Abstract
With increasing availability of data, in many situations it is now possible to reasonably estimate the probability density function (pdf) of a random variable. This is far more informative than using a few summary statistics like mean or variance. In this paper, we propose a method of forecasting the density function based on a time series of estimated density functions. The proposed method uses kernel estimation to pre-process the raw data followed by dimension reduction using functional principal components analysis (FPCA). Then we fit Vector ARMA models to the reduced data to make a prediction of the principal component scores, which can then be used to obtain the forecast for density function. We need to transform and scale the forecasts to ensure non-negativeness and integration to one. We compared our method to [1] for histogram forecasts, on simulated data as well as real data from S&P 500 and the Bombay Stock Exchange. The results showed that our method performed better on both the datasets and the simulation using uniform and Hilbert distance. The time dependence and complexity of density function are different for the two markets, which is captured by our analysis.
Highlights
Contemporaneous aggregation is often the only way to analyze temporal data, for example, considering the observations of a variable measured through time in a population, e.g. the monthly output of firms in a country
Ma displayed a time series of the weekly returns of the firms in the Standard & Poor’s 500 (S&P 500), summarized by histograms. This is an interesting precedent which shows that in some cases, histograms are preferable to averages or totals. [1] used smoothing and non-parametric method to estimate and predict histogram time series on S&P 500 data
Simulation was carried out to compare the performance of the proposed Functional Data Analysis (FDA) method to the method of [1] for prediction with a histogram time series using uniform and Hilbert norm distances
Summary
Contemporaneous aggregation is often the only way to analyze temporal data, for example, considering the observations of a variable measured through time in a population, e.g. the monthly output of firms in a country. If one is interested in the overall evolution of all firms, histograms or densities can still be studied. How to cite this paper: Sen, R. and Ma, C. (2015) Forecasting Density Function: Application in Finance. Ma displayed a time series of the weekly returns of the firms in the S&P 500, summarized by histograms. This is an interesting precedent which shows that in some cases, histograms are preferable to averages or totals. [1] used smoothing and non-parametric method to estimate and predict histogram time series on S&P 500 data. We follow-up on this idea, but replace histograms by density estimates. Kernel density estimates are smooth estimates of the probability density function and do not depend on the choice of end-points as opposed to histograms
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