The value of the Durbin-Watson d-statistic always falls in the interval from 0 to 1.
A) True
B) False
The Correct Answer and Explanation is :
The correct answer is B) False.
Explanation:
The Durbin-Watson (DW) statistic is used to test for autocorrelation in the residuals from a regression analysis, particularly in time series data. Autocorrelation refers to the correlation of a time series with its own past values. The DW statistic is valuable for checking whether the residuals from a regression model exhibit patterns over time, which can violate the assumption of independence in linear regression.
The value of the Durbin-Watson statistic actually ranges between 0 and 4, not 0 to 1. Here’s a breakdown of the interpretation:
- DW = 2: No autocorrelation. This suggests that there is no significant correlation between the residuals.
- DW < 2: Positive autocorrelation. This indicates that residuals from earlier periods tend to be similar to residuals from later periods.
- DW > 2: Negative autocorrelation. This suggests that residuals from earlier periods tend to be opposite in sign from those of later periods.
Values closer to 0 indicate strong positive autocorrelation, while values closer to 4 suggest strong negative autocorrelation.
The DW statistic is computed using the following formula:
[
DW = \frac{\sum_{t=2}^{n} (e_t – e_{t-1})^2}{\sum_{t=1}^{n} e_t^2}
]
Where (e_t) represents the residuals at time (t). The statistic measures the squared differences between consecutive residuals, which is why it provides information about autocorrelation.
If the statistic is closer to 0, it indicates a higher level of autocorrelation between consecutive residuals. A value close to 4 suggests the residuals are negatively correlated.
The Durbin-Watson test is commonly used in regression analysis to determine the validity of the assumption of no autocorrelation. If there is autocorrelation, model adjustments or transformations may be needed.
I will now generate an image to visually explain the range of the Durbin-Watson statistic.
Here is the visual representation of the Durbin-Watson statistic range. It shows the values from 0 to 4, where 0 indicates strong positive autocorrelation, 2 indicates no autocorrelation, and 4 indicates strong negative autocorrelation.
Let me know if you need further details or clarification!
