A multivariate try series

A Multivariate Time-Series Prediction Model for Cash-Flow DataAuthor(s): Kenneth S. Lorek and G. Lee WillingerSource: The Accounting Review, Vol. 71, No. 1 (Jan., 1996), pp. 81-102Published by: American Accounting AssociationStable URL: http://www.jstor.org/stable/248356 .Accessed: 27/10/2014 01:30

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THE ACCOUNTING REVIEW Vol. 71, No. 1 January 1996 pp. 81-101

A Multivariate Time-Series

Prediction Model For

Cash-Flow Data

Kenneth S. Lorek Florida State University

G. Lee Willinger University of Oklahoma

ABSTRACT: This paper provides evidence on the time-series properties and predictive ability of cash-flow data. It employs a sample of firms on which the accuracy of one-step-ahead cash-flow predictions is assessed during the 1989- 1991 holdout period. We develop a new multivariate, time-series prediction model that employs past values of earnings, short-term accruals and cash-flows as independent variables in a time-series regression. Our predictive results indicate that this model clearly outperforms firm-specific and common-structure ARIMA models as well as a multivariate, cross-sectional regression model popularized in the literature. These findings are robust across alternative cash-flow metrics (e.g., levels, per-share, and deflated by total assets) and are consistent with the viewpoint espoused by the FASB that cash-flow prediction is enhanced by consideration of earnings and accrual accounting data.

Key Words: Cash-flow, Time-series models, ARIMA.

Data Availability: A list of the sample firms may be obtained from the authors. The remainder of the data is available from the sources indicated in the text.

We appreciate the comments and suggestions of the referees and the associate editor. Elton Scott provided useful suggestions as well. Ken Lorek acknowledges the financial support provided by the Florida State University College of Business Summer Research Grant Program. Lee Willinger appreciates the financial support provided by the Samuel R. Noble Foundation.

Submitted August 1993. Accepted September 1995.

81


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82 The Accounting Review, January 1996

I. INTRODUCTION

R ecent empirical findings by Lorek et al. (LSW) (1993) suggest that univariate autoregressive-integrated-moving-average (ARIMA) models for quarterly cash flows provide more accurate cash-flow (CF) predictions than the multivariate cross-sectional

models used in prior research (e.g., Wilson 1986,1987; Rayburn 1986; Bernard and Stober 1989). LSW propose two common-structure ARIMA models ((000)x(100) and (000)x(O1 1)) for CFs that are structurally different from the ARIMA models that have been popularized for quarterly earnings. I We extend this previous work by including multivariate, time-series regression models as well as firm-specific ARIMA models in our tests of predictive ability.

Multivariate, cross-sectional regression models make use of accrual-based, independent variables in addition to past values of the quarterly CF series. At first glance, the dominance of simple ARIMA models over richer multivariate ones appears counterintuitive and contrary to the very reasons for accrual accounting. In fact, the Financial Accounting Standards Board (FASB 1984) asserts that accrual earnings should provide a better basis for the assessment of firms' future CFs than the information contained in past CFs alone (SFAC No.5, par. 24). The fact that LSW' s univariate ARIMA models outperformed richer, multivariate ones that employ accrual accounting variables is counter to the FASB's perspective on CF prediction.

Our first objective in this study is to provide descriptive evidence regarding the time-series properties of quarterly CFs. We identify a great diversity in firm-specific ARIMA structures for CFs, but such models do not provide enhanced predictive performance relative to the alternative models that we assess. This finding is reminiscent of results in the earnings forecast literature (e.g., Foster 1977) where firm-specific ARIMA models have not systematically outperformed common-structure or premier ARIMA models.

We assess the inter-temporal stability of the common-structure ARIMA models identified in a cross-sectional fashion by LSW. The cross-sectional sample autocorrelation function (SACF) of our CF series exhibits purely seasonal characteristics, in marked contrast to the quarter-by- quarter (seasonal) and quarter-to-quarter (adjacent) patterns documented in quarterly earnings work. Moreover, the descriptive fit of the (000)x(100) and (000)x(O1 1) common-structure ARIMA models for quarterly CF data is confirmed. We employ both of these common-structure ARIMA models as well as the firm-specific ARIMA models in our predictive ability tests.

In addition, we compare the predictive performance of these ARIMA models directly with the cross-sectional regression model developed originally by Wilson (1986, 1987). When LSW made similar comparisons, they used CF per-share (CFPS) as the input series for their time-series modeling and a different series, CF deflated by total assets (CFTA), for the regression model. Unfortunately, use of different input series for time-series and cross-sectional CF prediction models reduces the comparability of LSW' s predictive ability results2 and may be one reason why Wilson's cross-sectional regression model did not perform as well as the time-series models. In

IUsing customary (pdq)x(PDQ) notation, quarterly earnings time-series models popularized by Brown and Rozeff (1979) and Griffin (1977), among others, are (100)x(Ol 1) and (01 1)x(O1 1), respectively. The (p,P) variables represent the number of autoregressive or seasonal autoregressive parameters; (d,D) represent the levels of consecutive or seasonal differencing and (q,Q) represent the number of moving-average or seasonal moving-average parameters. Finally, a common-structure ARIMA model is one in which all sample firms use the identical model structure, but the parameters are estimated on a firm-specific basis.

2 Initially, LSW' s analyses were based entirely on the CFPS series. At a later stage in the project, LSW added the cross- sectional regression model of Wilson (1986, 1987). Since Wilson estimated his model using the CFTA series, LSW did as well.



