11
Aug 17

## Violations of classical assumptions

This will be a simple post explaining the common observation that "in Economics, variability of many variables is proportional to those variables". Make sure to review the assumptions; they tend to slip from memory. We consider the simple regression

(1) $y_i=a+bx_i+e_i.$

One of classical assumptions is

Homoscedasticity. All errors have the same variances$Var(e_i)=\sigma^2$ for all $i$.

We discuss its opposite, which is

Heteroscedasticity. Not all errors have the same variance. It would be wrong to write it as $Var(e_i)\ne\sigma^2$ for all $i$ (which means that all errors have variance different from $\sigma^2$). You can write that not all $Var(e_i)$ are the same but it's better to use the verbal definition.

Remark about Video 1. The dashed lines can represent mean consumption. Then the fact that variation of a variable grows with its level becomes more obvious.

Video 1. Case for heteroscedasticity

Figure 1. Illustration from Dougherty: as x increases, variance of the error term increases

Homoscedasticity was used in the derivation of the OLS estimator variance; under heteroscedasticity that expression is no longer valid. There are other implications, which will be discussed later.

Companies example. The Samsung Galaxy Note 7 battery fires and explosions that caused two recalls cost the smartphone maker at least $5 billion. There is no way a small company could have such losses. GDP example. The error in measuring US GDP is on the order of$200 bln, which is comparable to the Kazakhstan GDP. However, the standard deviation of the ratio error/GDP seems to be about the same across countries, if the underground economy is not too big. Often the assumption that the standard deviation of the regression error is proportional to one of regressors is plausible.

To see if the regression error is heteroscedastic, you can look at the graph of the residuals or use statistical tests.