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 𝑒𝑒𝑡𝑡 and 𝑒𝑒𝑡𝑡′ = 1

𝑒𝑒𝑡𝑡2 , 𝑓𝑓∗(𝑥𝑥) 𝑎𝑎𝑎𝑎𝑎𝑎 ( )h y were compared for some values of

𝜎𝜎 within the interval[0.01,0.25], and it was found that the distribution of the transformed variable loses its symmetry when 𝜎𝜎 > 0.08. Rolle’s Theorem was also used to find the region where ℎ(𝑦𝑦) satisfies the bell-shaped condition; and this is met when𝜎𝜎 ≤ 0.094. Use of the simulated error terms shows that the transformed variable is normal for 𝜎𝜎 < 0.08. Finally, from the functional expressions for 𝐸𝐸(𝑒𝑒𝑡𝑡′) and 𝑉𝑉(𝑒𝑒𝑡𝑡′), it was observed that the mean of 𝑒𝑒𝑡𝑡′ is one and the increased variance is approximately 4 times the variance of 𝑒𝑒𝑡𝑡 for 𝜎𝜎 ≤ 0.070. Therefore, the condition for successful inverse square transformation with respect to the error component of the multiplicative time series model is 𝜎𝜎 ≤ 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: 𝑋𝑋𝑡𝑡 = 𝑇𝑇𝑡𝑡𝑆𝑆𝑡𝑡𝐶𝐶𝑡𝑡𝑒𝑒𝑡𝑡 (1)

Additive Model: 𝑋𝑋𝑡𝑡 = 𝑇𝑇𝑡𝑡+𝑆𝑆𝑡𝑡+𝐶𝐶𝑡𝑡+𝑒𝑒𝑡𝑡 (2)

Mixed Model: 𝑋𝑋𝑡𝑡 = 𝑇𝑇𝑡𝑡𝑆𝑆𝑡𝑡𝐶𝐶𝑡𝑡+𝑒𝑒𝑡𝑡 (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 𝑋𝑋𝑡𝑡 = 𝑀𝑀𝑡𝑡𝑆𝑆𝑡𝑡𝑒𝑒𝑡𝑡 (4) 𝑋𝑋𝑡𝑡 = 𝑀𝑀𝑡𝑡+𝑆𝑆𝑡𝑡+𝑒𝑒𝑡𝑡 (5) 𝑋𝑋𝑡𝑡 = 𝑀𝑀𝑡𝑡𝑆𝑆𝑡𝑡 + 𝑒𝑒𝑡𝑡 (6)

respectively, where 𝑀𝑀𝑡𝑡 is the trend cycle component. Consider a random variable, 𝑋𝑋 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 © 2013 Scientific & Academic Publishing. All Rights Reserved

𝑓𝑓(𝑥𝑥) = �1

𝜎𝜎√2𝜋𝜋𝑒𝑒𝑥𝑥𝑒𝑒 �− (𝑥𝑥−1)2

2𝜎𝜎2 � ,−∞ < 𝑥𝑥 < ∞,𝜎𝜎2 > 00, 𝑜𝑜𝑡𝑡ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒

� (7)

Most often, the random variable 𝑋𝑋, which is normally distributed with mean, 1 and 𝜎𝜎2 < ∞ do not admit values less than or equal to zero. This usually leads to the truncation of all values of 𝑋𝑋 ≤ 0 to take care of the admissible region of 𝑋𝑋 > 0.

The resulting distribution after truncation was given by[3] as

𝑓𝑓∗(𝑥𝑥) = �𝑘𝑘

𝜎𝜎√2𝜋𝜋𝑒𝑒𝑥𝑥𝑒𝑒 �− (𝑥𝑥−1)2

2𝜎𝜎2 � , 0 < 𝑥𝑥 < ∞0 , −∞ < 𝑥𝑥 ≤ 0

� (8)

where 𝑘𝑘 is the normalizing quantity. [4]obtained the value of k to be

𝑘𝑘 = 11−𝜑𝜑�−1

𝜎𝜎� (9)

( ) ( )2

2

*

0, 0

11 exp ,0212 1

x

xx xfσσ π ϕ

σ

− ∞ < ≤

− = − < < ∞ − −

∴

(10) 𝑓𝑓∗(𝑥𝑥) was shown to be a proper probability density

function[4]with mean and variance

𝐸𝐸∗(𝑋𝑋) = 1 + 𝜎𝜎𝑒𝑒− 1

2𝜎𝜎2

√2𝜋𝜋�1−𝜑𝜑�−1𝜎𝜎��

(11)

