Correlation and the time interval in multiple regression models

Stochastics and Statistics

Correlation and the time interval in multiple regression models

Rong Jea

, Jin-Lung Lin

, Chao-Ton Su

a_{Department of Management Information Systems, Yuanpei University of Science and Technology, Hsinchu, Taiwan} b_{The Institute of Economics, Academia Sinica, Nankang, Taipei, Taiwan}

c_{Department of Economics, National Chengchi University, Mucha, Taipei, Taiwan}

d_{Department of Industrial Engineering and Management, National Chiao Tung University, Hsinchu, Taiwan}

Received 24 March 2003; accepted 30 July 2003 Available online 13 December 2003

Abstract

In this paper we investigate the time interval effect of multiple regression models in which some of the variables are additive and some are multiplicative. The effect on the partial regression and correlation coefficients is influenced by the selected time interval. We find that the partial regression and correlation coefficients between two additive variables approach one-period values as n increases. When one of the variables is multiplicative, they will approach zero in the limit. We also show that the decreasing speed of the n-period correlation coefficients between both multiplicative variables is faster than others, except that a one-period correlation has a higher positive value. The results of this paper can be widely applied in various fields where regression or correlation analyses are employed.

Ó 2003 Elsevier B.V. All rights reserved.

Keywords: Correlation coefficient; Partial regression coefficient; Time interval

1. Introduction

In time series analysis of a given set of variables, practitioners often have to decide whether to use monthly, quarterly, or annual data. They usually try to use the time series data of the higher fre-quency in order to increase the number of obser-vations. However, the data for such analyses are sometimes limited and available for different peri-odicities and different time spans. The standard

approach is to change them to a common time interval through temporal aggregation or system-atic sampling, depending on whether the variables are flow variables or stock variables respectively (Abeysinghe, 1998). This approach, apart from losing information, may defeat the purpose of using the association between variables so as to make a correct decision or to forecast a key vari-able of interest. Thus, we are concerned with the question of whether the regression and the corre-lation coefficients are affected by the selected time interval.

The effect of the differencing interval on several economic indices has been studied by Schneller (1975), Levhari and Levy (1977), Levy (1972, 1984), and Lee (1990). In addition, Bruno and *_{Corresponding author. Address: Department of Industrial}

Engineering and Management, National Chiao Tung Univer-sity, Hsinchu, Taiwan. Tel.: 5731857; fax: +886-3-5722392.

E-mail address:ctsu@cc.nctu.edu.tw(C.-T. Su).

0377-2217/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2003.07.020

Easterly (1998) explain that the inflation-growth correlation is only present with high frequency data and with extreme inflation observations. There is no cross-sectional correlation between long-run averages of growth and inflation. Souza and Smith (2002) show that decreasing the sampling rate will bias the estimation of the long memory parameter towards zero for all estimation methods. All these studies make it clear that the time interval cannot be selected arbitrarily.

Many studies employ some additive variables and some multiplicative simultaneously (Easton and Harris, 1991; Elton et al., 1995; Tang, 1992, 1996; Chance and Hemler, 2001; McAvinchey, 2003), but these are not our present concern. In general, flow variables and stock variables are additive (e.g., gross domestic product (GDP), industrial production, population, inventories, etc.). Examples multiplicative variables include the growth rates of GDP, industrial production, population, etc. Levy and Schwarz (1997) show that when two random variables are multiplicative over time, the coefficient of determination de-creases monotonically as the differencing interval increases, approaching zero in the limit. Levy et al. (2001) write that when one of the variables is additive and the other is multiplicative, the squared multi-period correlation coefficient de-creases monotonically as n inde-creases and ap-proaches zero when n goes to infinity. Thus far, we have seen the importance of analyzing the time interval effect on the regression coefficients when some of the variables are additive and some mul-tiplicative.

