sp(t)= A1S1(t+t1) + A2S2(t+t2) + A3t +A4
where sp(t) is the stock price at time t, A1 through A4 are empirical coefficients, S1 and S2 are components of headline CPI, t1 and t2 are time difference (negative or positive) between the change in the stock price and relevant changes in the CPI components. Term A3t is introduced to compensate linear trends observed in the difference between the components.
For Microsoft, an accurate empirical model is as follows:
MSFT(t)= -3.61*R(t-1) – 1.22*MSC(t-11) + 23.5*(t-2000) + 680.5.
where t is the calendar time, R(t-1) is the index for recreation leading MSFT(t) by one month, MCS(t-11) is the index for medical service leading the MSFT(t) by eleven months. The linear trend term 23.5*(t-2000) compensates the trend term in the difference between R and MCS. Standard deviation between the curves is $1.75 for the period between July 2003 and August 2009.