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Combining forecasts to improve earnings per share prediction
1nttmutiom.I Journal of Forecasting 3 (1987) 229-238 Node-HoB~d
229
CO~INING FORECASTS TO IMPROVEEARNINGS PER SHARE PREDICTION An Examinationof Ektric UtiMies Paul NEWBOLD and J. Kenton ZIJMWALT ~niv~rsi~ ufIi~inois,~hurnp~i~ IL ~~8~~, USA Srinivasan KANNAN Texas A&W University, CollegeStation, TX 77843, USA
Abstract:
The purpose of this study is to investigate the efficacy of combining forecasting models in order to improve earnings per shme forecasts. The utility industry is used because regulation causes the accounting procedures of the firms to be more homogen=as than other industries. Three types of forecasting models which use historical data are compared to the forecasts of the Value Line Investment Survey. It is found that predictions of the analysts of Value Line are more accurate than the predictions of the models which use only historical data. However the study also shows that forecasts of eqmings per share can be improved by combining the predictions of Value Line with the predictions of other models. §p~ifi~~ly, the forecast error is the least when the Value Line forecast is combined with the forecast of the Brown-Rozeff ARIMA model.
Ki?ywi?r&; Application, earning per share, firm: utilities, Evaluation, profit earnings ex-ante, Com-
bining forecasts.
1. Introduction The purpose of this paper is to assess the potential of combining several forecasting models to obtain better EPS forecasts than any of the individual forecasting models. Four different types of fore~ast~g models are used: (1) extrapolation of ~sto~~al growth rates, (2) an expected return model based on the Sharpe (1964)-Lintner (1965)-Mossin (1966) Capital Asset Pricing Model and the Gordon-Shapiro (1956) constant growth model, (3) the Brown-Rozeff (1979) and the Griffin (1977)-Watts (1975) time series ARIMA models, and (4) estimates of the financial analysts of the Value Line rnvest~ent Survey. After compa~ng the forecasting ability of all the models, alternative models are combined in an OLS regression in an attempt to provide more accurate forecasts. This study which focuses on the electric utility industry finds that the forecasts by Value Line are consistently better than the other individual models. This indicates that information from sources other than historical data is important in forecasting earnings. However, it is also found that the forecasts can be improved by ~ombi~ng Value Line forecasts with the forecasts of the Brown-Rozeff ARIMA model, This suggests Value Line does not efficiently incorporate a utility’s historical EPS record in generating the forecast. 0169-2070/87/$3.50 0 1987, Elsevier Science Publishers B.V. (North-Holland)
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P. Newbold et al. / Combining forecasts to improve earnings prediction
2. Research in forecasting Research into the predictability of corporate earnings generally falls into one of two categories. The first category involves efforts to model the process by which earnings are generated by comparing the forecasting abilities of alternative models. In one early paper, Lintner and Glauber (1972) described earnings per share as ‘Higgledly-Piggledly’ and suggested that the earnings series could best be described as random walk processes. Since that time, studies by Ball and Watts (1972), Albrecht et al. (1977), and Watts and Leftwich (1977), among others, have provided supporting evidence that annual accounting earnings can best be approximated by a random walk with drift or the submartingale process. ’ The second category of forecasting studies involves the comparison of forecasts by financial analysts with forecasts generated by extrapolative and/or time-series models. The evidence from tnese studies is mixed. Early studies by Cragg and Malkiel (1968) and Elton and Gruber (1972) found that analysts’ forecasts were not superior to forecasts based only on past earnings. However, two studies by Brown and Rozeff (1978,1979) indicate that Vdue Line’s financial analysts are better forecasters than individual ARIMA models and that analysts’ forecast revisions could be partially explained by using an adaptive expectations technique. The Brown-Rozeff study also indicates analysts use other information than updated EPS data in their forecast revisions. Collins and Hopwood (1980) also compared Value Line forecasts with forecasts of ARIMA models. Their conclusions supported those of Brown and Rozeff: that financial analysts can outperform the models based on past earnings information and that the forecasts of analysts more quickly adjust to reflect changing economic conditions. More recently, Rozeff (1983) concluded that expected return models based on the Capital Asset Pricing outperformed the submartingale model, but that the Value Line forecasts were significantly superior to the forecasts of the other models. In a summary article, Armstrong (1983) found that in general managements’ forecasts were more accurate than analysts’ forecasts which in turn were more accurate than extrapolative forecasts. Finally, Moyer, Chatfield and Kelley (1986) report that Value Line outperformed several extrapolative models in forecasting long-term earnings growth for electric utilities. In summary, the studies indicate that annual EPS tend to nearly approximate a random walk with drift but that financial analysts use information in addition to historical earnings or EPS in order to outperform models based on only historical data.
3. Electric utility earnings This study focuses on the electric utility industry in order to reduce the problems associated with different firms using different accounting procedures. Generally accepted accounting principles allow firms much leeway in their accounting practices. For example, differences in accounting for depreciation, inventory, R&D, etc. can cause two firms in the same industry to report substantially different earnings. When examining or comparing firms in several industries, the problem is exacerbated. As with all studies concerned with time series data, temporal changes in the structure of the earniligs generation process can result in substantial estimation error and, hence, reduce the forecasting ability of the models. During the period covered by this study the electric utility industry enctiuntered rapidly increasing ,‘uel costs, high inflation and interest rates, long construction lead ’ For a comprehensive summary of time r r.;esstudies, see Bao et al. (1983).
