# Autocorrelation An Introduction (2020)

The term autocorrelation may be defined as a “correlation between members of a series of observations ordered in time (as in time series data) or space (as in cross-sectional data)”. Autocorrelation is most likely to occur in time-series data. In the regression context, the CLRM assumes that covariances and correlations do not exist in the disturbances $u_i$. Symbolically,

$$Cov(u_i, u_j | x_i, x_j)=E(u_i u_j)=0, \quad i\ne j$$

In simple words, the disturbance term relating to any observation is not influenced by the disturbance term relating to any other observation. In other words, the error terms $u_i$ and $u_j$ are independently distributed (serially independent). If there are dependencies among disturbance terms, then there is a problem of autocorrelation. Symbolically,

$$Cov(u_i,u_j|x_i, x_j) = E(u_i, u_j) \ne 0,\quad i\ne j$$

Suppose, we have disturbance terms from two different time series say $u$ and $v$ such as $u_1, u_2, \cdots, u_{10}$, and $v_1,v_2,\cdots, v_{11}$, then the correlation between these two different time series is called serial correlation (that is, the lag correlation between two series).

Suppose, we have two-time series $u$ ($u_1,u_2,\cdots, u_{10}$) and the lag values of this series are $u_2, u_3,\cdots, u_{12}$, then the correlation between these series is called auto-correlation (that is the lag correlation of a given series with itself, lagged by a number of time units).

The use of OLS to estimate a regression model results in BLUE estimates of the parameters only when all the assumptions of the CLRM are satisfied. After performing regression analysis one may plot the residuals to observe some patterns when results are not according to prior expectations.

### Plausible Patterns of Autocorrelation

Some plausible patterns of autocorrelation and non-autocorrelation are:

Figure $a$–$d$ shows that there is a discernible (قابل دریافت، عیاں، قابل فہم) pattern among the $u$’s.
ٖFigure (a) shows a cyclical pattern.
Figure (b) suggests an upward linear trend in the disturbances
Figure (c) suggests a downward linear trend in the disturbances
Figure (d) indicates both the linear and quadratic trend terms are present in the disturbances
Figure (e) shows no systematic pattern. Therefore, supporting the assumption of CLRM of no autocorrelation.

The importance of autocorrelation can be described as follows:

• Identifying Patterns: Autocorrelation measures the correlation between a variable and its lagged versions, essentially checking how similar past values are to present values. Therefore, it helps identify trends or seasonality within the data. For instance, positive auto-correlation in stock prices might suggest momentum, where recent gains could indicate continued increase.
• Validating Models: Many statistical models, especially related to time series forecasting, assume independence between errors term. Autocorrelation helps to assess this assumption. If data exhibits autocorrelation, it can mislead the model, and further may lead to inaccurate forecasts. Accounting for autocorrelation through appropriate techniques improves model accuracy.
• Understanding Dynamic Systems: Presence of auto-correlation indicates that the dependence of system on its past states. This is valuable in various fields, like finance or engineering, where system behavior is influenced by its history.

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