Covarince Stationary Time Series

(Chapter 6, Section 1)

The cyclical component of a time series is, in contrast to our assumptions about the trend and seasonal components, a “stochastic” (vs. “deterministic”) process. That is, its time path is, to some extent, fundamentally unpredictable.

Our model of that component must take this into account.

We will assume that the cyclical component of a time series can be modeled as a covariance stationary time series.

What does that mean?

Consider our data sample, y1,…,yT

We imagine that these observed y’s are the outcomes of drawings of random variables.

In fact, we imagine that this data sample or sample path, is just part of a sequence of drawings of random variables that goes back infinitely far into the past and infinitely far forward into the future.

In other words, “nature” has drawn values of yt for t = 0, + 1, +2,… :

{…,y-2,y-1,y0,y1,y2,…}

This entire set of drawings is called a realization of the time series. The data we observe, {y1,…,yT} is part of a realization of the time series.

(This is a model. It is not supposed to be interpreted literally!)

We want to describe the underlying probabilistic structure that generated this realization and, more important, the probabilistic structure that governs that part of the realization that extends beyond the end of the sample period (since that is part of the realization that we want to forecast).

Ideally, there are key elements of this probability structure that remain fixed over time. This is what will enable us to use the sample path to use inferences drawn about the proababilistic structure that generated the sample to draw inferences about the probabilistic structure that will generate future values of the series.

For example, suppose the time series is a sequence of 0’s and 1’s corresponding to the outcomes of a sequence of coin tosses, with H = 0 and T = 1: {…0,0,1,0,1,1,1,0,…}

What is the probability that at time T+1 the value of the series will be equal to 0?

If the same coin is being tossed for every t, then the probability of tossing an H at time T+1 is the same as the probability of tossing an H at times 1,2,…,T. What is that probability? A good estimate would be the number of H’s observed at times 1,…,T divided by T. (By assuming that future probabilities are the same as past probabilities we are able to use the sample information to draw inferences about those probabilities.)

Suppose however that a different coin will be tossed at T+1 than the one that was tossed in the past. Then, our data sample will be of no help in estimating the probability of an H at T+1. All we can do is make a blind guess!

Covariance stationarity refers to a set of restrictions/conditions on the underlying probability structure of a time series that has proven to be very especially valuable in this regard.

A time series, yt, is said to be covariance stationary if it meets the following conditions –

Constant mean
Constant (and finite) variance
Stable autocovariance function

1. Constant mean

Eyt = μ {vs. μt}

for all t. That is, for each t, yt is drawn from a population with the same mean.

Consider, for example, a sequence of coin tosses and set yt = 0 if H at t and yt = 1 if T at t. If Prob(T)=p for all t, then Eyt = p for all t.

Consider, for example, the HEPI time series.

Does that time series look like a time series whose values have been drawn from a population with a constant mean?
What about the deviations from trend?
What about the growth rate?

What about your detrended series?

Caution – the conditions that define covariance stationarity refer to the underlying probability distribution that generated the data sample rather than to the sample itself. However, the best we can do to assess whether these conditions hold is to look at the sample and consider the plausibility that this sample was drawn from a stationary time series.

2. Constant Variance

Var(yt) = E[(yt- μ)2] = σ2 {vs. σ2t} for all t.

The dispersion of the value of yt around its mean is constant over time.A sample in which the dispersion of the data around the sample mean seems to be increasing or decreasing over time is not likely to have been drawn from a time series with a constant variance.

Consider, for example, a sequence of coin tosses and set yt = 0 if H at t and yt = 1 if T at t. If Prob(T)=p for all t, then

Eyt = p

and

Var(yt) = E[(yt – p)2] = p(1-p)2+(1-p)p2

for all t.

Detrended HEPI? HEPI growth rate? Your detrended series?

Digression on covariance and correlation

Recall that

Cov(X,Y) = E[(X-EX)(Y-EY)]

and

Corr(X,Y) = Cov(X,Y)/[Var(X)Var(Y)]1/2

measure the relationship between the random variables X and Y.

A positive covariance means that when

X > EX, Y will tend to be greater than EY (and vice versa). A negative covariance means that when X > EX, Y will tend to be less than EY (and vice versa).

The correlation between X and Y will have the same sign as the covariance but its value will lie between -1 and 1. The stronger the relationship between X and Y, the closer their correlation will be to 1 (or, in the case of negative correlation, -1).

If the correlation is 1, X and Y are perfectly positively correlated. If the correlation is -1, X and Y are perfectly negatively correlated. X and Y are uncorrelated if the correlation is 0.

Independent random variables are uncorrelated. Uncorrelated random variables are not necessarily independent.

End Digression

3. Stable autocovariance function

The autocovariance function of a time series refers to covariances of the form:

Cov(yt,ys) = E[(yt - Eyt)( ys – Eys)]

i.e., the covariance between the drawings of yt and ys.

Note that

Cov(yt,yt) = Var(yt)
Cov(yt,ys) = Cov(ys,yt)

For instance,

Cov(yt,yt-1)

measures the relationship between yt and

yt-1.

Your textbook refers to this as the autocovariance at displacement 1.

We expect that Cov(yt,yt-1) > 0 for most economic time series: if an economic time series is greater than normal in one period it is likely to be above normal in the subsequent period – economic time series tend to display positive first order correlation.

In the coin toss example (yt = 0 if H, yt = 1 if T), what is Cov(yt,yt-1)?

What about the sign of Cov(yt,yt-1) for the detrended HEPI?

Suppose that

Cov(yt,yt-1) = γ(1) for all t

where γ(1) is some constant.

That is, the autocovariance at displacement 1 is the same for all t:

…Cov(y2,y1)=Cov(y3,y2)=…=Cov(yT,yT-1)=…

In this special case, we might also say that the autocovariance at displacement 1 is stable over time.

For example, in the coin toss example,

Cov(yt,yt-1) = γ(1) = 0 for all t

The third condition for covariance stationarity is that the autocovariance function is stable at all displacements.

That is –

Cov(yt,yt-τ) = γ(τ) for all integers t and τ

The covariance between yt and ys depends only t and s only through t-s (how far apart they are in time) not on t and s themselves (where they are in time).

Cov(y1,y3)=Cov(y2,y4) =…=Cov(yT-2,yT) = γ(2)

and so on.

It is hard to make this condition intuitive. The best that I can offer – If we break the entire time series up into different segments, the general behavior of the series looks roughly the same for each segment.

Notes –

stability of the autocovariance function (condition 3) actually implies a constant variance (condition 2); set τ = 0.

γ(τ) = γ(-τ) since

γ(τ) = Cov(yt,yt-τ) = Cov(yt- τ,yt) = γ(-τ)

If yt is an i.i.d. sequence of random variables then it is a covariance stationary time series.

{The “identical distribution” means that the mean and variance are the same for all t. The independence assumption means that γ(τ) = 0 for all nonzero τ, which implies a stable autocovariance function.}

The conditions for covariance stationary are the main conditions we will need regarding the stability of the probability structure generating the time series in order to be able to use the past to help us predict the future.

It is important to note that these conditions do not

imply that the y’s are identically distributed (or, independent).
place restrictions on third, fourth, and higher order moments (skewness, kurtosis,…). [Covariance stationarity only restricts the first two moments and so it also referred to as “second-order stationarity.”]