Lorek and Willinger-A Multivariate Time-Series Prediction Model For Cash-Flow Data 83

this study, we provide descriptive and predictive evidence on the time-series properties of three cash flow series: undeflated CF, CFPS and CFTA. Therefore, all descriptive and predictive comparisons across prediction models are confined to within-series comparisons.

Our second objective is to identify a multivariate, time-series regression model for CFs that circumvents the cross-sectional restriction on the model coefficients invoked in previous work. We incorporate the information contained in past earnings series, in addition to short-term accruals such as accounts receivable, accounts payable and inventory. The predictive results indicate that this new multivariate, time-series model for CFs provides significantly more accurate one-step-ahead CF predictions than either the univariate ARIMA models or the multivariate cross-sectional regression model developed by Wilson (1986, 1987). These results are consistent with the FASB's assertion that accrual accounting variables are useful in a CF prediction context.

The rest of this paper proceeds as follows. The following section details the motivation for the analysis followed by a background section that reviews extant work on CFs. Next, we include a discussion of the research method. Finally, we present the predictive results and provide some concluding remarks.

II. MOTIVATION

CF prediction is an important task relevant in diverse decision-theoretic contexts. Bowen et al. (1986) have argued that CF data are potentially useful in any decision-theoretic context in which accrual accounting data are employed such as: (1) distress prediction, (2) risk assessment with respect to the size and timing of business loans, (3) predicting credit ratings, (4) valuing closely-held companies and (5) provision of incremental information to security markets.3 Recently, financial analysts have also recommended that investors pay increasing attention to CF analysis in stock selection activities. Dorfman (1987) relates that, while financial analysts do not rely solely upon CF analysis, they view it as an important supplementary tool useful in avoiding misleading inferences in the patterns of accrual-based earnings numbers. Berton (1994) relates that Fitch Investors Service Inc. recently published a CF-based corporate rating system for prospective investors.

Standard setting bodies have also emphasized the notion that CF prediction represents the underlying rationale for the very existence of accrual accounting. Specifically, SFAC No. 1 (FASB 1978, par. 37) states that the primary objective of financial reporting is to: "provide information to help investors, creditors, and others assess the amounts, timing, and uncertainty of prospective net cash inflows to the related enterprise." This underscores the need to delineate precisely the time-series properties and predictive ability of CF expectation models. In the next section we review the work that has been undertaken in this area and summarize the methodologi- cal improvements we make relative to such work.

III. LITERATURE REVIEW

In marked contrast to the extensive literature on the time-series properties and predictive ability of accrual-based earnings numbers, much less attention has been focused on CF data. Bowen et al. (1986) predicted CF from operations one and two years into the future by employing a set of alternative predictor variables including current net income, net income plus depreciation,

I See Bowen et al. (1986) for specific literature citations of extant work in each of the above areas.




working capital from operations and past values of CF from operations. They confined their data analyses to annual numbers and employed a limited set of simple linear forecasts. They did not attempt to identify multivariate CF prediction models. In general, Bowen et al. (1986) found that the differences in relative forecast errors of the net income and CF from operations predictor variables are not significant. Using a strict interpretation of the FASB's statements, Bowen et al. (1986, 722) conclude: "...none of the results are consistent with the FASB's assertion of the superiority of earnings as predictors of future cash flows." However, the use of relatively short, annual data bases in conjunction with naive expectation models limits the generalizability of this study.

In a related work, Greenberg et al. (1986) used either current CF from operations or net income as predictors in regression models where the dependent variable was CF from operations. Rather than assessing predictive ability directly, they used a ranking procedure based on the coefficient of determination (e.g., R2) for each of the two models (CF or net income) per sample firm. Their results indicated that in four of five prediction intervals, net income outperformed CF in a significant manner. Unfortunately, Watts and Leftwich (1977), among others, have documented that superiority in goodness of fit tests (e.g., R2) does not necessarily translate into superiority in predictive ability. This descriptive-predictive paradox, which has also been referred to as the regression fallacy, suggests that Greenberg et al.' s (1986) findings must be interpreted cautiously.

Hopwood and McKeown (1992) investigated the time-series properties of quarterly operating CFs per-share and earnings per-share for a sample of manufacturing companies. They find that the time-series properties of CFs are quite different from those of earnings. Specifically, their results indicate a pattern of autocorrelation that is much stronger in the earnings series than in the CF series. In their CF predictive ability tests, they compared firm-specific ARIMA models to the premier ARIMA models attributed to Brown and Rozeff (1979) and Griffin (1977). The former models exhibited a slight advantage over the premiers. Unfortunately, these premier ARIMA models, which were originally identified on earnings data, may not have served as a useful benchmark because Hopwood and McKeown' s (1992) autocorrelation patterns for the CF series differed from the typical patterns for earnings series. Moreover, Hopwood and McKeown (1992) did not examine multivariate CF prediction models for comparison against their ARIN4A CF models.

LSW provide a recent examination of the time-series properties and predictive ability of quarterly CF data. They used a multivariate, cross-sectional model for CF prediction that was developed originally by Wilson (1986, 1987) and employed by Bernard and Stober (1989). Despite the fact that Wilson's multivariate model is consistent with a wide variety of time-series behavior, the simpler, univariate ARIMA models that LSW identified provided more accurate CF predictions. It seems premature, however, to conclude that the optimal CF prediction model is a simple, common-structure ARIMA model without investigating the propriety of alternative multivariate structures. Additionally, if identification of a multivariate CF structure proves futile, it may still be possible to enhance CF predictions by considering firm-specific ARIMA models, as opposed to ARIMA models based on a common structure.

