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A comparison of econometric, time series, and composite forecasting methods in predicting accounting variables
J EC0 BUSN
301
w&3:35301-3 I I
A Comparison of Econ etric, Time Series, and Composite Forecasting Methods in lBn%iictin Accounting Variables James S. Ang, Jess H. Chua, and Ali M. Fatemi
This study comparesfive forecastingmodels in predicting seven accounting variables for a fum in the rubber and plastics industry. The five models consistof an econometric model (CXS), two time series mod& (univariate, and multivariate), and two combined models (combinations of the ecoimmetric model and one of the time series models). Seventy-six monthly obsetvations were used lo estimate the models. Each estimated model was usedto forecast each of the seven variables over ;’ Wmonth period. These forecasts wefe then tested for accuracy The test involves acomparison of Theil’s “inequality coefficient .‘* The multivariate time series model snd one composite model, i.e., the OLS-multivariate time series maiel, performed well. The multivariate time series model, however, achieved bener results than the other models.
Introductiwt The need for $:oodaccounting and financial forecasts can not be overemphasized. Such forecasts provide the basis for budgeting, control. financial planning, investment decision, timing and amount of new financing, working capital management, etc. For example. Horprgren ( 1977, p. 129) suggests “the sales forecast is the starting point for budgeting, because inventory levels and production are generally geared to the rate of sales activity.” in the search for good forecasts, research has resulted in new forecasting models, Few studies. however, have directed their attention to combining &e forecasting models presently available to investigate whether improved forecasts can be obtained for accounting variables. Pindyck and Rubinfeld (1976) obtained improve j forecasts of short-term interest rates when they combined regression analysis with a time series rnok-4. Ludwig (1974) also found that combining regression analysis with a time series mo&l results in improved forecasts of short-term savings deposit flows. This study contributes to the current literature by studying the predictive abilities of combined econometric and time series models. It compares five forecasting models in forccastrng accounting variables for a single firm. The models used are* I) an ordinary
James S. An8 IS Wdham 0. Cullp,n Professor of Finance, Florlda State Uruversity, Tallahassee, Florida. less H. Chue ISAssociate Profeswr of Finance, The University of Calgary, Calgary, Alberta Canada. Ali M Fatemt is Aswtant Professor of Ftnance. Kansas StqteUnwerstty. Manhattan, Kansas tb502 Address repnt requests to Prof. Ah M. Fateml.
016-61%/83/503.00
1. S. Anget al.
302
Tabie 1. Ecanometric Variables
C.4SH N’rR
Cash = Net
Tl = Total
trade receivables inventories
NS = Net sale:: CGS = a-‘ost of goods sold SELL
=. Selling expenses
GP = Gross profit
,4,PR = Average prime rste charged by banks E-1 = Business
investment
(seasonail)
adjusted at annual rates) ;billions
Lc‘ = index of labor cost per unit of output.
manufacturing
P’l = f’ersonal
in 1972 dollars
income less trlnsfcr
PRP
= Industrial
SRP
= Shipments:
VMNO ItPRP
production:
payments.
rubber and plastics products
rubber and plastics products
= VdlUc of manufacturers’ = H’~wlw4c
(millions
of dollars)
(I 967 = 100) of dollars)
new orders, durable goods mdustries.
prices: rubber and plastics products
in 1’172 dollars (billions
ofdollar<)
(I 967 = I(I(Q
least-squares econometric model (OLS); 2) a univariate time series model (LINI): multivarlatc
tinre series model (MLILT):
series model (OLSmodel (OLS-
l_JNI); and 5) an ordinary
least squares
univeriate
least squares- multivariate
3) a time
lime series
M LJ LT).
Do ta aad Methodology Data for the study consist of and plastrcs industry April
4) an ordinary
monthly financiA stsiement
from Januar!
ltcrns for a lirm in the rubber
1971 to June 1978.’ Data from January
1971 to
1977 are used to estimate the models. Data from May i977 to June 197s ;~re
reserved f
in Table
1 are the firm, industry.
and economy VU iables used. Endogenous (firm) variables are those most often used in short-term
business financidl planning. Exogenous variables are industry and economy
variables. Data for the rubber and plaskcs industry C:trrtwr
Busintw.
National
are cbtained from the Surcq~ (!I
economic data arc obtained from
Brcsiwss
Ccwslilirw
Di@‘Sf.
Draw ‘ng on the available literature
in ticcounting and lknce.
endogenous varmbles
are specified as functions of exogenous and other cndogcrtous variables. The resu’ting structural
equations are presented in Table 2:
1. Cash is expressed as a function A.. ’ Inventories shipments
iSlY
of net sales and inverltories.
expressed as it function
of rubkr
the value of ni~nufac‘turers’
Ned orders,
and plastic products, and the prime rate of interest.
