Trading days, seasonal unit root, and variance change

International North-Holland

Journal

of Forecasting

8 (1992) 61-67

Trading days, seasonal unit root, and variance change Carlos Henrique

Motta

Institute for Applied

Economic

Coelho

and Moyses Tenenblat

Research,

Brazil

Abstract: A time series model is developed for the Brazilian index of total industrial production, including a trading days effect well accounted for by only one variable. Estimation using the exact likelihood function produces a unitary MA seasonal root and thus a common factor in the equation. This is dealt with first by using dummies and later by transforming the original data using difference equation solution properties to obtain a simplified model. The residuals show a variance change at a well defined point and this is confirmed with an F test. A new simplified model is built that allows for a variance it is reestimated and its forecasting performance tested. change, and after a suitable data transformation Keywords: Trading Transfer function.

days, Seasonal

time

series,

Seasonal

unit

root,

Common

1. Introduction

Y, is the non-observed and

In this paper a time series model is derived for the Brazilian index of industrial production plotted in fig. 1 ‘. A variable for the trading days effect is included in the mode1 as, e.g., in Bell and Hillmer (1983). The mode1 can thus initially be written:

@( B)Y, = O(B)a,.

Z,=R,+Y,, where t = 1, 2, 3,. . . ,120, and Z, is the monthly observed value of the index, R, is the trading days effect in month t,

Correspondence to: C.H.M. Coelho, IPEA (Institute for Applied Economic Research), Brazil; and M. Tenenblat, IPEA and UnB (University of Brasilia), Brazil. ’ The series used here from 1979Ml to 1988M12 - is the aggregate of the indices on consumer, capital, and intermediate goods published by 1BGE (Brazilian Institute of Geography and Statistics) under the title Index of Total Industrial Production (data in the Appendix). 0169-2070/92/$05.00

C 1992 - Elsevier

Science

Publishers

factor,

Variance

true value

change,

of the index,

In the latter equation Q(B) and O(B) are polinomials in the backshift operator B having its roots on or outside the unit circle, and a, is a white noise term, i.e., {a,} is a sequence of i.i.d. NO, a*) random variables. The explanation of the trading days effect is greatly simplified as a result of the identification and estimation stages. Also, a common factor 1 -B’* and a variance change in the last third part of the series were found to be present. The probable cause of the latter is the uncertainty brought about by the interventionist economic policy of the government which came into power in March of 198.5. In sections 2 and 3 the identification of the basic model and the simplification resulting from the existence of the common factor 1 - B” are discussed. In section 4 we try to analyse and correct for the variance change, and finally in section 5 forecasting results are presented and analysed.

B.V. All rights reserved

62

15Or

130

110

90

70 1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

Month Fig. I. Brazil - index of Industrial

2. Identification

and estimation

of the basic model

Production

using the SCA system with the exact likelihood method [see Liu et al. (1986)], we found:

From the sample autocorrelation function of Z, we conclude that a first and a seasonal difference are necessary to obtain stationarity. To identify a model for Y,, we examine the residuals of the regression: (1 -B)(l

-B”)Z,=(l

-B)(l

could be written:

,=I

Estimating

this model

- OB”)a,. as a transfer

~ 0.170 I.156

(1) function,

Standard

error

0.45X 0.482 0.4x4 0.473 0.51 1 0.51x 0.513 osKl3

and Q(30)

= 22.2:

s = 2.531,

where Q(30) is the Box-Ljung statistic for lag 30 and s is the residual standard error. If we drop W(,, and W,, from (1) due to lack of statistical significance and once more follow Hillmer (1982), only this time reparameterizing with only five variables, we obtain &,Y,= ,=I

(1-B)(1-B”)Z,=(1-B)(l-B’2)~~,M/,, +(l

1.835 2.413 1.I36 2.402

0.386

-8B”)u,.

At this stage, the model

I .900

-B”)R,+e,,

where K, = Ci= ,t,Wil, and M/r,, i = 1,. . ,7 are, respectively, the number of Mondays through Sundays in month t, with the exception of holidays, as suggested by Hillmer (1982), and where E, = (1 - B)(l - B”)y,. The sample autocorrelation function of the residuals indicates that e,=(l

Estimate

&D;,+BX,. i= I

In this equation D,, = I+(- W,, for i= I,..., and X, = ?I,‘= ‘W,,, henceforth the number working days in month t.

