Read Section 5.1, “Stationarity through differencing,” on pages 88-92. Know equation 5.1.10 on page 90 and its derivation. Distinguish between ó²_ε and ó²_e in this equation.

 

Read again the last paragraph on page 90 and review Exhibit 5.4 on page 91. Most actuarial time series are not stationary. For your student project, you take first and second differences, and you might also take logarithms. The homework assignment shows how a loss cost trend is made stationary by logarithms and first differences.

 

Cryer and Chan do not stress changes in the time series over time. The authors know how to judge if the parameters are stable, but they keep the statistics at a first year level.

 

For the student project, ask yourself whether the time series itself has changed. The module on the interest rate time series on the NEAS web site stresses the three interest rate eras affecting the time series.

 

Happy Monday, fellow followers of the NEAS.

I would like to walk through the derivation of 5.1.10 as I had to walk myself though this mystery, the autocorrelation at lag 1 of the first difference time series.  First, as the book compares this to an MA(1) model, I decided I need to find the autocovariance function at lag zero, A.K.A., the variance of the time series:

Var(delta-Y_t) = Var(eplison_t+e_t-e_t-1) = Var(epsilon_t)+Var(e_t)+Var(e_t-1) (due to our given assumption of independence)

Var(delta-Y_t) = gamma₀ = sigma_epsilon² + 2sigma_e²

Next, in order to find the autocorrelation at lag 1, I need to know the autocovariance at lag 1:

Cov(delta-Y_t,delta-Y_t-1) = Cov(eplison_t+e_t-e_t-1,eplison_t-1+e_t-1-e_t-2) (which very simply reduces due to independence)

Cov(delta-Y_t,delta-Y_t-1) = gamma₁ = -Var(e_t-1) = -sigma_e²

Finally, recall the autocorrelation at lag 1 is simply gamma₁/gamma₀:

(-sigma_e²)/(sigma_epsilon² + 2sigma_e²) = (-sigma_e²)/{sigma_e²[(sigma_epsilon²/sigma_e²)+2]}

=-{1/[2+(sigma_epsilon²/sigma_e²)]}

 

Thanks for your time!

RDH

Can anybody explain how "M(t)-hat = [Y(t) + Y(t-1)] / 2" makes any sense?  Why are we estimating M at time T to be the average value of Y between T and T-1?  If we take expectations, this doesn't appear to make any sense...

On page 89, we are told Corr(Y_t, Y_t-k) = 3^k * SQRT{[9^(t-k) - 1] / [9^k - 1]}.

Now I'm confused!

I thought rho_k = Corr(Y_t, Y_t-k) = gamma_k / gamma_0 = 3^k * [9^(t-k) - 1] / [9^k - 1].

Where did Cryer and Chan get the SQRT?

To CalLady,

Keep in mind that Var(Y_t) has a t in its formula, so when you're calculating the correlation, the denominator doesn't simplify to Var(Y_t) but is the square root of sigma_Y_t times sigma_Y_t-k...so once you have the two std devs under the square roots, it's a matter of factors cancelling.