P. Newbold et al. / Combining forecasts to improve earn’rgs prediction
231
time, and higher construction expense. It should also be noted that the earnings of most electric utilities are quite seasonal. All of these factors underly the earnings reported by electric utilities. 4. The data The sample for this study consists of all electric and electric/gas utilities which had sufficient data in the Value Line Investment Survey and on the Compustat and CRSP tapes. The most current seven years were used as forecast periods. That is, one-year-ahead earnings per share were forecast annually for each year beginning with 1978 and ending with 1984. Because some utilities did not have complete data for all years, the number of utilities available in each year varied from 65 firms in 1978 to 73 firms in 1984. Different forecasting models have different data needs. For example, the three historical growth models required five years of annual data. The data necessary for these models are dividends per share, earnings per share, the return on equity, and the retention rate. The Sharpe-Lintner-Mossin capital asset pricing model required 15 years of monthly return data in order to estimate an adjusted beta. In addition, the risk free rate was taken as l-year constant maturity U.S. Treasury Notes and Bonds as reported in the Federal Reserve Bulletin. The expected market return was taken as the arithmetic mean of common stock returns as reported by Ibbotson and Sinquefield (1985). Only one observation per utility per year was needed from Value Line. This observation was gathered from the Value Line issue nearest December 31 for each year. The ARIMA models utilized quarterly EPS data beginning in 1962. This resulted in 64 quarterly observations per utility for the 1978 forecast, 68 observations for the 1979 forecast, and so on. 5. The models
The three extrapolative models are based on the latest five years of annual data. These models were selected because of their superior performance in the electric utility study by Moyer, Chatfield and Kelly. These historical growth models are
(1) The 5-year historical compound dividend per share growth rate. For example the 1978 growth forecast is
and the 1978 EPS forecast is (I)
EPSl, = EPs770 + 9078)*
(2) The 5-year historical compound earnings per share growth rate. This variable is handled exactly like the previous variable. The 5-year implied earnings growth. This growth rate is calculated as the product of the 5-year (3) average return on equity and the 5-year average retention rate (i.e., g = br). The Sharpe-Lintner-Mossin (SLM) expected return model was Insed by Rozeff (1983). First, the expected return is estimated as (following Rozeff) Wir)
= R,, +
[ wLJ
- &I]
Pi9
(2)
P. Newboldet al. / Combiningforecasts to improve earningsprediction
232
where R, is interest rate on a l-year constant maturity U.S. Treasury Not. d and Bonds, E( R,,) is a.&hmeticmean for common stocks as reported by Ibbotson and Sinquefieid, & is beta coefficient of stock i expected to prevail over the period. The beta has been adjusted to reflect its regression tendencies. 2 After the expected return for each security is estimated, Rozeff used the estimates with the constant growth dividend model developed by Gordon and Shapiro
DiO(l+ git) p
E(Ri*)=
(3)
gitm
and solving f@rgit gives
Rearranging
git =
+
i0
EC RitJ - DiO/piO 1 + Dio/Pio
(4)
’
The git from this formulation is then used as in eq. (1) to generate a forecast of EPSit. The time series ARIMA models used in this study are known as the Griffin-Watts and Brown-Rozeff models. The Griffin-Watts model is, in the notation of Box and Jenkins (1970), (0,1,l) (0,1,l); that is, it is assumed that earnings per share, Et, are generated by the model (1 - B)(l - B4)Et = (1 - dB)(l
- B,B4)r,,
(5)
where B is the back-shift operator, E, is assumed CObe white noise, and 8 and e4 are parameters estimated from the historical time series data. Using this model, forecasts were made for each quarter of the following year, and summed to give the annual EPS forecast. The second ARIMA model, employed by Brown and Rozeff (1979) is (1, 0, 0) (0, 1, 1); that is, earnings per share are taken to be generated by (1 - +B)(l - B4)E, = (1 - B,B4)rt, where + and e4 are fixed parameters to be estimated forecasts were summed to give the annual forecast.
(6) from the data. Again, the four quarterly
6. Measurement of forecast errors The mean squared error (MSE >was utilized as the measure of forecast error and was calculated as follows: MSE=;
k
(Pi-AJ2,
i=l
where Pi and Ai are the predicted and actual earnings, respectively, for company i and n is the number of companies. Thus, in any given year, we are averaging squared forecast errors over all utilities. In this particular case this is not unreasonable, since earnings per share levels and their * The beta adjustment is accomplished by using a multiple rootlinear model. ISera and Kannan (1985) have shown this model outperfmns the Blume (1975) adjustment procedure.
P. Newbold et al. / Combining forecasts to improve earnings prediction
233
volatilities are of roughly comparable magnitudes for our sample of firms. The MSE was calculated for each of the seven forecasting models for each year 1978 through 1984. Obviously, if actual earnings equal predicted earnings, then the MSE will be zero and the model can be used to predict exactly the earnings in the succeeding year. However, if the MSE is not zero, the reason for the error may be determined by dividing the MSE into its components of bias, inefficiency, and random error. MSE = (ij -x)2
+ (1 - b,)2S; + (1 - R;.&j,
where b, is the slope coefficient of actual earnings, A, regressed on predicted earnings, P, S#.fand Sj are the sample variances of the predicted and actual earnings respectively, and R$,A is the coefficient of determination for the regression of actual earnings on predicted earnings. The first component term, (F -x)‘, is the bias and it represents the portion of the MSE due to over- or underestimation of the mean. The inefficiency component (1 - bl)ZS,2 indicates the departure of the coefficients from the expected value of 1.0 in the regression of actual on predicted earnings. The last term, (1 - Ri A)S’, is a random error component. In addition to the MSE, the mean absolute error and median absolute error were determined for each model for each year.
7. The results The mean and range of the forecast errors for the 19781984 period for the seven forecasting models are shown in table 1. As can be seen, K&e Line (VL) consistently outperformed the competing models. VL exhibits the lov+vestaverage mean absolute error, median absolute error and mean squared error. When examined on a year-by-year basis, VL forecasts have the lowest MSE in five of the seven years and the lowest mean and median absolute errors in four of the years. 3 The sources of the forecasting errors are dominated by the random error component which is frequently a factor of 10 or greater than the bias and inefficiency components. The mean VL random error component is the lowest of the seven models. The BR model has the second lowest average random error term followed by the Sharpe-Lintner model. The Griffin-Watts and earnings per share models have the highest random error components. The BR and VL models exhibit the lowest inefficiency components while the GW and EPS models exhibit the highest inefficiency terms. While being the two best performers with respect to random error and inefficiency, the BR and VL models are the two worst performers with respect to the bias component. In summary, the Vdue Line forecasts are more accurate as measured by the MSE because of its low random error and inefficiency terms. The VL forecasts also dominate when the forecast errors are measured by the mean absolute error and median absolute error. Over the long run the greater forecast accuracy of Value Line relative to the other models would be economically significant for institutional investors. Not only is the forecast error less for the VL 3 The Brown-Rozeff (BR) model performed well in the first three years, having the second lowest MSE in 1978 and the lowest MSEs in 1979 and 1980. After 1980, the BR model is never above the fifth lowest MSE. Conversely, the Sharpe-Lintner (SL) expected return model performs poorly in the first three years but exhibits the second lowest MSE each of the last four years. The dividend growth (DG) model appears to exhibit thg third lowest MSEs over the seven year period. This model has the third lowest MSE for three of the years and fourth and fifth best twice each. Interestingly, the earnings per share growth (EG) model is the worst performer of all the models. (The year-by-year results may be obtained from the authors.)
