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FORECASTING BETA: HOW WELL DOES THE ‘FIVE-YEAR RULE OF THUMB’ DO by Nicolaas Groenewold* Department of Economics, University of Western Australia, Nedlands, WA 6009 Australia and Patricia Fraser, Department of Accountancy, University of Aberdeen, Dunbar St., Aberdeen, Scotland. Draft 3: January 1999 *Corresponding author. This research was partly funded by a small ARC grant awarded at the University of Western Australia. We are grateful to Gino Rossi for research assistance. 2Abstract CAPM betas are widely used in practice. While they are estimated from historical data, they are generally applied to a future period. This is appropriate only if the betas are stable over time. However, there is widespread evidence that the CAPM betas vary considerably over time. This raises two questions: is the time-variation in the betas systematic and can systematic variation in the betas be used to generate forecasts which improve on the usual practice of using the betas estimated over the most recent five-year period (the “five-year rule of thumb”) We address both of these questions. We estimate time-varying betas using both overlapping sub-periods (five-year rolling regressions) and non-overlapping two-year periods. We proceed to explain the time-variation in the betas using two different regression models which we subsequently use for forecasting. We find that, despite the forecasting equations having relatively high within-sample explanatory power, the forecasts generated by these equations are dominated, on average, by the five-year rule of thumb. Further experimentation shows that the five-year rule of thumb is itself dominated by two-, three-, four-, six- and seven-year rules of thumb, with the three-year rule being optimal for our sample. 31 Introduction CAPM betas are widely used in practice. In general, they are used for some future period, e.g. for calculating required returns or expected future returns or for cost of capital computations as an input into project evaluation. However, since future betas are not known , it is common practice to use betas estimated over some recent period. Underpinning this application of the CAPM betas is the assumption that the betas estimated over a recent period are appropriate as an input into calculations that require projections into the future. While this procedure would be acceptable if the betas were known to be stable over time, there is widespread evidence to the contrary – rather than being constant, the betas are time-varying. This is reflected in the very fact that in practice the betas are generally estimated over relatively short periods, five years being common. Using recently-estimated betas to apply to some future period presumes that the historical betas are good predictors of the future betas and, in particular, that they outperform alternative forecasting schemes. What little evidence there is on this question suggests that this is generally not so. However, beta-forecasting methods have generally been quite ad hoc and have not used the information contained in the time-variation in estimated betas. We set out in this paper to assess the forecasting performance of regression models estimated using time-varying betas. We begin by estimating the time-varying betas using a rolling-regression procedure with a five-year window and use the estimated betas as dependent variables in three different forecasting models. The first and simplest model is an AR(1) model. The second model is based on the regression of the time-varying betas on a time trend, allowing for a break in level and in trend at October 1987. The regressors in this forecasting equation are deterministic which makes the model attractive for forecasting because the future values of the explanatory variables are known with certainty and so do not themselves need to be forecast before the betas can be forecast. The third forecasting model is based on the regression of the time-varying betas on a set of economy-wide factors or macroeconomic variables which may affect systematic risk as measured by the betas. The model is intuitively appealing given the growing evidence of the effects of aggregate shocks on share markets. However, it has the disadvantage from a forecasting perspective that the regressors are stochastic so that future values of the regressors themselves must be forecast before the model can be used to forecast the betas. We compare the performance of these three models 4to the use of historical betas estimated over the most recent five-year period (the ‘five- year rule of thumb’). To anticipate our results, we