The purpose of this paper is to complement and extend the results in Levy and Schwarz (1997) and Levy et al. (2001). Both studies con-sider the time interval effect when two random variables are additive or multiplicative. They use the correlation and the regression coefficient to demonstrate the importance of analyzing the time interval effect and provide us with a very good concept. However, using two random variables, we can only construct a simple regression model; that is, a model with a single regressor that has a relationship with a response. Unfortunately, very often we move to the situation with more than one independent variable such that the inferential

possibilities increase more or less exponentially. Therefore, it always behooves the investigator to make the underlying rationale and the goals of the analysis as explicit as possible. For practical reasons we study the time interval effect by using the multiple-regression model that can be widely applied in many fields where regression or corre-lation analyses are employed.

The paper proceeds as follows. Section 2 briefly describes prior research and presents the numerical results with some discussion. Section 3 shows the time interval effect on the partial correlation and the regression coefficients in the multiple-regres-sion model, and gives a numerical example corre-sponding to the US stock market. Section 4 offers concluding remarks.

2. The correlation coefficients between two random variables

Let ðY11; X11; X21Þ; . . . ; ðY1n; X1n; X2nÞ and ðY21; X11; X21Þ; . . . ; ðY2n; X1n; X2nÞ be sequences of inpendent, identically distributed variables. We de-fine four new variables to denote an n-fold increase of the differencing interval two multiplicative and two additive variables.

The additive variables, denoted by Y1ðnÞand X ðnÞ 1 , are given by Y₁ðnÞ¼ Y11þ Y12þ


þ Y1n and X₁ðnÞ¼ X11þ X12þ


þ X1n:

The multiplicative variables, denoted by Y₂ðnÞ and X₂ðnÞ, are given by

Y₂ðnÞ¼ Y21
Y22


Y2n and

X₂ðnÞ¼ X21
X22


X2n:

Using the above four variables, denoted by Y₁ðnÞ, Y₂ðnÞ, X1ðnÞ, and X

ðnÞ

2 , we can study a few different cases depending on the types of variables and the number of independent variables in the regression models.

2.1. The additive–additive case

Using two random variables, we can construct a simple regression model. If the independent vari-able X₁ðnÞ and the dependent variable Y₁ðnÞ are both additive, then the regression coefficients corre-sponding to the model and the correlation coeffi-cient between them will be unaffected by the selected time interval.

Proof. Let X1j and Y1j be identically independent distributed variables (i.i.d.), j¼ 1; 2; . . . ; n. We have

EðX1jÞ ¼ lx; VarðX1jÞ ¼ r2x; EðY1jÞ ¼ ly and VarðY1jÞ ¼ r2y: The one-period correlation coefficient is q1¼ CovðX1t; Y1tÞ rxry ¼ rxy rxry :

Because X11; X12; . . . ; X1nare i.i.d., we have EðX1ðnÞÞ ¼ E Xn j¼1 X1j ! ¼X n j¼1 lx¼ nlx ð1Þ and VarðX1ðnÞÞ ¼ Var Xn j¼1 X1j ! ¼X n j¼1 r2_x ¼ nr2 x: ð2Þ

Similarly, we can obtain

EðY1ðnÞÞ ¼ nly ð3Þ

and

VarðY1ðnÞÞ ¼ nr 2

y: ð4Þ

The n-period covariance is CovðX₁ðnÞ; Y₁ðnÞÞ ¼ Cov X n j¼1 X1j; Xn j¼1 Y1j ! ¼ n CovðX1t; Y1tÞ ¼ nrxy: ð5Þ Using Eq. (5), so the n-period correlation coeffi-cient can be easily written as follows:

qn¼ CovðX1ðnÞ; Y ðnÞ 1 Þ r_xðnÞ 1 r_yðnÞ 1 ¼ _ffiffiffinrxy n p rx ffiffiffin p ry ¼ rxy rxry ¼ q1: ð6Þ Hence, the correlation coefficient between X₁ðnÞ and Y1ðnÞ is independent of the differencing interval. Using Eq. (6) and the relationship between the correlation coefficient and the regression coeffi-cient, we can easily obtain the same result. That is, the regression coefficient is also unaffected by the time interval employed.

