What is the difference between two stage least squares and instrumental variable regression?

I'm doing independent study and I am having trouble understanding the difference between these two estimators.

I get that 2SLS is predicting the endogenous variable, and that instrumental variables are similar to proxy variables, but I don't get how one differs from the other.

IV estimators are 2SLS estimators.

An IV estimator is the sample analog of the form: $$\beta = \frac{Cov(Y, Z)}{Cov(X, Z)}$$, where $$Y$$ is the outcome variable, $$X$$ is the endogenous variable, and $$Z$$ is the instrumental variable.

It can be shown that the 2SLS is of the above form. The advantage of 2SLS estimators over other IV estimators is that 2SLS can easily combine multiple instrumental variables, and it also makes including control variables easier.

The meaning of the words first

Some people use the word "IV estimator" to refer to any estimator that uses instrumental variables. To them, IV estimators contain 2SLS, LIML, k-class estimators, and others, so 2SLS is a special case of IV. For example, the title of Bekker's (1994, Econometrica) paper is "Alternative approximations to the distribution of instrumental variable estimators".

More traditional people mean by IV the particular instrumental variable estimator $(Z'X)^{-1}Z'y$ for the exactly identified case ($Z$ = instrument matrix, $X$ = regressor matrix, $y$ = regressand vector), and 2SLS is a generalization of IV to the overidentified case. But, as Paul says, 2SLS can be expressed as an IV estimator of this second sense because it is $(\hat{X}'X)^{-1} \hat{X}'y$, where $\hat{X} = Z(Z'Z)^{-1}Z'X$ is the instrument matrix.

I personally think it is very fine to leave the meaning of IV estimators ambiguous because the meaning is usually clear in the context and we need not rigorously distinguish them.

It seems to me that the sentence "2sls is predicting the endogenous variable" means the first stage regression of the endogenous regressor on the instrumental variables (to get $\hat{X}$). The expression "instrumental variables are similar to proxy variables" looks more casual. Proxy variables (e.g., IQ for ability) can be used to solve the endogeneity problem. Instrumental variables are another way of solving the endogeneity problem. In that sense they are "similar".
• In the exactly identified case, we assume each instrument has a correlation of 1 with a regressor: $Z(Z'Z)^{-1}Z'X = Z\ \impliedby\ (Z'Z)^{-1}Z'X = I$. This shows that 2SLS is equivalent to IV, up to the order of the instruments. If the instruments are not in the correct order, the projection $Z(Z'Z)^{-1}Z'X$ will re-order the variables so that $\hat X$ has a different ordering to $Z$ but the same as $X$. – ahorn Sep 19 '19 at 12:03
• In case $dim(Z) = dim(X)$, $Z'X$ is invertible, and $Z'Z$ is nonsingular, we have $(Z'X)^{-1} Z'y = [X'Z(Z'Z)^{-1} Z'X]^{-1} X'Z(Z'Z)^{-1} Z'y$ by cancellation. But it's not true that $(Z'Z)^{-1}Z'X = I$ unless $Z=X$; it's not true even when $Z=-X$. – chan1142 Aug 2 '20 at 2:56
• Exactly my point. I was saying that if $Z=X$ (i.e. if the ordering is the same), then they are equivalent. Note that I did not claim that $(Z'Z)^{-1}Z'X = I$; I merely claimed that the implication is true. – ahorn Aug 2 '20 at 7:21
• I think it's common for mathematicians to say "$A\implies B$", when they really mean "$A$ is true. Therefore $B$ is true." However, I'm the type of mathematician who prefers to evaluate the fundamental truth of the implication, regardless of whether the antecedent is true or not. – ahorn Aug 2 '20 at 7:27