find that the estimated time-varying betas show considerable evidence of instability and that the fluctuations are well explained by both our regression models. On the forecasting front, we find that the five-year rule of thumb, on average, outperforms all other models. Thus the common application of the most recently estimated beta to some future time period is vindicated, at least relative to the alternative models we consider and for our data set. This result does not apply to all sectors uniformly, however; there are some sectors for which the rule of thumb can be improved upon by a variety of alternative forecasting models. An important qualification of the results outlined above is that they are conditional on the particular method of estimating the time-varying betas. While the rolling regression procedure with a five-year window is a natural extension of the common five-year rule of thumb, it does produce estimated betas which have a high degree of autocorrelation due to the large number of common observations used to estimated successive betas. This is bound to make the betas behave very much like a random walk and it is well known that for this type of variable the current value is the best predictor of any future value and it is therefore not altogether surprising that the five-year rule of thumb outperforms the regression-based models. To assess the influence of this overlapping-observations problem on the outcome of the forecasting comparisons, we also experimented with betas estimated over non-overlapping sub-periods. The resulting betas also show considerable variation over time which can be well explained by the regression models used previously. There is, however, no evidence of autocorrelation in the betas, confirming our conjecture that it was caused by the overlapping observations problem in the rolling-regression betas. Despite this, the forecasting performance of the five-year rule of thumb was superior to that of the other models providing some evidence of the robustness of our original conclusion. Finally, we address the issue of whether 5 is an optimal value for N in the N-year rule of thumb by experimenting with various values of N. We find that, while the five-year rule of thumb dominated the forecasting performance of the regression-based models as well as that of the AR(1) model, the five-year rule is, in turn, dominated by two-, three-, four-, six- and seven-year rules of thumb, with the three-year rule of thumb being optimal for our sample. 5The practical implications of our results are simple. While we confirm that the CAPM betas are, indeed, time-varying, the nature of the time-variation is such that for most sectors the standard N-year rule of thumb works well relative to more sophisticated alternatives although for some sectors the rule of thumb can be improved upon by alternative forecasting models. The best value for N in the rule of thumb is an empirical matter and will possibly change from one sample to another. The structure of the paper is as follows. We begin in section 2 by reviewing three strands of literature. The first deals with the evidence for instability of CAPM betas. The second strand deals with methods of estimating the market model with time-varying betas. The third deals with methods for forecasting betas. In section 3 we describe the data used in our empirical work and in section 4 we report the results of testing constancy of the betas using our data. In section 5 we present our time- varying betas estimated using both the rolling-regression procedure and the two-year non-overlapping sub-samples. We briefly report the results of estimating the regression models for the rolling-regression betas in section 6. In section 7 we use each of the models, separately and together, as well as the AR(1) model and the five- year rule of thumb to forecast betas and compare the forecasting performance of the different models. In section 8 we present the time-varying betas estimated using the two-year sub-periods and the models used for forecasting them. Forecasts of the two- year betas are presented in section 9 and we report the results of our experimentation with alternative rules of thumb in section 10. Conclusions are presented in section 11. 