2.2. The multiplicative–multiplicative case

Levy and Schwarz (1997) explain that when two random variables are multiplicative, their correla-tion coefficient will not be independent of the dif-ferencing interval even when each of the random variables is a product of i.i.d. variables over time. They show that unless Y ¼ kX ; k > 0, the coeffi-cient of determination (q2_{) decreases}

monotoni-cally as the differencing interval increases,

approaching zero in the limit. 2.3. The additive–multiplicative case

Levy et al. (2001) study the time interval effect when one of the variables is additive and one is multiplicative. They show that the squared multi-period correlation coefficient (q2

n) monotonically decreases in n, and approaches zero when n goes to infinity.

2.3.1. Numerical example

Fig. 1 indicates the change of the correlation coefficients by the selected time interval in the

0 0.1 0.2 0.3 0.4 0.5 n=1 n=300 n=600 n=900 r^2_aa r^2_mm r^2_am 2 ρ

above cases. The data used are the monthly rates of returns of IBM stock and the S&P500 index from January 1926 to December 1999. There are 888 observations and the returns include dividends (Tsay, 2002). As the figure indicates, the squared n-period correlation coefficient in the additive– additive case (i.e., r^_{2 aa) shows a horizontal line.} That is, the correlation coefficient between two additive i.i.d. variables is independent of the dif-ferencing interval. The squared n-period correla-tion coefficient in the multiplicative–multiplicative case and additive–multiplicative case are denoted by r^_{2 mm and r}^_{2 am, respectively. Fig. 1 reveals}

that they indeed decrease as n increases. Hence, the results of Levy and Schwarz (1997) and Levy et al. (2001) are demonstrated simultaneously.

Table 1 reveals the relationship between qn and q1. The corresponding parameters of the two return series are EðX Þ ffi 1:0148, r2

xffi 0:0046, EðY Þ ffi 1:0070, r2

y ffi 0:0032 and rxyffi 0:0024, where fxtg and fytg are the monthly rates of re-turns of IBM stock and the S&P500 index, respectively. Table 1 illustrates, for various values of a one-period correlation, that the squared cor-relation coefficients monotonically decrease as n increases. For example, if q₁¼ 1 corresponding

Table 1

The multi-period correlation coefficient between additive or multiplicative variables