American Journal of Mathematics and Statistics 2013, 3(6): 362-374 363

Table 1. Bartlett’s transformation for some values of 𝛽𝛽

𝛽𝛽 0 12

1 32

2 3 −1

Transformation No transformation �𝑋𝑋𝑡𝑡 𝑙𝑙𝑜𝑜𝑙𝑙𝑒𝑒𝑋𝑋𝑡𝑡 1

�𝑋𝑋𝑡𝑡 1

𝑋𝑋𝑡𝑡

1𝑋𝑋𝑡𝑡2

𝑋𝑋𝑡𝑡2

𝑉𝑉𝑎𝑎𝑒𝑒∗(𝑋𝑋) =𝜎𝜎2

2 �1 − 𝜑𝜑�−1𝜎𝜎��

�1 + 𝑃𝑃𝑒𝑒 �𝜒𝜒(1)2 < 1

𝜎𝜎2��

− 𝜎𝜎𝑒𝑒− 1

2𝜎𝜎2

√2𝜋𝜋�1−𝜑𝜑�−1𝜎𝜎��

− � 𝜎𝜎𝑒𝑒− 1

2𝜎𝜎2

√2𝜋𝜋�1−𝜑𝜑�−1𝜎𝜎���

2

(12)

1.1. Data Transformation and Classification Data transformation is the application of a non-linear

function such as 𝑙𝑙𝑜𝑜𝑙𝑙𝑒𝑒(𝑋𝑋𝑡𝑡),�𝑋𝑋𝑡𝑡 ,1𝑋𝑋𝑡𝑡

,𝑋𝑋𝑡𝑡2, 1𝑋𝑋𝑡𝑡2 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 𝑌𝑌 = √𝑋𝑋 would be used. If the data follow log-normal distribution, then the logarithmic transformation 𝑌𝑌 = 𝑙𝑙𝑜𝑜𝑙𝑙𝑒𝑒𝑋𝑋 is appropriate. For binomial data expressed as fractions the arcsine transformation 𝑌𝑌 = 𝑎𝑎𝑒𝑒𝑎𝑎𝑒𝑒𝑒𝑒𝑎𝑎𝑋𝑋 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 (𝛽𝛽) in the linear relationship between the natural log of the periodic standard deviations (𝑙𝑙𝑜𝑜𝑙𝑙𝑒𝑒𝜎𝜎𝑒𝑒) and natural log of the periodic means (𝑙𝑙𝑜𝑜𝑙𝑙𝑒𝑒𝜇𝜇𝑒𝑒)given as

𝑙𝑙𝑜𝑜𝑙𝑙𝑒𝑒𝜎𝜎𝑒𝑒 =∝ +𝛽𝛽𝑙𝑙𝑜𝑜𝑙𝑙𝑒𝑒𝜇𝜇𝑒𝑒 (13) For the inverse square transformation, β 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 (𝑒𝑒𝑡𝑡) of a multiplicative time series model where 𝑒𝑒𝑡𝑡~𝑁𝑁(1,𝜎𝜎2) and discovered that the logarithmic transformation 𝑌𝑌𝑡𝑡 = 𝑙𝑙𝑜𝑜𝑙𝑙𝑒𝑒𝑒𝑒𝑡𝑡 isnormally distributed with mean zero and the same variance, 𝜎𝜎2 if 𝜎𝜎 < 0.1

[7]studied the effect of inverse transformation on the error component of the same multiplicative model and established that the inverse transformation 𝑌𝑌𝑡𝑡 = 1

𝑒𝑒𝑡𝑡 is normally

distributed with mean, one and same variance provided 𝜎𝜎 ≤ 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 σ< 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 𝑌𝑌𝑡𝑡 = �𝑒𝑒𝑡𝑡 is normally distributed with unit mean and variance, 1