Finger (1994) assesses whether earnings, used alone or together with CF, are a significant predictor of future CF. She employs a sample of 50 firms and assesses predictive ability one through eight years ahead using annual data from 1935-1987. Her results, with respect to out-of- sample annual CF forecasts, do not support the FASB viewpoint that earnings are a better predictor of CF than CF alone. However, Finger does not employ ARIMA models and confines her analysis to relatively simple, time-series regression models estimated on a firm-specific basis.

Univariate ARIMA models simply employ past values of CF data to predict future CF, and do not rely on accrual accounting information. In contrast, Wilson' s multivariate, cross-sectional model employs numerous independent variables derived from the accrual accounting process and




is generally consistent with the FASB' s assertions on CF prediction.4 The performance of LSW' s common-structure ARIMA models may be partially attributable to Wilson's restriction that parameters in the multivariate model be identical across firms. To the extent that the cross- sectional restriction on model coefficients is inappropriate, the predictive ability of Wilson's model may be adversely affected. In this paper, we develop a multivariate, time-series model that allows firm-specific parameter estimation and compare its predictive performance against Wilson's cross-sectional model as well as common-structure and firm-specific ARIMA CF prediction models. Since this multivariate, time-series model employs independent variables derived from the accrual accounting process, we test the utility of employing such variables in predicting CF. Our findings suggest that knowledge of accrual accounting variables enhances CF predictions relative to those derived from ARIMA CF prediction models.

IV. RESEARCH METHOD

Cash-Flow Series

We calculated undeflated quarterly CF from operations according to the procedures employed by Bernard and Stober (1989) and LSW by using information obtained from both the annual and quarterly COMPUSTAT data bases. These data were constructed by using the following income statement and balance sheet accounts:

Operating Income before depreciation - Interest expense - Current portion of income tax expense - Increase in net working capital other than cash and securities, net of short-term debt

Cash flow from operations

There are two benefits to employing this particular proxy for CF from operations. It is calculated consistently across firms and it has been employed extensively in previous work.5 Moreover, actual CF measures are not available throughout our extensive identification and test periods. We also assess whether alternative deflators affect the time-series properties and predictive ability of CF series. Three forms of the CF series were tested: undeflated cash-flow, cash-flow per-share and cash-flow deflated by total assets. Predictive results are presented separately for each CF series.

We constructed a time series of quarterly observations for each CF series beginning in the second quarter of 1979 and ending in the fourth quarter of 1991. Three sampling criteria were employed. First, we required sample firms to have complete time-series data for interest expense, the current portion of income tax expense and increases in net working capital to enable construction of the proxy for CF from operations. Second, all data necessary to operationalize Wilson's cross-sectional regression model were required, including contemporaneous values of sales and earnings as well as lagged values of sales, CF, current and noncurrent accruals and capital expenditures. Third, we imposed a December 31 year-end requirement to properly align quarters for seasonal differencing. Such data requirements resulted in a relatively small sample

4 The specific independent variables included in Wilson's multivariate model are detailed in section V on CF prediction models.

5We observe that interest in developing statistically-based expectation models for this quarterly CF variable is heightened further by the unavailability of financial analyst forecasts of the CF series used in this paper.




of firms in each year. Table 1 indicates that 62, 61 and 51 firms had complete quarterly CF time series through 1989, 1990 and 1991, respectively. The initial model identification period included the 39 quarterly CF numbers beginning with the second quarter of 1979 and ending in the fourth quarter of 1988. We withheld the 12 quarters within the 1989-1991 interval to serve as a holdout period for predictive ability tests.

Due to the need to obtain extensive time-series data across the 1979-1991 interval, newly formed firms and failed firms were excluded from the sample. In fact, the sample is dominated by relatively large and successful firms.6 Therefore, the findings only pertain to firms that have similar characteristics. Additional research is necessary to assess the robustness of the findings to other populations of firms.

Table 1 provides a two-digit SIC breakdown of the sample firms for each year in the holdout period. In 1989, the heaviest concentration of sample firms pertained to the following industries: #29 (petroleum refining, n=9), #33 (primary metal industries, n=6), and #28 (chemicals and allied products, n=5). The SIC two-digit codes show that a broad spectrum of businesses is represented within the sample for each year in the holdout period.

Behavior of Sample Autocorrelation Function

Table 2 provides information on the cross-sectionally derived SACF and partial autocorrelation (PACF) functions for the undeflated CF, CFPS and CFTA series. For each of the 62 firms for which predictions are generated in 1989, SACF and PACF values were computed over 39 quarters

6 The median market value of common equity for the sample firms at December 31, 1988 was $1,734.6 million which is larger than the median of the upper strata (large) firms in Bathke et al. (1989).