4, of

63

Model (1 -B)(l

model

(1) could now be written: -P)Z,

is equivalent

z,-px,=

to

&J4,,+u,, i=l

= (1 -B)(l

-P)

; i

+(1

Results

-

r,L+,+PX,

eB’*)a,.

for this model

are:

Parameter

Estimate

Standard

YI

0.225 0.275 ~ 0.391 0.325 2.014 1.134

0.283 0.358 0.361 0.347 0.218 0.062

Y2

Y? Y4

P 0

Q(30)

where M,, are dummies defined to be 1 when observation t is in month i, and to be zero otherwise. Multiplying by (1 -B) to bring a, back in the equation, we get

i=l

= 22.5;

error

(1 -B)Z,-p(l

s = 2.576.

-P)Zt=/3(l

-B)(l f(1

for which results

-P)X, (2)

are: Estimate

Standard

P

1.958 1.145

0.175 0.063

s = 2.622.

= 29.7:

2_
12, (1 -B)ai=ai-a,_,

Thus, to simplify notation and in order to separate the two series, we can write (Y,- ai_ 1=-rPS,, where yi is the mean of (1 - B)Z, and 6, is the mean of (1 - B)X, in month i. This allows new variables to be defined as deviations from their monthly means, since eq. (3) can now be written:

error

and Q(30)

We estimated this equation using one dummy as the intercept, in order to avoid perfect multicollinearity, obtaining /3 = 1.963, Q(30) = 28.3, s = 2.703, and 4 statistically non-significant coefficients out of a total of 12. Instead of insisting on this model, we will adopt an equivalent version estimated in two stages with a significant reduction in computation, as follows. Taking expectations of eq. (3), we can see that for the series (1 - B)Z, - /3(1 - p)X, the mean in month i is i= 1, a, -aIz

- HP)a,,

Parameter 0

Thus, although using only X,, instead of the original seven variables of model Cl), we can explain the trading days effect with a residual standard error only 3.6% higher.

(l-B)Z,-

&M,, 1=I

(I-B)X,-

=p

i 3. Simplification

of the model

where

-PX,)

U, = a,/(1

by

= (1 -P)u,,

- B). Following

&,M,, I=1

+a, 1

or simply

If we assume 0 = 1, model (2), after dividing 1 - B and rearranging terms, reduces to: (1 -P)(Zr

; (I -B)a,M,,+a,. /=I (3)

We see that the only statistically significant parameters in this equation are 0 and /3, the coefficient of X, the number of working days in month t. We thus conclude that the model could, in the name of parsimony, be: (1 -B)(l

-B)X,=

Bell (1987), this

z, = px, + a,

t=l,2

,...,

119,

(4)

where z, and x, are the deviations from the monthly means of the variables (1 - B)Z, and (1 - B)X,, respectively. Thus, the final model is a simple linear regression.

Estimation of this model with the newly fined variables leads to the results: f, = 1.963X . (0.167)

Q(30)

= 28.3;

s = 2:703

”

which are naturally the same results almost identical to those in (2).

4. Variance

de-

as in (3), and

change

Examination of the plot of residuals of the simplified model (4) discloses a change in variance beginning in April 198.5. The reason for this could be stated as follows. That year, on the 16th of March, a new government came into power and severe price controls, particularly for oligopolies, were adopted in order to reduce inflation. Price controls were soon to be lifted but were enforced again later, this pattern being repeated successively over the next five years, during which time there were also periods of price freezes. Since this is a physical production series, a higher volatility (variance) in quantity produced would reflect the reaction from producers facing an on-and-off policy of severe price controls. To construct a one sided test for the variance change, we divide the differenced series into two parts, the first one (subindexed 1) having 74 observations and the second 45. Consequently, we test H,, : a,Z = a,2 against H, : ut > mf. After discarding two additive outliers (Ohs. 75 and 113) that would bias the test, and since Fc = s,‘/sf = 2.514, with a ‘p value’ of 5.04 x lo-‘, we reject H,,. The model for Y, with variance change, [see Tsay (1988)], would be:

5. Forecasting

performance

of the final model

A final test of our model (that is, the simplified model with the variance change correction) is, of course, its forecasting ability as measured by the root mean square error (RMSE) and the mean absolute percentage error (MAPE) [see Makridakis (1984)]. To do this we cut the original series at 1986M12, estimate the coefficients of the model, and compare the forecasts with the observed values of the index for the 24 month period 1987Ml-1988M12. The resulting one-step-ahead errors are presented below. Year 1987 1988

1987 1988

Jan.

