Statistics Definitions > Serial Correlation / Autocorrelation

## What is Serial Correlation / Autocorrelation?

Serial correlation (also called Autocorrelation) is where error terms in a time series transfer from one period to another. In other words, the error for one time period *a* is correlated with the error for a subsequent time period *b*. For example, an underestimate for one quarter’s profits can result in an underestimate of profits for subsequent quarters. **This can result in a myriad of problems**, including:

- Inefficient Ordinary Least Squares Estimates and any forecast based on those estimates. An
*efficient estimator*gives you the most information about a sample; inefficient estimators can perform well, but require much larger sample sizes to do so. - Exaggerated goodness of fit (for a time series with positive serial correlation and an independent variable that grows over time).
- Standard errors that are too small (for a time series with positive serial correlation and an independent variable that grows over time).
- T-statistics that are too large.
- False positives for significant regression coefficients. In other words, a regression coefficient appears to be statistically significant when it is not.

## Types of Autocorrelation

The most common form of autocorrelation is **first-order serial correlation**, which can either be positive or negative.

- Positive serial correlation is where a positive error in one period carries over into a positive error for the following period.
- Negative serial correlation is where a negative error in one period carries over into a negative error for the following period.

**Second-order serial correlation **is where an error affects data two time periods later. This can happen when your data has seasonality. Orders higher than second-order do happen, but they are rare.

## Testing for Autocorrelation

You can test for autocorrelation with:

- A plot of residuals. Plot e
_{t}against t and look for clusters of successive residuals on one side of the zero line. You can also try adding a Lowess line, as in the image below. - A Durbin-Watson test.
- A Lagrange Multiplier Test.
- Ljung Box Test.
- A correlogram. A pattern in the results is an indication for autocorrelation. Any values above zero should be looked at with suspicion.
- The Moran’s I statistic, which is similar to a correlation coefficient.

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