P. Newbold et al. / CLmbining forecasts to improve earnings prediction
234
Table 1 Average and range of forecast errors for alternative forecasting models (7-year average: 1978-1984). Dividend growth
Earnings growth
Implied growth
SharpeLintner
BrownRozeff
GriffinWatts
Value Line
0.350 0.389 0.279
0.322 0.357 0.252
0.329 0.394 0.299
0.369 0.504 0.245
0.360 0.429 0.320
0.294 0.364 0.245
0.352 0.313 0.186
0.245 0.304 0.165
0.240 0.329 0.192
0.296 0.403 0.214
0.257 0.308 0.222
0.226 0.280 0.160
0.242 0.402 0.134
0.211 0.372 0.116
0.212 0.321 0.157
0.246 0.100
0.267 Q.474 0.179
0.156 0.273 0.103
0.016 0.061 0.001
0.018 0.057 0.000
0.020 0.047 0.001
0.070 0.222 0.007
0.023 0.097 0.000
0.020 0.028 0.003
0.027 0.071 0.008
0.011 0.401 0.001
0.017 0.060 0.004
0.002 0.008 0.000
0.026 0.053 0.005
0.003 0.008 0.000
0.183 0.365 0.110
0.175 0.310 0.112
0.173 0.430 0.088
0.220 0.455 0.125
0.132 0.243 0.084
Mean absolute error Mean High Low
0.321 0.357 0.261
Median absolute error Mean High Low
0.246 0.321 0.163
Mean squared error Mean High LOW
0.208 0.378 0.125
0.444
MSE component: Bias Mean High Low
0.011 0.034 0.000
MSE component: heffit iency Mean
High Low
0.016 0.050 0.006
MSE component: Random error Mean High Low
0.182 0.359 0.114
0.200 0.369 0.120
forecasts, but the range of forecast errors is smaller economically valuable for institutional investors.
also. This reduced
error range is also
e combin The final step in this study involves combining the best forecasting model, Value Line, with other models in order to improve the ability to forecast earnings per share for electric utilities. A regression model is developed using actual earnings per share as the dependent variable and the forecasts of Value Line, the SL expected return model, the historical dividend growth model, and the Brown-Rozeff ARIMA model as the independent variables. 4 It is expected that the addition of these models will result in a model which will result in better EPS forecasts. The MSE is used to determine the accuracy of the forecasts. 4 Since the dividend grostith, earnings growth, and implied growth models are rather similar. we chose to include only one of them in the combining exercise. Similarly, only one of the two time series models was employed.
P. Newbold et al. / Combining forecasts to improve earnings prediction
235
Table 2 Regression estimation of combined models. a
Year
Intercept
Dividend growth
SharpeLintner
BrownRozeff
Value Line
Adjusted R2
Number observations
1978
-0.184 ( - 0.999) - 0.210 (- 1.138) - 0.175 (0.973) - 0.125 ( - 0.827) - 0.137 ( - 0.917) - 0.120 ( - 0.816) 0.121 (0.652) 0.133 (0.718) 0.2178 (1.2222) 0.296 (1.773) 0.310 (1.852) - 0.127 ( - 0.695) - 0.108 ( - 0.612) - 0.130 ( - 0.719) 0.039 (0.216) 0.160 (1.006)
0.944 (1.047) - 0.202 ( - 0.902)
- 1.118 (- 1.312)
0.653 (2.203) 0.645 (2.262) 0.449 (2.437) 0.573 (3.074) 0.557 (3.025) 0.485 (3.255) 0.709 (2.482) G.671 (2.387) 0.3508 (1.9020) 0.362 (1.506) 0.269 (1.161) 0.633 (2.421) 0.657 (2.578) 0.340 (1.700) 0.090 (0.402) 0.095 (0.421)
0.650 (3.927) 0.641 (3.852) 0.610 (3.753) 0.638 (3.611) 0.609 (3.562) 0.555 (3.691) 0.634 (3.206) 0.649 (3.307) 0.5386 (2.9320) 0.384 (2.511) 0.430 (2.867) 0.896 (4.250) 0.876 (4.279) 0.750 (3.783) 0.901 (4.468) 0.814 (4.233)
0.771
65
1979
1980
1981
1982
1983
0.326 (0.494) - 0.116 ( - 0.672)
- 0.470 ( - 0.812)
0.929 (1.688) 0.254 (1.197) - 0.248 ( - 0.450)
- 0.876 (-1.351)
-0.467 ( - 0.694)
0.072 (0.124) - 0.360 (- 1.498)
- 0.693 (- 1.327)
- 0.152 (-0.273) - 0.412 ( - 1.945)
0.934 (1.442) 0.102 (0.505)
0.768 0.769 0.840
65
0.841 0.843 0.731
66
0.733 0.7276 0.790
68
0.787 0.829
68
0.831 0.824 0.858
71
0.856
a ?gures in parentheses beneath parameter estimates are the associated t-ratios.
The procedure employed for determining the weights of the individual forecasts in the combination is as follows. For any year, regress over all utilities actual earnings per share on the set of predictions made in the previous year. In principle, the resulting linear combination of forecasts can then be used to predict earnings per share of all utilities in the next year. However, as indicated by the results in table 2, such a strategy would invariably imply the assignment of negative weights to some constituent forecasts. We feel that this is hard to justify as a practical prescription, and contrary to the spirit of forecast combination. Accordingly, we proceeded by dropping from the regression that individual forecast with the largest associated negative t-ratio, re-estimating the regression, and continuing in this way until a regression is achieved in which all remaining forecasts have positive weights. Table 2 shows the full derivation, in this manner, of our six combination models, based on data for the years 19784983. Notice that some variable other than the Value Line forecast often appears in these regressions with an estimated coefficient that is statistically significant at the usual levels.