TABLE 1

S.I.C. Industry Breakdown of Sample S.!. C. Two-digit Industry Code Industry Title 1989 1990 1991

13 Crude oil and natural gas 2 2 2 20 Food and kindred products 2 2 2 26 Paper and allied products 4 4 4 27 Newspapers: publishing-print 4 4 3 28 Chemicals and allied products 5 5 5 29 Petroleum refining 9 9 7 33 Primary metal industries 6 6 4 34 Hardware and metal products 3 3 2 35 Machinery 2 2 2 38 Photographic equipment and supplies 2 2 2 45 Air transportation 3 3 3 49 Electric, gas and sanitary services 3 3 2 50 Wholesale materials 2 2 2

Other industries 15 14 11 Total Sample Firms 62 61 51




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beginning with the second quarter of 1979 and ending with the last observation in the identification period, the fourth quarter of 1988. In accordance with the methodology of Foster (1977), these firm-specific values were summed across firms and averaged to obtain the values reported in table 2.7 Panels A, C and E contain the SACFs and PACFs of the levels of the undeflated CF, CFPS, and CFTA series, respectively. The three series exhibit nearly identical time-series behavior with a virtual absence of autocorrelation at the non-seasonal lags (n= 1, 2 and 3). All three series have a monotonic decline in the SACF at the seasonal lags (n=4, 8 and 12) that is supportive of the seasonal autoregressive model (SAR): (000)x(100). Panels B, D and F pertain to seasonal differences of the three series. They depict a singular spike at lag 4 consistent with a seasonally differenced, seasonal moving-average process (SMA): (000)X(O11). These descriptive findings suggest that alternative CF deflators (e.g., no deflation, per-share, total assets) do not affect the underlying time-series properties of CFs.

Descriptive findings on a firm-specific basis complement the cross-sectional SACFs and PACFs reported in table 2. Since the aggregation of firm-specific SACF and PACF values may mask idiosyncratic firm-specific behavior, we also analyzed each firm's individual SACF and PACF values and identified firm-specific ARIMA models. These models were identified based on the patterns in firm-specific autocorrelation functions computed during the identification period of 1979-1988. No data from the holdout period were used in the identification of the firm- specific ARIMA structures.8 It is possible that the CF time-series properties of an "average" firm may not necessarily be similar to those of individual firms. In fact, table 3 shows considerable diversity in firm-specific (pdq)x(PDQ) ARIMA models.9

Firm-Specific ARIMAs

To ease exposition, our remarks pertain primarily to the CF series (e.g., column 2 in table 3), however, the firm-specific ARIMA models identified for the CFPS and CFTA series were very similar in structure. First, 38 of 62 firm-specific ARIMA models for the CF series were seasonal in nature exhibiting non-zero values in either the seasonal autoregressive parameter (P), seasonal differencing (D) or the seasonal moving-average parameter (Q). Similar results, 33 and 31 of 62 firms, pertain to the CFPS and CFTA series, respectively. Second, the most frequently identified firm-specific ARIMA structure was the (000)x(O 11) SMA model appearing 10 of 62 times for the CF series. This provides additional support for the propriety of the SMA structure that complements the cross-sectional evidence reported above. Third, the firm-specific ARIMA models for quarterly CFs are dissimilar from ARIMA models popularized on quarterly earnings. Specifically, the ( 100)x(O 10) Foster model was not identified for the CF series of any sample firm while the (00)x(O411) Brown-Rozeff model was only identified once and the (011)x(O 11) Griffin-Watts model was identified for four firms. These results underscore the differences in time-series properties of quarterly CFs and earnings. Finally, whether the diversity in firm- specific ARIMA structures for CF series exhibited in table 3 will enhance predictive ability is an empirical issue upon which we provide evidence in a later section of the paper. We now discuss the CF prediction models against which the predictions of the ARIMA models are compared.

7 This process may control for sampling variation, measurement error and noise in the data when working with finite samples.

8 ARIMA models were selected on the basis of minimum residual autocorrelation as measured by the Box-Ljung Q statistic. ARIMA modeling employs an iterative process that ends when the residuals of a candidate model behave as a white-noise series.

9 We report ARIMA models for the 62 firm sample. Results for the 61 and 51 firm samples were quite similar.




TABLE 3 Frequency of Firm-Specific ARIMA Model Structures Identified for Sample Firms

(pdq)x (PDQ) CF Series CFPS Series CFTA Series

(000) x (000) 4 4 6 (001) x (000) 6 6 6 (002)x(000) 5 5 4 (003) x (000) 0 1 2 (100) x (000) 2 3 4 (200) x (000) 1 1 0 (000) x (001) 1 3 3 (000) x (100) 4 3 2 (0()1) x (001) 1 1 1 (001) x (100) 0 0 2 (002) x (100) 1 1 1 (100) x (001) 1 0 0 (100) x (002) 1 0 0 (01 1) x (000) 2 4 5 (012) x (000) 2 3 2 (013)x(000) 0 1 1 (lO0)x(000) 2 1 1 (000) x (011) 10 8 10 (000) x (012) 0 0 1 (OO0)x(I1o) 0 0 1 (01 1) x (001) 2 0 0 (012)x(001) 1 0 0 (01 1) x (100) 0 1 1 (001) x (010) 2 1 0 (002) x (010) 0 1 0 (003) x (010) 0 2 0 (001) x (01 1) 5 4 2 (002) x (01 1) 0 1 0 (003)x(011) 2 1 1 (001) x (1 10) 0 0 1 (100) x (01 1) 1 1 1 (010) x (012) 1 1 1 (011)x(011) 4 3 2 (013) x (01 1) 1 1 0 (011)x(O10) 0 _ 1

62 62 62

where: (pdq)x(PDQ) = Customary ARIMA notation defined in footnote number 1.