’ The Dada arc from unpuhhshed cornpan! owned prlsate sources. but the\ are the data a\all,lhle to an lntcrnal anal\~.t or a planner of lhc titm
Methods in Pmdicting Accoming Recc~tables
m
Tabk 2. StructuralEquatims in Their General Form CASH
= f(P(S. TI)
TI
= f(VMNO. SRP. APR) = j(SRP. NS, 4PR) = ((SELL. PRP. SRP. Bl. PI) = f(PRP. VMNO, LC. NS. WPRP) = f(NS. GP. WPRP. SRPI = f(WPRP. NS. SELL)
W-R NS CGS SELL GP
3. Receivables are specified as a function of shipments of rubber and plastic produc , rlet sales, and the prime rate of interest. 4. Net sales is expressed as a function of selling expenses, rubber and plastics production, shipments, business investment and persona! income. 5. Cost of goods sold is specified as a function of rubber and plastics productions, value of manufacturers’ new orders. labor cost, net sales, and wholesale prices. 6. Selling expenses is specified as a function of net sales, gross profit, wholesale prices. and rubber and plastics shtpments. 7. Finally. gross profit is expressed as a function of wholesale prices, net sales, and selling expenses.
Univariate models and a multivariate model were used for comparison. The univariate model incoporates past history of only the forecasted variable. while the mult;variate method allows a cross-sectiat.al correlation of one variable upon others. L~~rit~toru Mod&s. Univariate time series models have received wide attention, e.g., Albrecht. Lookabili, and McKeown 1977; Box and Jenkins 1970; Brown and Rozeff 1978. 1979: Dopuch and Watts 1972: Foster 1977; Griffin 1977; Kinney 1978; Lorek 1976; Lorck.
McDonald.
and Patz 1976; Mabert
1974: Maberr
and Radcliffe
1974;
N&on 1973; Watts 1975; Watts and Leftwich 1977. The univariare integrated autoregressive-mo4ng average process of order (p, d, 4). or simply ARIMA (Q,d, q), for th,: series )’ can be written as’
(1)
&(B)A“r,=ii+t),(B)~ where @,(Sl== I --&,B-c#J~B~---.~. pth order autorcgrcs;ive B is the backward that BY,= \‘,
-$,BPdenotes
shi’i operator
,. B+.,=
‘!
process;
y,
such
.‘, etc.;
Ad denotes the dth diflerence for the serves J., such that A)*,= )I, - !‘, ,, A=y, = Ay, -As, _ ,, etc.; ____
-. 2 We shah no1 go Inlo de1 ui about Ibe model. ror that. see Box and Jer.kms (1970) Nelson Pmdyck and Rubinfeld (1976).
i 1973). and
S is a constant term which represents the mean for the series; O,(R)= 1 -tI,B-8:BL-,,,-t),Bq denotes a yth order moving average process: and E,’ f,-I. . . series represent a white noise process. Time series modeling involves both specification and estimation problems. The estimation procedure used typically is least-squares error minimization or likelihood function maximization with errors ( es) assumed to be normally distributed. The specification problem has often been solved by the Box-Jenkins (1970) technique. In this procedure, the model is specified subjectively through an iteration process. The model obtained is nonunique in the sense that two forecasters could arrive B?different models when modeling the same series. Subjective specification also implies that the model obtained may not be the “best.” However, the specification problem has been solved to a certain extent by the development of the Akaike Information Criterion (1971. 1972, 1974. 1976b The Akaike Information Criterion (AK‘) provides for time series models to be constructed objectively and uniquely. The AIC is a maximum likelihood criterion with significance level ndju.X*ments for iterative testing. It is defined as: AIC= - 2 lop (maximum likelihoc>d) -t-2 (number of independently
adjusted parameters)
,
(11)
The criterion has been subject*>d to exter sive testing. Monte Carlo experiments to test its ability in rccapturmg the true order? of simulated time series were performed bj Akaike (1971. 1974) and Jones (1975). and the results were excellent. The criierion has also been app:;ed in the time series studies by Gersch anu Sharpe (19l.l). Jo les (1974). and Landers and Lacoss (1975) who all reported good results. The procetiure for constructing the minimum AK time series model is is follows: 1.
construct time series models of increasing order up to a reasonable limit, say number of observations divided by ten; 2. compute the AIC for each of these models; and 3. select the minimum AIC model as the optimal model. The procedure is used in this study to construct
univariate autoregressive
models.
,Vulrir~ritifcp i2fodtBl.s.Ur;variate models have one seriuus limitation in that they do not consider the interrelationships among variables. For example, in a univariate time series analysis of net trade receivables (NTR). the current level of NTR is related only to its own past levels. It is likely, however, that the current level of NTR relates not only to past levels of NTR but aiso to past levels of sales and past levels of inventories. Multivariate models allow for such cross-sectional correlaticns. Unlike univariate models, multivariate time series models have received limited attention (Cramer and Miller 1978, Pierce 1977. Tiao et al. 1979) The generalized autoregressive movin, lverage model for li series {I,\ takes the form (3)
Methais
inPwdkting
Accounting Receivables
305
where f&tLJ=I-
,B-.
.-+pi?p.
and
8,(@=/--8,B.-8,&F are matrix poljmomials in 8: 4s and A are k.lekmatrices; &=_Vn-- S, is the vector of deviations from some origin S (the mean if the series is stationary): and { (~~1with c,=( cl,. ....
ekt)’ is a sequence of random shock vectors with zero mean and covariance matrix X. Multivariale time &es modeling. as with univariate modeling, involves both specification and esttmation. 1 he AK approach is used to specify rhe multivar;aie ml;del. btimation of parametez followed tte recursive method developed by Durbin (1960) and Whittle (1963). Composite Models Cornposit: models were developed in two php,es. Phase I developed the econometric model by utilizing an ordinary least-squares approach. In phase II, time series models of the residuals were cons,tructed. Composite forecasts *Nerethen obtained by adding time series forecasts of the residuals to the predictions of the ordinary least-squares econometric model.’