4𝜎𝜎2 for

𝜎𝜎 ≤ 0.30 where 𝜎𝜎2 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 𝑌𝑌𝑡𝑡 = 𝑒𝑒𝑡𝑡2 can be assumed to be normally distributed with unit mean and same variance for 𝜎𝜎 ≤ 0.027 they observed that 𝑉𝑉𝑎𝑎𝑒𝑒(𝑒𝑒𝑡𝑡2) >𝑉𝑉𝑎𝑎𝑒𝑒(𝑒𝑒𝑡𝑡) 𝑓𝑓𝑜𝑜𝑒𝑒 𝑎𝑎𝑙𝑙𝑙𝑙 𝜎𝜎 and that 𝑒𝑒𝑡𝑡2~𝑁𝑁(1.0,1.0),𝑎𝑎𝑎𝑎𝑎𝑎 𝜎𝜎 ≤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 𝛽𝛽 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 𝛽𝛽 = 3, we adopt inverse square transformation on the multiplicative time series model given in equation (4) to obtain

𝑌𝑌𝑡𝑡 =1𝑋𝑋𝑡𝑡2

=1𝑀𝑀𝑡𝑡

2 .1𝑆𝑆𝑡𝑡2

.1𝑒𝑒𝑡𝑡2

= 𝑀𝑀𝑡𝑡′𝑆𝑆𝑡𝑡′𝑒𝑒𝑡𝑡′ (14)

where 𝑀𝑀𝑡𝑡′ = 1

𝑀𝑀𝑡𝑡2 , 𝑆𝑆𝑡𝑡′ = 1

𝑆𝑆𝑡𝑡2 and 𝑒𝑒𝑡𝑡′ = 1

𝑒𝑒𝑡𝑡2

Thus, it will be of interest to find the distribution of 𝑒𝑒𝑡𝑡′ 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; 𝜇𝜇 = 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, 𝑒𝑒𝑡𝑡′ = 𝑌𝑌 = 1𝑋𝑋2, then

𝑋𝑋 = �1𝑌𝑌�

12 = 𝑌𝑌−

12 (15)

Adapting equations (10 & 15) and using the transformation of variable technique,

h(y)=𝑓𝑓∗(𝑥𝑥) �𝑎𝑎𝑥𝑥𝑎𝑎𝑦𝑦� [11] (16)

hence 21

2 112

32

y ,012 2 1

0 , 0

y

eh yy

y

σ

σ π ϕ σ∞

−−−

= < < ∞− −

− < ≤

（ ） (17)

Observe that h(y) is a proper probability density function (pdf).

2.2. Plot of the Probability Density Functions 𝒇𝒇∗(𝒙𝒙) 𝒂𝒂𝒂𝒂𝒂𝒂 𝒉𝒉(𝒚𝒚)

Using the probability density functions of the two variables as given in Equations, (10) and (17), the functions 𝑓𝑓∗(𝑥𝑥) 𝑎𝑎𝑎𝑎𝑎𝑎 ℎ(𝑦𝑦) were plotted for some values of 𝜎𝜎 (see Figures (1) to (5))

Figure 1. Curves Shapes for 𝜎𝜎 =0.02

Figure 2. Curves Shapes for 𝜎𝜎 =0.04

American Journal of Mathematics and Statistics 2013, 3(6): 362-374 365

Figure 3. Curves Shapes for 𝜎𝜎 =0.06

Figure 4. Curves Shapes for 𝜎𝜎 =0.08

Figure 5. Curves Shapes for 𝜎𝜎 =0.10

2.3. Region of Normality for 𝒉𝒉(𝒚𝒚)

Since h(y) has one maximum point 𝑦𝑦𝑚𝑚𝑎𝑎𝑥𝑥 (mode) hence one maximum value ℎ(𝑦𝑦𝑚𝑚𝑎𝑎𝑥𝑥 ) for all values of σ.

Rolle’s Theorem was used to find the values of σ that satisfy the bell-shaped and symmetric condition of mode≈1≈mean.

