Multicollinearity Problems in Modeling Time Series with Trading-Day VariationAuthor(s): Teresita S. Salinas and Steven C. HillmerSource: Journal of Business & Economic Statistics, Vol. 5, No. 3 (Jul., 1987), pp. 431-436Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/1391619Accessed: 19/09/2008 13:39

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Journal of Business & Economic Statistics, July 1987, Vol. 5, No. 3

Multicollinearity Problems in Modeling Time Series With Trading-Day Variation

Teresita S. Salinas School of Business, Washburn University, Topeka, KS 66621

Steven C. Hillmer School of Business, University of Kansas, Lawrence, KS 66045

This article discusses the multicollinearity problems associated with the estimation of time series models influenced by trading-day variation. An analysis of the design matrix is performed, and measures of the degree of multicollinearity are computed. The characteristics of the design matrix of a popular parameterization are also analyzed, and it is shown that in some cases use of this reparameterization significantly alleviates the multicollinearity problem.

KEY WORDS: Condition indexes; Variance decomposition proportions; Eigenvalues.

1. INTRODUCTION

Many time series that are reported monthly and that represent monthly totals (flow series) contain effects caused by the composition of the calendar. This phe- nomenon is known as trading-day variation. Young (1965) defined trading-day variation as the monthly variation in a series caused by within-month variation or calendar composition. Examples of monthly series that may be affected by trading-day variation can be found in retail sales, wholesale sales, and levels of activities in service industries, among others.

Several authors (e.g., Bell and Hillmer 1983) have shown that attempts to model series affected by trading day effects using pure autoregressive integrated moving average (ARIMA) models may lead to unsatisfactory results. Liu (1980), Hillmer (1982), Bell and Hillmer (1983), and Cleveland and Grupe (1983) considered using a regression model with ARIMA errors when modeling series containing trading-day effects. In this article, we follow Bell and Hillmer (1983), who consid- ered the model

Z, = TD, + N,, TDt = E PiXti, i=l

Z = Xp + N, (1)

where Z, is the observed series, Z = (Z, ... , Z,)', N = (N1, ..., Nn)', and 1 = (],l, . . . , 7)'. TD,t represents the trading-day component with X,i being the number of times the day-of-the-week i occurs in month t, X is an n x 7 matrix whose tth row is (X,1, . . . , Xt7), and fi (i = 1, . . . , 7) are parameters to be estimated. Nt is a noise term following an ARIMA model of the form

O(B)6(B)N, = 0(B)a,,

where a, is white noise; +(B), 6(B), and 0(B) are poly- nomials of the backshift operator B with no common roots; and the zeros of 4(B) and 0(B) lie outside the unit circle, whereas the zeros of 6(B) are on the unit circle. Bell and Hillmer (1983) discussed methods to identify and estimate models in (1) from monthly data.

In this article we will study the design matrix X as- sociated with the trading-day part of model (1) and investigate the nature of the multicollinearity associated with that matrix. To illustrate why this is an important consideration, consider the time series of the logarithms of nationwide monthly wholesale dollar sales of paper and paper products in the United States from January 1967 to November 1979. These data were obtained from the U.S. Bureau of the Census, and characteristics of the trading-day part of this time series is similar to that of many other time series that we have studied. Fol- lowing Bell and Hillmer (1983), the model

7 = ? yt + (1 - 012B12)

z, = f ixti + a. i=1 (1 - S )(1 - S'2)

t (2)

was identified and estimated for this time series. The correlation matrix of the parameter estimates obtained from the 1981 BMDP time series program is reported in Table 1. Observe from Table 1 that, as expected from theoretical considerations (see Pierce 1971), the esti- mate of 012 is uncorrelated with estimates of the f's, but the estimates of the f's are highly correlated among themselves.