Results In a seriesof regressions,dependent variables were regressedon different combinations of contemporaneous and lagged values of (their respective) independent variables. The combinations with the highest explanatory power !R’) were defined as the “best fit” models. The predictive abilities of these best fit models *Merethen compared to those of time series and composite models. Best fit models were also used to construct ;he composite models. Best-fit OLS estimates of the a!ructural equations are presented in 1 able 3. CoelKcients of determination (R2s) range between 4%; and 84% and all F-statistics are significant at lpb. The Durbin---Watscrr! d-statistics for CGS equation indicates that the null hypothesis of no atltocorrelatior of the residuals cannot be rejected at !o/,. For all other equations, however, the null hypothesis of no serial correlation is rejected at cO,O. Examining the results reported in Table 3, we observe thar the estimated coefficients have rhe correct sign vis-&vis the assumed theorP!icai role of the variables in each equation: 1. Net sales has a significant positive effect on cash holdings. The effect of inventory
holdings. although insignificant, is negative. ’ In the presence of autocorrelatlon
~CPCS forecasts
m the OLS model, a more eficlent
of the residuals to the residuals obtamed
approach
from a GLS model.
would k
;o add
llme
=
=
=
=
=
-
Tl,
M-R,
NS,
t c;s,
SkLL,
GP,
Id
u
(6.1 l)b
+ ~J.2~A%s~
Econometnc
and Statrsrtcs
0.31T1, i- 1.21)
k.quarionQ
Equations
=
.
.
--
(- 3.29)b
_.
-
_
(2.4l)b
_” __ _. _
(11.51)b
(-3
-
65)b
--
-
___
.~
3197 8sPI, (-1.11) - 2194.4IWPRPt (-0.81)
___~~ _._._
1239086C 249302 3V.%lN0,_3 346.XlSRP,_. 3 161356.4APR, I-iz.olP ( -0.95) (10.37)6 (-3.58)b 532524 21.81SRP,_3 + 0.729!‘+ + 1597?56APR, (-0.62j (i .06) i1.53) (7.93)b 1225362 + 5_80StLL, + 9856.66PRP,_ 2 105.69SRP,_2 + 6470.84B1, (0.52) 13.47)b (-Q.62) (1.9O)c (0.57) 1458763 + 4?88.16PRP, + 1 151.54V~fS0,__2 + 10124.77LCt + 0.38NSt (2.69)t C I.C4)C (0.73) (1.18) (4.18)b 184633.7 + O.O94NS, O.l6GP, + 1367.33WPRP, + 21.13SRP, (3.68)b (4.63)b (1.21) t-2.84$’ &.23)b 274598.6 + 1852.62WPRP, + 0.38NS, I .09SkLL,
! 3rJ838 j (1 I .98;6
Least-Squares
a ( ) indicates I . u!urs. b Significant at <‘I,. <’Stgnificant a: ! 07
=
c AW,
variat;k
lkpcndent
_..
Ordinary
3.
T&e
R2
F-Stat
_ _
I.1 0.98
2.02
35.47
f4.56 17.30 39.66 81.91
0.70
0.67 0.78 0.84
-
0.99
0.76
1
II..-_ 67
0.49
~__~
~..~..._-.-----
52.44
0.63
~-. Durbm-%‘;ttwn Statistic
0.77
0.49
22.24
_-.____.._.,~-.----__--_
-.--. -.-.-
Methods in Predicting AccountingReceivabks
M?
2. Total inventories is negativeiy akcted by the value of manufacturers’ new orders, prime rate of interest. and shipments of rubber and plastics products. Thz last factor’s effect. however. is insignificant. 1 _. Net sales is the only variable that has a sigmfkant positive efkct on net rnde reckvabhes. Although ksi@nifkant, the e#kt ofthe prime rate is positi4reand that 4.
value of) industrial production both have a si~~~~t ~s~tiv~ t cmnet s&s. Perwnal income and (the two-period value of) ~i~rne~ nt negative effect, whife btlsiness tment has an insi~~i~~~nt positive e&et on net sales. a. Industrial pr~~~tios‘~ net sates, value aif manufacturers’ new orders, and labor cost all have B positive effect an the cost c~Fgmds sold. Only :he first two factors. hmwver, me statisticalEy significost. Wholesale prices have an insignificant n~~~ti~e e&43 0 St Qf @Kds Wk!. 6. Net sales, whoI e prices, and shtpments have a positive effect on selling cxpnsees. but the efkt ofthe last variable is insignificant. Gross profit and selling expnses ;~re s~~n~~~~~~t~~ cot related negatrvely. 7. f ‘,nally, gross profit is significantly afkcted by wholesale prices, net sates, and seiling expenses.with the first IWOfactor? having positive effects and the last factor having a negative effect.
