Regression assumptions and residual analysis
Regression can be computed on any data — the button won't ask whether a straight line is appropriate. But the coefficients and their "significance" are honest only if the assumptions hold. You check them not by r itself but by the RESIDUALS — the line's misses. Drag the points and watch the residuals.
A good regression leaves the residuals as structureless noise around zero. Any pattern in the residuals is a signal the line cannot be trusted, whatever the R². That is why this chart is always checked after a regression.
A residual is the model's miss (actual minus predicted). The main diagnostic tool is the plot of residuals against x. In the "all fine" mode they form a structureless cloud around zero: the straight line is appropriate. Any pattern in the residuals is a signal the model cannot be trusted, whatever the R².
The working ritual: computed a regression — plotted the residuals. An arc → the relationship is nonlinear. A fan → heteroscedasticity. A lone distant point → check its influence. Only after that do you read the coefficients and their significance.
A high R² does not save you from broken assumptions: a model can fit well on average while giving wrong intervals and a deceptive slope. The residual picture is more honest than a single number.
Economic and financial data are almost always heteroscedastic and time-dependent — which is why robust standard errors and dedicated time-series models are the default there.
In product data nonlinearity is everywhere (saturation, thresholds), so blind linear regression often understates or distorts the relationship — a look at the residuals and transformations save the day.
Definitions
Residual diagnostics themselves assume you have enough data for structure to be visible; on a handful of points a pattern in the residuals is hard to spot.
Fixing one violation (say, a log transform for heteroscedasticity) changes how the coefficients read — they are now about percentages, not absolute units.