This example illustrates that the potential of a prob- lem with multicollinearity exists for this set of data. The purpose of this article is to investigate the structure implicit in the design matrix X to better understand the nature of this problem. In this analysis we show that the multicollinearity problem will generally occur in the analysis of time series influenced by trading-day vari-

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? 1987 American Statistical Association

432 Journal of Business & Economic Statistics, July 1987

Table 1. Correlation Matrix of Parameter Estimates From Model (2)

012 11 (2 (3 (4 5 A f7

012 1.000 fi, -.032 1.000 (2 -.016 .888 1.000 (3 .001 .916 .884 1.000 (4 -.041 .938 .917 .887 1.000 f5 -.014 .924 .939 .911 .886 1.000 36 -.010 .924 .930 .935 .918 .883 1.000 fi7 -.013 .885 .925 .928 .937 .922 .887 1.000

ation and that a common reparameterization helps to alleviate this problem.

2. PRELIMINARY RESULTS

In situations in which the noise model in Equation (1) is nonstationary [6(B) # 1] the usual approach in- volves estimation of the parameters based on the dif- ferenced time series ((B)Z, using differenced trading- day variables 6(B)Xit. The model that is estimated be- comes

7 O(B)

5(B)Z, = fli[(B)Xti] + (B) a,. i=1 4(B)

(3)

Therefore, the design matrix corresponding to any par- ticular time series will be dependent on the differencing operator 6(B) in N,. In this article we will study prop- erties of the design matrix Xo, the matrix whose ith column contains elements ((B)Xti = Xti - 1X(t_ 1)i - ..* - 6dX(t-d)i [d is the degree of the polynomial 6(B)]. We will restrict our study to the cases in which 5(B) = 1, 6(B) = (1 - B), 6(B) = (1 - B12), and S(B) =

(1 - B)(1 - B12), because these are the most common

degrees of differencing we expect to find in series in- fluenced by trading-day effects. We also restrict our study to the situation in which

7

6(B)Zt = f,i[((B)Xti] + at (4) i=i

so that the series 6(B)Zt follows a multiple-regression model. The model in (4) may not be appropriate for many time series affected by trading-day variation; however, the analysis of (4) is a starting point from which a general understanding of the problem can be obtained.

In our analysis of the design matrix associated with (4), we will make use of the following lemma.

Lemma. Let A be a real m x p matrix with rows a' (i = 1,.. ,m). Let R be a k x p matrix (k _ m) whose rows are the rows of A with al appearing ni 2

1 times in R. Then (a) R'R = A'A + ;m= (ni- l)aial, and (b) R'R is singular iff A'A is singular.

Proof. (a) is proved by the results from the prop- erties of multiplication of partitioned matrices and the fact that changing the order of the rows in R does not affect R'R. To show (b), assume that A'A is singular

so that there exists a p x 1 vector w 7 0 such that A'Aw = 0 and w'A'Aw = 0, implying alw = 0 for all i. These in turn imply R'Rw = 0, which means that R'R is singular. Conversely, if R'R is singular, then there exists a vector w : 0 such that R'Rw = 0, which implies that

m

w'A'Aw + (n, - l)w'aia/w = 0. i=l

Both terms of this sum are nonnegative numbers, im- plying that w'A'Aw = 0; thus A'A is singular.

3. ANALYSIS OF THE DESIGN MATRIX

The design matrix X6 in (4) is derived from the dif- ferencing operator 6(B) and from the characteristics of the Gregorian calendar. It is well known that months vary in length, and the composition of each type of month varies from year to year. Young (1965) gave the 22 different types of months that occur in the 28-year cycle of the calendar. (The cycle is broken at the be- ginning of each century not divisible by 400.)

The ith row of the design matrix X6 consists of seven numbers that depend on the differencing operator 6(B) and the composition of the ith month and possibly the months preceding the ith month. For instance, if S(B) = 1, each row of X6 will consist of the numbers 4 and 5, depending on how many of each type of day there is in the month corresponding to that row. If 6(B) = 1 - B or (1 - B12), each row of XO will contain the numbers -1, 0, or 1. If 5(B) = (1 - B)(1 - B12), each row of X6 will comprise the numbers -2, -1, 0, 1, or 2. All of the possible rows were reported by Young (1965) for the case 5(B) = 1 and by Salinas (1983) for the case ((B) = (1 - B), (1 - B12), and (1 - B)(1 - B12).