n q1¼ 1 q1¼ 0:6 q1¼ 0:1 q1¼ 0:2

M&M A&M M&M A&M M&M A&M M&M A&M 2 )0.9962 )0.9992 )0.5982 )0.5995 )0.0998 )0.0999 0.1997 0.1998 3 )0.9925 )0.9984 )0.5964 )0.5991 )0.0996 )0.0998 0.1994 0.1997 4 )0.9887 )0.9976 )0.5946 )0.5986 )0.0994 )0.0998 0.1991 0.1995 5 )0.9850 )0.9969 )0.5928 )0.5981 )0.0992 )0.0997 0.1988 0.1994 6 )0.9812 )0.9961 )0.5910 )0.5976 )0.0990 )0.0996 0.1985 0.1992 7 )0.9775 )0.9953 )0.5892 )0.5972 )0.0987 )0.0995 0.1982 0.1991 8 )0.9738 )0.9945 )0.5874 )0.5967 )0.0985 )0.0995 0.1979 0.1989 9 )0.9702 )0.9937 )0.5856 )0.5962 )0.0983 )0.0994 0.1976 0.1987 10 )0.9665 )0.9929 )0.5838 )0.5958 )0.0981 )0.0993 0.1973 0.1986 50 )0.8306 )0.9618 )0.5166 )0.5771 )0.0901 )0.0962 0.1853 0.1924 100 _{)0.6871 )0.9233 )0.4426 )0.5540 )0.0808 )0.0923} 0.1709 0.1847 500 _{)0.1500 )0.6426 )0.1197 )0.3856 )0.0303 )0.0643} 0.0807 0.1285 1000 _{)0.0223 )0.3771 )0.0204 )0.2263 )0.0071 )0.0377} 0.0255 0.0754 5000 0.0000 _)0.0015 0.0000 _)0.0009 0.0000 _)0.0002 0.0000 0.0003 10000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 q1¼ 0:4 q1¼ 0:5 q1¼ 0:8 q1¼ 1 2 0.3995 0.3997 0.4995 0.4996 0.7997 0.7994 1.0000 0.9992 3 0.3991 0.3994 0.4990 0.4992 0.7994 0.7987 0.9999 0.9984 4 0.3986 0.3991 0.4985 0.4988 0.7990 0.7981 0.9999 0.9976 5 0.3982 0.3987 0.4981 0.4984 0.7987 0.7975 0.9999 0.9969 6 0.3977 0.3984 0.4976 0.4980 0.7984 0.7969 0.9999 0.9961 7 0.3972 0.3981 0.4971 0.4976 0.7981 0.7962 0.9998 0.9953 8 0.3968 0.3978 0.4966 0.4973 0.7977 0.7956 0.9998 0.9945 9 0.3963 0.3975 0.4961 0.4969 0.7974 0.7950 0.9998 0.9937 10 0.3958 0.3972 0.4956 0.4965 0.7971 0.7944 0.9997 0.9929 50 0.3775 0.3847 0.4763 0.4809 0.7838 0.7694 0.9985 0.9618 100 0.3549 0.3693 0.4522 0.4617 0.7667 0.7386 0.9967 0.9233 500 0.1980 0.2571 0.2754 0.3213 0.6167 0.5141 0.9772 0.6426 1000 0.0795 0.1508 0.1261 0.1885 0.4356 0.3017 0.9465 0.3771 5000 0.0000 0.0006 0.0001 0.0008 0.0176 0.0012 0.7445 0.0015 10000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0000 0.5544 0.0000

to the data, then the correlation coefficient be-tween both multiplicative variables is decreasing in n and approaches zero as n¼ 5000. Similarly, in the additive–multiplicative case, the correlation coefficient decreases to )0.0015 as n ¼ 5000 and q1¼ 1. In addition to these, we also find that the correlation coefficient between two multiplicative time series decreases faster than other cases, except that the one-period correlation has a higher posi-tive value.

As we can see, the correlation coefficient be-tween both multiplicative variables (M&M) is relatively small for all n and q₁6_{0:5. On the other}

hand, the correlation coefficient is relatively large for q1¼ 0:8 and 1. The reduction in qn is rather minor, particularly for q1¼ 1. However, this table tells us that the multi-period correlation jqnj in-deed decreases as n increases even when q1¼ 1 or q1¼ 1. Hence, there is generally good evidence to show that the correlation decreases in n.

3. The partial regression and correlation coefficients in multiple regression models

From what has been mentioned above, we know the effect of the selected time interval when two random variables are additive or multiplica-tive. Here, we would like to focus on an extension to the multiple regression models. We may con-sider the subject under the following cases: (1) the dependent variable is additive; (2) the dependent variable is multiplicative.

3.1. The dependent variable is additive

In the multiple regression model, the dependent variable is additive and the regressors are com-posed of one additive and one multiplicative var-iable simultaneously. We can then construct the following n-period multiple regression model: Y₁ðnÞ¼ a0nþ a1nX1ðnÞþ a2nX2ðnÞþ e; ð7Þ where Y₁ðnÞ, X₁ðnÞ, and X₂ðnÞ are as defined in Section 2. Terms a0n, a1n, and a2n are the regression coef-ficients corresponding to the n-period multiple regression model. The error term e is assumed to

be normally and independently distributed. We additionally assume that the errors have mean zero and unknown variance r2_.

