StD
V
ar
Med
ian
πΎπΎ 1
πΎπΎ 2
AD
P-
Val
ue
ππ(ππ π‘π‘β²
)ππ(ππ π‘π‘
)
1 0.
1 0.
01
1.00
16
0.01
-0
.05
0.18
3 0.
908
1.03
11
0.21
64
0.04
68
0.99
69
0.89
1.
12
1.14
2 0.
005
5
1 0.
1 0.
01
1.00
03
0 0.
2 0.
195
0.88
9 1.
0313
0.
2189
0.
0479
0.
9993
1.
07
2.11
1.
165
<0.0
05
5
1 0.
1 0.
01
1.00
49
0 0.
22
0.23
4 0.
79
1.03
14
0.21
88
0.04
79
0.99
04
1.03
1.
74
1.28
3 <0
.005
5
1 0.
1 0.
01
1.00
62
0 -0
.03
0.17
8 0.
918
1.03
12
0.21
67
0.47
0.
9878
0.
89
1.04
1.
266
<0.0
05
5
1 0.
1 0.
01
1.00
74
0.1
0.05
0.
435
0.29
4 1.
0309
0.
2152
0.
463
0.98
54
0.95
1.
8 0.
958
0.01
5 5
1 0.
1 0.
01
1.00
62
0 -0
.03
0.17
8 0.
918
1.03
12
0.21
67
0.47
0.
9878
0.
89
1.04
1.
266
<0.0
05
5
1 0.
1 0.
01
1.00
22
0.07
-0
.04
0.13
7 0.
976
1.03
09
0.21
44
0.04
6 0.
9955
0.
81
0.77
1.
037
0.01
5
1 0.
1 0.
01
0.99
02
0.05
0.
1 0.
196
0.88
8 1.
031
0.21
53
0.04
64
1.02
0.
84
0.81
1.
184
<0.0
05
5
1 0.
1 0.
01
1.00
07
0.01
0.
06
0.2
0.88
1.
0312
0.
2169
0.
0471
0.
9987
0.
91
1.07
1.
299
<0.0
05
5
1 0.
1 0.
01
1.00
74
0.1
0.05
0.
435
0.29
4 1.
0309
0.
2152
0.
0463
0.
9854
0.
95
1.8
0.95
8 0.
015
5
1 0.
1 0.
01
0.99
84
-0.0
1 -0
.05
0.18
3 0.
908
1.03
13
0.21
8 0.
0475
1.
0032
1
1.77
0.
951
0.01
6 5
1 0.
1 0.
01
0.99
71
0.1
0.1
0.25
0.
739
1.03
09
0.21
55
0.04
64
1.00
58
0.97
2.
04
0.78
9 0.
039
5
1 0.
1 0.
01
1.00
16
0.18
0.
05
0.20
9 0.
859
1.03
04
0.21
05
0.04
43
0.99
67
0.66
0.
34
0.97
0.
014
5
1 0.
1 0.
01
1.00
47
0.03
0
0.19
5 0.
889
1.03
11
0.21
64
0.04
68
0.99
07
0.92
1.
43
1.19
4 <0
.005
5
1 0.
1 0.
01
1.00
52
0.05
-0
.12
0.14
1 0.
972
1.03
1 0.
2147
0.
0461
0.
9898
0.
81
0.82
1.
009
0.01
1 5
1 0.
1 0.
01
0.99
59
0.27
0.
18
0.31
0.
55
1.03
01
0.20
88
0.04
36
1.00
83
0.68
0.
92
0.71
4 0.
06
5
1 0.
1 0.
01
1.00
11
-0.1
4 -0
.47
0.26
2 0.
699
1.03
16
0.21
91
0.04
8 0.
9979
0.
89
0.71
1.
288
<0.0
05
5
1 0.
1 0.
01
0.99
71
0.03
-0
.04
0.18
2 0.
911
1.03
11
0.21
54
0.04
64
1.00
58
0.82
0.
69
1.25
5 <0
.005
5
1 0.
1 0.
01
0.99
31
0.02
0.
27
0.15
0.
962
1.03
13
0.21
89
0.04
79
1.01
4 1.
09
2.33
1.
112
0.00
6 5
1 0.
1 0.
01
0.98
97
0.25
0.
04
0.29
0.
606
1.03
02
0.20
88
0.04
36
1.02
08
0.64
0.
