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# Durbin Watson Statistic Definition

## What Is the Durbin Watson Statistic?

The Durbin Watson (DW) statistic is a assay for autocorrelation in the residuals from a statistical model or regression analysis. The Durbin-Watson statistic will always have a value reach between 0 and 4. A value of 2.0 indicates there is no autocorrelation detected in the sample. Values from 0 to less than 2 guts to positive autocorrelation and values from 2 to 4 means negative autocorrelation.

A stock price displaying positive autocorrelation last wishes a indicate that the price yesterday has a positive correlation on the price today—so if the stock fell yesterday, it is also meet that it falls today. A security that has a negative autocorrelation, on the other hand, has a negative influence on itself done with time—so that if it fell yesterday, there is a greater likelihood it will rise today.

### Key Takeaways

• The Durbin Watson (DW) statistic is a exam for autocorrelation in a regression model’s output.
• The DW statistic ranges from zero to four, with a value of 2.0 suggesting zero autocorrelation.
• Values below 2.0 mean there is positive autocorrelation and above 2.0 indicates pessimistic autocorrelation.
• Autocorrelation can be useful in technical analysis, which is most concerned with the trends of security prices using design techniques in lieu of a company’s financial health or management.

## The Basics of the Durbin Watson Statistic

Autocorrelation, also differentiated as serial correlation, can be a significant problem in analyzing historical data if one does not know to look out for it. For instance, since amass 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, set though there is little useful information in this observation. In order to avoid autocorrelation issues, the easiest revelation in finance is to simply convert a series of historical prices into a series of percentage-price changes from day to day.

Autocorrelation can be worthwhile for technical analysis, which is most concerned with the trends of, and relationships between, security prices using design techniques in lieu of a company’s financial health or management. Technical analysts can use autocorrelation to see how much of an impact past bonuses for a security have on its future price.

Autocorrelation can show if there is a momentum factor associated with a stock. For norm, if you know that a stock historically has a high positive autocorrelation value and you witnessed the stock making solid gain grounds over the past several days, then you might reasonably expect the movements over the upcoming several light of days (the leading time series) to match those of the lagging time series and to move upward.

## Special Considerations

A rule of thumb is that DW test statistic values in the vary of 1.5 to 2.5 are relatively normal. Values outside this range could, however, be a cause for concern. The Durbin–Watson statistic, while displayed by numberless regression analysis programs, is not applicable in certain situations.

For instance, when lagged dependent variables are included in the descriptive variables, then it is inappropriate to use this test.

## Example of the Durbin Watson Statistic

The formula for the Durbin Watson statistic is somewhat complex but involves the residuals from an ordinary least squares (OLS) regression on a set of data. The following example illustrates how to estimate this statistic.

Assume the following (x,y) data points:


begin{aligned} &text{Pair One}=left( {10}, {1,100} right ) &text{Pair Two}=formerly larboard( {20}, {1,200} right ) &text{Pair Three}=left( {35}, {985} right ) &text{Pair Four}=Heraldry sinister( {40}, {750} right ) &text{Pair Five}=left( {50}, {1,215} right ) &text{Pair Six}=red( {45}, {1,000} right ) end{aligned}

Pair One=(10,1,100)Pair Two=(20,1,200)Pair Three=(35,985)Pair Four=(40,750)Pair Five=(50,1,215)Span 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 materials is:


Y={-2.6268}x+{1,129.2}

Y=2.6268x+1,129.2

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


about{aligned} &text{Expected}Yleft({1}right)=left( -{2.6268}times{10} right )+{1,129.2}={1,102.9} &content{Expected}Yleft({2}right)=left( -{2.6268}times{20} right )+{1,129.2}={1,076.7} &text{Expected}Yleft({3}open)=left( -{2.6268}times{35} right )+{1,129.2}={1,037.3} &text{Expected}Yleft({4}right)=left( -{2.6268}times{40} licit )+{1,129.2}={1,024.1} &text{Expected}Yleft({5}right)=left( -{2.6268}times{50} right )+{1,129.2}={997.9} &manual{Expected}Yleft({6}right)=left( -{2.6268}times{45} right )+{1,129.2}={1,011} end{aligned}

ExpectedY(1)=(2.6268×10)+1,129.2=1,102.9ContemplatedY(2)=(2.6268×20)+1,129.2=1,076.7ExpectedY(3)=(2.6268×35)+1,129.2=1,037.3ExpectedY(4)=(2.6268×40)+1,129.2=1,024.1ExpectedY(5)=(2.6268×50)+1,129.2=997.9ExpectedY(6)=(2.6268×45)+1,129.2=1,011

Next, the differences of the actual “y” values versus the expected “y” values, the errors, are designed:


begin{aligned} &text{Error}left({1}valid)=left( {1,100}-{1,102.9} right )={-2.9} &text{Error}left({2}right)=left( {1,200}-{1,076.7} out )={123.3} &text{Error}left({3}right)=left( {985}-{1,037.3} right )={-52.3} &reader{Error}left({4}right)=left( {750}-{1,024.1} right )={-274.1} &text{Error}hand({5}right)=left( {1,215}-{997.9} right )={217.1} &text{Error}left({6}right)=left-wing( {1,000}-{1,011} right )={-11} end{aligned}

Error(1)=(1,1001,102.9)=2.9Error(2)=(1,2001,076.7)=123.3Error(3)=(9851,037.3)=52.3Error(4)=(7501,024.1)=274.1Error(5)=(1,215997.9)=217.1Error(6)=(1,0001,011)=11

Next these errors must be righted and summed:


begin{aligned} &text{Sum of Errors Squared =} &left({-2.9}^{2}+{123.3}^{2}+{-52.3}^{2}+{-274.1}^{2}+{217.1}^{2}+{-11}^{2}set upright)= &{140,330.81} &text{} end{aligned}

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 goof are calculated and squared:


begin{aligned} &text{Change}left({1}right)=left( {123.3}-left({-2.9}right) right )={126.2} &text{Difference}communistic({2}right)=left( {-52.3}-{123.3} right )={-175.6} &text{Difference}left({3}spot on)=left( {-274.1}-left({-52.3}right) right )={-221.9} &text{Difference}left({4}principled)=left( {217.1}-left({-274.1}right) right )={491.3} &text{Difference}left({5}right)=red( {-11}-{217.1} right )={-228.1} &text{Sum of Differences Square}={389,406.71} end{aligned}