Let Tb denote the matrix of the different possible rows of X6 for a given b(B) with each possibility ap- pearing only once. Then, based on the lemma in Section 2, the rank of Xo is the number of nonzero eigenvalues of the matrix TET^. If all of these eigenvalues are non- zero, then X'Xb is nonsingular. The eigenvalues of T'T3 for selected values of 6(B) are reported in Table 2. Based on a principal-components interpretation of the variation in the seven trading-day variables, the sum of the eigenvalues represents the total amount of vari- ation in these variables. Based on these results, if the

Salinas and Hillmer: Multicollinearity Problems in Modeling Time Series

full 28 trading-day-year cycle is included in the data, the matrix X6 will have full rank for any choice of 6(B) considered. In particular, if 6(B) = 1, there is one

extremely large eigenvalue with the other eigenvalues being very small, so in this case one linear combination of the variables accounts for about 99% of the variation in the trading-day factors and would be the major factor in capturing the effect of the trading-day variables. If 6(B) = (1 - B)(1 - B'2), there is one eigenvalue that is smaller than the other six eigenvalues, so six linear combinations account for about 99% of the variation in the trading-day variables.

In practice, it is rare for one to have 28 years of

consistently defined data, so the practical significance of the results in Table 2 is not clear. It can be verified that each of the different compositions of the months occurs during a four-year span except for six of the seven types of leap-year February compositions. In other words, by ignoring leap-year Februaries, four years of data would be sufficient to guarantee that all of the different month compositions occur. Let S6 be the ma- trix of all of the different possible combinations of re- gressors in (4) except for those corresponding to leap- year Februaries. In other words, the rows of Sa are obtained from the rows of T6 by removing the rows derived from the presence of leap-year Februaries. The eigenvalues of S^S6 are reported in Table 3. Based on this table, it follows that, as long as we have at least four years of data, the matrix X^X6 will be nonsingular for 6(B) = 1 and 6(B) = (1 - B). On the other hand, if 6(B) contains a factor (1 - B12) and the effect of leap-year Februaries is ignored, part (b) of the lemma in Section 2 indicates that the matrix X^X6 is singular. The eigenvector associated with the 0 eigenvalue in both of these cases is proportional to e = (1, 1, 1, 1, 1, 1, 1)', reflecting the fact that, after taking twelfth differ- ences, the sum of any row of X6 is 0 when the influence of leap-year Februaries is ignored.

Consider the situation in which the data span a period of time longer than four years so that at least one leap- year February occurs. When 6(B) = 1 - B'2 or (1 -

B)(1 - B12), it can be shown that XAX6 is nonsingular. This is because the 0 eigenvalue of S'Sb has eigenspace {Ae : A E R}. If a row s' is added to the matrix S6 to form a matrix S and if v is a nonzero 7 x 1 vector, then

v'S'Sv = v'[S, Si] s v = v'SbS6v + (s;v)2. (5)

Table 2. Eigenvalues of T'T6 for Various Values of 6(B)

1 (1 - B) (1 - B12) B)(1 - B12)

1.84 9.81 2.00 4.00 1.84 9.81 5.83 36.94 2.86 11.11 5.83 36.94 2.86 11.11 13.64 45.53 9.30 28.00 13.64 45.56 9.30 70.08 28.53 195.53

2,814.00 70.08 28.53 195.53

Table 3. Eigenvalues of S'S, for Various Values of 6(B)

1 (1 - B) (1 - B'2) (1 - B)(1 - B'2)

.84 5.19 0.00 0.00

.84 5.19 3.83 25.57 1.86 5.83 3.83 25.57 1.86 5.83 11.64 30.08 8.30 20.00 11.64 30.08 8.30 55.98 26.53 154.35