Let V1n¼ Y₁ðnÞ YðnÞ1 ffiffiffiffiffiffiffiffiffiffiffi n 1 p S_yðnÞ 1 ; U1n¼ X₁ðnÞ XðnÞ1 ffiffiffiffiffiffiffiffiffiffiffi n 1 p S_xðnÞ 1 and U2n¼ X₂ðnÞ XðnÞ₂ ffiffiffiffiffiffiffiffiffiffiffi n 1 p S xðnÞ₂ : ð8Þ

To apply the above suitable transformation, standardized variables, the regression model be-comes V1n¼ a0nþ a1nU1nþ a2nU2nþ e; ð9Þ where a_0n¼ 0; a_1n¼ a1n S xðnÞ₁ S y₁ðnÞ and a_2n¼ a2n S xðnÞ₂ S yðnÞ₁ :

We can denote here a_n¼ a1n

a 2n

 

and Un¼ Uð 1n U2nÞ: The least-squares estimator of a

n can be ex-pressed as ^ a_n¼ ðU0 nUnÞ1ðU0nVnÞ ¼ 1 r ðnÞ 12 rðnÞ₂₁ 1 " #1 rðnÞ_1y 1 rðnÞ_2y 1 2 4 3 5 ¼ rðnÞ 1y1r ðnÞ 12r ðnÞ 2y1 1ðrðnÞ₁₂Þ2 rðnÞ 2y1r ðnÞ 21r ðnÞ 1y1 1ðrðnÞ₁₂Þ2 2 6 6 4 3 7 7 5; ð10Þ

where rijðnÞ is the simple correlation between regressor xðnÞi and x

ðnÞ

j (see Neter et al., 1989, p. 290). Similarly, rðnÞ_jy1 is the simple correlation be-tween the regressor xðnÞ_j and the response y₁ðnÞ. Proposition 1. Let ^a1n be the n-period partial regression coefficient of the regression as defined in (7). We obtain the following results:

1. As n approaches infinity, limn!1^a1n¼ ^a11 (for the properties of the partial regression coefficient ^

a2n, see Levy et al., 2001).

2. If the regressor variables, X₁ðnÞ and X₂ðnÞ, are inde-pendent, then ^a1n¼ ^a11.

Proof. 1. Applying the results of Section 2.3 to Eq. (10), we know that limn!1rðnÞ2y1¼ 0 and limn!1r

ðnÞ 12¼ limn!1rðnÞ21 ¼ 0. Hence, as n approaches infinity, the standardized regression coefficients ^an can be obtained lim n!1^a  n¼ rðnÞ_1y 1 0   ¼ r1yð1Þ1 0   ; ð11Þ

where rðnÞ_1y1¼ rð1Þ1y1is shown in Section 2.1. Using the relationship between the original and standardized regression coefficients, we achieve

^ ajn¼ ^ajn S_yðnÞ 1 S_xðnÞ j ; j¼ 1; 2 ð12Þ and ^ a0n¼
y1ðnÞ ^a1n
xðnÞ1  ^a2n
xðnÞ2 :

Using Eqs. (2), (4), (11) and (12), the n-period partial regression coefficient ^a1n is as follows:

lim n!1^a1n¼ limn!1^a  1n S_yðnÞ 1 S_xðnÞ 1 ¼ rð1Þ1y1 ffiffiffi n p S_yð1Þ 1 ffiffiffi n p S_xð1Þ 1 ¼ ^a11; which completes the proof.

2. BecauseX₁ðnÞ and X₂ðnÞ are independent, it is obvious that r₁₂ðnÞ¼ r₂₁ðnÞ¼ 0. Similarly, using Eqs. (2), (4) and (12), we obtain ^a1n¼ ^a11. h

Proposition 2. Let r_yðnÞ

11:2 and r

ðnÞ

y12:1 be the partial correlation coefficients of the regression as defined in (7). Therefore, 1. limn!1rðnÞy11:2 ¼ r ð1Þ y11 (if X ðnÞ 1 and X ðnÞ 2 are indepen-dent, then rðnÞ_y11:2 ¼ rð1Þy11). 2. limn!1rðnÞy12:1 ¼ 0.