Let 𝑘𝑘 = 1

2𝜎𝜎√2𝜋𝜋�1−𝜑𝜑(−1𝜎𝜎)�

in (17)

then, ℎ(𝑦𝑦) = 𝑘𝑘𝑦𝑦−32𝑒𝑒

− 12𝜎𝜎2�𝑦𝑦

−12 −1�

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

ℎ′(𝑦𝑦 ) =

𝑘𝑘 �𝑦𝑦−32 � 1

2𝜎𝜎2 �𝑦𝑦−1

21� 𝑦𝑦−32𝑒𝑒−

�𝑦𝑦−12−1�

2

2𝜎𝜎2 � �+𝑒𝑒− 1

2𝜎𝜎2�𝑦𝑦−1

2−1�2

�− 32𝑦𝑦−

52���(18)

Equating ℎ′(𝑦𝑦) = 0 gives 1𝑦𝑦𝜎𝜎2 �1 − 𝑦𝑦

12� − 3 = 0

3𝜎𝜎2𝑦𝑦 + 𝑦𝑦12 − 1 = 0 (19)

substituting 𝑦𝑦 = 𝑥𝑥2𝑓𝑓𝑒𝑒𝑜𝑜𝑚𝑚 ( 19), 𝑙𝑙𝑒𝑒𝑔𝑔𝑒𝑒𝑒𝑒 3𝜎𝜎2𝑥𝑥2 + 𝑥𝑥 − 1 = 0 (20)

𝑥𝑥 = −1±�1+12𝜎𝜎2

6𝜎𝜎2 , Since 𝑦𝑦𝑚𝑚𝑎𝑎𝑥𝑥 is positive then

𝑥𝑥 =−1+�1+12𝜎𝜎2

6𝜎𝜎2

But 𝑦𝑦 = 𝑥𝑥2, 𝑦𝑦𝑚𝑚𝑎𝑎𝑥𝑥 = �−1+�1+12𝜎𝜎2

6𝜎𝜎2 �2

The bell-shaped condition would imply 𝑦𝑦𝑚𝑚𝑎𝑎𝑥𝑥 ≈ 1, see Table 2 for the numerical computation of 𝑦𝑦𝑚𝑚𝑎𝑎𝑥𝑥 =

�−1+�1+12𝜎𝜎2

6𝜎𝜎2 �2

and Table 3 for the summary values of 𝑦𝑦𝑚𝑚𝑎𝑎𝑥𝑥

Table 2. Numerical Computation of

22

max 21 1 12

6y σ

σ

− + + =

𝜎𝜎 𝑦𝑦𝑚𝑚𝑎𝑎𝑥𝑥 = �−1 + √1 + 12𝜎𝜎2

6𝜎𝜎2 �2

1 − 𝑦𝑦𝑚𝑚𝑎𝑎𝑥𝑥 𝜎𝜎 𝑦𝑦𝑚𝑚𝑎𝑎𝑥𝑥 = �−1 + √1 + 12𝜎𝜎2

6𝜎𝜎2 �2

1 − 𝑦𝑦𝑚𝑚𝑎𝑎𝑥𝑥

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 ≈Mean ≈1, Where 𝒚𝒚𝒎𝒎𝒂𝒂𝒙𝒙 ≈ 𝟏𝟏

Decimal Places Mode ≈ mean ≈1,

2 0<𝜎𝜎≤0.028

1 0<𝜎𝜎≤0.094

Thus ℎ(𝑦𝑦) is symmetrical about one with Mode ≈

1 ≈

mean correct to two decimal places when 0 < 𝜎𝜎 ≤ 0.028

correct to one decimal place when 0 < 𝜎𝜎 ≤ 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 𝑁𝑁(1,𝜎𝜎2) for 𝑒𝑒𝑡𝑡 , subsequently transformed to obtain 𝑒𝑒𝑡𝑡′ = 1

𝑒𝑒𝑡𝑡2 for 0.01 ≤ 𝜎𝜎 ≤

0.10. Values of the required statistics were obtained for each of the variables, 𝑒𝑒𝑡𝑡 𝑎𝑎𝑎𝑎𝑎𝑎 𝑒𝑒𝑡𝑡′ as shown in Tables 4 to 8. For each configuration of (𝑎𝑎 = 100,0.01 ≤ 𝑥𝑥 ≤ 0.10), 1000 replications were performed for values of 𝜎𝜎 in steps of 0.01. For want of space the result of the first 25 replications are shown for the configurations, ( 𝑎𝑎 = 100,𝜎𝜎 = 0.05)

and .

