Time Series: Data Analysis and Theory

The front matter includes the title page, series page, copyright page, dedication, TOC, preface to the classics edition, preface to the expanded edition, and preface to the first edition

1.1 INTRODUCTION

In this work we will be concerned with the examination of r vector-valued functions [ X1 (t) ⋅⋅⋅ Xr (t) ] 1.1.1 where Xj (t) , j=1,… ,r is real-valued and t takes on the values 0, ±1, ±2,…. Such an entity of measurements will be referred to as an r vector-valued time series. The index t will often refer to the time of recording of the measurements.

An example of a vector-valued time series is the collection of mean monthly temperatures recorded at scattered locations. Figure 1.1.1 gives such a series for the locations listed in Table 1.1.1. Figure 1.1.2 indicates the positions of these locations. Such data may be found in World Weather Records (1965). This series was provided by J. M. Craddock, Meteorological Office, Bracknell. Another example of a vector-valued time series is the set of signals recorded by an array of seismometers in the aftermath of an earthquake or nuclear explosion. These signals are discussed in Keen et al (1965) and Carpenter (1965). Figure 1.1.3 presents an example of such a record.

2.1 INTRODUCTION

In this chapter we present portions of both the stochastic and deterministic approaches to the foundations of time series analysis. The assumptions made in either approach will be seen to lead to the definition of similar parameters of interest, and implications for practice are generally the same. In fact it will be shown that the two approaches are equivalent in a certain sense. An important part of this chapter will be to develop the invariance properties of the parameters of interest for a class of transformations of the series called filters. Proofs of the theorems and lemmas are given at the end of the book.

The notation that will be adopted throughout this text includes bold face letters A, B which denote matrices. If a matrix A has entries Ajk we sometimes indicate it by [ Ajk ]. Given an r × s matrix A, its s × r transpose is denoted by Ar , and the matrix whose entries are the complex conjugates of those of A is denoted by A ‾ . Det A denotes the determinant of the square matrix A; the trace of A, is indicated by tr A. ∣A∣ denotes the sum of the absolute values of the entries of A, and I, the identity matrix. An r vector is an r × 1 matrix.

We denote the expected value of a random variable X by EX generally, and sometimes, by ave X. This will reduce the possibility of confusion in certain expressions. We denote the variance of X by var X. If (X,Y) is a bivariate random variable, we denote the covariance of X with Y by cov {X, Y}. We signify the correlation of X with Y by cor {X, Y}.

3.1 INTRODUCTION

The principal analytic tool that we will employ with time series is the Fourier transform. In this chapter we present those portions of Fourier analysis that will be required for our discussion. All the functions considered in this chapter will be fixed, rather than stochastic. Stochastic properties of Fourier transforms will be considered in the next chapter.

Among the topics discussed here are the following: the degree of approximation of a function by the partial sums of its Fourier series; the improvement of this approximation by the insertion of convergence factors; the Fourier transform of a finite set of values; the rapid numerical evaluation of Fourier transforms; the spectrum of a matrix and its relation to the approximation of one matrix by another of reduced rank; mathematical properties of functions of Fourier transforms; and finally the spectral or harmonic representation of functions possessing a generalized harmonic analysis.

We begin by considering the Fourier series of a given function A(λ).

3.2 FOURIER SERIES

Let A (λ),−∞<λ< ∞ , be a complex-valued function of period 2π such that ∫ −π π |A (λ) |dλ < ∞ . 3.2.1

4.1 INTRODUCTION

Consider an r vector-valued sequence X(t), t=0,±1,… . In the previous chapter we considered various properties of the finite Fourier transform dX (λ) (T) = ∑ t=0 T−1 X (t) exp {−iλt} − ∞ <λ< ∞ 4.1.1 in the case that X(t) was a fixed, nonstochastic function. In this chapter we present a variety of properties of dX (λ) (T) , if X(t), t=0,±1,… is a stationary time series. We will also consider asymptotic distributions, probability 1 bounds, behavior under convolution as well as develop the Cramér representation of X(t).

In previous chapters we have seen that Fourier transforms possess a wealth of valuable mathematical properties. For example, in Chapter 3 we saw that the discrete Fourier transform has the important numerical property of being rapidly computable by the Fast Fourier Transform Algorithm, while in this chapter we will see that it has useful and elementary statistical properties. For all of the reasons previously given, the Fourier transform is an obvious entity on which to base an analysis of a time series of interest.

However, before developing stochastic properties of the transform (4.1.1) we first define two types of complex-valued random variables. These variables will prove important in our development of the distributions of various time series statistics.

5.1 POWER SPECTRA AND THEIR INTERPRETATION

Let X(t), t=0,±1,… be a real-valued time series with mean function EX (t) = cx t=0,±1,… 5.1.1 and autocovariance function cov {X (t+u),X (t) } = cxx (u) t,u=0,±1,…. 5.1.2 Suppose that the autocovariance function satisfies ∑ u=−∞ ∞ | cXX (u) | <∞, 5.1.3 then the power spectrum of the series X(t), t=0,±1,… at frequency λ is defined to be the Fourier transform ƒXX (λ) = (2π)−1 ∑ u=−∞ ∞ exp {−iλu} cXX (u) for−∞<λ<∞. 5.1.4 We have seen in Section 2.5 that this power spectrum is non-negative, even, and of period 2π with respect to λ. This evenness and periodicity means that we may take the interval [0, π] as the fundamental domain of definition of ƒXX (λ) if we wish.

