Interpretation of coefficients in polynomial regression for predictive modeling I am building a predictive model (binary target variable) in the financial services industry. One of the (many) potential predictors I am adding to the model is related to the customers checking account balance trend (longitudinal balance).
I'd like to capture if the balance is increasing or decreasing and how much. I have access to end of month balances going back a ways. One of the things I was considering is to, for each customer - fit a polynomial regression and include the coefficients into my predictive model.
In R, an example of a single customer:
balances <- c(657709,620729,713637,619224,558238,572402,536548,0,0,0)
time <- seq(1:10)
mod <- lm(balances~time+I(time*time))
mod$coefficients[2:3]

mod$coefficients[2:3]
time          I(time * time) 
61239.99      -13317.43 

Questions:


*

*Thoughts? Of course the fit can be very poor, but as a global process to include into a predictive model does it have merit? Is there a better way?

*It seems I have seen description of these coefficients in terms of velocity and acceleration, but I cant find it anywhere. Is this a true interpretation of them?
 A: Explanation at UCLA
Another link
I think the general answer is : not that easily.  There are ways to interpret the derivative, talk about which way the curve opens, etc.  But nothing simple and clear like in the linear model.  My hunch is that you shouldn't be modeling this as a quadratic, tho.  
I would also chuck out the zeros and call your model 'Balances of accounts which have not been closed'.
A: Like other polynomial models, your model is likely to be worse an a linear model if you extrapolate outside the time for which you have data as the time * time term is likely to dominate and the sign of its coefficient will determine whether you predict a large positive or large negative balances, when for many people balances are rather more stable. 
A: As Henry politely/correctly said the problem with this dated approach of fitting linear, squares, cubics, et cetera is that you are forcing/fitting using potentially ( always in my opinion ) unwarranted deterministic structure onto the model. You "pay" for the fitting when you either interpolate or predict. Consider what happens when you fit a cubic to 4 data points. The fit is perfect / all coefficients are significant / the r-square is 1.00 but the forecast is more than likely ridiculous. A simpler and much more correct approach is to model the y variable as a function of it's past and as a function of user-suggested input series ( including any necessary lags ) and also to incorporate any empirically identified Intervention Series such as Pulses, Level Shifts, Seasonal Pulses and or Local Time Trends. This is known by many names e.g. Transfer Function and ARMAX to name two.