2. Literature Review Betas are typically estimated as a constant parameter in the market model using OLS or similar estimation techniques. Yet, it is a stylised fact of empirical finance that betas are not stable over time, a fact reflected in the universal practice of estimating betas only over short time periods. Early evidence on beta-instability for the US dates from the early 1970s: Blume (1971, 1975), Gonedes (1973), Meyers (1973), Levy (1974) and Baesel (1974). More recent evidence using more sophisticated tests is reported in, e.g., Bos and Newbold (1984) and Gonzalez-Rivera (1997); recent Australian evidence is reported in Faff, Lee and Fry (1992), Brooks, Faff and Lee (1992, 1994) and Brooks, Faff and Josev (1997). In addition to the instability of the betas over time, there is a general finding that the extent of the instability depends on the length of the period over which the 6betas are estimated and on the length of the period to which they are applied, the “forecast period”. Although there is no unanimity on the optimal length of the estimation period, five years of monthly data appears to be the most common choice. There is also strong evidence that the betas are less stable for individual securities than they are for portfolios and that as the size of the portfolio increases, betas become more stable reflecting the effects of diversification; see, e.g., Alexander and Chervany (1980) using US data. Brooks, Faff and Lee (1994), however, find little evidence of this effect using Australian data. Given that the betas are generally found to be unstable over time, it is desirable to accommodate this instability in the procedure used to estimate the betas. Numerous methods have been used. The most straightforward and most common is to assume that betas are constant over relatively short periods of time and to estimate them over these periods using an estimation technique (such as OLS) suitable for a constant-coefficient model. As stated above, five years of monthly data is a common sample period to use although others have been used – Chang and Weiss (1991), for example, use daily data over three and six-monthly time-spans. A simple extension of five-year estimation periods is to use a rolling regression technique which shifts the five-year period forward by one month at a time to obtain a monthly time series for beta. A related technique is recursive regression which progressively increases the length of the sample by one observation at a time; see, e.g., Groenewold and Fraser (1999) for an application of these two techniques to Australian data. A model which explicitly departs from the assumption of a constant coefficient is the random-coefficient model which has been applied to US data by Fabozzi and Francis (1978), in Bayesian form by Chen and Lee (1982) and McDonald (1983) and, to Australian data, by Brooks, Faff and Lee (1992, 1994) and Brooks, Faff and Josev (1997). This method does not, however, produce time-series estimates of the betas but is focussed more on measuring the extent of the randomness in the estimated coefficients. A more general time-varying coefficients model is that of Cooley and Prescott (1976) which has been applied to the estimation of betas by Dotan and Ofer (1984) who assume that beta is generated by a random walk and estimate the resulting time- varying coefficients model using Bayesian methods. 7Finally, an alternative and general technique for estimating time-varying coefficient models is the Kalman Filter procedure which has been applied to US data by Fisher and Kamin (1985), to UK data by Black, Fraser and Power (1992), to Swedish data by Wells (1994) and to Australian data by Brooks, Faff and McKenzie (1998) and Groenewold and Fraser (1999). A related technique which assumes the coefficients to be generated by a general linear stochastic process is applied to US data in Gonzalez-Rivera (1997). Generally these techniques have been used not so much to obtain time series for the betas as to provide further evidence of the time- varying nature of the betas. In practical applications of the CAPM, betas are applied to future periods and, given the evidence that betas are not constant over time, it is not suprising that various authors have attempted to forecast betas in some way. The simplest forecasting procedure is to assume that betas are constant over relatively short periods so that recently-estimated betas can be used for application to the near future. A common procedure is to use a beta estimated over the most recently available five-year period. For future reference we call this the “five-year rule of thumb”. Early empirical work found that there was considerable sampling variability in estimated betas, particularly for individual securities and attempts have been made to moderate the influence of this variability on forecast betas by using historical betas in cross-section regressions and using the generated values of the betas from these regression equations as forecasts; see Blume (1971) and Vasicek (1973) who