Trying to use a linear regression model to forecast the CPI. I noticed that when I took a moving average of the residuals, though homoscedatisc and nonautocorrelated (i.e. they squiggle up&down with no uniform pattern), that they seemed to move in the same direction as the CPI. That is the moving average of the residuals and the depended variables correlated. What does this imply? Is this a case of omitted variable bias? Are my coefficients biased? What are some common prognoses for a such a problem?
The reason you're seeing the bias is because of the adjustment, commonly referred to the "fudge factor" that they government applies to basically all of their published statistics. This is almost always in the direction to make the numbers rosier, and therefore your error term likely will exhibit signs of positive serial correlation.