Proof. 1. The partial correlation coefficient r_yðnÞ 11:2 can be expressed by rðnÞ_y 11:2 ¼ rðnÞ_y 11 r ðnÞ y12r ðnÞ 12 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðrðnÞy12Þ 2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðr12ðnÞÞ 2 q :

Because limn!1rðnÞ12 ¼ 0 and limn!1rðnÞy12¼ 0 (see Section 2.3), we achieve limn!1ryðnÞ11:2¼ r

ðnÞ y11. Using the relationship rðnÞ_y11¼ ryð1Þ11 (see Section 2.1), we

obtain limn!1rðnÞy11:2 ¼ r

ð1Þ

y11. In particular, if X

ðnÞ 1 and X₂ðnÞ are independent, then rðnÞ_y

11:2¼ r

ðnÞ y11¼ r

ð1Þ y11. 2. The partial correlation coefficient rðnÞ_y12:1 can be expressed by rðnÞ_y 12:1¼ rðnÞ_y 12 r ðnÞ y11r ðnÞ 12 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðryðnÞ11Þ 2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðrðnÞ12Þ 2 q :

Since limn!1rðnÞ12 ¼ 0 (see Section 2.3) and r ðnÞ y11¼ rð1Þ_y

11 (see Section 2.1), we directly obtain that limn!1rðnÞy2:1¼ 0, which completes the proof. h 3.2. The dependent variable is multiplicative

When the dependent variable is multiplicative, the regression model is as follows:

Y₂ðnÞ¼ b0nþ b1nX ðnÞ 1 þ b2nX

ðnÞ

2 þ e; ð13Þ

where Y₂ðnÞ, X₁ðnÞ, and X₂ðnÞ are as defined in Section 2. Terms b0n, b1n, and b2n are the regression coef-ficients corresponding to Eq. (13). Here, e is a random error component.

We similarly let: V2n¼ Y₂ðnÞ YðnÞ2 ffiffiffiffiffiffiffiffiffiffiffi n 1 p S y₂ðnÞ ; U1n¼ X₁ðnÞ XðnÞ1 ffiffiffiffiffiffiffiffiffiffiffi n 1 p S xðnÞ₁ and U2n¼ X₂ðnÞ XðnÞ₂ ffiffiffiffiffiffiffiffiffiffiffi n 1 p S xðnÞ₂ : ð14Þ The regression model then becomes

V2n¼ b0nþ b  1nU1nþ b2nU2nþ e; ð15Þ where b_0n¼ 0; b_1n¼ b1n S_xðnÞ 1 S_yðnÞ 2 ; and b_2n¼ b2n S_xðnÞ 2 S_yðnÞ 2 : We can denote b_n¼ b  1n b_2n   and Un¼ Uð 1n U2nÞ:

The least-squares estimator of b_ncan therefore be expressed as

^ b_n¼ ðU0 nUnÞ1ðUn0VnÞ ¼ 1 rðnÞ₁₂ rðnÞ₂₁ 1 " #1 r_1yðnÞ 2 r_2yðnÞ 2 " # ¼ rðnÞ 1y2r ðnÞ 12r ðnÞ 2y2 1ðrðnÞ₁₂Þ2 rðnÞ 2y2r ðnÞ 21r ðnÞ 1y2 1ðrðnÞ₁₂Þ2 2 6 6 4 3 7 7 5:

Proposition 3. Let ^b2n be the n-period partial regression coefficient of the regression as defined in (13). As n approaches infinity, limn!1b^2n¼ 0 (for the properties of the partial regression coefficient ^

b1n, see Levy et al., 2001).

The proof for Proposition 3 appears in Appendix A.

Proposition 4. Let rðnÞ_y

21:2 and r

ðnÞ

y22:1 be the partial correlation coefficients of the regression as defined in (13). Therefore

