Question about what is causing endogeneity

Question

I have the following model:

$r_t=\beta_1+\beta_2\ div_t + \beta_3 \ result_t + u_t$

$r$ is the return of a stock index, $div$ are the dividend gains in percentage of the market price, $result$ are the results of the companies included in the stock index. Also:

$r_t=\frac{p_t+ \ d_t-p_{t-1}}{p_{t-1}}=\frac{p_t+ \ d_t}{p_{t-1}}-1$

$div_t=\frac{d_t}{p_{t-1}}$

$p$ is the market price of the index, $d$ are the dividends.

I am told that it is reasonable to consider that $div$ is endogenous. Is it because of the presence of $p_{t-1}$ in both $div$ and $r$? Is it a problem of simultaneity that causes endogeneity?

Answer 1

Both $r_t$ and $div_t$ are functions of $p_{t-1}$, hence the two variables will be correlated with the past price. However, $p_{t-1}$ does not occur in your regression and therefore is left in the error, meaning that you will have a correlation between $div_t$ and $u_t = \epsilon_t + p_{t-1}$. So if instead of running $$r_t = \beta_1 + \beta_2 div_t + \beta_3 result_t + \beta_4 p_{t-1} + \epsilon_t$$ you choose to run the short regression $$r_t = \beta_1 + \beta_2 div_t + \beta_3 result_t + u_t$$ the bias for your estimated coefficient on $div_t$ is going to be $$\widehat{\beta}_2 = \beta_2 + \beta_4 \frac{Cov(div_t, p_{t-1})}{Var(div_t)}$$ and this bias is not going to be zero. From the expressions provided for $r_t$ and $div_t$, both are decreasing in $p_{t-1}$ meaning that it is likely that $\beta_4 < 0$ and $Cov(div_t,p_{t-1})<0$, in which case $\widehat{\beta_2}$ will be biased upward, i.e. it is going to be too large.

However, even if you include $p_{t-1}$ in your regression, there might be other issues such as simultaneity or reverse causality. Right now the model posits that an increase in the dividend increases the return of a stock. Perhaps it could also be though that an increase in the return of a stock increases the dividend, e.g. more successful stocks pay out higher dividends (I'm only speculating here because this isn't my field of expertise but it might be worth thinking about it).