1,973.00 55.98 26.53 154.35

Thus S'S is singular iff (5) is 0, which occurs iff v is in the eigenspace of S^S6 and s'v = 0. This will happen iff the sum of the elements of the added row, s', equals 0. But it is easily verified (see Salinas 1983) that the sum of any row of trading-day variables associated with 6(B) = (1 - B'2) or (1 - B)(1 - B12) and with leap- year Februaries does not sum to 0. Thus, by applying the second part of the lemma in Section 2, in most practical situations (those with more than four years of data) the matrix XX<, will be nonsingular.

The analysis of this section shows that, when we are estimating a model of the form (4) in the case in which the X,i are trading-day variables, the matrix X^X5 may be ill-conditioned. When 6(B) contains a twelfth dif- ference, Equation (4) is nearly overparameterized, since the only reason that the full seven parameters are re- quired is because of the influence of leap-year Febru- aries, which occur infrequently. This indicates that the characteristics of the correlation matrix of the param- eter estimates in the example of Section 1 is a problem that will occur more generally.

The preceding analysis suggests the potential for mul- ticollinearity problems in the design matrix Xg. To more fully understand the extent of these problems, we follow the analysis of Belsley, Kuh, and Welsch (1980, chap. 3) to detect and assess multicollinearity. They developed a number of measures to detect multicollinearity prob- lems. In particular, they defined a set of condition in- dexes, which are the largest singular value of X6 divided by each of the singular values of X6. Based on some practical experiences, they advised that weak linear de- pendencies are associated with condition indexes of about 5-10 and moderate to strong relationships are associated with condition indexes of about 30-100. The presence of one or more large condition indexes only signals the potential of multicollinearity problems. Belsley et al. (1980) also derived a decomposition of the variance of the parameter estimates var(bi) into a sum of components in which each component is asso- ciated with exactly one of the singular values. They argued that for a collinearity problem to exist there must be an unusually high proportion of the variance of two or more of the bi's concentrated in the components associated with one small singular value. These consid- erations led them to a diagnostic procedure for multi- collinearity that requires both (a) a singular value with a high condition index, which is associated with (b) a

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434 Journal of Business & Economic Statistics, July 1987

Table 4. Variance Decomposition Proportions and Condition Indexes (data from 1976-1985)

Variance decomposition proportions Condition

index var(b ) var(b2) var(b3) var(b4) var(b5) var(b6) var(b7)

6(B)= 1

1.00 14.98 .01 .05 .04 - .03 .07 .02 15.26 .06 .01 .02 .07 .03 - .05 35.34 .13 .19 .07 .41 .01 .31 .14 36.20 .21 .06 .23 .02 .46 .06 .26 43.99 .58 .59 .25 .02 .02 .13 .29 45.78 .01 .10 .38 .48 .44 .44 .24

6(B) = (1 - B)

1.00 .01 .04 .03 - .03 .04 .01 1.01 .04 - .02 .05 .02 - .04 2.56 .03 .15 .16 .06 .28 .03 .35 2.75 - .06 .14 .41 .05 .27 .28 2.82 .40 .01 .10 .01 .49 .17 .14 3.01 .40 .51 .31 .05 .07 .08 - 3.25 .11 .23 .25 .43 .06 .40 .18

6(B) = (1 - B'2)

1.00 .02 .01 - - .02 .01 1.03 - - .01 .01 - .01 .02 1.53 - .02 - .04 .02 .01 .04 1.59 .02 - .04 .01 .03 .04 1.81 .08 .07 .01 - - .01 .02 1.97 - .01 .04 .06 .03 .05 .02

10.50 .88 .89 .89 .87 .90 .87 .90

6(B) = (1 - B)(1 - B12)

1.00 .01 - - .01 - - 1.03 - - .01 .01 - .01 .01 1.70 .01 - .01 .02 - .03 .04 1.82 - .01 .01 .01 .06 .02 - 1.91 .06 .05 - .01 - .01 2.10 - .01 .06 .03 .01 .02 .02

13.13 .91 .91 .91 .90 .92 .91 .92

NOTE: A dash indicates a value less than .01.

high variance decomposition proportion for two or more estimated coefficients.

