# 2.4 Endogenous explanatory variables  (Page 3/10)

 Page 3 / 10

The thick red line shows the regression that would result from using OLS to estimate either of the two structural equations. As illustrated, an OLS estimate of the slope estimate will be biased. We need to use some other estimation technique than OLS.

## Estimation

As noted earlier, the basic problem created by the endogeneity problem is that the endogenous explanatory variable is correlated with the error term. The most logical approach would be to replace this variable with one that is not correlated with the error term but highly correlated with the endogenous variable. Consider the value of the price predicted by the reduced form equation (5):

${\stackrel{⌢}{p}}_{t}^{w}={\stackrel{⌢}{\gamma }}_{0}+{\stackrel{⌢}{\gamma }}_{1}{I}_{t}+{\stackrel{⌢}{\gamma }}_{2}{p}_{t}^{c}+{\stackrel{⌢}{\gamma }}_{3}{W}_{t}$

where ${\stackrel{⌢}{\gamma }}_{i}$ is the OLS estimate of ${\gamma }_{0}=\frac{{\beta }_{0}+{\beta }_{1}{\alpha }_{0}}{1-{\alpha }_{1}{\beta }_{1}},$ ${\gamma }_{1}=\frac{{\alpha }_{2}{\beta }_{1}}{1-{\alpha }_{1}{\beta }_{1}},$ ${\gamma }_{2}=\frac{{\alpha }_{3}{\beta }_{1}}{1-{\alpha }_{1}{\beta }_{1}},$ and ${\gamma }_{3}=\frac{{\beta }_{2}}{1-{\alpha }_{1}{\beta }_{1}}.$

Clearly, ${\stackrel{⌢}{p}}_{t}^{w}$ is correlated with ${p}_{t}^{w}.$ It also is true that the covariance between ${\stackrel{⌢}{p}}_{t}^{w}$ and ${\epsilon }_{t}$ goes to zero as the sample size increasing. Thus, we can use (8) to construct a variable that will produce a consistent estimator of ${\alpha }_{1}.$ It is this conclusion that underlies the strategy of both two-stage least squares (TSLQ) and instrumental variable (IV) estimators.

## Two-stages least squares

The easiest way to understand two-stage least squares is to think of the estimation process as being in the following two steps (although the computer programs calculate the estimators in one step):

Stage 1: obtain a OLS predictions for any endogenous variable on the right-hand side of the equation to be estimated using as the explanatory variables all of the exogenous variables in the system.

Stage 2: estimate the parameters of the equation using OLS and replacing the endogenous variable on the right-hand side of the equation by the its predictions as obtained in step 1.

For obvious reasons he TSLS method works best when the full model is specified or when you know and can measure all of the exogenous variables in the system.

## Instrumental variables (iv)

While the use of instrumental variable (IV) estimators is appropriate in a large number of situations, the two situations where they are most commonly used are (1) in the presence of endogenous explanatory variables and (2) in cases when errors arise in the measurement of an explanatory variable (or the errors-in-variables problem). Since I have already described the endogeneity problem, I now turn to a brief discussion of errors-in-variables.

Consider the following simple model:

${y}_{i}={\beta }_{1}{x}_{i}^{\ast }+{\epsilon }_{i}\text{and}{x}_{i}={x}_{i}^{\ast }+{\mu }_{i}.$

In this model the researcher observes ${x}_{i}$ but not the desired ${x}_{i}^{\ast }$ because of some random measurement error. Using OLS to estimate (9) using the observable ${x}_{i}$ instead of the correct ${x}_{i}^{\ast }$ is equivalent to estimating:

${y}_{i}={\beta }_{1}{x}_{i}+\left({\epsilon }_{i}-{\beta }_{1}{\mu }_{i}\right).$

The important thing to note in estimating (10) using OLS is that the explanatory variable, ${x}_{i}$ , is correlated with the error term, $\left({\epsilon }_{i}-{\beta }_{1}{\mu }_{i}\right).$ As was the case with the endogeneity problem, the OLS estimate of ${\beta }_{1}$ is biased. Murray (2006) summarizes the situation as follows:

In both examples, ordinary least squares estimation is biased because an explanatory variable in the regression is correlated with the error term in the regression. Such a correlation can result from an endogenous explanator, a mismeasured explanator, an omitted explanator, or a lagged dependent variable among the explanators. I call all such explanators “troublesome.” Instrumental variable estimation can consistently estimate coefficients when ordinary least squares cannot—that is, the instrumental variable estimate of the coefficient will almost certainly be very close to the coefficient’s true value if the sample is sufficiently large—despite troublesome explanators. [Murray (2006a): 112]

how do I set up the problem?
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Abdullahi
hi mam
Mark
find the value of 2x=32
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
how do we prove the quadratic formular
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years? Kala Reply lim x to infinity e^1-e^-1/log(1+x) given eccentricity and a point find the equiation Moses Reply A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place. Kimberly Reply Jeannette has$5 and \$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
What is the expressiin for seven less than four times the number of nickels
How do i figure this problem out.
how do you translate this in Algebraic Expressions
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Got questions? Join the online conversation and get instant answers! By OpenStax By Saylor Foundation By OpenStax By OpenStax By Mackenzie Wilcox By Madison Christian By OpenStax By OpenStax By Anindyo Mukhopadhyay By OpenStax