Autoregressive coefkients for the UNI, MULT, OLS-UN1 and OLS-NULT models are. respectively. presented in Tables 4 7.J The MULT and OLS-MULT -m&k arc all of order one (p = I). The order of the autoregressive prxesses for UNI and OLS 1lNl models vary between a minimum of one (UN1 model for CASH) and a rllaximum of seben (UN1 model for GP and OLS.-l tN1 model for NS). Each r:sttmatcd modci was u!cd to forecast the endogenous variables over the 14month period from May 1977 to June 197% These forecasts were then tested for accutacq. The test Involves a comparisorr of Thetl’s “inequality coefftc ,t.”
ukre P, and A, are the pairs cr predicted and actual values, respectively. U reaches a value of zero as a lower houndarq when all predictions are correct, that is when P, = A,. It assumes a value of I when the forecasting procedure is lesseffective than a naive FIOchange forecast.5 L’ values for all five models *c reported in Table 8. Two moaels-- OLS MU LT and MU LT models produced 2 values below ! for all scben rndogcnous variables. The other three models produced L’ values greater thzq I for at least two of the \ariablcs. The MU LT model. however. produced better forecasts than ihu OLS MIJLI’ modet, for all seven varlubles. I lsmg the rniibns and rhc 4t;md;trd cktlatlonh ol L’ values as the perl’orrnancecrltcria
- . - _--.
___I_“_____.--
cot%cten~s
’ POSIIIV~ ~mplgthat the current value of the vaitahte IS pos~tivcfy related 10 I& own other vartables’ prwous values. Ncpatwc ccdikxnts ~rnplithe opposite For example, accordmg MULT model. current devlatmns from mean for CASH arc negatIveI> related to one-period devlatlons CASH. TI. VTR. <‘
of
and/or
to the lagged related
I
*
1
I, I
J
i
e
1
=
= _
SELL,
GPt _
__.
__.,_ ,,__ ._--_.-
.-
~._ ._ --_I
_Ip__.-==3s1-
l.O203OGP,_~ - 0.74603GPt_2 _ _-.- -.------.---II_ - 0.69694GP,
J
r( W)Y ICP,_4 _.-- .- ._I-
4.37090 4.29260
-0 $7206 __ __________ ~___.
1.25290
-0.26391
-0.74516
-0.71160
0 13105
-9.58190
n.05705
-0.041i 1-I
2 8974@
0.97212
-0.?:451 -O.lO!fZ
0.53850 -1.19390 1.88210 - 1.22600 -1.73770 .-___
-n.o1125 0.11398 -0.03691 -0.‘?3588 -0.19891 0.10442
.-0.02381 0.33416 -0.77199 1.61520 -n.c5959 . _ 050446
-0.07983 -0.32970 -1.16410 1.20890
-1.19740 -.-_
-0.363!3
-0.26350
-0.036 14
0.01507
0.08673
._ ______xII______-.-_~-
+ 0.83082GP,_, - _._ __ ____ll__.-_-____--_
i .2492OSELL,_ 1 - 1.9146OSELL,_2 - I .87450SELLr_~ -- 2.06000SELLI_4 + 1.3O63WELL, _ 5 ...--._-
____
-O~.~-
!r 881198CASH, _i - 0.248CICASH,_, - 0.21853CASff,_J - 0.01830CASf~,_4 - k34469CASff,_~ e 0.79793T1,_2 - 0.0087TI,- 3 - 0.77294Tf,_4 - 0.90149T1,_5 - ~.~6157~~~_6 : .02>8Ml,_ I 0.70517~~R~_~ 0.60686NTR,_, - 0.744 17NTR,_2 - 0.875!%!NTR,_3 -- O.?8151NTR,_~ -- 0.33110NTR,_s ~.~~464~~~_~ i .Or)460Nb,_, - : .Of59QNS,_2 .- 0.94435?&_; 1.16230NS,_4 - 0,i8625NSt, 4 0.5912RNS,_6 0.94573Ccs,_ 1 0.51009CGS,_, - 0.0Y1146CGS,_J - 0.46Y2OU&.4
Uruvanatc P uuxegre?rsive Model for Residual Scncs from OLS Models ..~_ _--._... ..-__._ _.
Tabk 7. Multivanate Autoregressive Model for Residual Series from OIS hkniels: (Order = 1) ~ ____--__--l_l_,-.__-
= = = = =
CASH, &I, NTR, NS, CC-S,
Tat& i. --.
.-
-...
--..
.
_ _._--l
J. S. Anget al.
310
Table 8. Theil U Values
time series m4els.
comparing
the M1II.T
mode: produced better forecasts than
the L’N1 model. C‘ompariug
combinations
model, OLS- ML1I.T Overall,
the MULT
of a time series mode’ amt the ordiilary
performed
better than the OLS
lJNl
least squares
model.
model produced beiter forecasts than the other four models.
Conclusions This study con3 ,,art‘c an econometric and two combined
econometric-
and twc’. time series models in their singular form.
time series models
IO forecast seven accounting
varmbles for a single firm. The test prriod for all five models was the same. The ordinary least-quares
multivariate
time series composite model and ths multivariatc time sews
model both produced acceptable achieved
better
Multivariatu
recciicd
results than time
the kind
U values. The multivariate
of attention
model. however,
models.
series and combined
forecasting and plannninp. potential
Theil
the other
time series econometric.
they cJ:serve. Such models ma!
models have nc)t
bc valuable
tools kr
This study may stimulate furt!lcr research in esploring
the
of such models.
