Regression Analysis and Time Series Modelling
Regression Analysis in Legal Scenario
Multiple regression and time series analysis have played an important role in legal context also. For instance, in the famous supreme court case of McCleskey v/s Kemp, the appellant, who was sentenced to death, claimed that race of the criminal plays a very important role when the judge decides the punishment. Accordingly, David Baldus, a professor at the University of Iowa, ran regression on the number of death penalty cases, while keeping the race of the criminal, his past criminal record, method of killing and motive of killing as independent variables. Ironically, the outcome of regression analysis confirmed that the race of the criminal is a significant factor while deciding the death penalty.
Henceforth, in this paper, we will discuss a hypothetical legal case where multiple regression and time series analysis can jointly produce a better regression model.
Defining Time Series
Time series is a set of observations for a variable over a successive period of times. Such series have a pattern, either positive or negative, and can be graphed.
The Case
Assuming that the shareholders of Company XYZ have filed a lawsuit against the company for duping them and eventually going bankrupt. The lawyer representing the shareholders is interested in evaluating the factors that forced the company’s management to opt for fraudulent accounting and thus use the net income for the year of bankruptcy as dependent variable and sales growth as independent variables. The regression equation with given variables was expressed as:
Net Incomet = b0 + b1Sales Growtht + ƹt
Following the regression analysis, the lawyer found that sales growth was significant in explaining the variation in net income and with r-square of the model at 0.55 and p-value less than 0.05, it was confirmed that model was useful.
However, in order to improvise the model, the lawyer used Autoregressive model(AR) as part of which, the net income of the current year can was regressed against the net income of the previous year. As expected, the r-square of the model improved to 0.85 while even the p-value of lagged net income was significant at 5% level of confidence. In order to make sure that AR model is not autoregressive, the lawyer also tested the model by performing t-test for autocorrelation’ to test if autocorrelation is different from zero. Here, the t-statistic was the estimated autocorrelation divided by the standard error.The t-test confirmed that none of the autocorrelation of the residual terms were different from zero, which confirmed that error terms were not serially correlated.
Henceforth, by including the time variable and transforming the model to autoregressive model, the lawyer was able to prove that both sales growth and most importantly, past year net income of the company was the influential factor in manipulating the current year net income. In addition, the AR model was also very much valid as even the outcome of the autocorrelation test confirmed that residual error terms in the model are not serially correlated.
Conclusion
Therefore, considering the above example, it can be seen that including time series factor in an appropriate scenario can improve the validity of the statistical model.
References
Paul Brest, Linda Hamilton Krieger. "Scores, Dollars and Other Quantitative Variables." Paul Brest, Linda Hamilton Krieger. Problem Solving, Decision Making, and Professional Judgmen. Oxford, 2010. 153-165.
