# Longitudinal Methods, from Victor Niederhoffer

April 26, 2007 |

An important area of research in many fields is a study over time of repeated measurements of outcome and predictor variables, often with a view to seeing whether the relations between predictor and outcome are different between groups. For example, males and females are measured with respect to blood pressure, weight, cholesterol count, respiratory efficiency at 12 separate times, and their longevity and morbidity are noted and related to their sex, smoking behavior, and changes in baseline predictor variables.

This is a very important area of research in health. The book, "Applied Longitudinal Data Analysis for Epidemiology," by Joseph Twisk, which I have read and find useful and provocative, has many modern techniques for the analysis of such data starting with repeated multivariate analysis, categorical analysis, dichotomous variable analysis, and missing data analysis. Ultimately two sophisticated methods are introduced to take account of all the problems of differentiating the cross sectional effects from the time series effects. These techniques are generalized estimating equations and random coefficient analysis.

The development of epidemiological methods has been a mix of practical and theoretical work in their own fields with many papers developing ad hoc solutions based on these methods in their own field, for example attitude surveys towards condom use, or efficacy of various diet studies.

The old methods of analyzing such data were to look at the final outcome and first outcome and relate the changes to the changes in the predictor variables and take account of various aspects of the clustering and correlation structure for the individuals in the study. The problem is that it threw out all the changes in the variables between the beginning and end points, and applied techniques that didn't differentiate between the time structures of the data.

A similar set of problems comes in market studies. For example, a researcher wishes to know whether value of growth stocks are better and measures the returns over 10 consecutive years. How are the two groups affected by changes in interest rates, and earnings growth during the period, and past market movements?

Every study of a system over time where the effects of individual markets and changes in the backdrop variables is affected by similar factors as the longitudinal studies.

The techniques of generalized estimating equations, based on maximum likelihood methods and random coefficients analysis, based on multivariate repeated regression analysis and comparisons, would seem to serve an important role in most market studies of individual stocks and all other studies where the path of the return structure is important.

There has been little cross fertilization between the fields and the Twisk book, with its references and applications of software, is highly recommended to get some tangible and useful work done in the fields without throwing out intermediate data or fitting the results into a glorified series of cross sectional analyses.

This book discussed methods of generalized randomized estimation equations for coefficient analysis. Two articles that I have found [1 ,2] give good introductions to the same methods. Both articles are technical but a pencil and paper and some simple numerical examples might open up good lines of thought and analysis.

## Andrew West replies:

As far as I can tell, longitudinal data analysis is far underused in the world of finance, considering so many data are available in this form.

A few years ago I taught myself R and bought the Pinheiro & Bates book Mixed-Effects Models in S and S-PLUS to learn how to use the NLME package. This was so that I could try to model valuation ratios of all companies within an industry. Most relative valuation analysis is done using either a recent industry cross section, or one company's historical ratios. Using NLME, I could take 15 years of annual data for 10 companies within an industry, and model how the next year's price/sales ratios were associated with current price/sales, next years' operating profit margins, debt to assets, and the yield curve. Much better to use 150 observations than either 15 or 10. So if I forecast that a company would see higher sales, margins, and lower debt, I had a tangible idea of how much the market might reward that, based on how the market responded to changes in that company and nine other similar companies over the prior 15 years. For some reason, I've seen zero studies of this sort published in finance.

The models certainly didn't predict most variation, but seemed to work better than alternative naive models available. I later applied the same longitudinal data technique to calculate Fama-French three-factor models for multiple stocks within an industry. The goal was to allow some warranted individual variation for companies, but to educate the guess by considering the experience of peers (e.g. reduce some of the random non-stationarity of a company's beta).

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