# Instrumental variables regression Vs Linear Regression for price elasticity

Based on comment here that What do "endogeneity" and "exogeneity" mean substantively? should we use linear regression or instrumental regression for price elasticity? As price and demand might have some endogeneity?

Linear regression cannot handle endogeneity phenomenon if or goal is a causal model with endogenous variables. i.e. if we want to predict if my price is x what will be demand in a real world scenario. To cure this, instrumental regression is used. It makes logical sense that Instrumental regression SHOULD be used in this situation. However, i still wanted some expert comment if this is the right thing to do as i didn't find any paper which advocates using IV regression for price elasticity causal model.

• Please, don't cross post on multiple sites. Since an answer has been provided on CV, I would suggest to delete the duplicate.
– chl
Commented Feb 22, 2016 at 11:01
• @chl I think price elasticity is a multi-discipline subject and not restricted to statistics or economics alone. Hence posts in both statistics and economics forum. I think economists might have different approach or opinion. Let me know if you still think the other post needs to be deleted. Thanks Commented Feb 22, 2016 at 11:31
• I agree that this might of interest for different communities, but see our recent Meta discussion (and related links) about cross-posting.
– chl
Commented Feb 22, 2016 at 11:57

In general, you cannot use plain OLS on market data to estimate demand elasticity. The one, unusual, exception would be if supply was not price sensitive (the supply curve was a straight up and down line with the same price for every quantity). The same goes for supply elasticity. See this link which explains it well. You can also see Greene's Econometric Analysis textbook (edition 5 or 7, 5 is available for free in pdf online) which covers this topic thoroughly. I will make an informal attempt at explaining below.

Suppose we want to estimate a linear supply and demand curve, something like; $$\begin{array}{llc} q^s_t &= \alpha_0 + \alpha_1 p_t + \alpha_2 w_t+ u_t&(Supply) \\ q^d_t &= \beta_0 + \beta_1 p_t +\beta_2 y_t+ v_t&(Demand) \end{array}$$ Where $q^s_t=q^s_t$ is log quantity, $p_t$ is log price, $w_t$ are the log input costs, and $y_t$ is log consumer income. (you can have other explanatory variables too if you like, but this is the basic case).

With the above specification, the demand and supply price elasticities are $$\frac{\partial q^s_t}{\partial p_t}= \alpha_1\;\;(Supply)\;\;\;\; \frac{\partial q^d_t}{\partial p_t} = \beta_1\;\;(Demand)$$

BUT THE ABOVE CANNOT BE ESTIMATED WITH OLS because price is endogenous. Why? because, for the market to clear, price and quantity are decided simultaneously with both demand and supply side mechanisms. For example, a firm may change it's prices in response to increases or decreases in it's competition. Such variation in prices is not exogenous but rather caused by shifts in $q^s_t$. This reverses the order of causality that would be implied by an OLS regression model of the above supply and demand functions.

Demand elasticity is generally upward bias when estimated with OLS and I am pretty sure supply elasticity is downward bias (but check the references above too, I think the bias is usually attenuating (toward 0) in all cases).

There are two prominent econometric methodologies for estimating demand/supply curves. These are

1. Structural equation modeling: $$\begin{bmatrix}q_t \\ p_t \end{bmatrix} = \begin{bmatrix}\pi_0 \\ \omega_0 \end{bmatrix} + \begin{bmatrix}\pi_1 & \pi_2 \\ \omega_1&\omega_2 \end{bmatrix}\begin{bmatrix}w_t \\ y_t \end{bmatrix}+\begin{bmatrix}e_{1,t} \\ e_{2,t} \end{bmatrix}$$ which is a multivariate likelihood.

2. Instrumental Variables (as suggested in your question). In the case of demand this is: $$\begin{array}{llc} p_t &= \omega_0 + \omega_1 w_t + \omega_2 y_t + e_t&(First\;Stage) \\ q^d_t &= \beta_0 + \beta_1 \hat p_t +\beta_2 y_t+ v_t&(Second\;Stage) \end{array}$$

$w_t$ works as an instrument because it acts like an exogenous “shifter” in supply. The basic idea is that by moving the supply curve up and down and recording the equilibrium quantities and prices, we can trace out the demand curve. The assumption here is that the suppliers cannot control their input costs (this is not always true when suppliers can negotiate their input costs as in oligopsony but we assume it is here). For supply we would use the same methodology but with $y_t$ as the instrument.

• Interesting to know use of SEM. I believe IV regression might be most frequently used than SEM given the challenge it is to build an SEM model. Thanks for sharing the assumption around 'input cost' Commented Feb 22, 2016 at 9:26

The original application of the instrumental variable approach was in the estimation of elasticities! Wright (1928) referred to IVs as 'curve shifters' in his use of them to estimate the elasticity of flaxseed (referred to here).