A.     If this rate change occurs immediately, how should you change the model? Do you add an autoregressive or a moving average term? (In practice, rate filings take several months to be approved. Assume this rate change takes effect immediately.)

B.     If this rate change has a one month lag, how should you change the model? (For this part of the homework assignment, assume the insurer compares its actual vs expected policy count at the end of each month and changes the rate beginning either the next day or one month later. Use whichever assumption you prefer.)

D. Since a low rate in the previous month will tend to be correlated with a low rate in the current month, then higher than expected amounts of new business in the previous month will tend to be correlated with higher than expected amounts of business in the current month.

I believe that implies using an autoregressive term from the prior month so that

Y_t = .9Y_t-12 + (phi)(e_t-1) + n + e_t

Where n is the expected number of new business policies written and

e_t is an independently distributed random variable with mean = 0

[NEAS: Correct]

F. I’m not as certain about this, but I believe that the form of the model would not change from part D, but the actual values you get for your parameter would change.  Assume that the insurer always changes rates by less than what would be needed to return to an equilibrium state in which the prior month’s writings were the long term average writings.  In this case I think we would have the same model as before, but the magnitude of phi would be smaller.  If the insurer always overcorrected with the rate change, then the sign of phi would be reversed.  If the insured always changed rates by exactly the right amount, then phi would be zero.  But I think that in all cases, the model stays the same.

G. I don’t think I fully understand this question.  Since we’re dealing in discrete time intervals of one month, I don’t understand the distinction between the assumptions in F and the optional assumption in G that “the insurer compares its actual vs expected policy count at the end of each month and changes the rate beginning […] the next day”  How could any rate change in F be more instantaneous than that?  Or are we not actually dealing with discrete intervals of 1 month?  In which case, how do we use time series to approximate this process?

Using the other assumption, I think you would have an additional autoregressive term of (phi₂)e_t-2.  But since I admit I don’t really understand the question, I’d take that with a large grain of salt.

After thinking about this more, I think that I was wrong about using an autoregressive term to model the effects of a previous month, I think it should be a moving average term.  I'm editing my previous post to make what I think should be the corrections, the changes will be in red.

[NEAS: Statisticians have used both moving average and autoregressions terms to model seasonality, depending on the reasons for the seasonal effects. Both methods are acceptable for the homework.]

This homework assignment does not ask for numerical results. It asks you to relate ARIMA processes to business practice. Say how each market attribute might affect the time series model. Statisticians differ; there is no single solution. You receive credit for a solution that explains how the seasonal parameters work, whether or not your business interpretation is correct.