In an attempt to determine the extent of the multi- collinearity problems associated with modeling time se- ries affected by trading-day variation, the condition indexes and associated variance-decomposition pro- portions for the matrix X6 are reported in Table 4. The values were computed using the data for the years 1976- 1985; however, the general characteristics of any typical set of trading-day variables are the same as those ex- hibited in Table 4. Based on these results, we have the following conclusions:

1. When modeling a time series affected by trading- day variation with a stationary noise component (6(B) = 1), the corresponding design matrix has six moderate to large condition indexes, and several of these have two or more components with large variance-decom- position proportions. Thus it appears in this case as if multicollinearity is a problem.

2. If S(B) = (1 - B) so that the first difference of the time series affected by trading-day variation has a stationary noise component, there does not appear to be a problem with multicollinearity.

3. In cases in which a twelfth difference is part of the factor required to make the noise component station- ary, 6(B) = (1 - B12) or (1 - B)(1 - B12), there is one moderately large condition index, and associated with this condition index there are very large variance- decomposition proportions for all of the components. This indicates that a multicollinearity problem exists for both of these cases.

4. REPARAMETERIZATIONS

It has been pointed out by several authors that in the presence of multicollinearity it is still possible to esti- mate precisely some linear combinations of the param- eters (see, e.g., Malinvand 1970, pp. 216-221; Theil 1971, pp. 153-154). Because of this fact, it is sometimes possible to alleviate a multicollinearity problem by a re- parameterization of the model being estimated. Belsley et al. (1980, p. 177) discussed the possibility of repa- rameterization and pointed out that the appropriate re- parameterization depends on the design matrix, which in most cases is determined by chance outcomes of the data. In the case of modeling time series affected by trading-day factors, however, the design matrix X6 is,

Salinas and Hillmer: Multicollinearity Problems in Modeling Time Series

Table 5. Variance Decomposition Proportions and Condition Indexes-Reparameterization (data from 1976-1985)

Variance decomposition proportions Condition

index var(b,) var(b) var(b3) var(b4) var(bs) var(b6) var(b7)

6(B)= 1

1.00 - - -- -1.00 21.64 - .01 .02 .02 .01 - 39.73 .03 .06 .02 .01 .06 .04 72.51 .24 .05 .10 .12 .04 .20 94.37 .19 - .34 .23 .09 .42

107.23 .38 .45 .03 .20 .39 .14 121.04 .16 .42 .49 .42 .40 .19

6(B) = (1 - B'2)

1.00 .01 .02 .03 .03 .02 .01 2.09 .06 .12 .06 .02 .14 .11 - 2.70 .27 .03 .07 .24 .01 .26 3.11 .21 - .37 .11 .12 .35 3.37 .32 .39 .01 .25 .31 .06 3.97 .13 .42 .42 .31 .36 .19 8.11 - .02 .04 .03 .04 .02 1.00

6(B) = (1 - B)(1 - B'2)

1.00 .01 .01 .03 .03 .02 .01 2.04 .05 .10 .05 .02 .14 .10 2.79 .32 - .03 .14 .01 .36 - 3.47 .05 .20 .33 .04 .48 .21 - 3.63 .28 .07 .23 .67 .11 .05 4.16 .30 .60 .29 .09 .23 .26 9.91 -.01 .04 .02 .02 .01 1.00

NOTE: A dash indicates a value less than .01.

aside from the differencing, determined largely by the calendar, and consequently it is relatively stable across different sets of data. This suggests that the reparam- eterization possibility is feasible in this situation.