uses a Bayesian adjustment procedure. These and similar ad hoc methods are compared for the US by Klemkosky and Martin (1995), for Belgium by Hawawini, Michel and Corhay (1985) and for Australia by Brooks and Faff (1997). Another strand of the literature has used accounting variables as a basis for the forecasting of betas; see Beaver, Kettler and Scholes (1970) for an early application and Young, Berry, Harvey and Page (1991) for a more recent application of this method to macroeconomic risk factors obtained from an APT model of the sort first estimated by Chen, Roll and Ross (1986). The evidence suggests that these ad hoc adjustment methods generally produce forecasts which are superior to the five-year rule of thumb, in many cases to a considerable extent. All the forecasting methods discussed above are ad hoc and do not take advantage of the information in the betas generated by techniques of estimating time- varying betas. Thus the literature which estimates time-varying coefficient models for 8the betas generally uses the results simply as a basis for tests of time-variation and not for forecasting purposes. We propose to combine the time-varying beta estimation results and forecasting. We estimate a time series for the beta for each of a set of portfolios using a time-varying estimation technique and use these estimated betas as inputs into the estimation of regression models which are then used for (out-of- sample) forecasting of the betas. This has the advantage of using as much information on the nature of time-variation as possible in the forecasting process but is, naturally, conditional on the method used to estimate the time-varying betas. In our empirical work, the results of which are reported below, we use a rolling- regression technique with a five-year window for estimating time-varying betas. We choose this procedure because it is a natural extension of estimating betas over five- year samples. Experimentation in an earlier paper, Groenewold and Fraser (1999), with the use of recursive estimation and the Kalman Filter produced beta estimates which varied greatly at the beginning of the sample period when few observations are used and became progressively less variable as more observations were used in the estimation process so that by the end of the sample the estimated beta was effec

tively constant. This effect adds an artificial element to the resulting parameter estimates which is overcome by the use of the rolling-regression technique which uses a constant window size (of 60 observations in our case). A disadvantage of all three methods is the overlapping observations problem – each successive estimated beta is based on substantially the same observations as the previous betas causing autocorrelation in the resulting time-series for beta. We therefore also experiment with betas estimated over relatively short non-overlapping sub-periods. 3. The Data We used monthly data for 18 Australian industrial sectors for the period January 1973 to June 1998 obtained from Datastream. For each sector we calculated the continuously-compounded rate of return which includes both capital gains and dividends. The sectors for which data are available for the entire sample period are listed in Table 1. Datastream has a further 10 sectors at this level of disaggregation but for none of them are data available for the whole sample period. Since we wished to estimate betas over a set of non-overlapping sub-periods as well as by rolling regression techniques, it was important to have as long a sample period as possible so that we used only the 18 sectors for which data were available from 1973. To 9calculate the market return we used Datastream’s Total Market index, again using an index which includes both capital gains and dividends. Table 1 contains summary statistics for the returns for the 18 sectors. The skewness and excess kurtosis statistics are both asymptotically standard-normally distributed under the null hypothesis of normality of returns. The normality statistic is the goodness-of-fit version; it is 2 )27(c distributed under the null of normality. The next three columns in the table report the previous three statistics for the sample omitting the October 1987 observation. It is often found that the events surrounding the Crash have a disproportionate effect on the properties of the data. The final two columns of the table report ARCH(6) and ADF statistics; the ARCH (6) statistic is 2 )6(c distributed in the absence of heteroskedasticity. The results in Table 1 show widespread departures from normality in the returns – there is significant skewness in 16 of the 