CF = Cash-flow from operations. CFPS = Cash-flow from operations on a per-share basis. CFTA = Cash-flow from operations deflated by total assets.




V. CASH-FLOW PREDICTION MODELS

Univariate Time-Series Models

We employed the (000)x(100) SAR, (000)x(O1 1) SMA, and firm-specific ARIMA models discussed above in our predictive ability tests. Both the SAR and SMA ARIMA models are simple univariate models that are based primarily on four-quarter-old CF data due to their purely seasonal structures. Specifically, the SAR model employs a seasonal autoregressive parameter while the SMA model has a seasonal moving-average parameter applied on seasonally differenced data. Although the firm-specific ARIMA models employ diverse model structures, they also rely solely upon past values of the CF series.

Multivariate Cross-Sectional Model

We also operationalized a fourth CF prediction model, Wilson' s cross-sectional regression model (W), to provide a comparative assessment of the predictive ability of the ARIMA CF prediction models. The W-model may be characterized as a pooled time-series, cross-sectional regression model. It employs a vector of 15 independent variables for each firm in the cross- section. Briefly, these independent variables are comprised of current and lagged values of sales revenues, net earnings, CF from operations, current and noncurrent accruals, and the most recent annual capital expenditures (e.g., Wilson 1986, 1987; Bernard and Stober 1989; LSW 1993). In sharp contrast to the ARIMA models, the W-model employs a rich set of independent variables from which CF predictions are generated. However, such CF predictions rely on the restrictive assumption that all model parameters are constant across firms and time.

Wilson (1987, 303) provides the underlying rationale for the specific independent variables incorporated in the W-model. Lobo and Song (1989,336), among others, have criticized Wilson's estimation of the cross-sectional regression model because observations of the independent variables are used in subsequent years to predict prior years' CFs (i.e., an in-sample forecasting process). Therefore, we estimated the W-model in a true ex ante fashion without allowing future values of the independent variables in the pooled cross-section. As mentioned earlier, we restricted the set of 15 independent variables to contemporaneous and lagged values. Wilson' s (1986) lag structure included variables from periods t-1 and t-4 as well as the sum of periods t-2 and t-3. Finally, capital expenditures were measured over the most recent annual period.10

Multivariate Time-Series Model

Univariate ARIMA CF prediction models can result in more accurate CF predictions than the W-model if the cross-sectional restriction on the W-model' s coefficients is violated. To the extent that CF behavior is contextual, parameters must be estimated on a firm-specific basis. Therefore, we operationalized MULT to be estimated on a time-series rather than cross-sectional basis without Wilson's restriction that parameter values must be constant across firms.

Although ARIMA models also allow firm-specific parameter estimation, they do not employ the potentially rich set of independent variables included in the W-model.1 l The specific structure of MULT that we test is stipulated below:

?0We also estimated the W-model on an in-sample basis by allowing future values of the independent variables in the holdout period (1989-1991) to be used for parameter estimation. Since the overall tenor of the predictive findings was unaffected, we only provide out-of-sample predictive results for all forecast models on a consistent basis. See LSW (1993, 158) for a thorough discussion of such matters.

IIWe observe that Neill et al. (1991) and LSW criticize Wilson on several factors related to the construction of the W- model. In particular, there is little theoretical justification provided by Wilson for the lag structure or level of aggregation of the 15 independent variables.




CFt = a + b1(CFt_,) + b2(CFt) + b3(OIBDt_1) + b4(OIBDt_) +

b5(REC, ) + b6(INVt_) + bPAY) + et(1)

where: CFt = cash flows from operations at time t

OIBDti = operating income before depreciation at time t-i

RECtJ = accounts receivable at time t-1

INV,_1 = inventory at time t-1

PAYt_ = accounts payable at time t-l

et= current disturbance term

The selection of independent variables in MULT was influenced by a desire to construct a more parsimonious, multivariate model for CF prediction than Wilson (1986, 1987). There is considerable evidence from the forecasting literature that complex models often fail in out-of- sample comparisons,'2 so the large number of independent variables used in the W-model (n= 15) may have contributed to that model's lackluster predictive performance. Therefore, we deliber- ately selected a smaller set of independent variables than Wilson.

The first set of independent variables consisted of lagged values of the dependent variable at time t-1 (CFt_,) and t-4 (CF t_). Intuitively, the selection of lagged CF variables is consistent with ARIMA modeling procedures that rely on past values of the dependent variable to predict future values. It appears reasonable to include CFt-, to capture adjacent effects and CF t- to capture seasonal effects in the CF series.

The second set of independent variables included lagged values of accrual-based earnings. We selected OIBD as the proxy for accrual-based earnings since Wilson (1986) and Rayburn (1986) have determined that long-term accruals possess little information content in a capital market setting. Our results indicate that OIBD provides a better descriptive fit in MULT and results in superior CF predictions than when net income is employed. We employed OIBDt 1 to capture adjacent effects and OIBT t- to capture seasonal effects between CF and the proxy for earnings.

The employment of the final set of independent variables (REC,PAY,INV) is consistent with Wilson's use of current accruals in his multivariate, cross-sectional CF regression model. We disaggregate the current accruals variable into REC, PAY and INV to allow firm-specific parameter estimation for each subcomponent. Each variable was lagged one period, implicitly invoking a random walk assumption. We stress that the selection of independent variables is based on our intuition regarding possible ways to improve extant CF prediction models. We do not claim that MULT represents an optimal CF prediction model.