It is interesting that a number of authors who have been involved with modeling time series influenced by trading-day factors have used a common parameteri- zation. For example, Bell and Hillmer (1983) used the parameterization

7

TD, = yiVti, (6) i=l

where Vi = X,i - Xt7, the number of a given day of the week in month t minus the number of Sundays in month t, and Vt7 = 7_=1 Xt, the number of days in month t. Young (1965) used a multiplicative form of this parameterization, Cleveland and Devlin (1982) used the form (6) subject to the constraint that 7=, y, = 0, and Cleveland and Grupe (1983) used a similar repa- rameterization.

In an attempt to understand the nature of the mul- ticollinearity problems associated with the parameter- ization in (6), we created the matrix V?, whose ith col- umn contains the elements 3(B)V,i, and performed the Belsley et al. (1980) analysis on the matrix V6. The condition indexes and associated variance-decomposi- tion proportions based on data for the years 1976-1985 are reported in Table 5 for 6(B) = 1, (1 - B12), and

(1 - B)(1 - B12). Based on the results reported in Table 5, we make the following observations:

1. If the noise component is stationary, there are six condition indexes that are large, so that the reparam- eterization does not alleviate the multicollinearity.

2. When the noise model has differencing terms 6(B) = (1 - B'2) or (1 - B)(1 - B12), then the design matrix associated with the reparameterized model has one moderate-sized condition index, and associated with that index is a single component with a high variance de- composition proportion.

Thus, according to the criteria set up by Belsley et al. (1980), there is no evidence of a serious multicollin- earity problem in the reparameterized model. There- fore, one virtue of the parameterization in (6) is that inference about individual parameters can be more eas- ily made. This gives one reason to explain why this particular reparameterization may be popular.

5. CONCLUSIONS

We have analyzed the extent of the multicollinearity problems associated with time series affected by trad- ing-day factors. Based on our results, there are multi- collinearity problems inherent when the noise compo- nent is stationary or requires a twelfth difference to make it stationary. One way to greatly alleviate the multicollinearity problems when the noise component

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436 Journal of Business & Economic Statistics, July 1987

has a factor (1 - B12) is to use a popular reparame- terization.

ACKNOWLEDGMENTS

The comments of two referees played an important role in improving the article. This research is based on work supported by National Science Foundation Grant SES-8219336.

[Received August 1986. Revised November 1986.]

REFERENCES

Bell, W. R., and Hillmer, S. C. (1983), "Modeling Time Series With Calendar Variation," Journal of the American Statistical Associa- tion, 78, 526-534.

Belsley, D. A., Kuh, E., and Welsch, R. E. (1980), Regression Di- agnostics Identifying Influential Data and Sources of Collinearity, New York: John Wiley.

Cleveland, W. P., and Grupe, M. R. (1983), "Modeling Time Series

When Calendar Effects Are Present," in Applied Time Series Anal- ysis of Economic Data, ed. Arnold Zellner, Washington, DC: U.S. Department of Commerce, pp. 57-67.

Cleveland, W. S., and Devlin, S. (1982), "Calendar Effects in Monthly Time Series: Modeling and Adjustment," Journal of the American Statistical Association, 77, 520-528.

Hillmer, S. C. (1982), "Forecasting Time Series With Trading Day Variation," Journal of Forecasting, 1, 385-395.

Liu, L. M. (1980), "Analysis of Time Series With Calendar Effects," Management Science, 26, 106-112.

Malinvand, E. (1970), Statistical Methods of Econometrics, Amster- dam: North-Holland.

Pierce, D. A. (1971), "Least Square Estimation in the Regression Model With Autoregressive-Moving Average Errors," Biometrica, 64, 419-421.

Salinas, T. S. (1983), Modeling Time Series With Trading Day Vari- ation, unpublished Ph.D. dissertation, University of Kansas, School of Business.

Theil, H. (1971), Principles of Econometrics, New York: John Wiley. Young, A. H. (1965), "Estimating Trading-Day Variation in Monthly

Economic Time Series," Technical Paper 12, U.S. Bureau of the Census, Washington, DC.