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P. Newbold et al. / Combining forecasts to improve earnings prediction
Table 3 Average mean squared error and MSE components
for Value Line and combined forecasts (averages for 1979-1984 period).
Unwei.ghted average forecasts
Value Line forecasts
Combined forecasts
Mean Squared Error (MSE)
0.196
0.161
0.153
Portion of MSE due to Bias Inefficiency Random., Error
0.01 Q O.G97 0.154
0.020 0.003 0.138
0.015 0.003 0.134
Accordingly, in the terminology of Granger and Newbold (1973), we conclude that, although the Value Line forecasts are the best of the individual predictions, they do not appear to be conditionally efficient with respect to the others. The weights given in the individual regressions to the Value Line forecasts differ somewhat from year to year. However, when related to their estimated standard errors these differences in weights are generally not statistically significant at the usual levels. For example, in comparing, from table 2, the estimated Value Line weights for 1981 and 1983 - where there is the largest numerical difference in the point estimates - the usual t statistic for testing equality of the population parameters is 1.575, so that the hypothesis of equality cannot be rejected at the 10% level. For our population of utilities, table 3 sets out the mean squared error characteristics of the combined forecasts for the six years 1979-1984. For every comparison, the corresponding figures for the Value Line forecasts are also shown. It can be seen that, in terms of mean squared error, the combined forecast outperforms the Value Line forecast in four years out of six. In total in table 3, information is reported on the prediction of 411 earnings per share observations. Over all these observaGons, the averages of the squared forecast errors were 0.161 for Value Line and 0.153 for the combined fcrecasts. Therefore, if a combined forecasting model rather than Value Line had been used for forecasting over this period, then squared forecast errors would have been reduced on average by 4.97 percent. Error reduction of this size could be economically significant to institutional investors. Returning to table 2, it can be seen that in every year the Brown-Rozeff forecast is included with positive weight in the combination with Value Line. Moreover, in four years these two forecasts are the only constituents of the combination. We might therefore ask about thn merits of the predictor that combines just these two forecasts. Re-estimating the 1981 and 1983 regressions, we found for 1982 and 1984 mean squared errors of 0.14096 and 0.25374 for this predictor. Hence, Value Line and Brown-Rozeff in combination outperform Value Line alone in five of the six years. Over all 411 observations, the average of the squared errors for this predictor was 0.15287. This represents, on average, a decrease of 5.95 percent in squared error compared with the Value Line forecasts. The gains in forecast accuracy achieved through combining forecasts suggests the possibility for modest improvements in the Value Line forecasts. Moreover, the fact that it is the Brown-Rozeff time series modrl that appears to be the major source of these gains argues that the Value Line forecasts do not efficiently incorporate the information in the historical records of the utilities’ earnings per share. Following Clemen and Winkler (1986), we also include in table 3 results for the combined forecast that is simply the unweighted average of the individual forecasts. As can be seen, this performs quite poorly in comparison with Value Line. In retrospect, t situations such as the present when the quality of a set of forecasts is domina members.
P. Newbold et al. / Combining forecasts to improve earnings prediction
237
9. Conclusion The paper has examined the efficacy of combining earnings per share forecasts from several forecasting models in order to improve overall forecast accuracy. The study first compared the forecast errors of the Value Line Investment Suruey and three types of models which used historical data. The results were consistent with previous studies in that the Value Line forecasts were more accurate than the other models. These results support the previous research that contends financial ar,nlvsts react more quickly to changmg economic events than can the other models. This study indicates, however, that Value Line does not make use of all the information inherent in historical earnings per share. Specifically, forecast errors are reduced when the forecasts of Value Line are combined with the Brt;wn-Rozeff ARIMA model. Future research must consider other industries and other time periods to further assess the effectiveness of combining earnings per share forecasting models.
References AIbrecht, W.S., L.L. LookabiII and J.C. McKeown, 1977, The time series properties of arm& earnings, Journal of Accounting Research, 15226-244. Armstrong, J.S., 1983, Relative accuraby of judgemental and extrapolative methods in forecasting annual earnings, Jam-d of Forecasting 2,437-447. Ball, R. and R. Watts, 1972, Some time series properties of accounting income _,umbers, Journal -C Finance 27.663681. Baa, D-H., M.T. Lewis, W.T. Lin and J.G. Manegold, 1983, Applications of time-series anaIys. : accounting: A review, JoumaI of Forecasting 2,405-423. Bera, A.K. and S. Kannan, 1985, An adjustment procedure for predicting systematic risk, Working paper no. 1184 (University of Illinois, Urbana, IL). Blume, M.E., 1975, Betas and their regression tendencies, Journal of Finance 30, 785-: Box, G.E.P. and G.M. Jenkins, 1970, Time series analysis, forecasting and control (Holden Day, San Francisco, CA). Brown, L. and M. Rozeff, 1978, The superiority of analyst forecasts as measures of expectations: Evidence from earnings, Journal of Finance 33, 1-16. Brown, L.D. and M.S. Rozeff, 1979a, Adaptive expectations, time-series models, and analyst forecast revision, Journal of Accounting Research 17, 341-351. nodels of quarterly accounting earnings per share: A proposed Brown, L.D. and M.S. Rozeff, 1979b, Univariate time-s model, Journal of Accounting Research 17,179-189. Clemen, R.T. and R.L. WinkIer, 1986, Combining economic forecasts, Journal of Business and Economic Statistics 4, 39-46. Collins, W.A. and W.S. Hopwood, 1980, A multivatiate analysis of annual earnings forecasts generated from quarterly forecasts of financial analysts and univariate time-series models, Journal of Accounting Research 18, 390-406. Cragg, J.G. and B.G. Malkiel, 1968, The consensus and accuracy of some redictions of the growth of corporate ea.rings, Journal of Finance 23,67-84. Elton, E.J. and M.J. Gruber, 1972, Earnings estimates and the accuracy of expectational data, Management Science, 19, 409-424. Foster, G., 1977, Quarterly accouting data, time-series properties and predictive-ability results, The Accounting Review 52, 1-21. Gordon, M.J. and E. Shapiro, 1967, Capital equipment analysis: The required rate of profit, Management Science 3, 102-110. Granger, C.W.J. and P. Newbold, 1973, Some comments on the evaluation of economic forecasts, Applied Economics 5. 35-47. Griffin, P.A., 1977, The time-series behavior of quarterly earnings: Preliminary evidence, Journal of Accounting Research 15, 71-83. Ibbotson, R.G. and R.A. Sinquefield, 1985, Stocks, bonds, bills and inflation, 1984 yearbook (lbbotson Association). Lintner, J., 1965, The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets9 Review of Economics and Statistics 47, 13-37. iggledy-piggledy growth in America, in: J. Lorie and R. Erealey, eds., Modern Lintner, J. and R. Glauber, 1972, developments in investment management (Praeger, New York).
