Serial Correlation

Multiple Regression Model

Let the multiple regression model be

$$y_i=x_i\beta +{\epsilon }_i,i=1,2,\text{}\dots n,$$

where yi is the response and xi is a 1 × (k + 1) vector consisting of a 1 and the values of the k predictors in the ith observation. The assumptions of this model are (a) the expectation of the error $\epsilon$i is 0; (b) $\epsilon$i has constant variance $\sigma$2; and (c) $\epsilon$i and $\epsilon$j are uncorrelated if i ≠ j. Let $\widehat{\beta }$ be the least squares (LS) estimator of β and SSE be the sum of squared errors. When the assumptions are valid, $\widehat{\beta }$ is the best linear unbiased estimator and $s^2=$ $SSE/(n-k-1)$ is an unbiased estimator of σ2. When the errors are correlated (violating assumption c), although $\widehat{\beta }$ is still unbiased, s2 and the estimated standard error of $\widehat{\beta }$ are biased. Consequently, the F or t statistic in testing the significance of $\widehat{\beta }$ is misleading. Therefore, it is important to test for the presence of serial correlation. The most widely used test is the DurbinWatson d test, which tests for first-order serial correlation $\rho$ = Corr($\epsilon$i, $\epsilon$i1) using the autoregressive model

$${\epsilon }_i=\eta {\epsilon }_{i-1}+e_i,-1\text{ <}\eta <1.$$

Obviously, $\rho$ = 0 if η = 0. To test the null hypothesis H0 : η = 0, the test statistic d is formulated as

$$d={\displaystyle \sum }_{i=2}^n({\widehat{\epsilon }}_i-{\widehat{\epsilon }}_{i-1})2/{\displaystyle \sum }_{i=1}^n{\widehat{\epsilon }}_i^2,$$

where ${\widehat{\epsilon }}_i$ are the LS residuals with fitting Equation 1. Let $\widehat{\eta }={\displaystyle {\sum }_{i=2}^n{\widehat{\epsilon }}_i}{\widehat{\epsilon }}_{i-1}/{\displaystyle {\sum }_{i=2}^n{\widehat{\epsilon }}_{i-1}^2}$ be the LS estimate of · in Equation 2. It follows from ${({\widehat{\epsilon }}_i-{\widehat{\epsilon }}_{i-1})}^2={\widehat{\epsilon }}_i^2-2{\widehat{\epsilon }}_i{\widehat{\epsilon }}_{i-1}+{\widehat{\epsilon }}_{i-1}^2$ that $d\approx 2(1-\widehat{\eta })$. Thus, if there is no serial correlation, d $\approx$ 2; if the serial correlation is close to 1, d $\approx$ 0; and if the serial correlation is close to 1, d $\approx$ 4. The critical values of d (denote the lower bound as dL and the upper bound as dU) depend on n, k, and the significance level of the test. Tables of dU and dL can be found in the Appendix of Arnold Studenmund’s book. The appropriate decision rules of testing for positive serial correlation are

$$\begin{array}{l}\text{reject}{H}_{\text{0}}\text{if}d<d_L;\\ \text{do}\text{not}\text{reject}{H}_{\text{0}}\text{if}d>d_U;\\ \text{inconclusive}\text{otherwise}.\end{array}$$

The decision rules of testing for negative serial correlation are

$$\begin{array}{l}\text{reject}{H}_{\text{0}}\text{if}d>4-d_L;\\ \text{do}\text{not}\text{reject}{H}_{\text{0}}\text{if}<4-d_U;\\ \text{inconclusive}\text{otherwise}.\end{array}$$

The inconclusive region is one main disadvantage of the DurbinWatson d test. Moreover, the test ignores serial correlation beyond the first order. It does not allow earlier observed y to predict later y in the regression model either.

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