Feb.

3.46 1.12

Mar.

0.22 0.58

Apr.

0.x2 1.52

1.Y7 0.08

May

Jun

- 4.70 ~ x.02

~ 3.83 Y.09

Jul.

Aug.

Sep.

Oct.

Nov.

DK.

-5.52 - 1.30

1.97 1.09

5.52 - 1.h3

~ 0.38 - 7.46

- 1.49 ~ 1so

- 4.85 0.17

As a means of comparison, the same procedure was applied to three other models: (a) a model with no trading days correction. This was an AROMA (0, 1, 1X0, 1, l),* model for Z,, i.e. the ‘airline model’ [see Box and Jenkins (1976)], (b) the model with trading days correction (2), and (c) the simplified model (4). The values of RMSE and MAPE for these models are: Model

RMSE

MAPE

‘Airline’ Model (2) Simplified model Final model

s.115 3.968 3.864 3.X64

3.415 2.394 2.329 2.328

where i =s,/s2 = 0.631. Estimation of this model again produces an estimate of 0 around 1. Thus we once more use the simplified model, allowing for the variance change, and the results are:

As we can see, the main improvement in forecast error reduction is due to the inclusion of the trading days effect. On the other hand, the variance change correction, although it increases the efficiency in estimating parameters and eventually reduces the numbers of outliers [see Tsay (1988)1, seems to have no effect on forecast error reduction. The test for the adequacy of the final model [see Abraham (1985)], using the F distribution, leads to its non rejection, since

i, = 1920x,;

Fc = 3.864*/12.055

@( B)Y, = O( B)a,

for

t=

@( B)Y, = O( B)a,/h

for

t = 76, 77,. . . ,120,

(0.149)

Q(30)

= 24.7;

1, 2,...,75

s = 2.102

where

the

= 1.239,

denominator

is the

estimated

error

65

variance for the second for 75
part

of the sample,

i.e.

6. Conclusion We have developed a very simple model to forecast Brazilian industrial production, using a single variable for the trading days effect. A further improvement would be the extension of this simple framework to include economic variables in the model. Ideally, these economic variables would function as leading indicators for the forecasting of industrial production.

References Abraham, B., 1985, ‘Seasonal time series and transfer function modelling’, Journal of Business & Economic Statistics, 3, 3566361. Abraham, B. and G.E.P. Box, 1978, ‘Deterministic and forecast adaptative time-dependent models’, Applied Statistics, 27, 120-130. Abraham, B. and J. Ledolter, 1983, Statistical Methods for Forecasting (Wiley, New York). Bell, W., 1987, ‘A note on overdifferencing and the equivalence of seasonal time series models with monthly means and models with (0, 1, l),, seasonal parts when 0 = I’, Journal of Business & Economic Statistics, 1987, 5, 3. Bell, W.R. and S.C. Hillmer, 1983, ‘Modeling time series with

calendar variation’, Journal of the American Statistical Association, 78, 52-534. Bowerman, B.L., A.B. Koehler, and D.J. Pack, 1989, ‘Forecasting time series with increasing seasonal variation’, Department of Decision Sciences, Miami University (Ohio). Mimeo. Box, G.E.P. and G.M. Jenkins, 1976, Time Series Analysis: Forecasting and Control (Holden Day, San Francisco). Harvey, A.C. and P.L.V. Pereira, 1988, ‘Trend, seasonality, and seasonal adjustment’, Texto Para Discussao lnterna n. 154, INPES/IPEA, Brazil. Hillmer, S.C., 1982, ‘Forecasting time series with trading day variation’, Journal of Forecasting, 1, 385-395. Hillmer, S.C. and G.C. Tiao, 1979, ‘Likelihood function of stationary multiple autoregressive moving average models’, Journal of the American Statistical Association, 74, 652660. Liu, L.-M., G. Hudak, G.E.P. Box. M.E. Muller, and G.C. Tiao, 1986, The SCA Statistical System: Reference Manual for Forecasting and Time Series Analysis, Scientific Computing Associates, P.O. Box 625, DeKalb, Illinois 60115, USA. Liu, L.-M., 1987, ‘Sales forecasting using multi-equation transfer function models’, Journal of Forecasting, 6, 223238. Makridakis. S., 1984, ‘Forecasting: State of the Art’, in: S. Makridakis et al. eds., The Forecasting Accuracy of Major Time Series Methods (Wiley & Sons). Tsay, RX, 1988, ‘Outliers, level shifts, and variance changes in time series’, Journal of Forecasting, 7, l-20.

