Introduction to autocorrelation

Time series basically refer to time oriented data. It is applicable in many cases, such as in economics and weather forecasting.

The assumption of independent errors that we have when analyzing non- time oriented data is usually appropriated for time series data. Errors in time series can have what we call autocorrelation:  errors are no longer independent.

The errors can be positively correlated, which is more common, or negative correlated.  In positively correlated series if one variable increase the other will increase, and if one decrease, the other will also decrease. The correlation is in the same direction.  In contrast, negative autocorrelation is characterized by opposite tendencies between the variables.

The presence of autocorrelation has several implications in our models:

1.  Regression coefficients are no longer minimum-variance.
2. In case of positively correlated errors we can have an estimate of the variability that is underestimated
3. Hypothesis tests and confidence intervals are no longer exact: confidence and prediction intervals they are shorter than they really should be, and hypothesis tests may indicate that some predictors are important to the model when they really are not.

There are three main approaches to remedy the autocorrelation problem:

1. Missing predictors can be identified and included in the model
2. Generalized least squares can be used in case the autocorrelation structure is known
3. Use a model that specifically incorporates the autocorrelation structure