Serial Correlation

Serial correlation exists when successive values of a time-series process exhibit nonzero covariances. The term serial correlation is often used synonymously with autocorrelation, which refers to the more general case of correlation with self and may be either over time, as when the president's approval rating today is correlated with his approval rating yesterday (serial correlation), or across space, as when gang membership is correlated with physical distance of residence from an inner city (spatial correlation). Spatial correlation is less pervasive than serial correlation and is seldom discussed at any length in textbook treatments of autocorrelation. Serial correlation is a common feature of time-series processes and has implications for dynamic model specification. The presence of serial (or spatial) correlation in the errors of a regression model violates the Gauss-Markov regression assumptions and invalidates inference from OLS estimates.

A time-series process is serially correlated if its value at time t is a linear function of some number, p, of its own past values:

If yt is correlated only with yt−1, then we say that we have first-order serial or autocorrelation, often denoted ρ1. If yt is correlated with yt−1 and also with yt−2, we have second-order autocorrelation, denoted ρ2, and so on. The autocorrelations themselves, the ρ, give the strength of correlation. Serial correlation does not depend on the time period itself, only on the interval between the time periods—the length of the lag—for which the correlation is being assessed. The correlation between yt and yt is perfect or 1 (ρ0 = 1). The estimated serial correlations give us a sense of the nature and extent of the time dependence inherent in the yt process. For first-order serial correlation, by far the most common case, the correlations at successive lags will decay at a rate determined by ρ1, with large correlations indicating stronger time dependence and slower decay.

If a series is covariance stationary (the mean, variance, and lagged covariances are constant over time), we can define the autocorrelations between yt and yt−p as the covariance between yt and yt−p divided by the autocovariance (variance) of yt. Furthermore, for a stationary series, the correlations range from −1.0 to 1.0, with larger magnitudes characteristic of stronger over-time dependence.

Examples

Typically, social processes will exhibit positive rather than negative serial correlation: Successive [Page 1028]observations will tend to move in the same direction—either up or down—over time. Consider some examples.

• Government or agency budgets are based on the previous budget(s) and are not created a new each year, so that changes in the process are incremental. Correlations in adjacent periods will be strong and positive.
• The aggregate partisan balance (percent of Democrats, for example) tends to move only sluggishly, presumably because features of the political system keep the balance of partisanship from varying wildly over time. This produces strong positive serial correlation in the partisan balance.
• The percent of women working in professional or managerial positions has tended to increase slowly over time as mean levels of education of women increased, producing strong positive correlations between adjacent values.

In contrast, negative serial correlation occurs when the current value of a process is related to its past in such a way that the signs on the b above oscillate. A process that adjusts over time for some sort of error in the past realization would create negative correlation, where in one time period, the process goes up, and then in the next, it goes down.

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