2.5. Derivation of the Mean and Variance of

By definition,

𝐸𝐸(𝑌𝑌) = (21)

= 𝑎𝑎𝑦𝑦

= 1

2𝜎𝜎√2𝜋𝜋�1−𝜑𝜑�−1𝜎𝜎��

𝑒𝑒−1

2�𝑦𝑦−

12−1𝜎𝜎 �

2

𝑎𝑎𝑦𝑦 (22)

Let u = 𝑦𝑦−12 −1𝜎𝜎

, 𝑡𝑡ℎ𝑒𝑒𝑎𝑎 𝑦𝑦 = (𝜎𝜎𝜎𝜎 + 1)−2 and 𝑎𝑎𝑦𝑦 = −2𝜎𝜎(𝜎𝜎𝜎𝜎 + 1)−3𝑎𝑎𝜎𝜎 for ∞< u<

∴ 𝐸𝐸(𝑌𝑌) =1

2𝜎𝜎√2𝜋𝜋 �1 − 𝜑𝜑�−1𝜎𝜎��

𝑒𝑒−𝜎𝜎22 (−2𝜎𝜎(𝜎𝜎𝜎𝜎 + 1)−3𝑎𝑎𝜎𝜎)

= 1

√2𝜋𝜋�1−𝜑𝜑�−1𝜎𝜎��

(23)

Using the binomial expansion[12]

(1 + 𝑥𝑥)𝑎𝑎 = 1 + 𝑎𝑎1!𝑥𝑥 + 𝑎𝑎(𝑎𝑎−1)

2!𝑥𝑥2 + 𝑎𝑎(𝑎𝑎−1)(𝑎𝑎−2)

3!𝑥𝑥3 + ⋯ (24)

∴ (1 + 𝜎𝜎𝜎𝜎)−2 = 1 − 2(𝜎𝜎𝜎𝜎) + 3(𝜎𝜎𝜎𝜎)2 − 4(𝜎𝜎𝜎𝜎)3 + ⋯ and

𝐸𝐸(𝑌𝑌) = 1

√2𝜋𝜋�1−𝜑𝜑�−1𝜎𝜎��

E(Y) = 1

�1−𝜑𝜑�−1𝜎𝜎��

⎣⎢⎢⎢⎡

⎦⎥⎥⎥⎤

(25)

∴ 𝐸𝐸(𝑌𝑌) =1

�1 − 𝜑𝜑(−1𝜎𝜎�� −

2𝜎𝜎𝑒𝑒−1

2𝜎𝜎2

√2𝜋𝜋−

3𝜎𝜎𝑒𝑒−1

2𝜎𝜎2

√2𝜋𝜋+

3𝜎𝜎2

2�1 + Pr � < 1

𝜎𝜎2�� −4𝜎𝜎𝑒𝑒−

12𝜎𝜎2

√2𝜋𝜋−

8𝜎𝜎3𝑒𝑒−1

2𝜎𝜎2

√2𝜋𝜋+ ⋯�

= 1 − 9𝜎𝜎𝑒𝑒− 1

2𝜎𝜎2

√2𝜋𝜋�1−𝜑𝜑(−1𝜎𝜎�

+3𝜎𝜎2�1+Pr�𝜒𝜒2

(1)< 1 𝜎𝜎2��

2�1−𝜑𝜑(−1𝜎𝜎�

− 8𝜎𝜎3𝑒𝑒− 1

2𝜎𝜎2

√2𝜋𝜋�1−𝜑𝜑(−1𝜎𝜎�

+… (26)

To find the variance, we first obtain the second moment;

( )100, 0.08n σ= = ( )100, 0.10n σ= =

( )h y

( )0

yh y dy∞

∫21

21 12

30 212 2 1

y

eyy

σ

σ π ϕσ

− −

− ∞

− −

∫

12

0

y∞

−

∫

1σ

−

( )( )1

2 2

1

1uσ

σ−−

−

∞+∫

( )2

2 2

11

u

u e du

σ

σ−−

∞

−

+∫

( ) ( ) ( )2

2 3 2

1(1 2 3 4 ) du

u

u u u e

σ

σ σ σ−

∞

−

− + − +…∫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

σ σ σ σ

σ σ σ

π π π π

−− − −

∞ ∞ ∞ ∞

− − − −

− + − + …∫ ∫ ∫ ∫

( )11 ϕ

σ− −

( )21χ

368 G. C. Ibeh et al.: Study on the Error Component of Multiplicative Time Series Model Under Inverse Square Transformation

𝐸𝐸(𝑌𝑌2) =

= 12𝜎𝜎√2𝜋𝜋�1−𝜑𝜑(−1

𝜎𝜎� (27)