18 sectors and excess kurtosis in all 18 sectors. The goodness-of-fit test rejects normality for 10 of the 18 sectors. There is a noticeable effect on these results of the October 1987 observation, however; the incidence of skewness falls to 6/18 while the excess kurtosis statistics fall markedly in magnitude although there is still evidence of excess kurtosis for all sectors. The normality statistic is still significant in nine of the 18 sectors. There is some evidence of heteroskedasticity, at least of the ARCH type, with six of the series exhibiting significant ARCH(6) effects. The ADF results indicate that the null hypothesis of non-stationarity in the returns can be rejected for all sectors. In addition to share-price index and dividend-yield data, we also used macro variables in our empirical work. As explained in the introductory section, one approach to modelling the time-variation in the betas and generating forecasts is to use aggregate macroeconomic variables in our forecasting equations. Our choice of macro variables for this purpose was based on the hypothesis that at the aggregate level, risk is influenced by three classes of factors – real domestic activity, nominal domestic factors and foreign variables. Changes in any of these variables may conceivably influence agents’ risk perceptions and therefore the betas. Table 1: Summary Stataistics No. Sector Name Mean Variance Skewness Kurtosis Normality Skewness Kurtosis Normality ARCH (6) ADF (ex. Oct ‘ 87) (ex. Oct ‘ 87) (ex. Oct ’87) 1 Other Mining 0.0086 0.0062 -5.819 16.711 33.178 -1.004 3.327 46.993 22.134 -5.299 2 Building Materials & Merch. 0.0080 0.0046 -4.786 13.577 26.759 -0.546 2.469 26.923 5.217 -4.565 3 Chemicals 0.0132 0.0058 -4.382 15.557 38.912 0.466 2.864 51.049 24.530 -3.986 4 Diversified industries 0.0129 0.0043 -4.985 16.411 42.703 -0.227 3.858 27.062 0.607 -4.341 5 Electronic & Elect. 0.0169 0.0075 -5.597 23.634 41.764 1.122 4.477 47.856 8.748 -4.007 6 Engineering 0.0082 0.0056 -4.499 7.145 46.235 -2.680 3.463 32.010 5.340 -4.533 7 Paper & Print 0.0104 0.0046 -2.696 12.050 36.648 1.227 3.075 32.420 16.989 -4.036 8 Brewers 0.0117 0.0085 -5.543 30.303 81.023 -0.034 18.501 59.067 1.318 -3.714 9 Food Producers 0.0096 0.0044 -5.847 14.065 32.603 -2.052 4.103 35.933 9.202 -3.964 10 Health Care 0.0158 0.0062 4.115 11.649 72.946 4.313 11.941 75.213 3.165 -4.199 11 Pharmaceuticals 0.0113 0.0053 -4.070 16.524 64.583 -0.217 8.075 60.368 6.842 -4.661 12 Tobacco 0.0164 0.0067 -8.164 20.504 31.219 -2.490 2.181 27.794 2.746 -4.017 13 Media 0.0180 0.0158 -8.398 21.745 69.429 -3.951 9.192 74.542 5.684 -4.519 14 Support Services 0.0144 0.0051 -1.000 5.219 40.309 0.288 3.517 34.634 5.880 -5.328 15 Transport 0.0124 0.0044 -3.166 11.888 23.267 0.145 4.756 25.521 16.796 -3.849 16 Banks Retail 0.0121 0.0048 -1.669 11.197 29.637 -0.227 9.739 25.982 56.112 -4.864 17 Other Financial 0.0101 0.0030 -27.430 148.592 77.524 0.903 5.449 61.300 2.511 -5.380 18 Property 0.0140 0.0082 -8.401 28.985 79.000 -2.222 11.053 70.014 19.344 -4.776 Critical Values (5%): skewness, kurtosis (N(0,1)): 1.96; Normality (c227): 40.11; ARCH (c 2 6):12.59; ADF(10%): -.2.57 11 Data limitations restricted the choice of macro variables since several obvious choices (such as GDP, CPI and average weekly earnings) are not available at the monthly frequency of our index data. Therefore, in the first group of macroeconomic factors we experimented with an index of production, employment and the unemployment rate; for the nominal domestic influences we used an index of manufacturing prices, award wages, M3, M6 and the 90-day bank-accepted bill rate; foreign influences were captured by three alternative exchange-rate measures (in terms of the US dollar, the Japanese Yen and a trade-weighted basket of currencies) and the deficit on the current account of the balance of payments. In our estimates of beta based on non-overlapping sub-samples, we used two-year averages of macro variables and extended our set to include real GDP and the CPI inflation rate. The variables chosen were broadly similar to those used in studies of the macro-factor arbitrage-pricing models such as Chen, Roll and Ross (1986) for the US, Clare and Thomas (1994) for the UK, Ariff and Johnson (1990) for Singapore, Martikainen (1991) for Finland and Groenewold and Fraser (1997) for Australia. 