2SLS Without Mixing The instruments

Question

Suppose I have the following strucutural equation in time-series:

$$y_t=\beta_0+\beta_1x_{1t}+\beta_2x_{t2}+\zeta w_t+\epsilon_t \quad (1)$$

In which both $x_{t1}$ and $x_{t2}$ are endogenous variables and $w_t$ is an exogenous variable. Suppose that $z_{1t}$ and $z_{2t}$ are the elected instruments for a 2SLS estimation strategy, but you can only theoretically address one instrument to one endogenous variable, that is, $x_{t1}$ should be instrumented by $z_{t1}$ and $x_{t2}$ with $z_{t2}$. For example, that situation would arise if you're trying to estimate a demand with cross-price elasticities, in which one would have costs from different companies to instrument the different prices. One particular company cost should explain (instrument) the price of the same company.

In such a situation, a strategy in which you estimate a system of equations in the first stage, could be as follows:

$$x_{t1}=b_0+b_1z_{t1}+b_2w_t+e_t \quad(2)$$

$$x_{t2}=a_0+a_1z_{t2}+a_2w_t+u_t \quad(3)$$

By this system of equations, I believe that I cannot guarantee that the instruments won't be correlated with the residuals of both equations -- so, using the predicted values $\hat{x_{t1}}$ and $\hat{x_{t2}}$ in the structural equation would not be adequate. So, if this situations arises, what is the correct and more up-to-date approach?

Answer 1

For equations (2) and (3) I think you would be worried about potential correlation between the residuals $e_t$ and $u_t$ and the instruments $z_{t2}$ and $z_{t1}$ respectively. In the case that the instruments are correlated with the first stage errors, the 2SLS estimates will be inconsistent.

I think that in the context of your specific example of a demand equation, your first stage is not valid. Let $y_t$ be quantity demanded of good 1, $x_{1t}$ and $x_{2t}$ be the consumer prices of goods 1 and 2, and $z_{1t}$ and $z_{2t}$ be the input prices (or marginal costs) of goods 1 and 2 respectively.

Without loss of generality, assume goods 1 and 2 are complements. Then they exhibit negative cross-elasticity of demand in which case a rise in $z_{2t}$ will generally result in a rise of $x_{2t}$ and in turn a fall in $y_{t}$ holding all else fixed. However, there is an endogeneity here, because in Bertrand or Stackelberg competition or in another collusive game, the company pricing good 1 would react to a rise in $x_{2t}$ by changing $x_{1t}$ to maximize profit (see Tirole).

In such a case we expect there to be a causal relationship between $z_{2t}$ and $x_{1t}$. So a first stage of the form $$ \begin{bmatrix} x_{1t} \\ x_{2t} \end{bmatrix}=\begin{bmatrix} b_{1} \\ a_{1} \end{bmatrix}+\begin{bmatrix} b_2 & b_3 & b_4 \\ a_2 & a_3 & a_4 \end{bmatrix}\begin{bmatrix} z_{1t} \\ z_{2t} \\w_t \end{bmatrix}+\begin{bmatrix} e_t \\ u_t \end{bmatrix} \quad (1)$$ would be more appropriate.

Furthermore, for the more general case of IV regression, the consistency of the 2SLS estimates does not depend directly on correct specification of the first stage (Angrist and Pischke pg 191). So while you may still believe that the first stage equation (1) may be misspecified, it will still yield consistent 2SLS estimates so long as the 2 key assumptions (that the instruments are (1) exogenous and (2) correlated with $x$'s) hold true. For a formal proof of this see Greene ch 8.

Since structuring the first stage differently can yield different IV estimates, and because the above first stage is guaranteed consistency, IV with the first stage specified in equation (1) is generally accepted by the literature as the 2SLS estimate (see Wooldridge Ch 15).

• Angrist, Joshua D., and Jörn-Steffen Pischke. Mostly harmless econometrics: An empiricist's companion. Princeton university press, 2008.
• Greene, William H. "Econometric analysis, 5th." Ed.. Upper Saddle River, NJ (2003).
• Tirole, Jean. The theory of industrial organization. MIT press, 1988.
• Wooldridge, Jeffrey. Introductory econometrics: A modern approach. Cengage Learning, 2012.