6.1 INTRODUCTION

Let Y(t), ε(t), t=0,±1,… be real-valued stochastic series and let X(t), t=0,±1,… be an r vector-valued fixed series. Then suppose μ is a constant and that {a(u)} is a 1 × r filter. Hence, in this chapter we shall be concerned with the investigation of relations that have the form Y (t) =μ+ ∑ u=−∞ ∞ a (t−u) X (u) +ε (t). 6.1.1 We will assume that the error series, ε(t), is stationary with 0 mean and power spectrum ƒεε (λ) . This power spectrum is called the error spectrum, it is seen to measure the extent to which the series Y(t) is determinable from the series X(t) by linear filtering. We will assume throughout this text that values of the dependent series, Y(t), and values of the independent series, X(t), are available for t=0,… ,T−1 . Because E ε(t) =0 , EY (t) =μ+ ∑ u=−∞ ∞ a (t−u) X (u) . 6.1.2 That is, the expected value of Y(t) is a filtered version of X(t). Note from relation (6.1.2) that the series Y(t) is not generally stationary. However, for k>1 , cum {Y ( t1 ),… , Y ( tk ) }=cum {ε ( t1 ),… , ε ( tk ) } 6.1.3 and so the cumulants of Y(t) of order greater than 1 are stationary.

7.1 THE SPECTRAL DENSITY MATRIX AND ITS INTERPRETATION

In this chapter we extend the results of Chapter 5 to cover the case of the joint behavior of second-order statistics based on various components of a vector-valued stationary time series.

Let X(t), t=0,±1,… be an r vector-valued series with component series Xa (t) , t=0,±1,… for a=1,… ,r . Suppose EX (t) = cX 7.1.1 E [X (t+u) − cX ] [X (t) − cX ]τ =cov {X (t+u), X (t) } = cXX (u) for t,u=0,±1,…. 7.1.2 Indicate the individual entries of cX by ca , a=1,… ,r so ca =E Xa (t) is the mean of the series Xa (t) , t=0,±1,… . Denote the entry in row a, column b of cXX (u) by cab (u) , a,b=1,… ,r , so cab (u) is the cross-covariance function of the series Xa (t) with the series Xb (t) . Note that cXX (u)τ =cov {X (t),X (t+u) } = cXX (−u) for u=0,±1,…. 7.1.3

8.1 INTRODUCTION

Consider an (r + s) vector-valued stationary series [X (t) Y (t) ] 8.1.1 t=0,±1,… with X(t) r vector-valued and Y(t) s vector-valued. We assume the series (8.1.1) satisfies Assumption 2.6.1 and we define the means EX (t)= cX EY (t)= cY , 8.1.2 the covariances E { [X (t+u) − cX ] [X (t) − cX ]τ } = cXX (u) E { [X (t+u) − cX ] [Y (t) − cY ]τ } = cXY (u) E {[Y (t+u) − cY ] [Y (t) − cY ]τ } = cYY (u) u=0,±1,… , 8.1.3 and the second-order spectral densities fXX (λ) = (2π)−1 ∑ u=−∞ ∞ cXX (u) exp {−iλu} fXY (λ) = (2π)−1 ∑ u=−∞ ∞ cXY (u) exp {−iλu} fYY (λ) = (2π)−1 ∑ u=−∞ ∞ cYY (u) exp {−iλu} for−∞<λ<∞. 8.1.4

9.1 INTRODUCTION

In the previous chapter we considered the problem of approximating a stationary series by a linear filtered version of another stationary series. In this chapter we investigate the problem of approximating a series by a filtered version of itself, but restraining the filter to have reduced rank.

Specifically, consider the r vector-valued series X(t), t=0,±1,… with mean EX (t) = cX 9.1.1 absolutely summable autocovariance function E{ [X (t+u)− cX ] [X (t)− cX ]τ } = cXX (u) u=0,±1,… 9.1.2 and spectral density matrix fXX (λ) = (2π)−1 ∑ u=−∞ ∞ cXX (u) exp {−iλu} −∞<λ<∞. 9.1.3

Suppose we are interested in transmitting the values of the X(t) series from one location to another; however, only q⩽r channels are available for the transmission.

10.1 INTRODUCTION

In this chapter we consider the problem of approximating one stationary time series by a filtered version of a second series where the filter employed has reduced rank. Specifically consider the (r + s) vector-valued stationary series [X (t) Y (t) ] 10.1.1 t=0,±1,… with X(t) r vector-valued and Y(t) s vector-valued.

Suppose we are interested in reducing the series X(t) to be q vector-valued forming, for example, the series ζ (t) = ∑ u b (t−u) X (u) 10.1.2 t=0,±1,… with {b(u)} a q × r matrix-valued filter, and suppose we wish to do this so that the s vector-valued series Y ∗ (t) = ų ̜ + ∑ u c (t−u) ζ (u) 10.1.3 is near Y(t) for some s vector ų and s × q filter {c(u)}. If the series Y(t) were identical with X(t), then we would have the problem discussed in the previous chapter whose solution led to a principal component analysis of the spectral density matrix. If q=min (r,s) , then we are not requiring any real reduction in dimension and we have the multiple regression problem discussed in Chapter 8.

The back matter includes proofs of theorems, references, notation index, author index, subject index, and addendum