## What is the 'Durbin Watson Statistic'

The Durbin Watson statistic is a number that tests for autocorrelation in the residuals from a statistical regression analysis. The Durbin-Watson statistic is always between 0 and 4. A value of 2 means that there is no autocorrelation in the sample. Values from 0 to less than 2 indicate positive autocorrelation and values from more than 2 to 4 indicate negative autocorrelation.

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## BREAKING DOWN 'Durbin Watson Statistic'

Autocorrelation can be a significant problem in analyzing historical data if one does not know to look out for it. For instance, since stock prices tend not to change too radically from one day to another, the prices from one day to the next could potentially be highly correlated, even though there is little useful information in this observation. In order to avoid autocorrelation issues, the easiest solution in finance is to simply convert a series of historical prices into a series of percentage-price changes from day to day.

## Durbin Watson Statistic Calculation

The formula for the Durbin Watson statistic is rather complex, but involves the residuals from an ordinary least squares regression on a set of data. The following example illustrates how to calculate this statistic.

Assume the following (x,y) data points:

Pair one = (10, 1,100)

Pair two = (20, 1,200)

Pair three = (35, 985)

Pair four = (40, 750)

Pair five = (50, 1,215)

Pair six = (45, 1,000)

Using the methods of a least squares regression to find the "line of best fit", the equation for the best fit line of this data is:

Y = -2.6268x + 1,129.2

This first step in calculating the Durbin Watson statistic is to calculate the expected "y" values using the line of best fit equation. For this data set, the expected "y" values are:

Expected Y(1) = (-2.6268 x 10) + 1,129.2 = 1,102.9

Expected Y(2) = (-2.6268 x 20) + 1,129.2 = 1,076.7

Expected Y(3) = (-2.6268 x 35) + 1,129.2 = 1,037.3

Expected Y(4) = (-2.6268 x 40) + 1,129.2 = 1,024.1

Expected Y(5) = (-2.6268 x 50) + 1,129.2 = 997.9

Expected Y(6) = (-2.6268 x 45) + 1,129.2 = 1,011

Next, the differences of the actual "y" values versus the expected "y" values, the errors, are calculated:

Error(1) = (1,100 - 1,102.9) = -2.9

Error(2) = (1,200 - 1,076.7) = 123.3

Error(3) = (985 - 1,037.3) = -52.3

Error(4) = (750 - 1,024.1) = -274.1

Error(5) = (1,215 - 997.9) = 217.1

Error(6) = (1,000 - 1,011) = -11

Next these errors must be squared and summed:

Sum of errors squared = (-2.92 + 123.32 + -52.32 + -274.12 + 217.12 + -112) = 140,330.81

Next, the value of the error minus the previous error are calculated and squared:

Difference(1) = (123.3 - (-2.9)) = 126.2

Difference(2) = (-52.3 - 123.3) = -175.6

Difference(3) = (-274.1 - (-52.3)) = -221.9

Difference(4) = (217.1 - (-274.1)) = 491.3

Difference(5) = (-11 - 217.1) = -228.1

Sum of differences square = 389,406.71

Finally, the Durbin Watson statistic is the quotient of the squared values:

Durbin Watson = 389,406.71 / 140,330.81 = 2.77

A rule of thumb is that test statistic values in the range of 1.5 to 2.5 are relatively normal. Any value outside this range could be a cause for concern. The Durbin–Watson statistic, while displayed by many regression analysis programs, is not applicable in certain situations. For instance, when lagged dependent variables are included in the explanatory variables, then it is inappropriate to use this test.

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