Quarterly figures may be required when only a sequence of yearly totals is available. This article discusses some reasonable procedures to "interpolate" these quarterly figures.I. THE PROBLEM THE problem of deriving quarterly figures, given annual totals only, has been discussed by Lisman and Sandee (1964). A problem of this nature may very well arise if, in a multiple-regression analysis, quarterly data are available for all but one of the explanatory variables, for which only one yearly figure is available. Apart from the drastic but highly unsatisfactory procedure of simply deleting this variable, one can then proceed either by aggregating all other variables to yearly totals (at a loss of 75% of the observations) or by "reasonably" deriving the quarterly values from the annual totals. This is what Lisman and Sandee did. Their criteria for reasonableness were:(i) The sum of the quarterly figures should, for each year, equal the given yearly total.(ii) Symmetry considerations, in particular the requirement that if the yearly totals in three successive years are t 1 , t 2 and t 3 , the quarterly figures for year 2 are the same but in reverse order from what they would have been had the yearly totals been t 3 , t 2 , t 1 (that is, had the yearly totals been in reverse order).(iii) Trend considerations, in particular the desire that if the yearly totals in three successive years rise by equal steps (t 2 -t1 = t3 -t 2), the quarterly figures duringyear 2 should also rise by equal steps (of length n(t 2 -t1»'(iv) Cycle considerations, in particular the requirement that t 2 -t 1 = t 2 -t3 (for example, a sequence 80, 100, 80), the quarterly figures during year 2 should lie on a sinusoid.Lisman and Sandee (l964) showed that these four requirements-recorded here in what is felt to be decreasing order of reasonableness or increasing order of arbitrariness-lead uniquely to the result [ x s ] [0'073 0·198 -0'021] [t] X6 -0,010 0·302 -0,042 1 X7 = -0,042 0·302 -0,010 .~2 , X 8

The purpose of this paper is to simultaneously investigate several important issues that feature the dynamic and stochastic behavior of beta coefficients for individual stocks and affect the forecasting of stock returns. The issues include randomness, nonstantionarity, and shifts in the mean and variance parameters of the beta coefficient, and are addressed within the framework of variable-mean-response (VMR) random coefficients models in which the problem of heteroscedasticity is present.Estimation is done using a four-step generalized least squares method. The hypotheses concerning randomness and nonstationarity of betas are tested. The time paths, sizes, and marginal rates of mean shifts are determined. The issue of variance shift is examined on the basis of five special tests, called T*, B, S ' , G , and W . Then the impacts of the dynamic and stochastic instability on the estimation of betas is tested by a nonparametric procedure. Finally, the VMR models' ability of forecasting stock returns is evaluated against the standard capital asset pricing model. The empirical findings shed new light on the continuing debate as to whether the beta coefficient is random and nonstationary and have important implications for modeling and forecasting the measurement of performance and the determination of stock returns.

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