4. Time-Varying Betas: The Evidence Before turning to the matter of the estimation and evaluation of alternative forecasting models, we briefly assess the extent of beta instability for our data set since the motivation for forecasting betas is that they are unstable over time. The simplest way to examine and account for beta-instability is to estimate betas over short sub-samples and compare the resulting estimates across time-periods. Table 2 reports betas for each sector estimated for the full sample period, 1973(1) to 1998(6), as well as for five five-year sub-periods (although the last sub-period has an extra 6 observations) and two sub-samples divided by the October 1987 observation. All betas were obtained from OLS estimation of the market model, the linear relationship between sector returns and the return to the market portfolio. The betas reported in Table 2 show considerable variation over sub-samples. Focussing just on the five-year periods, we find that a third of the sectors have betas which change from less than one to greater than one or vice versa over the period. Some betas change by very large amounts; e.g. Food Producers changed from a figure in excess of 0.8 to approximately 0.2 and back to 0.8 in successive periods; the beta for the Health Care sector increased from 0.29 to 0.56 to 0.80 in successive five-year periods and the beta for the Other Financial sector increased from 0.2634 to 0.9108 in successive periods. More than half the sectors had betas which remained below one for all five five-year periods while only one sector (Mining) had betas which were greater than one for all sub-periods. The variability appears to be less over the two longer sub-periods divided by the October 1987 observation, reflecting the common 12 Table 2: Sub-Sample Betas Sample No. Sector Full 1973-77 1978-82 1983-87 1988-92 1993-98 Pre-Oct 87 Post-Oct 87 1 Other Mining 1.1188 1.0831 1.1781 1.0493 1.2848 1.1572 1.1206 1.2163 2 Building Materials & Merch. 0.9042 0.8603 0.9588 0.9170 0.8801 0.8753 0.9273 0.8901 3 Chemicals 0.7406 0.6898 0.4667 0.8623 0.8395 1.0615 0.6303 0.9173 4 Diversified industries 0.6924 0.6390 0.3855 0.7904 0.8351 1.0999 0.5769 0.9590 5 Electronic & Elect. 0.7757 0.7433 0.4993 0.9416 0.8451 0.9245 0.6387 0.8409 6 Engineering 0.7190 0.5703 0.5367 0.8742 0.6227 1.1846 0.6578 0.9154 7 Paper & Print 0.7383 0.6454 0.6471 0.7492 0.9236 0.9904 0.6667 0.9403 8 Brewers 0.8880 0.8171 0.8393 0.9859 0.9970 0.6864 0.8131 0.8292 9 Food Producers 0.6841 0.8244 0.2112 0.8111 0.8391 0.8228 0.6031 0.8495 10 Health Care 0.3316 0.3292 0.0936 0.2905 0.5624 0.8019 0.2214 0.6750 11 Pharmaceuticals 0.4482 0.5586 0.2704 0.6017 0.2399 0.1974 0.4149 0.1782 12 Tobacco 0.6672 0.7327 0.0933 0.9557 0.7053 0.7426 0.5301 0.7077 13 Media 1.2191 1.3816 0.6204 1.4831 1.3905 1.0268 1.1063 1.2255 14 Support Services 0.6108 0.6757 0.3567 0.6134 0.7859 0.9213 0.5455 0.8462 15 Transport 0.8102 0.8392 0.7397 0.7853 0.9078 0.9179 0.7984 0.9132 16 Banks Retail 0.8001 1.1801 0.4767 0.6401 0.9958 0.8891 0.8088 0.9528 17 Other Financial 0.5962 0.5315 0.2634 0.9108 0.5396 0.5643 0.4411 0.5538 18 Property 1.0658 1.4860 0.6657 1.1034 0.8990 0.8948 1.0593 0.8980 13 finding that there is less instability in betas estimated over longer periods. Nevertheless, nine of the 18 sectors show changes in betas of at least 0.2 from one sub-period to the other which, at an equity-premium of 5% pa (approximately the difference between the change in the aggregate accumulation index and the 90-day bill rate), makes for a change of 1 percentage point in the equilibrium return predicted by the CAPM. The informal evidence presented in Table 2, therefore, shows that there is considerable variability of the betas over time, particularly over shorter time-period. This suggests that the market model with constant coefficients is misspecified. Thus, we now turn to more formal diagnostic tests of the market model which are reported in Table 3. Column 2 contains the estimated betas and corresponds to column 2 of Table 2. All the betas were found to be significant and are clearly of a plausible magnitude. The market model generally explains a substantial proportion of the variation in returns over the sample period as measured by R2 in the following column. There are some exceptions, however: the Health Care and Pharmaceuticals sectors have returns which are clearly only weakly related to the return to the market as a whole. The Durbin-Watson statistic is often used as an informal indicator of omitted (autocorrelated) variables. For our sample there is no evidence of first-order autocorrelation in the errors, with the test statistic being in the uncertain range (at the 5% significance level) only for the Mining sector; otherwise the null hypothesis of no autocorrelation can not be rejected. The next two columns report statistics based on the recursive residuals which are commonly used to test model stability. The first of