Autocorrelation: an Introduction

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 and Non-Autocorrelation

Some plausible patterns of autocorrelation and non-autocorrelation are:

Patterns of Autocorrelation and Non-Autocorrelation

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.

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