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Biography:
Paul NEWBOLD is Professor of Economics, University of Illinois, Champaign. Hz is co-author (with C.W.J. Granger) of Forecasting Economic Time Series. He has published artices in Journal of American Statistical Association, Journal of Royal Statistical Society, Biometrika, Journal of Economics, International Economic Review, Journal of Accounting Research, Accounting Review, and Journal of Business. J. Kenton ZUMWALT is Associate Professor of Finance at the University of Illinois at Urbana-Champaign. Among his publications are articles in the Journal of Finance, Journal of Financial and Quantitative Analysis, Financial Management, the Journal of Portfolio Management, and Management Science. Srinivasan KANNAN is Assistant Professor of Finance at the Department College of Business Administration, Texas A & M University.
of Finance,
229
CO~INING FORECASTS TO IMPROVEEARNINGS PER SHARE PREDICTION An Examinationof Ektric UtiMies Paul NEWBOLD and J. Kenton ZIJMWALT ~niv~rsi~ ufIi~inois,~hurnp~i~ IL ~~8~~, USA Srinivasan KANNAN Texas A&W University, CollegeStation, TX 77843, USA
Abstract:
The purpose of this study is to investigate the efficacy of combining forecasting models in order to improve earnings per shme forecasts. The utility industry is used because regulation causes the accounting procedures of the firms to be more homogen=as than other industries. Three types of forecasting models which use historical data are compared to the forecasts of the Value Line Investment Survey. It is found that predictions of the analysts of Value Line are more accurate than the predictions of the models which use only historical data. However the study also shows that forecasts of eqmings per share can be improved by combining the predictions of Value Line with the predictions of other models. §p~ifi~~ly, the forecast error is the least when the Value Line forecast is combined with the forecast of the Brown-Rozeff ARIMA model.
Ki?ywi?r&; Application, earning per share, firm: utilities, Evaluation, profit earnings ex-ante, Com-
bining forecasts.
1. Introduction The purpose of this paper is to assess the potential of combining several forecasting models to obtain better EPS forecasts than any of the individual forecasting models. Four different types of fore~ast~g models are used: (1) extrapolation of ~sto~~al growth rates, (2) an expected return model based on the Sharpe (1964)-Lintner (1965)-Mossin (1966) Capital Asset Pricing Model and the Gordon-Shapiro (1956) constant growth model, (3) the Brown-Rozeff (1979) and the Griffin (1977)-Watts (1975) time series ARIMA models, and (4) estimates of the financial analysts of the Value Line rnvest~ent Survey. After compa~ng the forecasting ability of all the models, alternative models are combined in an OLS regression in an attempt to provide more accurate forecasts. This study which focuses on the electric utility industry finds that the forecasts by Value Line are consistently better than the other individual models. This indicates that information from sources other than historical data is important in forecasting earnings. However, it is also found that the forecasts can be improved by ~ombi~ng Value Line forecasts with the forecasts of the Brown-Rozeff ARIMA model, This suggests Value Line does not efficiently incorporate a utility’s historical EPS record in generating the forecast. 0169-2070/87/$3.50 0 1987, Elsevier Science Publishers B.V. (North-Holland)
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2. Research in forecasting Research into the predictability of corporate earnings generally falls into one of two categories. The first category involves efforts to model the process by which earnings are generated by comparing the forecasting abilities of alternative models. In one early paper, Lintner and Glauber (1972) described earnings per share as ‘Higgledly-Piggledly’ and suggested that the earnings series could best be described as random walk processes. Since that time, studies by Ball and Watts (1972), Albrecht et al. (1977), and Watts and Leftwich (1977), among others, have provided supporting evidence that annual accounting earnings can best be approximated by a random walk with drift or the submartingale process. ’ The second category of forecasting studies involves the comparison of forecasts by financial analysts with forecasts generated by extrapolative and/or time-series models. The evidence from tnese studies is mixed. Early studies by Cragg and Malkiel (1968) and Elton and Gruber (1972) found that analysts’ forecasts were not superior to forecasts based only on past earnings. However, two studies by Brown and Rozeff (1978,1979) indicate that Vdue Line’s financial analysts are better forecasters than individual ARIMA models and that analysts’ forecast revisions could be partially explained by using an adaptive expectations technique. The Brown-Rozeff study also indicates analysts use other information than updated EPS data in their forecast revisions. Collins and Hopwood (1980) also compared Value Line forecasts with forecasts of ARIMA models. Their conclusions supported those of Brown and Rozeff: that financial analysts can outperform the models based on past earnings information and that the forecasts of analysts more quickly adjust to reflect changing economic conditions. More recently, Rozeff (1983) concluded that expected return models based on the Capital Asset Pricing outperformed the submartingale model, but that the Value Line forecasts were significantly superior to the forecasts of the other models. In a summary article, Armstrong (1983) found that in general managements’ forecasts were more accurate than analysts’ forecasts which in turn were more accurate than extrapolative forecasts. Finally, Moyer, Chatfield and Kelley (1986) report that Value Line outperformed several extrapolative models in forecasting long-term earnings growth for electric utilities. In summary, the studies indicate that annual EPS tend to nearly approximate a random walk with drift but that financial analysts use information in addition to historical earnings or EPS in order to outperform models based on only historical data.
