Dr K Suresh Chandra, Dr T M Srinivasan and Dr D Karthikeyan

I . Introduction

Stock markets, which are in some sense laboratories to assess, continuously, the health of the country’s economy (both at macro as well as at sectoral levels), have attracted the attention of theoreticians as well as practitioners in social sciences. The time dependent nature of the Volume-Price statistics generated at these markets have induced researchers, seeking to develop models for the underlying phenomena, to adopt time series techniques, for analysing the empirical data. Although several attempts have been reported in the literature, it is a fact that not one of them has been able to adequately portray the vagaries of stock markets even for short-term prediction.

In contrast to other forms of economic Time Series, the amplitudes in the oscillations in stock market data generate a variety of economic activities, perhaps largely speculative in nature. We have used two words, namely, amplitude and speculation connoting their usual meaning. In Time Series, the amplitude usually refers to half the distance between a peak and the trough that follows in an oscillatory movement, after eliminating the trend. In the present context the amplitude refers the distance between a peak and the trough, and the prevalence of varying periods and varying amplitudes in stock market data, makes it necessary to differentiate between an upward amplitude and a downward amplitude, as distances between a trough and next peak and a peak and next trough respectively. Again, the short term and long term speculation induced by stock markets prevents the isolation of trend while defining the amplitude. By long term we mean the period of observations.

In view of these considerations, it appears that, for a comprehensive study of economic activities generated in and through stock markets, one should develop models for upward and downward amplitudes, with direct or indirect reference to the periods, incorporating the trend components.

II . A Simple Approach

As very little work seems to have been done on the distributional aspects of amplitudes in Time Series, especially in the area of business cycles, to begin with, one can assume a non-homogeneous but one-step Markovian dependence between consecutive upward and downward movements. That is, assuming that a fall is influenced only by the immediately preceding raise and vice versa, let p(d/u) and p(u/d) be the conditional probability density functions associated with downward and upward movements respectively. Using these densities one can evaluate E(D/U) and E(U/D), which give the expected fall (raise) given the raise (fall) in the previous stage. In the presence of an increasing trend one can expect E(U)=E{E(U/D)} to be larger than E(D)=E{E(D/U)}. These conditional expectations provide the rationale for the short term and long term speculations. The risks associated with decisions based on these values can be measured by the conditional and unconditional variances.

One can introduce the influence of the period on the amplitudes, by considering the rates of the raise or fall per unit of time of the oscillations, obtained be dividing these variables by the time taken for the Time Series to obtain these amplitudes. Let p(u*/d*) and p(d*/u*) be the conditional probability densities, associated with the rates. Then E(U*/D*) and E(D*/U*) will provide the average daily increase (decrease) given the average daily decrease (increase) in the previous stage. One can use these conditional expectations for identifying the short term speculation, and the corresponding unconditional expectations for long term speculations.

III . Empirical Data Used

Towards demonstrating the methodology suggested in section II, the data on Madras stock market index and All India daily index for the period of October 1, 1989 to September 30, 1991 was used for the analysis. A five day moving average was first adopted to remove the short term fluctuations.

IV . Concluding Remarks

This paper attempts to project the need and utility of statistical analysis of amplitudes of the oscillations in stock market data, without isolating the trend component, towards understanding the cyclical behaviour of the data.

From the empirical analysis, it was found that average amplitudes had more convenient statistical properties than absolute amplitudes. In fact, the exponential distribution was found to be a good fit for the data on U* and D*. Further, the distribution of number of days for a raise or for a fall also appeared to be exponential, based on the c² - goodness of fit.

The procedure discussed in this article is a simple beginning, and if found useful, that there is ample scope for refining the tools and techniques towards a more meaningful analysis.