56
0.67
3 0.
077
5
1 0.
1 0.
01
0.98
84
0.16
0.
04
0.45
0.
27
1.03
06
0.21
27
0.04
52
1.02
36
0.85
1.
57
0.82
6 0.
032
5
1 0.
1 0.
01
0.99
17
0.09
-0
.1
0.30
6 0.
559
1.03
07
0.21
28
0.04
53
1.01
67
0.71
0.
33
1.27
5 <0
.005
5
1 0.
1 0.
01
0.99
79
0.01
-0
.13
0.19
9 0.
882
1.03
11
0.21
58
0.04
66
1.00
43
0.82
0.
57
1.30
3 <0
.005
5
1 0.
1 0.
01
0.99
04
0.19
-0
.14
0.21
6 0.
841
1.03
03
0.20
94
0.04
39
1.01
95
0.61
0.
26
0.70
8 0.
063
5
1 0.
1 0.
01
0.99
08
0.25
-0
.09
0.31
1 0.
546
1.03
01
0.20
74
0.04
3 1.
0187
0.
52
-0.0
2 0.
837
0.03
5
NB
(i) 3
Ο2 = 0
.03
(
ii) E
(πππ‘π‘β² )
β 1
+3Ο2
American Journal of Mathematics and Statistics 2013, 3(6): 362-376
371
372 G. C. Ibeh et al.: Study on the Error Component of Multiplicative Time Series Model Under Inverse Square Transformation
2.6. Numerical Computation of the Mean and Variance of ππππβ²
Numerical computations of the means and variances of the truncated and the transformed probability density functions for ππ β [0.01,0.25]
The mean of the truncated probability density function is given as
πΈπΈβ(ππ) = 1 + ππππβ 1
2ππ2
β2πποΏ½1βπποΏ½β1πποΏ½οΏ½
,
πΈπΈβ(ππ)= 1 if Ο< 0.1 and the mean of the transformed probability density
function is given as
πΈπΈ(ππ) = 1 + 3ππ2
2οΏ½1βπποΏ½β1πποΏ½οΏ½οΏ½ 2
(1) 2
11 Pr ΟΟ
+ <
οΏ½Ο< 0.24
= 1 + 23
2BCΟ
where 1
1B ΟΟ
= β β
= 1,
2
(1) 2
11 PrC Ο
Ο= + <
= 2
= 1 + 23Ο (32) Note that Equation 32 is the relationship observed with
simulated data in Tables 4 to 6 From Equation 12 the variance of the truncated probability
distribution is given as
ππβ(ππ) =ππ2οΏ½1+Pr(ππ(1)
2 < 1ππ2)οΏ½
2οΏ½1βπποΏ½β1πποΏ½οΏ½
Ο< 0.1
= 2
22
CB
Ο Ο=
and the variance of the transformed probability distribution is given as
ππ(ππ) = 2ππ2 οΏ½1+Pr(ππ(1)2 < 1
ππ2)οΏ½
οΏ½1βπποΏ½β1πποΏ½οΏ½
β οΏ½3ππ2οΏ½1+Pr (ππ(1)
2 < 1ππ2)οΏ½
2οΏ½1βπποΏ½β1πποΏ½οΏ½
οΏ½2
Ο<
0.24
= 2 2 2
2 32
C CB B
Ο Ο β
= 4Ο2 β (3Ο2)2
From Table 7 i. πΈπΈ(ππ) = = 1 correct to 1 decimal place (dp)
when Ο< 0.13 ii. ππ(ππ)
ππβ(ππ) 4 correct to 2 dp when ππ β€ 0.020 correct
to 1 dp when ππ β€ 0.070
3. Summary of Results The following results were obtained from the
investigations carried out on inverse square transformation
of error component of the multiplicative model, 1. The curve shapes are bell-shaped and symmetric about
mean=1 for ππ β€ 0.08 2. Using Rolleβs theorem, mode β 1 β mean for
ππ β€ 0.08 to 2 dp to 1 dp
3. Using simulated random errors, a. Median β Mean β 1 when ππ β€ 0.10
b. + 3Ο2
c. when ππ β€ 0.08
4. The p-Value of the Anderson Darlingβs test statistic
strongly supports the non-normality of πππ‘π‘β²at ππ β₯ 0.08
5. Using the moments of πππ‘π‘β² a. πΈπΈ(ππ ) = πΈπΈ(πππ‘π‘β²)= 1 + 3Ο2 correct to 2 dp for ππ β€ 0.04
correct to 1 dp for Ο β€ 0.1
b. ππ(ππ)ππβ(ππ)
= = 4, correct to 2 dp for Ο β€ 0.02 correct to 1 dp for 0.070
Thus the inverse square transformation increases the error variance by four times that of the untransformed error.
