We continue testing our model againts bancruptcy cases. Lehman Brothers presents an old but crucial case. Before we start our quantitative analysis, we have to clarify the the term "predicting" has a narrow sense of quantitative description. In other words, we do not forecast the bankruptsy, but decompose relevant stock price, sp(t), into a sum of two CPI components, a linear and a free term:
sp(t) = A*CPI1(t+t1) + B*CPI2(t+t2) + C*(t-2000) + D (1)
where CPI1 and CPI2 are defining CPI components; t is the calendar time; A, B, C, and D are empirically determined coefficients. Because prices for good and services can lag or lead relevant share prices the time lags t1 and t2 are introduced. Both time lags can be positive or negative. In our model, we test all possible lags between 0 and +13 months. So, we do not pretend that one could see the fall of LEH months before it had happened.
So, the empirical model describing the time history of LEH share price is as follows:
LEH(t) = -9.75*F(t-4) – 8.13*PC (t-2) + 94.9*(t-2000) + 2802 (2)
, where t is the calendar time, F(t) is the (not seasonally adjusted consumer price) index for food and beverages, leading the LEH(t) by four months, and PC(t-13) in the index of personal care leading the LEH(t) by 2 months. Or this model we used LEH prices between July 2003 and September 2008; the CPI components were taken till December 2008, i.e. three months ahead of the last LEH reading. Figure 1 demonstrates that the model predicted the fall below zero but three months later. Standard deviation of the residual is $4.7.
Once again, this model could not be obtained in August 2008.
Figure 1. Observed and predicted LEH share price.