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301
w&3:35301-3 I I
A Comparison of Econ etric, Time Series, and Composite Forecasting Methods in lBn%iictin Accounting Variables James S. Ang, Jess H. Chua, and Ali M. Fatemi
This study comparesfive forecastingmodels in predicting seven accounting variables for a fum in the rubber and plastics industry. The five models consistof an econometric model (CXS), two time series mod& (univariate, and multivariate), and two combined models (combinations of the ecoimmetric model and one of the time series models). Seventy-six monthly obsetvations were used lo estimate the models. Each estimated model was usedto forecast each of the seven variables over ;’ Wmonth period. These forecasts wefe then tested for accuracy The test involves acomparison of Theil’s “inequality coefficient .‘* The multivariate time series model snd one composite model, i.e., the OLS-multivariate time series maiel, performed well. The multivariate time series model, however, achieved bener results than the other models.
Introductiwt The need for $:oodaccounting and financial forecasts can not be overemphasized. Such forecasts provide the basis for budgeting, control. financial planning, investment decision, timing and amount of new financing, working capital management, etc. For example. Horprgren ( 1977, p. 129) suggests “the sales forecast is the starting point for budgeting, because inventory levels and production are generally geared to the rate of sales activity.” in the search for good forecasts, research has resulted in new forecasting models, Few studies. however, have directed their attention to combining &e forecasting models presently available to investigate whether improved forecasts can be obtained for accounting variables. Pindyck and Rubinfeld (1976) obtained improve j forecasts of short-term interest rates when they combined regression analysis with a time series rnok-4. Ludwig (1974) also found that combining regression analysis with a time series mo&l results in improved forecasts of short-term savings deposit flows. This study contributes to the current literature by studying the predictive abilities of combined econometric and time series models. It compares five forecasting models in forccastrng accounting variables for a single firm. The models used are* I) an ordinary
James S. An8 IS Wdham 0. Cullp,n Professor of Finance, Florlda State Uruversity, Tallahassee, Florida. less H. Chue ISAssociate Profeswr of Finance, The University of Calgary, Calgary, Alberta Canada. Ali M Fatemt is Aswtant Professor of Ftnance. Kansas StqteUnwerstty. Manhattan, Kansas tb502 Address repnt requests to Prof. Ah M. Fateml.
016-61%/83/503.00
1. S. Anget al.
302
Tabie 1. Ecanometric Variables
C.4SH N’rR
Cash = Net
Tl = Total
trade receivables inventories
NS = Net sale:: CGS = a-‘ost of goods sold SELL
=. Selling expenses
GP = Gross profit
,4,PR = Average prime rste charged by banks E-1 = Business
investment
(seasonail)
adjusted at annual rates) ;billions
Lc‘ = index of labor cost per unit of output.
manufacturing
P’l = f’ersonal
in 1972 dollars
income less trlnsfcr
PRP
= Industrial
SRP
= Shipments:
VMNO ItPRP
production:
payments.
rubber and plastics products
rubber and plastics products
= VdlUc of manufacturers’ = H’~wlw4c
(millions
of dollars)
(I 967 = 100) of dollars)
new orders, durable goods mdustries.
prices: rubber and plastics products
in 1’172 dollars (billions
ofdollar<)
(I 967 = I(I(Q
least-squares econometric model (OLS); 2) a univariate time series model (LINI): multivarlatc
tinre series model (MLILT):
series model (OLSmodel (OLS-
l_JNI); and 5) an ordinary
least squares
univeriate
least squares- multivariate
3) a time
lime series
M LJ LT).
Do ta aad Methodology Data for the study consist of and plastrcs industry April
4) an ordinary
monthly financiA stsiement
from Januar!
ltcrns for a lirm in the rubber
1971 to June 1978.’ Data from January
1971 to
1977 are used to estimate the models. Data from May i977 to June 197s ;~re
reserved f
in Table
1 are the firm, industry.
and economy VU iables used. Endogenous (firm) variables are those most often used in short-term
business financidl planning. Exogenous variables are industry and economy
variables. Data for the rubber and plaskcs industry C:trrtwr
Busintw.
National
are cbtained from the Surcq~ (!I
economic data arc obtained from
Brcsiwss
Ccwslilirw
Di@‘Sf.
Draw ‘ng on the available literature
in ticcounting and lknce.
endogenous varmbles
are specified as functions of exogenous and other cndogcrtous variables. The resu’ting structural
equations are presented in Table 2:
1. Cash is expressed as a function A.. ’ Inventories shipments
iSlY
of net sales and inverltories.
expressed as it function
of rubkr
the value of ni~nufac‘turers’
Ned orders,
and plastic products, and the prime rate of interest.
’ The Dada arc from unpuhhshed cornpan! owned prlsate sources. but the\ are the data a\all,lhle to an lntcrnal anal\~.t or a planner of lhc titm
Methods in Pmdicting Accoming Recc~tables
m
Tabk 2. StructuralEquatims in Their General Form CASH
= f(P(S. TI)
TI
= f(VMNO. SRP. APR) = j(SRP. NS, 4PR) = ((SELL. PRP. SRP. Bl. PI) = f(PRP. VMNO, LC. NS. WPRP) = f(NS. GP. WPRP. SRPI = f(WPRP. NS. SELL)
W-R NS CGS SELL GP
3. Receivables are specified as a function of shipments of rubber and plastic produc , rlet sales, and the prime rate of interest. 4. Net sales is expressed as a function of selling expenses, rubber and plastics production, shipments, business investment and persona! income. 5. Cost of goods sold is specified as a function of rubber and plastics productions, value of manufacturers’ new orders. labor cost, net sales, and wholesale prices. 6. Selling expenses is specified as a function of net sales, gross profit, wholesale prices. and rubber and plastics shtpments. 7. Finally. gross profit is expressed as a function of wholesale prices, net sales, and selling expenses.
