7. a) What is instrumental variable(IV)? What is the rationale behind using IV?

Question

7. a) What is instrumental variable(IV)? What is the rationale behind using IV? Explain with an example b) Consider a 4 simple model to estimate the effect university education on wage, log(wage) Bo B1(education) u (316) (i) Why might education be correlated with u? Justify the usage of IV in the above model? What are the necessàry conditions in this case? (ii) Explain how education can be related to parent's education & distance from home. Does this mean parental education & distance from home is a good IV for education? Why or why not? c) Consider the following structural equations (4+4+2) Equation 1: yi- aiV2 + B1z1+u1 Equation 2: y2- a2y + B2z2+ u2 ) Assume, 2 * 1, then find out the reduced form for y2. ii) Assume, 2--1, then find out the reduced form for y ii) Can we use OLS in both of the reduced form equations? Why or why not?

Explanation / Answer

a) An instrumental variable is an exogenous variable that is used to correct for the endogeneity of an explanatory variable in the system. It is used because it is suspected that there may be correlation between the error term and the explanatory variable that leads to biased results when regression is performed. For example, when income is regressed on consumption, it may so happen that the causality may run in the opposite direction or there are omitted variables in the regression that may lead to endogeneity. Here, it may not just be consumption leading to increase in income but also increase in income causing increase in consumption. Also, income may be determined by other factors like availability of jobs, level of investment etc. which may be omitted in the regression, causing endogeneity. So, we introduce an instrumental variable that is correlated with consumption but is unrelated to income i.e. it is exogenous. It is usually difficult to find a good IV due data availability constraints.

b) i) Education may be related to ui since it may also be determined by the wages and there may be omitted variables in the system. Factors like labour union power, level of industrialization etc. may determine wage in addition to education. These may cause problems of simultaneity and omitted variable bias. This causes endogeneity of education and can be corrected through the usage of a suitable instrumental variable for education. The necessary conditions for the IV are that it should be correlated with education but must be exogenous and unrelated to wage.

ii) Education may be related to parent's education since highly educated parents are likely to invest in their children's education. Also, if the school is at a greater distance from home, it may lead to dropping out of school by children since it is costly and time consuming to travel to school. Parent's education is correlated with child's education but may also be related to wages as highly educated parents are likely to get higher wages. Therefore, it cannot be used as an IV. Distance from school, on the other hand, may be a suitable IV since it is correlated to education and not related to wages.

c) i) Substituting the value of y1 from equation 1 into equation 2, we get,

y_2 = alpha_2 (alpha_1 y_2 + beta_1 z_1 + u_1) + beta_2 z_2 + u_2

y_2 ( 1 - alpha_1 alpha_2) = alpha_2 beta_1 z_1 + alpha_2 u_1 + beta_2 z_2 + u_2

y_2 = (alpha_2 beta_1 z_1 + alpha_2 u_1 + beta_2 z_2 + u_2) / (1 - alpha_1 alpha_2)    ... (1)

(1) gives the reduced form equation for y_2.

ii) Similarly by substituting equation 2 in equation 1 , we get,

y_1 = (alpha_1 beta_2 z_2 + alpha_1 u_2 _ beta_1 z_1 + u_1) / (1 - alpha_1 alpha_2)  ... (2)

iii) OLS can be used to estimate both the reduced form equations separately since These equations can be consistently estimated through OLS.

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