If the residuals in time period 1, 2, 3 and 4 are 5, 4, 3 and 1, respectively, what would be the value of Durbin-Watson statistic? Round your final result to two decimal places.
Question
If the residuals in time period 1, 2, 3 and 4 are 5, 4, 3 and 1, respectively, what would be the value of Durbin-Watson statistic? Round your final result to two decimal places.
Solution
The Durbin-Watson statistic is a test statistic used to detect the presence of autocorrelation (a relationship between values separated from each other by a given time lag) in the residuals (prediction errors) from a regression analysis.
The Durbin-Watson statistic is defined as:
d = Σ(e[i] - e[i-1])² / Σ(e[i])²
where:
- e[i] is the residual at time period i
- e[i-1] is the residual at the previous time period
- Σ denotes the sum over all time periods
Given the residuals e[1] = 5, e[2] = 4, e[3] = 3, e[4] = 1, we can calculate the Durbin-Watson statistic as follows:
First, calculate the numerator of the Durbin-Watson statistic, which is the sum of the squared differences between consecutive residuals:
(4 - 5)² + (3 - 4)² + (1 - 3)² = 1 + 1 + 4 = 6
Next, calculate the denominator of the Durbin-Watson statistic, which is the sum of the squared residuals:
5² + 4² + 3² + 1² = 25 + 16 + 9 + 1 = 51
Finally, divide the numerator by the denominator to get the Durbin-Watson statistic:
d = 6 / 51 = 0.1176
Rounding to two decimal places, the Durbin-Watson statistic is 0.12.
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