3. Electric utility earnings This study focuses on the electric utility industry in order to reduce the problems associated with different firms using different accounting procedures. Generally accepted accounting principles allow firms much leeway in their accounting practices. For example, differences in accounting for depreciation, inventory, R&D, etc. can cause two firms in the same industry to report substantially different earnings. When examining or comparing firms in several industries, the problem is exacerbated. As with all studies concerned with time series data, temporal changes in the structure of the earniligs generation process can result in substantial estimation error and, hence, reduce the forecasting ability of the models. During the period covered by this study the electric utility industry enctiuntered rapidly increasing ,‘uel costs, high inflation and interest rates, long construction lead ’ For a comprehensive summary of time r r.;esstudies, see Bao et al. (1983).
P. Newbold et al. / Combining forecasts to improve earn’rgs prediction
231
time, and higher construction expense. It should also be noted that the earnings of most electric utilities are quite seasonal. All of these factors underly the earnings reported by electric utilities. 4. The data The sample for this study consists of all electric and electric/gas utilities which had sufficient data in the Value Line Investment Survey and on the Compustat and CRSP tapes. The most current seven years were used as forecast periods. That is, one-year-ahead earnings per share were forecast annually for each year beginning with 1978 and ending with 1984. Because some utilities did not have complete data for all years, the number of utilities available in each year varied from 65 firms in 1978 to 73 firms in 1984. Different forecasting models have different data needs. For example, the three historical growth models required five years of annual data. The data necessary for these models are dividends per share, earnings per share, the return on equity, and the retention rate. The Sharpe-Lintner-Mossin capital asset pricing model required 15 years of monthly return data in order to estimate an adjusted beta. In addition, the risk free rate was taken as l-year constant maturity U.S. Treasury Notes and Bonds as reported in the Federal Reserve Bulletin. The expected market return was taken as the arithmetic mean of common stock returns as reported by Ibbotson and Sinquefield (1985). Only one observation per utility per year was needed from Value Line. This observation was gathered from the Value Line issue nearest December 31 for each year. The ARIMA models utilized quarterly EPS data beginning in 1962. This resulted in 64 quarterly observations per utility for the 1978 forecast, 68 observations for the 1979 forecast, and so on. 5. The models
The three extrapolative models are based on the latest five years of annual data. These models were selected because of their superior performance in the electric utility study by Moyer, Chatfield and Kelly. These historical growth models are
(1) The 5-year historical compound dividend per share growth rate. For example the 1978 growth forecast is
and the 1978 EPS forecast is (I)
EPSl, = EPs770 + 9078)*
(2) The 5-year historical compound earnings per share growth rate. This variable is handled exactly like the previous variable. The 5-year implied earnings growth. This growth rate is calculated as the product of the 5-year (3) average return on equity and the 5-year average retention rate (i.e., g = br). The Sharpe-Lintner-Mossin (SLM) expected return model was Insed by Rozeff (1983). First, the expected return is estimated as (following Rozeff) Wir)
= R,, +
[ wLJ
- &I]
Pi9
(2)
P. Newboldet al. / Combiningforecasts to improve earningsprediction
232
where R, is interest rate on a l-year constant maturity U.S. Treasury Not. d and Bonds, E( R,,) is a.&hmeticmean for common stocks as reported by Ibbotson and Sinquefieid, & is beta coefficient of stock i expected to prevail over the period. The beta has been adjusted to reflect its regression tendencies. 2 After the expected return for each security is estimated, Rozeff used the estimates with the constant growth dividend model developed by Gordon and Shapiro
DiO(l+ git) p
E(Ri*)=
(3)
gitm
and solving f@rgit gives
Rearranging
git =
+
i0
EC RitJ - DiO/piO 1 + Dio/Pio
(4)
’
The git from this formulation is then used as in eq. (1) to generate a forecast of EPSit. The time series ARIMA models used in this study are known as the Griffin-Watts and Brown-Rozeff models. The Griffin-Watts model is, in the notation of Box and Jenkins (1970), (0,1,l) (0,1,l); that is, it is assumed that earnings per share, Et, are generated by the model (1 - B)(l - B4)Et = (1 - dB)(l
- B,B4)r,,
(5)
where B is the back-shift operator, E, is assumed CObe white noise, and 8 and e4 are parameters estimated from the historical time series data. Using this model, forecasts were made for each quarter of the following year, and summed to give the annual EPS forecast. The second ARIMA model, employed by Brown and Rozeff (1979) is (1, 0, 0) (0, 1, 1); that is, earnings per share are taken to be generated by (1 - +B)(l - B4)E, = (1 - B,B4)rt, where + and e4 are fixed parameters to be estimated forecasts were summed to give the annual forecast.
(6) from the data. Again, the four quarterly
6. Measurement of forecast errors The mean squared error (MSE >was utilized as the measure of forecast error and was calculated as follows: MSE=;
k
(Pi-AJ2,
i=l
where Pi and Ai are the predicted and actual earnings, respectively, for company i and n is the number of companies. Thus, in any given year, we are averaging squared forecast errors over all utilities. In this particular case this is not unreasonable, since earnings per share levels and their * The beta adjustment is accomplished by using a multiple rootlinear model. ISera and Kannan (1985) have shown this model outperfmns the Blume (1975) adjustment procedure.
P. Newbold et al. / Combining forecasts to improve earnings prediction
233
volatilities are of roughly comparable magnitudes for our sample of firms. The MSE was calculated for each of the seven forecasting models for each year 1978 through 1984. Obviously, if actual earnings equal predicted earnings, then the MSE will be zero and the model can be used to predict exactly the earnings in the succeeding year. However, if the MSE is not zero, the reason for the error may be determined by dividing the MSE into its components of bias, inefficiency, and random error. MSE = (ij -x)2
+ (1 - b,)2S; + (1 - R;.&j,
where b, is the slope coefficient of actual earnings, A, regressed on predicted earnings, P, S#.fand Sj are the sample variances of the predicted and actual earnings respectively, and R$,A is the coefficient of determination for the regression of actual earnings on predicted earnings. The first component term, (F -x)‘, is the bias and it represents the portion of the MSE due to over- or underestimation of the mean. The inefficiency component (1 - bl)ZS,2 indicates the departure of the coefficients from the expected value of 1.0 in the regression of actual on predicted earnings. The last term, (1 - Ri A)S’, is a random error component. In addition to the MSE, the mean absolute error and median absolute error were determined for each model for each year.