Appendix

month

Brazilian 1979

jan. feb. mar. apr may jun. jul. aug. sep. act. nov. dec.

94.99 88.62 98.81 95.27 103.59 103.60 105.20 111.71 102.81 115.04 105.88 98.05

index of industrial production 1981 1982 1980 101.24 99.24 109.58 101.94 112.04 113.51 119.04 117.73 120.74 123.01 114.09 103.87

101.40 100.91 102.02 95.28 98.76 102.53 106.29 103.49 100.64 103.34 96.83 88.49

86.17 85.36 101.16 95.10 100.95 105.93 109.69 112.06 108.32 106.21 99.70 89.73

1983

1984

1985

1986

1987

1988

83.90 80.80 95.42 86.43 95.81 96.52 97.57 105.26 101.55 103.75 99.98 91.23

87.20 90.93 93.30 89.96 102.47 105.43 109.43 112.52 106.67 115.87 107.84 97.42

100.10 92.61 103.45 92.85 104.93 108.26 119.70 122.01 120.03 131.09 118.59 108.92

111.22 104.66 107.04 111.43 116.18 122.99 133.09 131.86 138.85 144.99 128.54 116.31

118.30 117.54 122.04 120.84 122.30 125.05 124.69 125.64 131.42 134.59 125.55 111.98

107.81 107.41 122.16 111.44 115.31 127.17 127.21 134.91 129.72 123.90 116.76 108.21

66 Year

1979 1979 1979 1979 1979 1979 1979 1979 1979 1979 1979 1979 1980 1980 IYXO 1980 1980 1980 19x0 I’)80 1980 1980 1980 1980 1981 1981 1981 1981 1981 1981 1981 1981 1981 1981 1981 1981 1982 1YX2 1982 1982 1982 1982 1982 1982 1982 19X2 1982 1982 1983 1983 1983 1983 1983 1983 1983 1983

Trading Month

days Mon

01 02 03 04 05 06 07 08 09 10 11 12 01 02 03 04 OS 06 07 08 09 10 11 12 01 02 03 04 05 06 07 08 09 10 11 12 01 02 03 04 OS 06 07 08 09 10 11 12 01 02 03 04 05 06 07 08

4 3 4 5 4 4 5 4 4 5 4 5 4 3 5 3 4 5 4 4 5 4 4 5 4 4 4 4 4 5 4 5 3 3 4 4 3 3 5 4 5 4 4 5 4 4 4 4 5 3 4 4 5 4 4 5

Appendix Tue

Wed

Thu

Fri

Sat

Sun

X,

3 4 5 4 5 3 4 5 4 4 4 4 5 4 4 4 4 3 5 4 4 5 4 3 4 4 4 5 4 3 5 4 4 5 4 5 4 4 4 5 4 3 5 4 5 4 4 5 4 4 5 3 4 4 4 4

4 4 5 3 4 5 4 5 3 3 4 4 3 5 4 3 5 4 4 s 4 5 4 4 5 4 4 3 4 4 5 4 4 5 4 3 4 4 4 4 4 4 s 4 4 5 4 5 4 4 4 4 4 4 5 4

4 4 5 3 4 5 4 4 5 4 4 5 4 4 5 4 5 4 4 5 4 4 4 4 5 4 4 4 5 4 4 5 4 5 4 4 5 4 4 4 4 4 5 4 4 5 4 3 4 4 4 5 4 4 5 4

4 4 4 5 4 4 5 4 5 4 4 5 4 4 5 4 4 5 4 5 3 3 4 4 3 4 5 4 5 4 4 5 4 4 4 4 5 4 4 4 5 4 4 5 4 5 4 4 5 4 4 4 4 4 5 4