The results of this investigation together with findings from similar investigations with respect to the error term πππ‘π‘~ππ(1,ππ2) under other types of transformations are summarized in table 8
( )'tE e
β
'te
0.094Ο β€
( )' 1tE e β
( ) ( )' 4t tV e V eβ
( )( )*
'
t
tV
V
ee
American Journal of Mathematics and Statistics 2013, 3(6): 362-374 373
Table 7. Computations of πΈπΈβ(ππ),πΈπΈ(ππ),ππβ(ππ)ππππππ ππ(ππ) for ππ β [0.01,0.25]
ππ π΄π΄ = Οeβ1Ο2 1
1
B
ΟΟ
β β
=
( )21 2
11 Pr
C
ΟΟ
<+
=
πΈπΈβ(ππ) πΈπΈ(ππ) ππβ(ππ) ππ(ππ) ππ(ππ)ππβ(ππ)
0.010 0.0000000 1.00000 2.00000 1.00000 1.00030 0.000100 0.000400 3.99910 0.015 0.0000000 1.00000 2.00000 1.00000 1.00068 0.000225 0.000900 3.99797 0.020 0.0000000 1.00000 2.00000 1.00000 1.00120 0.000400 0.001599 3.99640 0.025 0.0000000 1.00000 2.00000 1.00000 1.00188 0.000625 0.002496 3.99437 0.030 0.0000000 1.00000 2.00000 1.00000 1.00270 0.000900 0.003593 3.99190 0.035 0.0000000 1.00000 2.00000 1.00000 1.00368 0.001225 0.004886 3.98897 0.040 0.0000000 1.00000 2.00000 1.00000 1.00480 0.001600 0.006377 3.98560 0.045 0.0000000 1.00000 2.00000 1.00000 1.00608 0.002025 0.008063 3.98177 0.050 0.0000000 1.00000 2.00000 1.00000 1.00750 0.002500 0.009944 3.97750 0.055 0.0000000 1.00000 2.00000 1.00000 1.00907 0.003025 0.012018 3.97277 0.060 0.0000000 1.00000 2.00000 1.00000 1.01080 0.003600 0.014283 3.96760 0.065 0.0000000 1.00000 2.00000 1.00000 1.01268 0.004225 0.016739 3.96198 0.070 0.0000000 1.00000 2.00000 1.00000 1.01470 0.004900 0.019384 3.95590 0.075 0.0000000 1.00000 2.00000 1.00000 1.01688 0.005625 0.022215 3.94938 0.080 0.0000000 1.00000 2.00000 1.00000 1.01920 0.006400 0.025231 3.94240 0.085 0.0000000 1.00000 2.00000 1.00000 1.02167 0.007225 0.028430 3.93498 0.090 0.0000000 1.00000 2.00000 1.00000 1.02430 0.008100 0.031810 3.92710 0.095 0.0000000 1.00000 2.00000 1.00000 1.02708 0.009025 0.035367 3.91878 0.100 0.0000000 1.00000 2.00000 1.00000 1.03000 0.010000 0.039100 3.91000 0.105 0.0000000 1.00000 2.00000 1.00000 1.03308 0.011025 0.043006 3.90078 0.110 0.0000000 1.00000 2.00000 1.00000 1.03630 0.012100 0.047082 3.89110 0.115 0.0000000 1.00000 2.00000 1.00000 1.03967 0.013225 0.051326 3.88097 0.120 0.0000000 1.00000 2.00000 1.00000 1.04320 0.014400 0.055734 3.87040 0.125 0.0000000 1.00000 2.00000 1.00000 1.04688 0.015625 0.060303 3.85938 0.130 0.0000000 1.00000 2.00000 1.00000 1.05070 0.016900 0.065030 3.84790 0.135 0.0000000 1.00000 2.00000 1.00000 1.05468 0.018225 0.069911 3.83598 0.140 0.0000000 1.00000 2.00000 1.00000 1.05880 0.019600 0.074943 3.82360 0.145 0.0000000 1.00000 2.00000 1.00000 1.06308 0.021025 0.080122 3.81078 0.150 0.0000000 1.00000 2.00000 1.00000 1.06750 0.022500 0.085444 3.79750
0.155 0.0000000 1.00000 2.00000 1.00000 1.07207 0.024025 0.090905 3.78378
0.160 0.0000000 1.00000 2.00000 1.00000 1.07680 0.025600 0.096502 3.76960