Let u = 𝑦𝑦−12 −1𝜎𝜎

, y = (σu + 1)−2and 𝑎𝑎𝑦𝑦 = −2𝜎𝜎𝑦𝑦32𝑎𝑎𝜎𝜎for ∞< u <

then

E( )=1

2𝜎𝜎√2𝜋𝜋�1−𝜑𝜑(−1𝜎𝜎�

=1

√2𝜋𝜋�1−𝜑𝜑(−1𝜎𝜎�

(28)

𝑏𝑏𝜎𝜎𝑡𝑡 (𝜎𝜎𝜎𝜎 + 1)−4 = 1 − 4(𝜎𝜎𝜎𝜎) + 10(𝜎𝜎𝜎𝜎)2 − 20(𝜎𝜎𝜎𝜎)3 + ⋯

∴E( )=1

√2𝜋𝜋�1−𝜑𝜑(−1𝜎𝜎�

= 1�1−𝜑𝜑(−1

𝜎𝜎��∫ 𝑒𝑒−

𝜎𝜎22

√2𝜋𝜋∞−1𝜎𝜎

𝑎𝑎𝜎𝜎 − ∫ 4𝜎𝜎𝜎𝜎𝑒𝑒−𝜎𝜎22

√2𝜋𝜋𝑎𝑎𝜎𝜎 + ∫ 10(𝜎𝜎𝜎𝜎 )2𝑒𝑒−

𝜎𝜎22

√2𝜋𝜋∞−1𝜎𝜎

∞−1𝜎𝜎

𝑎𝑎𝜎𝜎 − ∫ 20(𝜎𝜎𝜎𝜎 )3𝑒𝑒−𝜎𝜎22

√2𝜋𝜋𝑎𝑎𝜎𝜎∞

−1𝜎𝜎

+ ⋯� (29)

𝐸𝐸(𝑌𝑌2) =1

�1 − 𝜑𝜑(−1𝜎𝜎���1 − 𝜑𝜑(−1

𝜎𝜎� −4𝜎𝜎𝑒𝑒−

12𝜎𝜎2

√2𝜋𝜋−

10𝜎𝜎𝑒𝑒−1

2𝜎𝜎2

√2𝜋𝜋+ 5𝜎𝜎2 Pr �𝜒𝜒2

(1) < 1𝜎𝜎2� + 5𝜎𝜎2 −

20𝜎𝜎𝑒𝑒−1

2𝜎𝜎2

√2𝜋𝜋−

40𝜎𝜎3𝑒𝑒−1

2𝜎𝜎2

√2𝜋𝜋

+ ⋯�

= 1 − 34𝜎𝜎𝑒𝑒− 1

2𝜎𝜎2

√2𝜋𝜋�1−𝜑𝜑(−1𝜎𝜎�

+5𝜎𝜎2�1+Pr�𝜒𝜒2

(1)< 1𝜎𝜎2��

�1−𝜑𝜑(−1𝜎𝜎�

− 40𝜎𝜎3𝑒𝑒− 1

2𝜎𝜎2

√2𝜋𝜋�1−𝜑𝜑(−1𝜎𝜎�

+ ⋯ (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 σ− which

is zero for σ ≤ 0.23 Thus,

(31)

( )2

0

y h y dy∞

∫21

21 11 22

0

y

y e dyσ

− −

− ∞ ∫

⇒ 1σ

−

2Y2

11 32 2 22

u

y e y duσ

σ−

∞

−

−

∫

( )2

4 2

11

u

u e du

σ

σ − −∞

−

+∫

2Y ( ) ( ) ( )2

2 3 2

1(1 4 10 20 )

u

u u u e du

σ

σ σ σ−

∞

−

− + − +…∫

( )( ) ( )

2 2 2 21 12 2

21 15 1 Pr 3 1 Pr

1 11 11 ( 2 1 (

V Yσ χ σ χ

σ σ

ϕ ϕσ σ

+ < + < = + − +

− − − −

( )( ) ( )

22 2 2 2

1 12 2

1 12 1 Pr 3 1 Pr

1 11 ( 2 1 (V Y

σ χ σ χσ σ

ϕ ϕσ σ

+ < + < = −

− − − −

σ<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 𝜎𝜎

=0.

05

e tN

(1,𝜎𝜎

2 ),𝜎𝜎

=0.