7. The results The mean and range of the forecast errors for the 19781984 period for the seven forecasting models are shown in table 1. As can be seen, K&e Line (VL) consistently outperformed the competing models. VL exhibits the lov+vestaverage mean absolute error, median absolute error and mean squared error. When examined on a year-by-year basis, VL forecasts have the lowest MSE in five of the seven years and the lowest mean and median absolute errors in four of the years. 3 The sources of the forecasting errors are dominated by the random error component which is frequently a factor of 10 or greater than the bias and inefficiency components. The mean VL random error component is the lowest of the seven models. The BR model has the second lowest average random error term followed by the Sharpe-Lintner model. The Griffin-Watts and earnings per share models have the highest random error components. The BR and VL models exhibit the lowest inefficiency components while the GW and EPS models exhibit the highest inefficiency terms. While being the two best performers with respect to random error and inefficiency, the BR and VL models are the two worst performers with respect to the bias component. In summary, the Vdue Line forecasts are more accurate as measured by the MSE because of its low random error and inefficiency terms. The VL forecasts also dominate when the forecast errors are measured by the mean absolute error and median absolute error. Over the long run the greater forecast accuracy of Value Line relative to the other models would be economically significant for institutional investors. Not only is the forecast error less for the VL 3 The Brown-Rozeff (BR) model performed well in the first three years, having the second lowest MSE in 1978 and the lowest MSEs in 1979 and 1980. After 1980, the BR model is never above the fifth lowest MSE. Conversely, the Sharpe-Lintner (SL) expected return model performs poorly in the first three years but exhibits the second lowest MSE each of the last four years. The dividend growth (DG) model appears to exhibit thg third lowest MSEs over the seven year period. This model has the third lowest MSE for three of the years and fourth and fifth best twice each. Interestingly, the earnings per share growth (EG) model is the worst performer of all the models. (The year-by-year results may be obtained from the authors.)
P. Newbold et al. / CLmbining forecasts to improve earnings prediction
234
Table 1 Average and range of forecast errors for alternative forecasting models (7-year average: 1978-1984). Dividend growth
Earnings growth
Implied growth
SharpeLintner
BrownRozeff
GriffinWatts
Value Line
0.350 0.389 0.279
0.322 0.357 0.252
0.329 0.394 0.299
0.369 0.504 0.245
0.360 0.429 0.320
0.294 0.364 0.245
0.352 0.313 0.186
0.245 0.304 0.165
0.240 0.329 0.192
0.296 0.403 0.214
0.257 0.308 0.222
0.226 0.280 0.160
0.242 0.402 0.134
0.211 0.372 0.116
0.212 0.321 0.157
0.246 0.100
0.267 Q.474 0.179
0.156 0.273 0.103
0.016 0.061 0.001
0.018 0.057 0.000
0.020 0.047 0.001
0.070 0.222 0.007
0.023 0.097 0.000
0.020 0.028 0.003
0.027 0.071 0.008
0.011 0.401 0.001
0.017 0.060 0.004
0.002 0.008 0.000
0.026 0.053 0.005
0.003 0.008 0.000
0.183 0.365 0.110
0.175 0.310 0.112
0.173 0.430 0.088
0.220 0.455 0.125
0.132 0.243 0.084
Mean absolute error Mean High Low
0.321 0.357 0.261
Median absolute error Mean High Low
0.246 0.321 0.163
Mean squared error Mean High LOW
0.208 0.378 0.125
0.444
MSE component: Bias Mean High Low
0.011 0.034 0.000
MSE component: heffit iency Mean
High Low
0.016 0.050 0.006
MSE component: Random error Mean High Low
0.182 0.359 0.114
0.200 0.369 0.120
forecasts, but the range of forecast errors is smaller economically valuable for institutional investors.
also. This reduced
error range is also
e combin The final step in this study involves combining the best forecasting model, Value Line, with other models in order to improve the ability to forecast earnings per share for electric utilities. A regression model is developed using actual earnings per share as the dependent variable and the forecasts of Value Line, the SL expected return model, the historical dividend growth model, and the Brown-Rozeff ARIMA model as the independent variables. 4 It is expected that the addition of these models will result in a model which will result in better EPS forecasts. The MSE is used to determine the accuracy of the forecasts. 4 Since the dividend grostith, earnings growth, and implied growth models are rather similar. we chose to include only one of them in the combining exercise. Similarly, only one of the two time series models was employed.
P. Newbold et al. / Combining forecasts to improve earnings prediction
235
Table 2 Regression estimation of combined models. a
Year
Intercept
Dividend growth
SharpeLintner
BrownRozeff
Value Line
Adjusted R2
Number observations
1978
-0.184 ( - 0.999) - 0.210 (- 1.138) - 0.175 (0.973) - 0.125 ( - 0.827) - 0.137 ( - 0.917) - 0.120 ( - 0.816) 0.121 (0.652) 0.133 (0.718) 0.2178 (1.2222) 0.296 (1.773) 0.310 (1.852) - 0.127 ( - 0.695) - 0.108 ( - 0.612) - 0.130 ( - 0.719) 0.039 (0.216) 0.160 (1.006)
0.944 (1.047) - 0.202 ( - 0.902)
- 1.118 (- 1.312)
0.653 (2.203) 0.645 (2.262) 0.449 (2.437) 0.573 (3.074) 0.557 (3.025) 0.485 (3.255) 0.709 (2.482) G.671 (2.387) 0.3508 (1.9020) 0.362 (1.506) 0.269 (1.161) 0.633 (2.421) 0.657 (2.578) 0.340 (1.700) 0.090 (0.402) 0.095 (0.421)
0.650 (3.927) 0.641 (3.852) 0.610 (3.753) 0.638 (3.611) 0.609 (3.562) 0.555 (3.691) 0.634 (3.206) 0.649 (3.307) 0.5386 (2.9320) 0.384 (2.511) 0.430 (2.867) 0.896 (4.250) 0.876 (4.279) 0.750 (3.783) 0.901 (4.468) 0.814 (4.233)
0.771
65
1979
1980
1981
1982
1983
0.326 (0.494) - 0.116 ( - 0.672)
- 0.470 ( - 0.812)
0.929 (1.688) 0.254 (1.197) - 0.248 ( - 0.450)
- 0.876 (-1.351)
-0.467 ( - 0.694)
0.072 (0.124) - 0.360 (- 1.498)
- 0.693 (- 1.327)
- 0.152 (-0.273) - 0.412 ( - 1.945)
0.934 (1.442) 0.102 (0.505)
0.768 0.769 0.840
65
0.841 0.843 0.731
66
0.733 0.7276 0.790
68
0.787 0.829
68
0.831 0.824 0.858
71
0.856
a ?gures in parentheses beneath parameter estimates are the associated t-ratios.
