## 8.1 INTRODUCTION

This chapter is concerned with estimation of a function, *p*(*f*), that uses all of the covariances, *C*_{k}, to form a periodic function for all *f*, thought of as frequencies. Although it is hard to motivate this function and it seems to be pulled out of thin air, it will be important throughout the rest of the book. The (power) spectrum is the true function, *p*(*f*), while the periodogram is and is estimated from the data. While *p*(*f*) is based on all covariances (*k* = 0,1,2,3,…) and exists for all frequencies, *f*, the analogous sample estimate , must be limited by sample size, and uses the sample covariances *k* = 0, 1, …*n* − 1 and can only be computed at *f*_{j} = *j*/*n*, where *j* = 1, 2…*n*/2. Because *p*(*f*) is periodic, it need only be specified in the interval 0 ≤ *f* ≤ 1 or −0.5 ≤ *f* ≤ 0.5, furthermore the condition *C*_{k} = *C*_{− k} renders half of this information redundant, so it is sufficient to define the function in the interval 0 ≤ *f* ≤ 0.5.

The spectrum measures the intensity of periodic patterns of different frequencies. The periodic ripples could be transient and fading (as with AR(*m*) and MA(*l*) models) or persistent (as with periodic signals). Plots of versus the frequencies provide the second useful plot unique to time series and the rationale ...