05

𝑒𝑒 𝑡𝑡′=

1 𝑒𝑒 𝑡𝑡2~

N(1

,𝜎𝜎2 ),𝜎𝜎

=0.

05

mea

n m

edia

n St

D

Var

𝛾𝛾 1

𝛾𝛾 2

A

D

P-V

alue

m

ean

med

ian

Std

Var

𝛾𝛾 1

𝛾𝛾 2

A

D

P-V

alue

𝑉𝑉�𝑒𝑒𝑡𝑡′ �

𝑉𝑉( 𝑒𝑒𝑡𝑡)

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

σ2 = 3

(0.0

025)

= 0

.007

5

(ii) E

(𝑒𝑒𝑡𝑡′ )

≈ 1

+3σ2

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 𝜎𝜎

=0.

08

e tN

(1,𝜎𝜎

2 ),𝜎𝜎

=0.

08

𝑒𝑒 𝑡𝑡′=

1 𝑒𝑒 𝑡𝑡2~

N(1

,𝜎𝜎2 ),𝜎𝜎

=0.

08

Mea

n St

D

Var

M

edia

n 𝛾𝛾 1

𝛾𝛾 2

A

D

P-V

alue

M

ean

StD

V

ar

Med

ian

𝛾𝛾 1

𝛾𝛾 2

AD

P-

valu

e 𝑉𝑉(𝑒𝑒 𝑡𝑡′

)𝑉𝑉(𝑒𝑒 𝑡𝑡

)

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

σ2 = 3

(0.0

025)

= 0

.007

5

(ii) E

(𝑒𝑒𝑡𝑡′ )

≈ 1

+3σ2

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 𝜎𝜎

=0.

10

e tN

(1,𝜎𝜎

2 ),𝜎𝜎

=0.

10

𝑒𝑒 𝑡𝑡′=

1 𝑒𝑒 𝑡𝑡2~

N(1

,𝜎𝜎2 ),𝜎𝜎

=0.

10

Mea

n St

D

Var

M

edia

n 𝛾𝛾 1

𝛾𝛾 2

A

D

P-V

alue

M

ean

StD

V

ar

Med

ian

𝛾𝛾 1

𝛾𝛾 2

AD

P-

Val

ue

𝑉𝑉(𝑒𝑒 𝑡𝑡′

)𝑉𝑉(𝑒𝑒 𝑡𝑡

)

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

σ2 = 0

.03

(

ii) E

(𝑒𝑒𝑡𝑡′ )

≈ 1

+3σ2

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 𝒆𝒆𝒕𝒕′

Numerical computations of the means and variances of the truncated and the transformed probability density functions for 𝜎𝜎 ∈ [0.01,0.25]

The mean of the truncated probability density function is given as

𝐸𝐸∗(𝑋𝑋) = 1 + 𝜎𝜎𝑒𝑒− 1

2𝜎𝜎2

√2𝜋𝜋�1−𝜑𝜑�−1𝜎𝜎��

,

𝐸𝐸∗(𝑋𝑋)= 1 if σ< 0.1 and the mean of the transformed probability density

function is given as

𝐸𝐸(𝑌𝑌) = 1 + 3𝜎𝜎2

2�1−𝜑𝜑�−1𝜎𝜎��� 2

(1) 2

11 Pr χσ

+ <

�σ< 0.24

= 1 + 23

2BCσ

where 1

1B ϕσ

= − −

= 1,

2

(1) 2

11 PrC χ

σ= + <

= 2

= 1 + 23σ (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

𝑉𝑉∗(𝑋𝑋) =𝜎𝜎2�1+Pr(𝜒𝜒(1)

2 < 1𝜎𝜎2)�

2�1−𝜑𝜑�−1𝜎𝜎��

σ< 0.1

= 2

22

CB

σ σ=

and the variance of the transformed probability distribution is given as

𝑉𝑉(𝑌𝑌) = 2𝜎𝜎2 �1+Pr(𝜒𝜒(1)2 < 1

𝜎𝜎2)�

�1−𝜑𝜑�−1𝜎𝜎��

− �3𝜎𝜎2�1+Pr (𝜒𝜒(1)

2 < 1𝜎𝜎2)�

2�1−𝜑𝜑�−1𝜎𝜎��

�2

σ<

0.24

= 2 2 2

2 32

C CB B

σ σ −

= 4σ2 – (3σ2)2

From Table 7 i. 𝐸𝐸(𝑌𝑌) = = 1 correct to 1 decimal place (dp)

when σ< 0.13 ii. 𝑉𝑉(𝑌𝑌)

