1. Graphical Analysis

808 PART FOUR: SIMULTANEOUS-EQUATION MODELS

2. Autocorrelation Function (ACF) and Correlogram

One simple test of stationarity is based on the so-called autocorrelation function (ACF). The ACF at lag k, denoted by pk, is defined as

covariance at lag k variance where covariance at lag k and variance are as defined before. Note that if k = 0, po = 1 (why?)

Since both covariance and variance are measured in the same units of measurement, pk is a unitless, or pure, number. It lies between — 1 and + 1, as any correlation coefficient does. If we plot pk against k, the graph we obtain is known as the population correlogram.

Since in practice we only have a realization (i.e., sample) of a stochastic process, we can only compute the sample autocorrelation function (SAFC), pk. To compute this, we must first compute the sample covariance at lag k, yk, and the sample variance, Yo, which are defined as18

n where n is the sample size and Y is the sample mean.

Therefore, the sample autocorrelation function at lag k is

Yo which is simply the ratio of sample covariance (at lag k) to sample variance. A plot of pk against k is known as the sample correlogram.

How does a sample correlogram enable us to find out if a particular time series is stationary? For this purpose, let us first present the sample correlo-grams of a purely white noise random process and of a random walk process. Return to the driftless RWM (21.3.13). There we generated a sample of 500 error terms, the u's, from the standard normal distribution. The correlogram of these 500 purely random error terms is as shown in Figure 21.6; we have shown this correlogram up to 30 lags. We will comment shortly on how one chooses the lag length.

For the time being, just look at the column labeled AC, which is the sample autocorrelation function, and the first diagram on the left, labeled

18Strictly speaking, we should divide the sample covariance at lag k by (n — k) and the sample variance by (n — 1) rather than by n (why?) where n is the sample size.

CHAPTER TWENTY-ONE: TIME SERIES ECONOMETRICS: SOME BASIC CONCEPTS 809

Sample: 2 500 Included observations: 499

Autocorrelation

Partial Correlation

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