21 1X 22 20 22 20 22 23 19 22 20 20 21 19 21 20 21 20 23 21 22 23 20 22 21 20 20 20 20 21 23 21 21 21 20 22 19 18 23 20 21 21 22 22 21 20 20 23 20 18 23 19 22 21 21 23

(continued)

Year

Trading Month

days Mon

Tue

Wed

Thu

Fri

Sat

Sun

X,

1983 1983 19x3 1983 1984 1984 1984 1984 1984 1984 1984 1984 1984 1984 1984 1984 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 lY86 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1987 1988 1988 1988 1988 1988

09 10 11 12 01 02 03 04 05 06 07 08 09 10 11 12 01 02 03 04 05 06 07 08 09 IO 11 12 01 02 03 04 05 06 07 08 09 IO 11 12 01 02 03 04 05 06 07 08 09 10 11 12 01 02 03 04 05

4 5 4 4 5 4 3 5 4 4 5 4 4 5 4 5 4 3 4 5 4 4 5 4 5 4 4 5 4 3 5 3 4 5 4 4 5 4 4 5 4 4 4 4 4 5 4 5 3 3 4 4 3 3 4 4 5

4 4 4 4 5 4 3 4 4 4 5 4 4 5 4 3 4 3 4 5 4 4 5 4 4 5 4 5 4 3 4 5 4 4 5 4 5 4 4 5 4 4 4 3 4 5 4 4 5 4 4 5 4 3 5 4 5

3 3 4 4 3 5 4 4 5 4 4 5 4 5 4 4 5 4 4 4 4 4 5 4 4 5 4 3 4 4 4 5 4 4 5 4 4 5 4 5 4 4 4 5 4 4 5 4 5 4 4 5 4 4 5 4 4

5 4 4 5 4 4 5 4 5 3 4 5 4 4 4 4 5 4 4 4 5 3 4 5 4 5 4 4 5 4 4 4 3 4 5 4 4 5 4 3 4 4 4 5 4 3 5 4 4 5 4 5 4 4 5 3 4

5 4 4 5 4 4 5 3 4 5 4 5 3 3 4 4 3 4 5 3 5 4 4 5 4 4 4 4 5 4 3 4 5 4 4 5 4 5 4 4 5 4 4 3 4 4 5 4 4 5 4 3 4 4 4 4 4

4 5 4 5 4 4 5 3 4 5 4 4 5 4 4 5 4 4 5 4 4 4 4 5 3 3 4 4 3 4 5 4 5 4 4 5 4 4 4 4 5 4 4 4 5 4 4 5 4 5 4 4 5 4 4 5 4

4 5 4 3 4 4 4 5 4 4 5 4 5 4 4 5 4 4 5 4 4 5 4 4 5 4 4 5 4 4 5 4 4 5 4 5 3 3 4 4 3 4 5 4 5 4 4 5 4 4 4 4 5 4 4 4 4

21 20 20 22 21 21 20 20 22 20 22 23 19 22 20 20 21 18 21 21 22 19 23 22 21 23 20 21 22 18 20 21 20 21 23 21 22 23 20 22 21 20 20 20 20 21 23 21 21 21 20 22 19 18 23 19 22

67 Appendix Year

1988 1988 1988 1988 1988 1988 1988

(continued)

Trading Month

days Mon

Tue

Wed

Thu

Fri

Sat

Sun

X,

06 07 08 09 10 11 12

4 4 5 4 5 4 4

4 4 5 4 4 4 4

5 4 5 3 3 4 4

4 4 4 5 4 4 5

4 5 4 5 4 4 5

4 5 4 4 5 4 5

4 5 4 4 5 4 3

21 21 23 21 20 20 22

Biographies: Carlos Henrique M. COELHO is a senior researcher at IPEA (Institute for Applied Economic Research) under the Ministry of Economics in Brazil. He obtained his M.Sc. in Operations Research at the Federal University of Rio de Janeiro. His current research interest includes Time Series Analysis and Econometrics. Moyses TENENBLAT is also a senior researcher at IPEA, and assistant professor of econometrics at the University of Brasilia. He obtained his M.A. in economics at the University of Michigan, and the advancement to Ph.D. candidacy in economics at the University of California, Berkeley.