Some papers argue that OLS can produce less bias than IV estimation depending on the quality of your instruments. Suppose we consider a demand estimation equation.
Suppose the demand elasticity is negative in OLS. By my intuition weak instruments should produce biased estimates towards OLS, but no less negative. Can you guys produce an example? I cannot really grasp how it could lead to a more biased estimation with IV estimation.
Usually, $\hat{\beta_1^{IV}} = \beta_1 + \frac{cov(z,u)}{cov(z,x)}$. The denominator will go to zero.
That is true unless there is some correlation between the instrument and the error term, and the nominator is the strength of the relationship between the instrument and the endogenous variable. The smaller the denominator gets, the greater the bias $\left[\frac{cov(z,u)}{cov(z,x)}\right]$.
In addition, weak instrument will have no precision, so that the variance will have a big upward bias. \begin{eqnarray} var(\hat{\beta_1}) &\to_p& \frac{\sigma^2}{n \sigma^2_x} \nonumber \\ \hat{\beta_1^{IV}} &=& \frac{\sum (z_i - \bar{z})y_i}{\sum(z_i - \bar{z})x_i} = \beta_1 + \frac{\sum (z_i - \bar{z})u_i}{\sum(z_i - \bar{z})x_i} \nonumber \\ var(\hat{\beta_1^{IV}} &=& var \left( \frac{\sum (z_i - \bar{z})u_i}{\sum(z_i - \bar{z})x_i} \right) \nonumber\\ var(u | z) &=& \sigma^2 \nonumber\\ var(\hat{\beta_1^{IV}}) &=& \frac{\sigma^2 \frac{1}{n} \sum (z_i - \bar{z})}{n[ \frac{1}{n} \sum (z_i - \bar{z})(x_i - \bar{x})]^2} \nonumber \end{eqnarray}
As $n \to \inf$ \begin{eqnarray} var(\hat{\beta_1^{IV}}) &\to_p& \frac{\sigma^2 \sigma_z^2 }{\sigma^2_{zx}} \nonumber\\ var(\hat{\beta_1^{IV}}) &\to_p& \sigma^2 \frac{1}{n \sigma^2_x} \frac{1}{\rho^2_{xz}}\nonumber \\ \rho^2_{xz} &=& \frac{[\sigma^2 _{xz}]^2}{\sigma_x^2 \sigma_z^2} \textit{for} \rho \in [0,1]\nonumber \end{eqnarray}
That is why if your instrument is weak, then you may be better off running an OLS regression.
Weak instruments combined with slight instrumental endogeneity can lead to a larger bias than OLS. As Nox's answer shows, the probability limit of the IV estimator is $\beta_1 + cov(z,u)/cov(z,x)$. When $cov(z,u) \ne 0$ though small, if $cov(z,x)$ is small, then the bias can be large. See Bound, Jaeger and Baker's (1995, JASA) remark following equation (7) on page 444.
http://www.djaeger.org/research/pubs/jasav90n430.pdf
"It is clear from Equation (7) that a weak correlation between the potentially endogenous variable, $x$, and the instruments, $z_1$, will exacerbate any problems associated with the correlation between the instrument and the error, $\varepsilon$. If the correlation between the instrument and the endogenous explanatory variable is weak, then even a small correlation between the instrument and the error can produce a larger inconsistency in the IV estimate of $\beta$ than in the OLS estimate."
Without instrumental endogeneity, I don't think the IV estimator's bias (of the limit distribution, there may be no probability limit) is larger than OLS's inconsistency.
Another thing to consider is that the variance of the IV estimator using very weak instruments can be large even with a very large $n$, and thus you may have an IV estimate more nonsense than OLS for a data set just by chance.