Univariate models and a multivariate model were used for comparison. The univariate model incoporates past history of only the forecasted variable. while the mult;variate method allows a cross-sectiat.al correlation of one variable upon others. L~~rit~toru Mod&s. Univariate time series models have received wide attention, e.g., Albrecht. Lookabili, and McKeown 1977; Box and Jenkins 1970; Brown and Rozeff 1978. 1979: Dopuch and Watts 1972: Foster 1977; Griffin 1977; Kinney 1978; Lorek 1976; Lorck.
McDonald.
and Patz 1976; Mabert
1974: Maberr
and Radcliffe
1974;
N&on 1973; Watts 1975; Watts and Leftwich 1977. The univariare integrated autoregressive-mo4ng average process of order (p, d, 4). or simply ARIMA (Q,d, q), for th,: series )’ can be written as’
(1)
&(B)A“r,=ii+t),(B)~ where @,(Sl== I --&,B-c#J~B~---.~. pth order autorcgrcs;ive B is the backward that BY,= \‘,
-$,BPdenotes
shi’i operator
,. B+.,=
‘!
process;
y,
such
.‘, etc.;
Ad denotes the dth diflerence for the serves J., such that A)*,= )I, - !‘, ,, A=y, = Ay, -As, _ ,, etc.; ____
-. 2 We shah no1 go Inlo de1 ui about Ibe model. ror that. see Box and Jer.kms (1970) Nelson Pmdyck and Rubinfeld (1976).
i 1973). and
S is a constant term which represents the mean for the series; O,(R)= 1 -tI,B-8:BL-,,,-t),Bq denotes a yth order moving average process: and E,’ f,-I. . . series represent a white noise process. Time series modeling involves both specification and estimation problems. The estimation procedure used typically is least-squares error minimization or likelihood function maximization with errors ( es) assumed to be normally distributed. The specification problem has often been solved by the Box-Jenkins (1970) technique. In this procedure, the model is specified subjectively through an iteration process. The model obtained is nonunique in the sense that two forecasters could arrive B?different models when modeling the same series. Subjective specification also implies that the model obtained may not be the “best.” However, the specification problem has been solved to a certain extent by the development of the Akaike Information Criterion (1971. 1972, 1974. 1976b The Akaike Information Criterion (AK‘) provides for time series models to be constructed objectively and uniquely. The AIC is a maximum likelihood criterion with significance level ndju.X*ments for iterative testing. It is defined as: AIC= - 2 lop (maximum likelihoc>d) -t-2 (number of independently
adjusted parameters)
,
(11)
The criterion has been subject*>d to exter sive testing. Monte Carlo experiments to test its ability in rccapturmg the true order? of simulated time series were performed bj Akaike (1971. 1974) and Jones (1975). and the results were excellent. The criierion has also been app:;ed in the time series studies by Gersch anu Sharpe (19l.l). Jo les (1974). and Landers and Lacoss (1975) who all reported good results. The procetiure for constructing the minimum AK time series model is is follows: 1.
construct time series models of increasing order up to a reasonable limit, say number of observations divided by ten; 2. compute the AIC for each of these models; and 3. select the minimum AIC model as the optimal model. The procedure is used in this study to construct
univariate autoregressive
models.
,Vulrir~ritifcp i2fodtBl.s.Ur;variate models have one seriuus limitation in that they do not consider the interrelationships among variables. For example, in a univariate time series analysis of net trade receivables (NTR). the current level of NTR is related only to its own past levels. It is likely, however, that the current level of NTR relates not only to past levels of NTR but aiso to past levels of sales and past levels of inventories. Multivariate models allow for such cross-sectional correlaticns. Unlike univariate models, multivariate time series models have received limited attention (Cramer and Miller 1978, Pierce 1977. Tiao et al. 1979) The generalized autoregressive movin, lverage model for li series {I,\ takes the form (3)
Methais
inPwdkting
Accounting Receivables
305
where f&tLJ=I-
,B-.
.-+pi?p.
and
8,(@=/--8,B.-8,&F are matrix poljmomials in 8: 4s and A are k.lekmatrices; &=_Vn-- S, is the vector of deviations from some origin S (the mean if the series is stationary): and { (~~1with c,=( cl,. ....
ekt)’ is a sequence of random shock vectors with zero mean and covariance matrix X. Multivariale time &es modeling. as with univariate modeling, involves both specification and esttmation. 1 he AK approach is used to specify rhe multivar;aie ml;del. btimation of parametez followed tte recursive method developed by Durbin (1960) and Whittle (1963). Composite Models Cornposit: models were developed in two php,es. Phase I developed the econometric model by utilizing an ordinary least-squares approach. In phase II, time series models of the residuals were cons,tructed. Composite forecasts *Nerethen obtained by adding time series forecasts of the residuals to the predictions of the ordinary least-squares econometric model.’
