Unveiling the Durbin Watson Statistic
The Durbin Watson (DW) statistic serves as a critical test for autocorrelation in the residuals arising from a statistical model or regression analysis. The DW statistic ranges from 0 to 4, where a value of 2.0 implies no autocorrelation. Values below 2.0 signal positive autocorrelation, while higher values up to 4 indicate negative autocorrelation.
Suppose a stock price shows positive autocorrelation. This means that today’s price is positively correlated with yesterday’s price—hinting that the stock, if it falls today, is likely to fall tomorrow too. Conversely, negative autocorrelation implies an inverse relationship, where a fall in the stock price today increases the likelihood of a rise tomorrow.
Key Insights
- The Durbin Watson statistic tests for autocorrelation in regression model outputs.
- A DW statistic ranges from zero to four, with 2.0 suggesting no autocorrelation.
- Values below 2.0 indicate positive autocorrelation, while above 2.0 points to negative autocorrelation.
- Autocorrelation can be crucial in technical analysis, focusing on security price trends and using charting techniques in lieu of the financial health or management of a company.
Essentials of the Durbin Watson Statistic
Autocorrelation, also known as serial correlation, can be a significant issue in analyzing historical data. For example, stock prices typically do not vary radically from day to day, leading them to appear highly correlated while offering little meaningful information. To avoid autocorrelation, converting historical prices into day-to-day percentage changes is a simple yet effective strategy in finance.
In technical analysis, which emphasizes trends and relationships between securities using charting techniques over a company’s specific financial metrics, autocorrelation plays a key role. Technical analysts leverage it to ascertain past prices’ impact on future values. Autocorrelation can reveal if a stock has a momentum factor. For instance, if a stock historically shows high positive autocorrelation and has recently gained solidly, one might predict that this upward trend will likely continue.
The Durbin Watson statistic is named after statisticians James Durbin and Geoffrey Watson.
Important Considerations
A DW test statistic ranging between 1.5 and 2.5 is generally viewed as normal. Values outside this span could indicate potential concerns. However, the Durbin Watson statistic, often displayed in regression analysis outputs, isn’t always applicable. For instance, when lagged dependent variables are part of the explanatory variables, this test becomes inappropriate.
Example of Calculating the Durbin Watson Statistic
The Durbin Watson statistic’s formula is complex, involving residuals from an ordinary least squares (OLS) regression on a data set. Here’s a step-by-step calculation guide to clarify this process:
Initial Data Set Example
Assume the following data points \((x, y)\):
1Pair One: (10, 1,100)
2Pair Two: (20, 1,200)
3Pair Three: (35, 985)
4Pair Four: (40, 750)
5Pair Five: (50, 1,215)
6Pair Six: (45, 1,000)
Using least squares regression methods to find the line of best fit, we derive the following equation:
1Y = -2.6268x + 1,129.2
Expected Y-Values Calculation
The first step involves computing the expected y values using the best fit line:
1Expected Y(1) = (-2.6268 * 10) + 1,129.2 = 1,102.9
2Expected Y(2) = (-2.6268 * 20) + 1,129.2 = 1,076.7
3Expected Y(3) = (-2.6268 * 35) + 1,129.2 = 1,037.3
4Expected Y(4) = (-2.6268 * 40) + 1,129.2 = 1,024.1
5Expected Y(5) = (-2.6268 * 50) + 1,129.2 = 997.9
6Expected Y(6) = (-2.6268 * 45) + 1,129.2 = 1,011
Error Calculation
Next, we calculate the differences between the actual and expected y values:
1Error(1) = 1,100 - 1,102.9 = -2.9
2Error(2) = 1,200 - 1,076.7 = 123.3
3Error(3) = 985 - 1,037.3 = -52.3
4Error(4) = 750 - 1,024.1 = -274.1
5Error(5) = 1,215 - 997.9 = 217.1
6Error(6) = 1,000 - 1,011 = -11
Squaring the Errors and Summing Them Up
1Sum of Errors Squared = (-2.9)^2 + (123.3)^2 + (-52.3)^2 + (-274.1)^2 + (217.1)^2 + (-11)^2 = 140,330.81
Difference Calculation
We then determine the error differences, square them, and sum them up:
1Difference(1) = 123.3 - (-2.9) = 126.2
2Difference(2) = -52.3 - 123.3 = -175.6
3Difference(3) = -274.1 - (-52.3) = -221.9
4Difference(4) = 217.1 - (-274.1) = 491.3
5Difference(5) = -11 - 217.1 = -228.1
6Sum of Differences Square = 389,406.71
Final Calculation of Durbin Watson Statistic
Finally, the Durbin Watson statistic is calculated as:
1Durbin Watson = 389,406.71 / 140,330.81 = 2.77
Note: Minor discrepancies in the tenths place may arise due to rounding during the squaring process.
Related Terms: Autocorrelation, Regression Analysis, Least Squares Method, Serial Correlation, Technical Analysis.