The MULT model relies on time-series data for the CF, OIBD, REC, PAY and INV variables that were obtained over the same time period on which the ARIMA-based models were estimated. The MULT model possesses two distinguishing characteristics relative to the CF prediction models discussed earlier. First, the MULT model, unlike the W-model, allows firm-specific parameter estimation. Second, the MULT model, unlike the ARIMA models, incorporates a parsimonious set of accrual accounting variables in addition to past values of the CF series. Employment of the MULT model should help resolve the controversy about whether accrual accounting data are useful in formulating CF predictions.

"See Brown (1993) fora discussion on this matter. Also, Manegold (1981) and Wild (1987) discuss why the theoretical advantages of multivariate modeling may not be obtained in empirical work.




Diagnostic Tests

We employed a number of diagnostic tests for the MULT model.'3 First, we employed Ramsey's RESET test of linearity to assess the linear specification of MULT.14 This test assesses the incremental explanatory power associated with adding the squared powers of the fitted values of the dependent variable to the regression. For example, the median value of the adjusted R2 for the MULT model for the CF series is .316. When the squared powers of the fitted values of the dependent variable are added to the regression, the adjusted R2 improves modestly to .373. Of the 62 sample firms, 32 experienced increases while 30 experienced decreases to the adjusted R2. These findings are consistent with the overall propriety of the linear specification of MULT.

The second diagnostic test pertains to the reasonableness of the constant variance assumption in MULT. We employed White's test for heteroskedasticity. 15 In an otherwise properly specified linear model, heteroskedasticity leads to inefficient parameter estimates and inconsistent covariance matrix estimates. White's test for heteroskedasticity was insignificant (p=.Ol) for all 62 sample firms for the CF series, suggesting that the constant variance assumption is not violated by MULT.

The possibility of serious multicollinearity problems was dismissed by inspecting the correlation matrix of independent variables. Specifically, there are 21 pairwise correlation coefficients between the seven independent variables in MULT. The grand median of the 21 median correlation coefficients was a modest .402. The only median correlation coefficient exceeding .75 was between APt-1 and RECt-I which was .816. Since we are primarily interested in the predictive ability of MULT, we believe that the levels of multicollinearity do not adversely affect MULT's predictive power.16

Autocorrelation of residuals may result in misstatement of R2 values. MULT contains lagged values of the dependent variable (e.g., CFt-, and CFt_,) as independent variables, rendering the Durbin-Watson test invalid. Pindyck and Rubinfeld (1981), among others, recommend a two- stage least-squares test in such instances. Specifically, the residual value (et) from MULT is used as the dependent variable in a new regression which employs the original variables in MULT as well as the lagged residual value (et-) as independent variables. Autocorrelation is present if the estimated coefficient of the lagged residual is significantly different from zero based on a t-test. None of these particular coefficients was significant (p=.O 1) for the MULT models estimated on the CF data for the 62 sample firms.

Finally, we assessed the incremental explanatory power of MULT versus a regression model which suppresses the accrual-based independent variables (e.g., OIBDt-1, OIBD t,4 RECt-1, PAYt and INVt-1) and simply employs lagged cash-flow variables (e.g., CFt 1 and CF t-) as independent variables. The median adjusted R2 of the pure cash-flow model was .117 while the median adjusted R2 for MULT was .316. This 270 percent increase in explanatory power may be attributed to the set of accrual-based variables employed in MULT. In the next section, we assess whether this increase in explanatory power translates into enhanced predictive ability in a holdout period.

13To ease exposition, we discuss diagnostics for the CF series. The CFPS and CFTA series provide similar results. "See Ramsey (1969) for a thorough discussion of this test. s5See White (1980) for the derivation of this test statistic.

"Future research directed at specifying directional hypotheses about the coefficients in MULT is necessary to learn more about the contextual relationships among its variables. Such work would most likely be attempted on an industry or firm- specific basis.




VI. PREDICTIVE RESULTS

One-step-ahead quarterly CF predictions were generated in an ex ante fashion for the five CF prediction models. Initially, all models were estimated using data beginning with the second quarter of 1979 and ending with the fourth quarter of 1988 to generate CF predictions for the first quarter of 1989. All models were subsequently reestimated by adding the actual first quarter CF number to the data base prior to generating the second quarter CF predictions in 1989. This process was repeated and the models were sequentially reestimated until all one-step-ahead CF predictions over the three year holdout period (1989-1991) were obtained. We computed two error metrics, mean absolute percentage error (MAPE) and mean squared error. Since the overall tenor of the results is unaffected by the choice of error metric, only the MAPE values are disclosed.

Table 4 contains the MAPE metrics for the undeflated CF series across the five CF prediction models (SAR, SMA, FS, W, MULT) for each individual quarter (1 st, 2nd, 3rd, 4th), year (1989, 1990, 1991), and on a pooled basis across all quarters and years.17 The accuracy of the CF predictions was assessed by using the Friedman ANOVA ranks test (Hollander and Wolfe 1973). For each firm, the CF prediction model yielding the smallest absolute percentage error was given a rank of one, the next smallest error was given a rank of two and so on until the model yielding the largest error was given a rank of five. Table 4 also provides the average rank of each CF prediction model, Friedman's S-statistic and its associated level of significance.