Results In a seriesof regressions,dependent variables were regressedon different combinations of contemporaneous and lagged values of (their respective) independent variables. The combinations with the highest explanatory power !R’) were defined as the “best fit” models. The predictive abilities of these best fit models *Merethen compared to those of time series and composite models. Best fit models were also used to construct ;he composite models. Best-fit OLS estimates of the a!ructural equations are presented in 1 able 3. CoelKcients of determination (R2s) range between 4%; and 84% and all F-statistics are significant at lpb. The Durbin---Watscrr! d-statistics for CGS equation indicates that the null hypothesis of no atltocorrelatior of the residuals cannot be rejected at !o/,. For all other equations, however, the null hypothesis of no serial correlation is rejected at cO,O. Examining the results reported in Table 3, we observe thar the estimated coefficients have rhe correct sign vis-&vis the assumed theorP!icai role of the variables in each equation: 1. Net sales has a significant positive effect on cash holdings. The effect of inventory
holdings. although insignificant, is negative. ’ In the presence of autocorrelatlon
~CPCS forecasts
m the OLS model, a more eficlent
of the residuals to the residuals obtamed
approach
from a GLS model.
would k
;o add
llme
=
=
=
=
=
-
Tl,
M-R,
NS,
t c;s,
SkLL,
GP,
Id
u
(6.1 l)b
+ ~J.2~A%s~
Econometnc
and Statrsrtcs
0.31T1, i- 1.21)
k.quarionQ
Equations
=
.
.
--
(- 3.29)b
_.
-
_
(2.4l)b
_” __ _. _
(11.51)b
(-3
-
65)b
--
-
___
.~
3197 8sPI, (-1.11) - 2194.4IWPRPt (-0.81)
___~~ _._._
1239086C 249302 3V.%lN0,_3 346.XlSRP,_. 3 161356.4APR, I-iz.olP ( -0.95) (10.37)6 (-3.58)b 532524 21.81SRP,_3 + 0.729!‘+ + 1597?56APR, (-0.62j (i .06) i1.53) (7.93)b 1225362 + 5_80StLL, + 9856.66PRP,_ 2 105.69SRP,_2 + 6470.84B1, (0.52) 13.47)b (-Q.62) (1.9O)c (0.57) 1458763 + 4?88.16PRP, + 1 151.54V~fS0,__2 + 10124.77LCt + 0.38NSt (2.69)t C I.C4)C (0.73) (1.18) (4.18)b 184633.7 + O.O94NS, O.l6GP, + 1367.33WPRP, + 21.13SRP, (3.68)b (4.63)b (1.21) t-2.84$’ &.23)b 274598.6 + 1852.62WPRP, + 0.38NS, I .09SkLL,
! 3rJ838 j (1 I .98;6
Least-Squares
a ( ) indicates I . u!urs. b Significant at <‘I,. <’Stgnificant a: ! 07
=
c AW,
variat;k
lkpcndent
_..
Ordinary
3.
T&e
R2
F-Stat
_ _
I.1 0.98
2.02
35.47
f4.56 17.30 39.66 81.91
0.70
0.67 0.78 0.84
-
0.99
0.76
1
II..-_ 67
0.49
~__~
~..~..._-.-----
52.44
0.63
~-. Durbm-%‘;ttwn Statistic
0.77
0.49
22.24
_-.____.._.,~-.----__--_
-.--. -.-.-
Methods in Predicting AccountingReceivabks
M?
2. Total inventories is negativeiy akcted by the value of manufacturers’ new orders, prime rate of interest. and shipments of rubber and plastics products. Thz last factor’s effect. however. is insignificant. 1 _. Net sales is the only variable that has a sigmfkant positive efkct on net rnde reckvabhes. Although ksi@nifkant, the e#kt ofthe prime rate is positi4reand that 4.
value of) industrial production both have a si~~~~t ~s~tiv~ t cmnet s&s. Perwnal income and (the two-period value of) ~i~rne~ nt negative effect, whife btlsiness tment has an insi~~i~~~nt positive e&et on net sales. a. Industrial pr~~~tios‘~ net sates, value aif manufacturers’ new orders, and labor cost all have B positive effect an the cost c~Fgmds sold. Only :he first two factors. hmwver, me statisticalEy significost. Wholesale prices have an insignificant n~~~ti~e e&43 0 St Qf @Kds Wk!. 6. Net sales, whoI e prices, and shtpments have a positive effect on selling cxpnsees. but the efkt ofthe last variable is insignificant. Gross profit and selling expnses ;~re s~~n~~~~~~t~~ cot related negatrvely. 7. f ‘,nally, gross profit is significantly afkcted by wholesale prices, net sates, and seiling expenses.with the first IWOfactor? having positive effects and the last factor having a negative effect.
Autoregressive coefkients for the UNI, MULT, OLS-UN1 and OLS-NULT models are. respectively. presented in Tables 4 7.J The MULT and OLS-MULT -m&k arc all of order one (p = I). The order of the autoregressive prxesses for UNI and OLS 1lNl models vary between a minimum of one (UN1 model for CASH) and a rllaximum of seben (UN1 model for GP and OLS.-l tN1 model for NS). Each r:sttmatcd modci was u!cd to forecast the endogenous variables over the 14month period from May 1977 to June 197% These forecasts were then tested for accutacq. The test Involves a comparisorr of Thetl’s “inequality coefftc ,t.”
ukre P, and A, are the pairs cr predicted and actual values, respectively. U reaches a value of zero as a lower houndarq when all predictions are correct, that is when P, = A,. It assumes a value of I when the forecasting procedure is lesseffective than a naive FIOchange forecast.5 L’ values for all five models *c reported in Table 8. Two moaels-- OLS MU LT and MU LT models produced 2 values below ! for all scben rndogcnous variables. The other three models produced L’ values greater thzq I for at least two of the \ariablcs. The MU LT model. however. produced better forecasts than ihu OLS MIJLI’ modet, for all seven varlubles. I lsmg the rniibns and rhc 4t;md;trd cktlatlonh ol L’ values as the perl’orrnancecrltcria
- . - _--.