1. limn!1rðnÞy21:2¼ 0. 2. limn!1rðnÞy22:1¼ 0. Proof

1. The partial correlation coefficient r_yðnÞ_21:2 can be expressed by

Table 2

The multi-period partial regression and correlation coefficients n Corr (n) A&A (1) Corr (n) M&M (2) Corr (n) A&M (3) ^ a1n(4) rðnÞy11:2(5) r ðnÞ y12:1(6) ^ b2n(7) r ðnÞ y21:2(8) r ðnÞ y22:1(9) 1 0.62969 0.62969 0.62969 0.32102 0.38639 0.38639 0.52720 0.38639 0.38639 2 0.62969 0.62923 0.62920 0.32154 0.38702 0.38588 0.52664 0.38618 0.38626 3 0.62969 0.62878 0.62870 0.32206 0.38765 0.38538 0.52608 0.38596 0.38614 4 0.62969 0.62832 0.62821 0.32259 0.38828 0.38488 0.52553 0.38575 0.38602 5 0.62969 0.62787 0.62771 0.32311 0.38890 0.38438 0.52497 0.38554 0.38589 6 0.62969 0.62741 0.62722 0.32363 0.38953 0.38388 0.52441 0.38533 0.38577 7 0.62969 0.62695 0.62672 0.32414 0.39015 0.38338 0.52385 0.38512 0.38564 8 0.62969 0.62650 0.62623 0.32466 0.39077 0.38289 0.52329 0.38491 0.38552 9 0.62969 0.62604 0.62574 0.32517 0.39139 0.38239 0.52273 0.38470 0.38539 10 0.62969 0.62558 0.62524 0.32569 0.39201 0.38190 0.52218 0.38449 0.38526 11 0.62969 0.62512 0.62475 0.32620 0.39262 0.38140 0.52162 0.38428 0.38513 12 0.62969 0.62467 0.62426 0.32671 0.39324 0.38091 0.52106 0.38408 0.38501 13 0.62969 0.62421 0.62376 0.32721 0.39385 0.38042 0.52050 0.38387 0.38488 14 0.62969 0.62375 0.62327 0.32772 0.39446 0.37992 0.51994 0.38366 0.38475 15 0.62969 0.62329 0.62278 0.32823 0.39506 0.37943 0.51938 0.38345 0.38462 20 0.62969 0.62100 0.62031 0.33073 0.39808 0.37699 0.51659 0.38241 0.38395 25 0.62969 0.61871 0.61786 0.33319 0.40104 0.37457 0.51380 0.38137 0.38327 50 0.62969 0.60718 0.60561 0.34496 0.41521 0.36278 0.49983 0.37625 0.37966 75 0.62969 0.59558 0.59346 0.35589 0.42836 0.35148 0.48588 0.37122 0.37570 100 0.62969 0.58391 0.58140 0.36605 0.44060 0.34063 0.47196 0.36625 0.37144 500 0.62969 0.39989 0.40466 0.46291 0.55718 0.21094 0.26955 0.28973 0.28238 1000 0.62969 0.21934 0.23745 0.50477 0.60756 0.11652 0.11059 0.19559 0.17269 5000 0.62969 0.00072 0.00097 0.52315 0.62969 0.00046 0.00003 0.00097 0.00072 10000 0.62969 0.00000 0.00000 0.52315 0.62969 0.00000 0.00000 0.00000 0.00000 (1) The correlation coefficient in the additive–additive case.

(2) The correlation coefficient in the multiplicative–multiplicative case. (3) The correlation coefficient in the additive–multiplicative case. (4) The partial regression coefficient as defined in Proposition 1. (5) The partial correlation coefficient as defined in Proposition 2. (6) The partial correlation coefficient as defined in Proposition 2. (7) The partial regression coefficient as defined in Proposition 3. (8) The partial correlation coefficient as defined in Proposition 4. (9) The partial correlation coefficient as defined in Proposition 4.

rðnÞ_y 21:2¼ rðnÞ_y 21 r ðnÞ y22r ðnÞ 12 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðrðnÞ_y22Þ2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðr₁₂ðnÞÞ2 q :

Since limn!1rðnÞ12 ¼ 0 and limn!1rðnÞy21¼ 0 (see Section 2.3), and limn!1ryðnÞ22¼ 0 (see Section 2.2), we obtain that limn!1ryðnÞ21:2¼ 0.