The procedure employed for determining the weights of the individual forecasts in the combination is as follows. For any year, regress over all utilities actual earnings per share on the set of predictions made in the previous year. In principle, the resulting linear combination of forecasts can then be used to predict earnings per share of all utilities in the next year. However, as indicated by the results in table 2, such a strategy would invariably imply the assignment of negative weights to some constituent forecasts. We feel that this is hard to justify as a practical prescription, and contrary to the spirit of forecast combination. Accordingly, we proceeded by dropping from the regression that individual forecast with the largest associated negative t-ratio, re-estimating the regression, and continuing in this way until a regression is achieved in which all remaining forecasts have positive weights. Table 2 shows the full derivation, in this manner, of our six combination models, based on data for the years 19784983. Notice that some variable other than the Value Line forecast often appears in these regressions with an estimated coefficient that is statistically significant at the usual levels.
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P. Newbold et al. / Combining forecasts to improve earnings prediction
Table 3 Average mean squared error and MSE components
for Value Line and combined forecasts (averages for 1979-1984 period).
Unwei.ghted average forecasts
Value Line forecasts
Combined forecasts
Mean Squared Error (MSE)
0.196
0.161
0.153
Portion of MSE due to Bias Inefficiency Random., Error
0.01 Q O.G97 0.154
0.020 0.003 0.138
0.015 0.003 0.134
Accordingly, in the terminology of Granger and Newbold (1973), we conclude that, although the Value Line forecasts are the best of the individual predictions, they do not appear to be conditionally efficient with respect to the others. The weights given in the individual regressions to the Value Line forecasts differ somewhat from year to year. However, when related to their estimated standard errors these differences in weights are generally not statistically significant at the usual levels. For example, in comparing, from table 2, the estimated Value Line weights for 1981 and 1983 - where there is the largest numerical difference in the point estimates - the usual t statistic for testing equality of the population parameters is 1.575, so that the hypothesis of equality cannot be rejected at the 10% level. For our population of utilities, table 3 sets out the mean squared error characteristics of the combined forecasts for the six years 1979-1984. For every comparison, the corresponding figures for the Value Line forecasts are also shown. It can be seen that, in terms of mean squared error, the combined forecast outperforms the Value Line forecast in four years out of six. In total in table 3, information is reported on the prediction of 411 earnings per share observations. Over all these observaGons, the averages of the squared forecast errors were 0.161 for Value Line and 0.153 for the combined fcrecasts. Therefore, if a combined forecasting model rather than Value Line had been used for forecasting over this period, then squared forecast errors would have been reduced on average by 4.97 percent. Error reduction of this size could be economically significant to institutional investors. Returning to table 2, it can be seen that in every year the Brown-Rozeff forecast is included with positive weight in the combination with Value Line. Moreover, in four years these two forecasts are the only constituents of the combination. We might therefore ask about thn merits of the predictor that combines just these two forecasts. Re-estimating the 1981 and 1983 regressions, we found for 1982 and 1984 mean squared errors of 0.14096 and 0.25374 for this predictor. Hence, Value Line and Brown-Rozeff in combination outperform Value Line alone in five of the six years. Over all 411 observations, the average of the squared errors for this predictor was 0.15287. This represents, on average, a decrease of 5.95 percent in squared error compared with the Value Line forecasts. The gains in forecast accuracy achieved through combining forecasts suggests the possibility for modest improvements in the Value Line forecasts. Moreover, the fact that it is the Brown-Rozeff time series modrl that appears to be the major source of these gains argues that the Value Line forecasts do not efficiently incorporate the information in the historical records of the utilities’ earnings per share. Following Clemen and Winkler (1986), we also include in table 3 results for the combined forecast that is simply the unweighted average of the individual forecasts. As can be seen, this performs quite poorly in comparison with Value Line. In retrospect, t situations such as the present when the quality of a set of forecasts is domina members.
P. Newbold et al. / Combining forecasts to improve earnings prediction
237
9. Conclusion The paper has examined the efficacy of combining earnings per share forecasts from several forecasting models in order to improve overall forecast accuracy. The study first compared the forecast errors of the Value Line Investment Suruey and three types of models which used historical data. The results were consistent with previous studies in that the Value Line forecasts were more accurate than the other models. These results support the previous research that contends financial ar,nlvsts react more quickly to changmg economic events than can the other models. This study indicates, however, that Value Line does not make use of all the information inherent in historical earnings per share. Specifically, forecast errors are reduced when the forecasts of Value Line are combined with the Brt;wn-Rozeff ARIMA model. Future research must consider other industries and other time periods to further assess the effectiveness of combining earnings per share forecasting models.
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Biography:
Paul NEWBOLD is Professor of Economics, University of Illinois, Champaign. Hz is co-author (with C.W.J. Granger) of Forecasting Economic Time Series. He has published artices in Journal of American Statistical Association, Journal of Royal Statistical Society, Biometrika, Journal of Economics, International Economic Review, Journal of Accounting Research, Accounting Review, and Journal of Business. J. Kenton ZUMWALT is Associate Professor of Finance at the University of Illinois at Urbana-Champaign. Among his publications are articles in the Journal of Finance, Journal of Financial and Quantitative Analysis, Financial Management, the Journal of Portfolio Management, and Management Science. Srinivasan KANNAN is Assistant Professor of Finance at the Department College of Business Administration, Texas A & M University.
of Finance,