The main feature of table 4 is that there is a statistically significant difference (cx=.001) in the average ranks of the prediction models for each quarter as well as on a pooled basis. The best performing undeflated CF prediction model on the basis of the pooled MAPE metric was MULT (.556) outperforming the SMA (.654), FS (.658), W(.698) and SAR (.723) models. 18 The superior performance of the MULT model also pertains to each of the quarters and years individually. The SAR ARIMA model consistently provides the largest MAPEs. These results cast doubt upon the inter-temporal stability of the SAR ARIMA findings in LSW. Additionally, the SMA ARIMA model provides consistently smaller MAPEs than the W-model and performs as well as the FS ARIMA models.

Table 5 provides all possible, pairwise-comparisons based on the significant Friedman ANOVAs for each quarter (Panels A-D) and on a pooled basis (Panel E).'9 In all comparisons, the MULT model exhibits significantly (cx=.001) smaller ranks than the SAR, SMA, FS and W- models. The superior predictive ability of the MULT model is documented in each individual quarter as well as on a pooled basis. Such evidence clearly illustrates the dominance of the MULT model on undeflated CFs. On a pooled basis, the SMA, FS and W models all consistently outperform the SAR model which exhibited the worst performance.

Table 6 presents MAPEs for the deflated CF series: CFPS (Panel A) and CFTA (Panel B).20 The overall tenor of the results for the deflated CF predictions is quite similar to the undeflated CF results reported in considerably greater detail in tables 4 and 5.

7All forecast errors greater than 100 percent were truncated to 100 percent. Across predictions reported in table 4, the MULT model's predictions were truncated less frequently (21.4%) than those of the SAR (35.8%), SMA (35.9%), FS (36.5%) and W (47.7%) models. Additionally, we used a 300 percent truncation rule which also resulted in the fewest truncations for MULT (5.7%) compared to SAR (7.0%), SMA (11.4%), FS (11.5%) and W (23.0%). Finally, pooled untruncated median APEs for the undefeated CF predictions were: MULT (.523), SMA (.714), FS (.724), SAR (.893) and W (.933).

'8We also tested the predictive ability of the time-series regression model that simply employs CF1 and CF as the only independent variables. Its pooled MAPE for the CF series was .715 and it was significantly outperformed by MULT in each quarter, year, and on an overall basis. The superiority of MULT over this particular model is important because the differences between these two models can be attributed solely to the accrual components in MULT.

19 See Hollander and Wolfe (1973, 151-158). 20To ease exposition, the MAPEs for individual quarters as well as the detailed paired-comparisons of individual

prediction models have been suppressed. These results are available from the authors upon request.




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TABLE 5 Paired Comparisons Based on Ranks of Prediction Models

for Individual Quarters and on a Pooled Basis

Panel A: Ist Quarter

Model SMA W FS MULT (Avg Rank) (3.03) (3.24) (2.99) (2.46)

SAR FS** MULT**

(3.28)

SMA MULT**

(3.03)

W MULT** (3.24)

FS MULT**

(2.99)

Panel B: 2nd Quarter

SMA W FS MULT (2.98) (3.27) (3.09) (2.43)

SAR MULT** (3.22)

SMA SMA** MULT** (2.98)

W MULT** (3.27)

FS MULT**

(3.09)

Panel C: 3rd Quarter

SMA W FS MULT (2.94) (3.20) (3.00) (2.40)

SAR SMA** W* FS** MULT** (3.46)

SMA SMA* MULT** (2.94)

W MULT** (3.20)

FS MULT** (3.00)

(Continued)




TABLE 5 (Continued)

Panel D: 4th Quarter

SMA W FS MULT

(3.01) (3.12) (3.04) (2.32)

SAR SMA** W** FS** MULT**

(3.51)

SMA MULT**

(3.01)

W MULT**

(3.12)

FS MULT**

(3.04)

Panel E: Pooled

SMA W FS MULT

(2.99) (3.21) (3.03) (2.40)

SAR

(3.37) SMA** W** FS** MULT**

SMA

(2.99) SMA** MULT**

W

(3.21) FS** MULT**

FS

(3.03) MULT**

where: SAR = (OOO)x(100) ARIMA model.

SMA = (OOO)x(01 1) ARIMA model.

W = Wilson's multivariate cross-sectional regression model.

FS = Firm-specific, univariate ARIMA model.

MULT = Multivariate, time-series regression model. ** = Significant at .001.

* = Significant at .01. - Non-significant.

The best performing CFPS and CFTA prediction model on the basis of the pooled MAPE metric was MULT with MAPEs of .568 for the CFPS series and .612 for the CFTA series.2' The pooled MAPEs across models were significantly different for CFPS (ax=.001) and CFTA (ax=.003). Unreported paired comparisons on the pooled MAPEs indicate that the MULT model exhibits significantly (.01) smaller ranks than the SAR, SMA, FS and W-models. Although

21The pooled MAPE for the time-series regression model that simply employs CFP and CFP as independent variables was .724 for the CFPS series and .781 for the CPTA series. This model was significantly outperformed by MULT in all comparisons.




