How to determine autocolerration by using independent variables. Show even formulas if possible
Autocorrelation is a measure of the degree of similarity between a time series and a lagged version of itself over successive time periods. To determine the presence of autocorrelation, we can use the Durbin-Watson statistic.
The formula for the Durbin-Watson statistic is given by:
DW = ∑(ei – ei-1)2 / ∑ei2
where ei is the residual (difference between the observed and predicted value) at time i and ei-1 is the residual at time i-1.
The DW statistic ranges between 0 and 4, with a value of 2 indicating no autocorrelation. Values less than 2 indicate positive autocorrelation, indicating that the residuals at adjacent time periods are positively correlated. Values greater than 2 indicate negative autocorrelation, meaning that the residuals at adjacent time periods are negatively correlated.
To account for the effect of independent variables, we can use the following formula for the adjusted Durbin-Watson statistic:
DA = DW * (1 – ρ) / (1 – (1 / n))
where ρ is the correlation coefficient between the independent variables and n is the sample size.
The adjusted DW statistic is more appropriate when there are independent variables in the model, as it accounts for the potential correlation between the residuals and the independent variables.
In summary, to determine autocorrelation in the presence of independent variables, we can use the Durbin-Watson statistic with adjustments made using the formula above.