𝑉𝑉∗(𝑋𝑋) 4 correct to 2 dp when 𝜎𝜎 ≤ 0.020 correct

to 1 dp when 𝜎𝜎 ≤ 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 𝜎𝜎 ≤ 0.08 2. Using Rolle’s theorem, mode ≈ 1 ≈ mean for

𝜎𝜎 ≤ 0.08 to 2 dp to 1 dp

3. Using simulated random errors, a. Median ≈ Mean ≈ 1 when 𝜎𝜎 ≤ 0.10

b. + 3σ2

c. when 𝜎𝜎 ≤ 0.08

4. The p-Value of the Anderson Darling’s test statistic

strongly supports the non-normality of 𝑒𝑒𝑡𝑡′at 𝜎𝜎 ≥ 0.08

5. Using the moments of 𝑒𝑒𝑡𝑡′ a. 𝐸𝐸(𝑌𝑌 ) = 𝐸𝐸(𝑒𝑒𝑡𝑡′)= 1 + 3σ2 correct to 2 dp for 𝜎𝜎 ≤ 0.04

correct to 1 dp for σ ≤ 0.1

b. 𝑉𝑉(𝑌𝑌)𝑉𝑉∗(𝑋𝑋)

= = 4, correct to 2 dp for σ ≤ 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 𝑒𝑒𝑡𝑡~𝑁𝑁(1,𝜎𝜎2) under other types of transformations are summarized in table 8

( )'tE e

≈

'te

0.094σ ≤

( )' 1tE e ≈

( ) ( )' 4t tV e V e≈

( )( )*

'

t

tV

V

ee

American Journal of Mathematics and Statistics 2013, 3(6): 362-374 373

Table 7. Computations of 𝐸𝐸∗(𝑋𝑋),𝐸𝐸(𝑌𝑌),𝑉𝑉∗(𝑋𝑋)𝑎𝑎𝑎𝑎𝑎𝑎 𝑉𝑉(𝑌𝑌) for 𝜎𝜎 ∈ [0.01,0.25]

𝜎𝜎 𝐴𝐴 = σe−1σ2 1

1

B

ϕσ

− −

=

( )21 2

11 Pr

C

χσ

<+

=

𝐸𝐸∗(𝑋𝑋) 𝐸𝐸(𝑌𝑌) 𝑉𝑉∗(𝑋𝑋) 𝑉𝑉(𝑌𝑌) 𝑉𝑉(𝑌𝑌)𝑉𝑉∗(𝑋𝑋)

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 𝑒𝑒𝑡𝑡~𝑁𝑁(1,𝜎𝜎2)under different transformations

𝑒𝑒𝑡𝑡′ Distribution of 𝑒𝑒𝑡𝑡′ Condition for successful

transformation Relationship between 𝜎𝜎 and 𝜎𝜎1

𝑙𝑙𝑜𝑜𝑙𝑙𝑒𝑒𝑒𝑒𝑡𝑡 𝑒𝑒𝑡𝑡′~𝑁𝑁�0, 21σ � 𝜎𝜎 < 0.1 𝜎𝜎1 ≈ 𝜎𝜎

1𝑒𝑒𝑡𝑡

𝑒𝑒𝑡𝑡′~𝑁𝑁�1, 21σ � 𝜎𝜎 ≤ 0.1 𝜎𝜎1 ≈ 𝜎𝜎

�𝑒𝑒𝑡𝑡 𝑒𝑒𝑡𝑡′~𝑁𝑁�1, 21σ � 𝜎𝜎 ≤ 0.30 𝜎𝜎1 ≈

12𝜎𝜎

𝑒𝑒𝑡𝑡2 𝑒𝑒𝑡𝑡′~𝑁𝑁�1, 21σ � 𝜎𝜎 ≤ 0.027 𝜎𝜎1 > 𝜎𝜎

1𝑒𝑒𝑡𝑡2

𝑒𝑒𝑡𝑡′~𝑁𝑁�1, 21σ � 𝜎𝜎 ≤ 0.070 𝜎𝜎1 ≈ 2𝜎𝜎

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 𝜎𝜎 ≤ 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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