TABLE 6 Predictive Results: Deflated CF Series

Panel A: Mean Absolute Percentage Errors of One-Step-Ahead CFPS Predictions (1989-1991)

Model Year Pooled 1989 1990 1991 AvgRank MAPE

SAR .622 .663 .666 3.18 .650 SMA .634 .661 .651 3.11 .648 W .612 .627 .632 2.99 .623 FS .614 .656 .668 3.13 .644 MULT .550 .586 .568 2.59 .568

Friedman ANOVA S-Statistic 64.97

Significance Level .001

Panel B: Mean Absolute Percentage Errors of One-Step-Ahead CFTA Predictions (1989-1991)

Model Year Pooled 1989 1990 1991 AvgRank MAPE

SAR .618 .641 .653 3.03 .636

SMA .638 .676 .670 3.09 .660

W .647 .640 .609 2.98 .633

FS .612 .656 .665 3.10 .643

MULT .588 .635 .614 2.80 .612

Friedman ANOVA S-Statistic 16.39

Significance Level .003

where: CFPS = Cash flow from operations on a per-share basis CFTA = Cash flow from operations deflated by total assets All abbreviations are consistent

with table 4

paired comparisons between the ARIMA models and the W-model were insignificant at conventional levels, the W-model did have relatively smaller pooled MAPEs than the ARIMA models on the deflated CF series.

VII. CONCLUDING REMARKS

We provide evidence on the time-series properties and predictive ability of undefeated cash- flow, cash-flow per-share and cash-flow deflated by total assets, a matter of considerable importance to decision makers, policy makers and researchers in accounting. We identify a diverse set of firm-specific ARIMA models for cash-flow, however, the (O0O)x(O011) common-




structure ARIMA model performed about as well as the firm-specific models. On a cross- sectional basis, the purely seasonal autocorrelation patterns in quarterly cash-flow data originally identified by LSW are unaffected by the two deflators we tested. Such purely seasonal behavior stands in marked contrast to the well-documented time-series behavior of quarterly earnings data. Finally, the (000)x(4100) common-structure ARIMA model was significantly outperformed by all other prediction models on the cash-flow series, casting doubt upon its inter-temporal stability.

We shed additional light on the somewhat counterintuitive findings reported by LSW that simple ARIMA models (SAR and SMA) outperform multivariate cross-sectional regression models in one-step-ahead quarterly cash-flow predictions. By developing a multivariate, time- series cash-flow prediction model that employs lagged values of earnings (OIBD) and short-term accruals (REC, PAY and INV) and by estimating parameters on a firm-specific basis, we provide predictive results clearly superior to common-structure ARIMA models and the W-model. These findings are not altered by the employment of firm-specific ARIMA models or a simple, time- series regression model that employs lagged cash-flow values (CFt 1 and CF t ) as the only independent variables.

Finger (1994), among others, shows that the benefits of including additional variables in a model can sometimes be outweighed by the loss of degrees of freedom resulting in simpler models producing better forecasts. The MULT model is more complex than the annual cash-flow prediction models employed by Finger. Nevertheless, MULT provides enhanced levels of predictive ability compared to the other models we test. Finger restricts her independent variables to lagged values of annual earnings and cash-flow. In this study, we employ quarterly data which allows seasonal relationships among the variables to be considered. We also disaggregate current accruals into REC, PAY and INV to allow firm-specific parameter estimation for each subcomponent. While MULT is more complex than the models employed by Finger, it is still considerably more parsimonious than Wilson's cross-sectional regression model.

The predictive results in the current study are consistent with the views of the FASB that CF prediction is enhanced by consideration of earnings and accrual accounting data. However, due to the sampling criteria employed in data acquisition, the findings only pertain to firms that are relatively large and successful. Future research is necessary to determine the robustness of the findings to other populations of firms. Additional refinements of multivariate modeling structures on an industry-specific basis may further enhance the ability to predict CFs. Further extensions include disaggregating quarterly CFs into operating, investing and financing components in a manner similar to Livnat and Zarowin (1990) to determine the time-series properties and predictive ability of the disclosures prescribed by SFAS 95 (FASB 1987).22 It might also prove interesting to see whether the MULT CF expectation model can be used to resolve the conflict between Wilson (1987) and Bernard and Stober (1989) concerning the fundamental question of whether CFs convey information to the capital markets beyond that contained in earnings. Finally, analysis of considerably longer time-series data bases may pinpoint the impact of structural changes on the time-series properties of quarterly CF data similar to the work undertaken on quarterly earnings data by Lee and Chen (1990).

22Unfortunately, actual reported disclosures of these CF components are not presently available in sufficient quantity across time to allow time-series modeling similar to that contained herein.




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Article Contentsp.81p.82p.83p.84p.85p.86p.87p.88p.89p.90p.91p.92p.93p.94p.95p.96p.97p.98p.99p.100p.101p.102

Issue Table of ContentsThe Accounting Review, Vol. 71, No. 1 (Jan., 1996), pp. 1-136Front Matter [pp.129-130]Stewardship Value of Earnings Components: Additional Evidence on the Determinants of Executive Compensation [pp.1-22]Accounting System Management by Hospitals Operating in a Changing Regulatory Environment [pp.23-42]Auditors' Incentives and Their Application of Financial Accounting Standards [pp.43-59]U.S. Income Tax Transfer Pricing Rules for Intangibles as Approximations of Arm's Length Pricing [pp.61-80]A Multivariate Time-Series Prediction Model for Cash-Flow Data [pp.81-102]The Relation between a Prior Earnings Forecast by Management and Analyst Response to a Current Management Forecast [pp.103-115]Going Concern Opinions and the Market's Reaction to Bankruptcy Filings [pp.117-128]Book Reviewsuntitled [pp.131-132]Books Received June 1, 1995 to August 31, 1995 [p.132]

Back Matter [pp.133-136]

A multivariate try series

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