these, Harvey’s t-test, is based on the CUSUM and has a t-distribution with (N-2) degrees of freedom under the null hypothesis that the error process in the market model is normal. The second, Harvey’s F-test, is a similar test but based on the squared recursive residuals. It is distributed with an F (1, N-2) distribution under a similar null hypothesis. In both cases the null hypothesis cannot be reject for any of the sectors. There is therefore no indication from these statistics that the coefficients of the market model are unstable. The RESET test is a standard test of linearity and proceeds by regressing the dependent variable on the original regressor and the square of the predicted dependent variable from the original regression. The RESET(2) statistic is distributed F (1, N-2) under the null of linearity. It is rejected for 7 of the 18 sectors reported in the table, providing some evidence of mis- specification of functional form. On the whole, however, these four tests provide surprisingly little evidence of specification problems for the market model. 14 TABLE 3: Market Model Diagnostics No. Sector Beta R-squared DW Harvey’s Harvey’s RESET(2) Skewness Excess Normality t F Kurtosis 1 Other Mining 1.1188 0.8696 1.6597 -1.4149 1.7985 4.1866 -3.5172 6.6867 44.9338 2 Building Materials & Merch. 0.9042 0.7595 2.0554 0.2761 0.5791 1.7569 1.6597 1.8243 27.7553 3 Chemicals 0.7406 0.4091 2.3078 -0.3119 0.7716 1.9568 0.4319 7.6809 36.6344 4 Diversified industries 0.6924 0.4792 1.9958 0.5309 0.361 6.4604 2.8195 7.7959 24.8059 5 Electronic & Elect. 0.7757 0.3464 2.4597 -1.4226 0.4419 3.8898 1.7521 4.6098 33.0366 6 Engineering 0.7190 0.3969 2.2024 0.1052 0.4203 0.2202 0.3195 3.3927 34.1533 7 Paper & Print 0.7383 0.5154 2.2354 0.9049 0.5615 0.0161 1.5946 2.2249 27.4562 8 Brewers 0.8880 0.3989 2.3172 -0.9819 0.4798 2.8763 -2.4377 19.6270 60.1704 9 Food Producers 0.6841 0.4552 1.8575 0.7574 0.6794 3.3368 0.1103 3.5853 19.0844 10 Health Care 0.3316 0.0764 2.003 -0.0395 0.4552 0.0391 4.8302 13.6058 51.8526 11 Pharmaceuticals 0.4482 0.1640 2.1789 0.815 0.7324 16.6880 0.8044 7.3874 58.3396 12 Tobacco 0.6672 0.2852 1.9598 0.6005 0.9865 6.4857 -3.1891 2.0683 38.4607 13 Media 1.2191 0.4061 1.9902 0.9493 0.7674 9.3044 -2.9140 6.2663 43.5449 14 Support Services 0.6108 0.3135 2.1497 -0.2558 0.2767 0.1345 -0.8252 3.3809 32.1620 15 Transport 0.8102 0.6432 2.0123 0.3469 0.5757 2.1606 0.1712 0.7463 24.0023 16 Banks Retail 0.8001 0.5714 1.9953 0.2702 0.3687 4.2988 1.0380 3.2016 28.5581 17 Other Financial 0.5962 0.5199 2.0693 -0.9294 0.4307 54.0870 -10.3617 40.5282 23.4112 18 Property 1.0658 0.5980 1.9522 -0.6833 0.1999 3.4599 -0.1390 5.6741 37.2606 Critical
values (5%): DW; dL = 1.65, dU = 1.69; Harvey’s t: 1.96; Harvey’s F:
3.96; RESET(2): 3.96; Skewness and Excess Kurtosis: 1.96; Normality:
40.11. 15 The next three columns of Table 3 relate to the normality of the error process in the market model. The first tests for skewness, the second for excess kurtosis while the third is a goodness-of-fit test for normality. The skewness and kurtosis statistics are standard – normally distributed and the third is distributed c( )11 2 under the null of normality. The null hypothesis is rejected for seven, 16 and 17 of the 18 sectors respectively, indicating that there is considerable departure from normality in the errors of the market model. These rejection rates are similar to those of the original returns (corrected for the influence of October 1987) and indicate that the market model “explains” little of the non-normality in the original returns The results reported in Table 3 therefore suggest considerable departure from normality in the market model’s errors but not the widespread specification problems. This is surprisingly at variance with the strong evidence of beta instability reported in Table 2. A more direct test of parameter stability is the Chow test which, however, requires a prior specification of the break-point. We chose breakpoints at five-year intervals corresponding to the common practice of estimating betas over five-year periods as well as an alternative break- point at October 1987. The five-year breakpoints correspond to those used in Table 2. The results are reported in Table 4. We permit only the slope coefficient in the market model (i.e. the beta) to change at the break-point. In this situation the Chow test is equivalent to a t-test on a dummy variable interacted with the return to the market portfolio. The market model was, therefore, re- estimated with 4 dummy variables, D1,…,D4 interacted with the return to the market portfolio. The first dummy variable is zero except for the period