7. Show (2.28) is stationary.

8. The glacial varve record plotted in Figure 2.6 exhibits some nonstation- arity that can be improved by transforming to logarithms and some ad- ditional nonstationarity that can be corrected by differencing the loga- rithms.

1. Verify that the untransformed glacial varves has intervals over which

γ (0) changes by computing the zero-lag autocovariance over two dif-

�

ferent intervals. Argue that the transformation y _t = ln x _t stabilizes the variance over the series. Plot the histograms of x _t and y _t to see

whether the approximation to normality is improved by transform- ing the data.

2. Examine the sample ACF, � ( h ), of y _t

   ρ _y

and comment. Do any time

intervals, of the order 100 years, exist where one can observe behav-

ior comparable to that observed in the global temperature records in Figure 1.2?

3. Compute the first difference u _t = y _t − y _t − 1 of the log transformed

varve records, and examine its time plot and autocorrelation func- tion, � ( h ), and argue that a first difference produces a reasonably

ρ _u

stationary series. Can you think of a practical interpretation for u _t ?

4. Based on the sample ACF of the differenced transformed series com- puted in (c), argue that a generalization of the model given by Ex- ample 1.23 might be reasonable. Assume

u _t = µ _u + w _t − θ w _t − 1

is stationary when the inputs w _t are assumed independent with mean 0 and variance σ ² . Show that

w

^⎧ 2 2

⎨ σ _w (1 + θ ) if h = 0

w

γ _u ( h ) =

− θ σ ²

if h = ± 1

^⎩ 0 if | h |≥ 1.

Using the sample ACF and the printed autocovariance � (0), derive

γ _u

estimators for θ

and σ ² . This is an application of the method of mo-

ments from classical statistics, where estimators of the parameters are derived by equating sample moments to theoretical moments.