Imagine that you have survey data from teenagers who have been convicted of crime and are members of a neighborhood gang. The survey includes questions about (1) hours per seek spend in gang activity, H, (2) weekly earnings from gang activity, E. Based on economic theory, you suspect that earnings increase with hours of activity which leads you to estimate the following equation: E=α+βH+θX+ν
Where = E = earnings from crime, H = hours of gang activity, X = a full set of other variables so that you do not have to worry about omitted variable bias, ν = error term.
Because observations of E and H come from the same survey and the measurement error in E is likely positively correlated with the measurement error in H, show formally what problem of bias this is likely to create in ordinary least squares estimates of β? The relation you expect between estimates and truth is β* β?
1b. Let’s say that you had an unbiased estimate of β, how could you interpret this estimated coefficient?