Serial correlation (or autocorrelation) occurs when the error term is correlated **E ( ε**_{i} ε_{j} ≠ 0 across observations in a linear regression . When that is the case, standard errors and test statistics in a regression output will be incorrect unless they are adjusted for serial correlation.

Serial correlation generally arises in time-series regressions. There are two kinds of serial correlation:

When a positive error for one observation increases the chance of a positive error for another observation, and a negative error for one observation increases the chance of a negative error for another observation, we say that they are **positively serially correlated**.

When a positive error for one observation increases the chance of a negative error for another observation, and a negative error for one observation increases the chance of a positive error for another observation, we say that they are **negatively serially correlated**.

The most common test for serial correlation in a regression model is based on the Durbin-Watson statistic (1951). In this test, the number of degrees of freedom equal to the number of independent variables (K) in the regression.

\[ DW = \sum_{t=2}^T (\hat{ε}_t - \hat{ε}_t - 1 )^2 / \sum_{t=1}^T \hat{ε}^2_t \]

where, ε^_{t} is the regression residual for period t.

The Durbin-Watson statistic values ranges from 0 (in the case of serial correlation of +1) to 4 (in the case of serial correlation of -1). If the errors are homoskedastic and not serially correlated, then the DW statistic will be close to 2.

The calculated DW test statistic is the basis for deciding whether or not to reject the null hypothesis, comparing it with specified rejection or critical points. The comparison values we choose are based on the level of significance selected.

We can formulate the following sets of hypothesis.

**H**_{0}: No Serial Correlation versus H_{a}: Significant Positive Serial Correlation

If we find that the calculated value of the test statistics is less than DW critical lower value (dl), we reject the null hypothesis. When the DW statistic falls between dl and du *(upper value)*, the test results are inconclusive.

** H**_{0}: No Serial Correlation versus H_{a}: Significant Negative Serial Correlation

If we find that the calculated value of the test statistics is greater than DW critical (4- dl), we reject the null hypothesis. When the DW statistic falls between (4 - dl) and (4 - du), the test results are inconclusive.