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Autocorrelation: Causes and Consequences

Introduction

In CLRM, there are several assumptions are made which are difficult to fulfil in every data or regressional model. This article examines one of these assumptions that “Independence of the error terms”. Specifically, it will critically look at the following aspects.

            1.  Understanding of the problem of autocorrelation.

                  2.     Causes of autocorrelation

               3.     Consequences of autocorrelation

 What is Autocorrelation?

The term autocorrelation means “the correlation between the error term”. Or in other words, it is the violation of the one of crucial assumption of the OLS  that, the correlation between the two disturbance terms must be zero. Symbolically

\[cov(u{i},u_{j}|X_{i},X_{j})=0\]     (Equation (1)

Where cov means covariance and I and j shows two different observations.

The autocorrelation occurs when distortion in one time period affects the output in the next period. In such a situation, the assumption of the independent error terms violate, symbolically,

\[E(u{i},u_{j})\not\equiv 0\]     Equation (2)

Different terminologies for correlation between the disturbance terms

There are some different terminologies for this phenomenon. Autocorrelation is the correction between the disturbance term of the same series (as shown in equation 2). Whereas, serial correlation is the correlation between the disturbance term of two different series (as shown in equation 3).

\[E(u{i},v_{i})\not\equiv 0\]     Equation (3)

The autocorrelation is an issue of the time-series data. But there is an also possibility of the dependence in the cross-sectional data termed as “Spatial Autocorrelation”. Spatial correlation in the correlation between the space rather than time.

 Causes of Autocorrelation

There can be different causes of autocorrelation.

Intertia

The variables that follow the business cycle, for example, show an upward trend in recovery and downstream in recession. Thus, interdependence is likely to occur in such time-series data.

 Specification Bias

If some important variable is excluded from the model known as “Excludibility bias” that excluded variable appears in the error terms thus start following the specific pattern leads to autocorrelation.

Also, specification bias from wrong functional foam can create the problem of autocorrelation

  Cobweb Phenomenon

Cobweb is the phenomenon usually occurs in the agricultural sector in which supply reacts to prices changes with a lag of one period. Due to which disturbance term represents a systematic pattern

 Manipulation of Data

Manipulation of the data that smoothen the raw time series data often smooths the disturbance terms thereby introduce autocorrelation

Data Transformation

Consider the equation

\[Y_{t}=\beta_{1}+\beta_{2}X_{t}+u_{t}\]       Equation (4)

As this equation holds for every period. Taking a lag of one period, the new equation will be

\[Y_{t-1}=\beta_{1}+\beta_{2}X_{t-1}+u_{t-1}\]   Equation (5)

By subtracting equation 5 from 4, the resultant equation will be,

\[\Delta Y_{t}=\beta _{2}\Delta X_{t}+\Delta u_{t}\] Equation (6)

Or   \[\Delta Y_{t}=\beta _{2}\Delta X_{t}+v_{t}\]    Equation (7)

Where \[u_{t}-u_{t-1}=\Delta u_{t}=v_{t}\]

If disturbance term in equation 4 exhibit no autocorrelation then the error term in equation 6 is autocorrelated.

Non-Stationarity

If the dependent and independent variables are non-stationary and their corresponding error term would be stationary, thus, autocorrelation prevails.

Consider the following equation

\[u_{t}=\rho u_{t-1}+\varepsilon _{t}\]      Equation (8)

The stochastic error term satisfies the following OLS assumptions.

\[E(\varepsilon_{t})=0\]

\[var(\varepsilon_{t})=\sigma ^{2}\]

\[cov(\varepsilon _{t},\varepsilon _{t+s})=0\]

where, \[s\not\equiv 0\]

Equation (8) is known as Markov First-order Autoregressive Scheme or first-order autoregressive scheme\[\rho\] is the autocorrelation coefficient. It lies between -1 and +1. If the absolute value of  \[\rho\] is less than 1 equation (8) is stationary otherwise not. If \[|\rho |< 1\]then the disturbance term is homoscedastic but it is correlated.

Consequences of Autocorrelation 

In such a situation, by using the OLS will lead to the following consequences

1.       1.   Underestimation of true variance.

2.      2.       Overestimation of \[R^{2}\]

3.       3.        Underestimation of the variances of coefficients.

4.      4.         No longer validity of Fand T-test.

“Thus in the presence of autocorrelation, one should use GLS rather than OLS as it provides BLUE estimators"

 Conclusion

Autocorrelation is the fundamental issue of the existence of a correlation between the error terms. It can self exist or different econometric issue or function can lead to this problem. In the existence of autocorrelation, the OLS estimators remain no longer BLUE, thus, GLS is used to get efficient and reliable estimates.

Also Read:
Remedial Measures to Heteroscedasticity
Heteroscedasticity - Causes, Consequences and Detection
Multicollinearity Causes, Detection, Consequences and Remedial Measures

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