Autocorrelation or Serial Correlation

Whenever there is a time element in the regression analysis, there is a real danger of the dependent variable correlating with itself. In the literature of statistics, this phenomenon is termed autocorrelation or serial correlation in this text, we use the latter as descriptive of a situation in which the value, y is dependent on y, i, which, in turn, is dependent on y, 2. From a statistical perspective, this is problematic because the error term, e,-, is not independent—a requirement of the linear regression model. This interferes with least-squares calculation. 

The regression coefficients, bo and f i, although still unbiased, no longer have the minimum variance properties of the least-squares method for determining bo and b. Hence, the mean square error term MSg may be underestimated as well as both the standard error of bo, and the standard error of hi, si,. The confidence intervals discussed previously (Chapter 2), as well as the tests using the t and F distribution, may no longer be appropriate. 

Each Ci = y, — y, error term is a random variable that is assumed independent of all the other values. However, when the error terms are self- or autocorrelated, the error term is not but e, i + That is, e,- (error of the ith value) is composed of the previous error term e, i and a new value called a disturbance, do The di value is the independent error term with a mean of 0 and a variance of 1. 

Therefore, any time the y values are collected sequentially over time (x), the researcher must be on guard for serial correlation. The most common 

FIGURE 3.1 (a) Positive serial correlation of residuals. (The residuals change sign in gradual oscillation.) (b) Negative serial correlation of residuals. (The residuals bounce between positive and negative, but not randomly.) 

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