POL222 - weeks 3 and 4

Two main obstacles to empirically examining a causal relationship

Confounding variables - Other variables that influence both the independent (X) and dependent (Y) variables

Reverse causality - When the direction of cause and effect between X and Y is unclear

Spurious Relationship

When the relationship between X and Y disappears once a confounder, Z, is accounted for.

β* > 0 is positive, β* < 0 is negative, β* = 0 is nothing

EX. teacher finds correlation between height of students (X) and math scores (Y). Once grade level (Z) is controlled, the relationship disappears, taller students usually are in higher grades and heave learned more math, so there’s no real casual link between height and math scores.

Omitted variable bias

OVB occurs when a confounding variable (Z) is not included in the analysis, biasing the estimated relationship between X and Y

If Z is omitted → biased estimate (β*)

If Z is included → true relationship (β)

Can result in overestimation, underestimation, sign of the relationship reversed

Overestimation

Positive Relationship - True relationship: X and Y are positively related (β > 0). Z is positively related to both X and Y (a > 0, y > 0). If Z is omitted, the estimated relationship (β*) will be greater than the true β. This happens because β* = β + bias, and the bias is positive. We overestimate the strength of the positive relationship.

Negative relationship - True relationship: X and Y are negatively related (β < 0). Z is positively related to X and negatively related to Y. If Z is omitted, β* will be smaller than β — appearing more negative than the true effect. We overestimate the magnitude of the negative relationship (make it look “too negative”).

Underestimation

Positive relationship - True relationship: X and Y are positively related (β > 0). Z is positively related to X but negatively related to Y. If Z is omitted, the estimated β* will be smaller than the true β (β* < β). The negative bias counteracts the true positive relationship. We underestimate the magnitude of the positive effect.

Negative relationship - True relationship: X and Y are negatively related (β < 0). Z is positively related to both X and Y (a > 0, y > 0). If Z is omitted, β* will be greater (less negative) than the true β. We underestimate how negative the relationship really is.

Positive Relationship reversed

Same setup as underestimation - True β > 0, Z is positively related to X, negatively to Y.

If the negative bias is stronger than the true positive effect, the sign reverses and β* < 0. A positive relationship appears negative when Z is omitted.

Negative Relationship Reversed

Same setup as underestimation - True β < 0, Z is positively related to both X and Y.

If the positive bias is stronger than the true negative effect, the sign reverses and β* > 0. A negative relationship appears positive when Z is omitted.

Reverse Causality

Occurs when Y might cause X, instead of X causing Y, or both cause each other, threatens causal validity because we can’t be sure of the direction of influence

In order to detect it, ask:

• does our casual theory claim X → Y?

• Is there a credible casual mechanism that could also make Y→X? If so, there is a risk of reverse causality