What is endogeneity and how does it affect regression analysis?

Endogeneity is an often ignored problem may be because of lack of awareness that it can lead to serious mistakes in inferring the cause and effect estimates. To put it Statistically,when the predictor/s show a correlation with the error term (unexplained variation) we say Endogeneity is present in the data. It arises when the marginal distribution of the predictor is not independent of the conditional distribution of the?error. When we leave out relevant variables/ use faulty scale/ select non random sample or ‘Simultaneity’ dependent & independent variables influencing each other- we encounter endogeneity. one way of dealing with this is to use variables (instrumental) that allow us to isolate the exogenous variation in the predictor

The variables or explanatory variable chosen play such an important role. Their preprocessing appropriately can also help to eliminate the measurement errors. For e.g based on domain knowledge or context one knows that the height that was recorded was off by few cms across most of the samples.Having a through ubderstanding of the data and the context. Few transformation techniques before regression can be tried like PCA that de correlates the data i.e the principal components after PCA are decorrelated or orthogonal to each other.Such a regression after applying PCA on explanatory variables is called principal component regression.

Endogeneity arises when the error term is not iid (independent and identically distributed), which is a key assumption for many models. This means that the results will be biased in an unpredictable way, sometimes masking the effect, sometimes amplifying it. For example, an omitted variable can suggest an effect where there is none. A model regressing ice cream consumption on drownings may find a significant correlation, even though the link is that both happen more in summer. Questions of causality are even trickier, because there is no mathematical way to prove causal relationships. A regression cannot tell the difference between causality and reverse causality, for example.

Endogeneity occurs when the obvious independent variables are very correlated with the error term. Endogeneity can arise from simultaneity. When trying to determine variables jointly using simultaneous equations, it is very possible that the endogeneity in the dependent variables in one equation becomes the independent variable in another. Leaving out very relevant variables can lead to omitted variable bias and as such leading to spurious correlation outcomes.

Basically, endogeneity arises when there's a correlation between the independent variable and the unexplained error term in a statistical model. This can lead to misleading results, as the estimated relationship between the variables may not be truly causal. Some common causes of endogeneity are: 1. Omitted variable bias 2. Measurement error 3. Reverse causality 4. Sample selection bias

In any model, endogeneity will affect the consistency and bias of estimators. However, in causal models, including endogenous variables can mask the causal effect, leading to a breakdown in the model. For example, we know that weather conditions cause rain. But if we include the number of people umbrellas in the model, weather conditions have less of an effect, because many people check the forecast before leaving the house. Since the decision to carry an umbrella is based on weather, it is endogenous.

I always start with a DAG (directed acyclic graph), especially when I work with causal models. This allows me to make and check my assumptions about how my variables interact. I can then check for correlation between the variables I think should affect each other. For example, if I think that an advertising campaign should lead to more people in my store and higher sales, my DAG is ads->people->sales. If I find low correlation between ads and people, or between people and sales, but high correlation between ads and sales, I know that I have an endogeneity problem (here, and omitted variable).