Intercept Error

In this video, we examine one of the fundamental properties of ordinary least squares: when the intercept is correctly specified, the residuals sum to zero. Building on our earlier discussion of model approximation errors, we use conditional averages rather than individual observations to show why this property emerges and how it can be used as a diagnostic tool. By comparing observed conditional means to the values predicted by a regression line, we demonstrate that a correctly positioned intercept produces positive and negative residuals that exactly offset one another. We then explore what happens when the intercept is incorrect. Using a simple numerical example, we intentionally shift the regression line and observe how the residuals no longer sum to zero. This nonzero average residual provides a direct signal that the intercept is misplaced. From there, we develop a simple repair procedure that uses the average residual to adjust the intercept and recover the correct regression line. A recurring theme throughout the lesson is that these results hold whether we work with every individual observation or with conditional averages. The averaging approach dramatically reduces the amount of computation while preserving the key properties of the regression model, making it a useful stepping stone toward more advanced econometric concepts. Most importantly, this discussion introduces the idea of bias as a systematic upward or downward displacement of a regression line. While the examples in this video arise from deliberate mechanical mistakes, the intuition developed here will later help us understand more sophisticated forms of bias that appear throughout econometrics.