Simultaneous causality bias

Simultaneous causality bias is a common problem in econometrics that prevents us to draw causal interpretation from OLS estimate. The simultaneous causality bias, popularly known as reverse causality, occurs when dependent variable affects a key independent variable.

For example, one researcher may argue that education leads to better earnings, eventually higher income. But other researcher may propose that high income families are able to afford quality education. Both arguments are equally true. There are ample of examples, such as health expenditure and income, remittances and household welfare. Such situation germinates "reverse causality" problem.

`\text{Education} \rightarrow \text{Higher income}`

`\text{Higher income} \rightarrow \text{Quality education}`

Mathematically,

`Y=\alpha + \beta X + \epsilon ---(1)`

But,

`X=\gamma + \lambda Y + \nu---(2)`

Substituting Equation (2) in Equation (1),

`Y = \alpha + \beta (\gamma + \lambda Y + \varepsilon) + \epsilon`

`Y = \alpha + \beta\gamma + \beta\lambda Y + \beta \nu + \epsilon`

`Y (1-\beta\lambda) = \theta + \beta \nu + \epsilon---(3)`

 

Equation (4) shows that reverse causality distorts the relationship between `X` and `Y`. Specifically:

• The term $$`1-\beta\lambda` implies that the true relationship between `X` and $`Y`$ and the variables has been altered. If `\beta\lambda` is not 0, this leads to biased and inconsistent estimates of `\beta`.

In standard OLS regression, the estimate of `\beta` is:

`\hat{\beta} = \frac{\text{Cov}(X, Y)}{\text{Var}(X)}---(4)`

When reverse causality exists, `X` and `Y` influence each other. So, the covariance `\text{Cov}(X, Y)` includes the effect of `Y` on `X`, making the OLS estimate of `\hat{\beta}` biased.

Breaking down Equation (4), we get

`\text{Cov}(X, Y) = \text{Cov}(\gamma Y + \nu, Y) = \gamma \text{Var}(Y) + \text{Cov}(\nu, Y)`

Thus, the OLS estimate of `\beta` incorporates the influence of `Y` on `X`, leading to a biased estimation.

If `\nu` is independent of `Y`, then `\text{Cov}(\nu, Y) = 0`, and the covariance reduces to:

`\text{Cov}(X, Y) = \gamma \text{Var}(Y)`

However, if `\nu` is correlated with `Y`, the covariance `\text{Cov}(\nu, Y)` adds bias to the estimate, leading to problems in interpreting the OLS coefficient `\beta` correctly. This is the source of the bias in reverse causality.