0.165 0.0000000 1.00000 2.00000 1.00000 1.08167 0.027225 0.102229 3.75498
0.170 0.0000000 1.00000 2.00000 1.00000 1.08670 0.028900 0.108083 3.73990
0.175 0.0000000 1.00000 2.00000 1.00000 1.09187 0.030625 0.114059 3.72437
0.180 0.0000000 1.00000 2.00000 1.00000 1.09720 0.032400 0.120152 3.70840
0.185 0.0000001 1.00000 2.00000 1.00000 1.10268 0.034225 0.126358 3.69197
0.190 0.0000002 1.00000 2.00000 1.00000 1.10830 0.036100 0.132671 3.67510
0.195 0.0000004 1.00000 2.00000 1.00000 1.11407 0.038025 0.139087 3.65778
0.200 0.0000007 1.00000 2.00000 1.00000 1.12000 0.040000 0.145600 3.64000
0.205 0.0000014 1.00000 2.00000 1.00000 1.12607 0.042025 0.152205 3.62178
0.210 0.0000025 1.00000 2.00000 1.00000 1.13230 0.044100 0.158897 3.60310
0.215 0.0000043 1.00000 2.00000 1.00000 1.13868 0.046225 0.165669 3.58397
0.220 0.0000072 1.00000 1.99999 1.00000 1.14520 0.048400 0.172517 3.56440
0.225 0.0000116 1.00000 1.99999 1.00000 1.15188 0.050625 0.179434 3.54437
0.230 0.0000181 0.99999 1.99999 1.00001 1.15870 0.052900 0.186414 3.52390
0.235 0.0000275 0.99999 1.99998 1.00001 1.16568 0.055225 0.193452 3.50298
0.240 0.0000408 0.99998 1.99997 1.00002 1.17280 0.057600 0.200540 3.48160
0.245 0.0000591 0.99998 1.99996 1.00002 1.18008 0.060025 0.207673 3.45978
0.250 0.0000839 0.99997 1.99994 1.00003 1.18750 0.062500 0.214844 3.43750
374 G. C. Ibeh et al.: Study on the Error Component of Multiplicative Time Series Model Under Inverse Square Transformation
Table 8. Summary of this and similar research findings with respect to the error term πππ‘π‘~ππ(1,ππ2)under different transformations
πππ‘π‘β² Distribution of πππ‘π‘β² Condition for successful
transformation Relationship between ππ and ππ1
πππππππππππ‘π‘ πππ‘π‘β²~πποΏ½0, 21Ο οΏ½ ππ < 0.1 ππ1 β ππ
1πππ‘π‘
πππ‘π‘β²~πποΏ½1, 21Ο οΏ½ ππ β€ 0.1 ππ1 β ππ
οΏ½πππ‘π‘ πππ‘π‘β²~πποΏ½1, 21Ο οΏ½ ππ β€ 0.30 ππ1 β
12ππ
πππ‘π‘2 πππ‘π‘β²~πποΏ½1, 21Ο οΏ½ ππ β€ 0.027 ππ1 > ππ
1πππ‘π‘2
πππ‘π‘β²~πποΏ½1, 21Ο οΏ½ ππ β€ 0.070 ππ1 β 2ππ
4. Conclusions The results of this research show that the basic
assumptions of the error term of the multiplicative model which is normally distributed with mean 1 and finite variance can only be maintained in inverse square transformation of the error term if the standard deviation of the untransformed error term is less than or equal to 0.070. The study also reveals that the variance of the transformed error term is 4 times the variance of the untransformed for ππ β€ 0.070 , hence the inverse square transformation like square transformation leads to an increase in the error variance whereas the square root and the inverse square root transformations lead to a reduction in the error variance by a quarter while the logarithm and the inverse retain the value the error variance.
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