___I_“_____.--
cot%cten~s
’ POSIIIV~ ~mplgthat the current value of the vaitahte IS pos~tivcfy related 10 I& own other vartables’ prwous values. Ncpatwc ccdikxnts ~rnplithe opposite For example, accordmg MULT model. current devlatmns from mean for CASH arc negatIveI> related to one-period devlatlons CASH. TI. VTR. <‘
of
and/or
to the lagged related
I
*
1
I, I
J
i
e
1
=
= _
SELL,
GPt _
__.
__.,_ ,,__ ._--_.-
.-
~._ ._ --_I
_Ip__.-==3s1-
l.O203OGP,_~ - 0.74603GPt_2 _ _-.- -.------.---II_ - 0.69694GP,
J
r( W)Y ICP,_4 _.-- .- ._I-
4.37090 4.29260
-0 $7206 __ __________ ~___.
1.25290
-0.26391
-0.74516
-0.71160
0 13105
-9.58190
n.05705
-0.041i 1-I
2 8974@
0.97212
-0.?:451 -O.lO!fZ
0.53850 -1.19390 1.88210 - 1.22600 -1.73770 .-___
-n.o1125 0.11398 -0.03691 -0.‘?3588 -0.19891 0.10442
.-0.02381 0.33416 -0.77199 1.61520 -n.c5959 . _ 050446
-0.07983 -0.32970 -1.16410 1.20890
-1.19740 -.-_
-0.363!3
-0.26350
-0.036 14
0.01507
0.08673
._ ______xII______-.-_~-
+ 0.83082GP,_, - _._ __ ____ll__.-_-____--_
i .2492OSELL,_ 1 - 1.9146OSELL,_2 - I .87450SELLr_~ -- 2.06000SELLI_4 + 1.3O63WELL, _ 5 ...--._-
____
-O~.~-
!r 881198CASH, _i - 0.248CICASH,_, - 0.21853CASff,_J - 0.01830CASf~,_4 - k34469CASff,_~ e 0.79793T1,_2 - 0.0087TI,- 3 - 0.77294Tf,_4 - 0.90149T1,_5 - ~.~6157~~~_6 : .02>8Ml,_ I 0.70517~~R~_~ 0.60686NTR,_, - 0.744 17NTR,_2 - 0.875!%!NTR,_3 -- O.?8151NTR,_~ -- 0.33110NTR,_s ~.~~464~~~_~ i .Or)460Nb,_, - : .Of59QNS,_2 .- 0.94435?&_; 1.16230NS,_4 - 0,i8625NSt, 4 0.5912RNS,_6 0.94573Ccs,_ 1 0.51009CGS,_, - 0.0Y1146CGS,_J - 0.46Y2OU&.4
Uruvanatc P uuxegre?rsive Model for Residual Scncs from OLS Models ..~_ _--._... ..-__._ _.
Tabk 7. Multivanate Autoregressive Model for Residual Series from OIS hkniels: (Order = 1) ~ ____--__--l_l_,-.__-
= = = = =
CASH, &I, NTR, NS, CC-S,
Tat& i. --.
.-
-...
--..
.
_ _._--l
J. S. Anget al.
310
Table 8. Theil U Values
time series m4els.
comparing
the M1II.T
mode: produced better forecasts than
the L’N1 model. C‘ompariug
combinations
model, OLS- ML1I.T Overall,
the MULT
of a time series mode’ amt the ordiilary
performed
better than the OLS
lJNl
least squares
model.
model produced beiter forecasts than the other four models.
Conclusions This study con3 ,,art‘c an econometric and two combined
econometric-
and twc’. time series models in their singular form.
time series models
IO forecast seven accounting
varmbles for a single firm. The test prriod for all five models was the same. The ordinary least-quares
multivariate
time series composite model and ths multivariatc time sews
model both produced acceptable achieved
better
Multivariatu
recciicd
results than time
the kind
U values. The multivariate
of attention
model. however,
models.
series and combined
forecasting and plannninp. potential
Theil
the other
time series econometric.
they cJ:serve. Such models ma!
models have nc)t
bc valuable
tools kr
This study may stimulate furt!lcr research in esploring
the
of such models.
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Awwls ofrhr lnsrr~uteofSrurisricd Muthtmuncs
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In
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Cmr.
Accounting Receivables
R. H.. and Miller,
31t
R. R. Dec. 1978. Multivariate tim.: series analysis and w Anui~sis 13(S): 103-1017.
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te=h. W. and Sharp. D. 19?3. Estimation of power spectra and tinire order autoregressive model. 1EEE Tronsccrir ‘ns cm Autmuuic Commd. AC-18367469.
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rment
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Jourd
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