2. The partial correlation coefficient rðnÞ_y22:1 can be expressed by rðnÞ_y22:1¼ r ðnÞ y22 r ðnÞ y21r ðnÞ 12 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðrðnÞ_y21Þ 2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðr12ðnÞÞ 2 q :

Similarly, because limn!1rðnÞ12 ¼ 0, limn!1ryðnÞ21¼ 0 (see Section 2.3) and limn!1rðnÞy22¼ 0 (see Section 2.2), we obtain that limn!1rðnÞy21:2¼ 0, which com-pletes the proof. h

3.2.1. Numerical example

Table 2 illustrates the effect of the selected time interval on the partial regression and correlation coefficients in the multiple regression models cor-responding to the US stock market. We use the monthly rates of returns of IBM stock and the S&P500 index shown in Table 2 as a numerical example. The sample period is from January 1926 to December 1999. In Table 2, three of the corre-lation coefficients (Columns (1)–(3)), depending on the additive or multiplicative variables, seem to be helpful in attempting to sketch out the association between variables in the multiple regression mod-els. Using three distinct kinds of correlation coef-ficients corresponding to the two return series (corresponding to EðX Þ ffi 1:0148, r2

x ffi 0:0046, EðY Þ ffi 1:0070, r2

yffi 0:0032 and rxy ffi 0:0024), the other parameters (Table 2, Columns (4)–(9)) can be easily obtained.

To begin with, we claim that limn!1^a1n¼ ^a11in Proposition 1 where the dependent variable is addi-tive. Column (4) of Table 2 reveals that ^a1nbecomes closer to ^a11 (¼ 0.52315) as n increases and ^a1n¼ 0:52315 (i.e., ^a1n¼ ^a11) as n¼ 5000. Therefore, ryðnÞ11:2 approaches rð1Þ_y

11 and r

ðnÞ

y12:1 approaches zero as n in-creases (see Columns (5) and (6)). The results also conform with the claim of Proposition 2. Finally, we turn to the case where the dependent variable is multiplicative. Column (7) indicates that ^b2n ap-proaches zero and decreases monotonically as n

increases. This seems reasonable to support the claim of Proposition 3. The claim of Proposition 4 is shown in Columns (8) and (9).

4. Concluding remarks

We usually use a regression model to express the relationship between a variable of interest (the dependent variable) and a set of related indepen-dent variables. The association between variables is often measured by regression and correlation coefficients. The time interval of the data for such analyses cannot be selected arbitrarily. When two random variables are additive or multiplicative, the effect of the time interval employed is well documented in the literature.

In this paper we study the multiple linear regression models with two independent variables, where one of the variables is additive and the other variable is multiplicative. The dependent variable corresponding to the models is either additive or multiplicative. We show that the partial regression and correlation coefficients are affected by the se-lected time interval. When two variables are both additive, the partial regression and correlation coefficients between them approach one-period values as n goes to infinity. When one of the vari-ables is multiplicative, they approach zero as n in-creases. The longer time intervals will decrease the relevant association between variables, particularly for the multiplicative dependent variable. We should not overlook these phenomena in such empirical analyses or it might lead to making incorrect decisions and misguided actions. The power of the test for the correlation is also influ-enced by the differencing interval. We also find that the decreasing speed of the n-period correlation coefficients between both multiplicative variables is faster than others, except that the one-period cor-relation has a higher positive value. This subject in the case deserves more than a passing notice.

The results of this paper relate to a multiple regression analysis, which is one of the most widely used techniques for analyzing multifactor data. Its broad appeal and usefulness are applied to studies conducted in various fields where vari-ables are additive or multiplicative over time.

Appendix A. Proof of Proposition 3

Here we demonstrate the results of Proposition 3 from Levy and Schwarz (1997). Substituting the variable B (see Levy and Schwarz (1997) Eq. (1), p. 343) with the variable A, we get

qn¼ Cn_{ 1} ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðAn_{ 1Þ} p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðAn_{ 1Þ} p :

Using the above substitution, q_n can be regarded as the regression coefficient between two multipli-cative variables. Hence, the results are obtained directly from Levy and Schwarz (1997).

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