Simultaneous Equations Models. Sanjaya DeSilva

Transcription

1 Simultaneous Equations Models Sanjaya DeSilva

2 1 Reduced Form and Structural Models We will begin with definitions; 1. Exogenous variables are variables that are determined outside of the model. For the purposes of the model, they are treated as fixed and given. 2. Endogenous variables are variables that are determined within the model. They can be expressed as functions of other variables. 3. A reduced form regression equation contains only exogenous variables in the right hand side (as regressors). 4. A structural equation is an actual theoretical equation that represents all determinants of the dependent variables. The right hand side may contain both endogenous and exogenous variables as long as they are determinants of the dependent variables. 5. A structural model is a system of structural equations where the determinants of all endogenous variables in the system are explicitly modeled. Consider the following structural model that is fully and correctly specified. Y i = β 0 + β 1 X i + β 2 P i + β 3 Q i + ɛ1 i (1) X i = γ 0 + γ 1 Y i + γ 2 Z i + γ 3 P i + ɛ2 i (2) Z i = δ 0 + δ 1 P i + δ 2 Q i + δ 3 R i + ɛ3 i (3) In this model, Y, X, and Z are endogenous variables and P, Q and R are exogenous variables. Question: Using what criterion was this classification made? 1.1 Simultaneity Two variables, Y and X, have a simultaneous relationship if X is a determinant of Y, and Y is a determinant of X. In the above structural mode, Y and X have a 1

3 simultaneous relationship whereas Z does not have a simultaneous relationship with either X or Y. Question: If two variables have a simultaneous relationship, both of them must be endogenous variables in a structural model. However, if two variables are both endogenous variables in a structural model, they don t necessarily have to be simultaneously related. In other words, simultaneity is a necessary but not sufficient condition for endogeneity. Why? 1.2 OLS estimation of structural equations: The Problem of Bias Suppose we are only interested in the first equation of the structural model, and decide to use OLS to estimate it, ignoring the other two equations. Y i = β 0 + β 1 X i + β 2 P i + β 3 Q i + ɛ1 i (4) OLS assumes that this equation is a reduced form, and therefore all regressors and the error term are uncorrelated. However, in this model, we know that X is in fact an endogenous variable. The endogeneity of X can lead to biased coefficient estimates due to two reasons. 1. Simultaneity leads to bias because it implies that X and ɛ1 are correlated. We see from the first equation in the structural model that ɛ1 and Y are correlated. From the second equation, we see that Y and X are causally related due to the simultaneity. Therefore,ɛ1 and X are correlated, violating the classical assumption required to obtain unbiased coefficient estimates. 2. Even if there is no simultaneity, we could get biased coefficients if the error terms of the structural equations are correlated. For example, if cor(ɛ1, ɛ2) 0, it follows that cor(x, ɛ1) 0 because cor(x, ɛ2) 0 from the second structural equation. Therefore, the classical assumption needed to obtain unbiased coefficients is violated. 2

4 One might argue that the chances that the structural error terms,ɛ1, ɛ2, ɛ3 being correlated is quite slim. However, this argument doesn t makes sense when all endogenous variables are choice variables of the same decision-maker. For example, all three variables may be chosen by the same individual, household, firm or government. In this case, unobservable characteristics of the decision-maker that influences one of these structural equations is likely to influence the other equations as well. 1.3 An Example of a Structural Model Consider a regression equation that attempts to explain whether a country s expenditure on healthcare is adversely impacted by its defense expenditure. The researcher specifies the following reduced form cross-country model to be estimated with OLS. HE i = β 0 + β 1 DE i + β 2 GDP i + β 3 P OP i + ɛ1 i (5) This model overlooks the fact that there is another structural equation that needs to be considered. DE i = γ 0 + γ 1 HE i + γ 2 GDP i + γ 3 P OP i + ɛ2 i (6) Note also that the because both expenditure choices are made by the same government, the error terms of the two equations are correlated because they contain the same set of unobservable characteristics of the government (i.e. political regimes, incentives, values etc). There are two sources of bias in the coefficient estimate of β Suppose there is an exogenous shock that increases the health expenditure of the country (e.g. SARS epidemic). This leads to an increase in HE that in turns leads to a decrease in DE. Therefore,ɛ1 i (e.g. SARS) is correlated with defense expenditure DE due to simultaneity. The coefficient β 1 does not imply a causal relationship from DE to HE. 3

5 2. Suppose there are exogenous political values of the government that influence both health and defense expenditure, and because these values are not measurable, they get absorbed by the error terms which are correlated. For example, in the US, the Republicans have a set of values on the role of government that makes them less likely to see a large government role in the health sector but also more likely to be tough on national defense. Therefore, a Republican administration will have high DE and low HE, and a Democratic administration will have the opposite. Because the values that influence defense are correlated values that influence health policy, DE and ɛ1 i are correlated, causing estimation bias. Here again, a significant coefficient does not imply that high DE caused a decrease in HE. 2 Instrumental Variables: Solution for Endogeneity Bias The commonest solution to the problem of endogeneity bias is to use instrumental variables. An instrumental variable is an exogenous variable that can be used as a proxy for an endogenous variable. Specifically, an instrumental variable needs to have the following three criteria; 1. The instrumental variable must be exogenous. 2. The instrumental variable must be highly correlated with the endogenous variable that it intends to replace. This makes it a good proxy for the endogenous variable. 3. The instrumental variable must be able identify the structural equation. That is, the instrument cannot be a perfect linear function of the exogenous variables already included in the structural equation. 4

6 2.1 Two Stage Least Squares The two-stage least squares method proposes a specific way by which an instrument is constructed. Consider the original structural model we outlined, and suppose we are interested in obtaining unbiased estimates for the coefficients in the first structural equation. This equation has one endogenous variable X i in the right hand side. We need to construct an instrumental variable for X i that satisfied the three criteria for a suitable instrument. In the two stage least squares (2SLS) method, we carry out two steps; In the first stage, we construct the instrument. In the second stage, we use the instrument in the structural equation. Specifically, 1. Run a regression for the included endogenous variable in the structural equation, X i as a function of all the exogenous variables in the entire system. In our case, this is P, Q and R. X i = α 0 + α 1 P i + α 2 Q i + α 3 R i + ɛ i (7) 2. Obtain the predicted value of X i. The predicted value ˆX i is the instrumental variable. ˆX i = ˆα 0 + ˆα 1 P i + ˆα 2 Q i + ˆα 3 R i (8) 3. Run the first structural equation, replacing the endogenous variable X i with its instrument ˆX i Y i = β 0 + β 1 ˆXi + β 2 P i + β 3 Q i + ɛ1 i (9) Let us check whether the instrument ˆX i satisfies the three criteria for a good instrument. 1. ˆXi is weighted average of three exogenous variables P, Q and R. Therefore, ˆXi is exogenous and cannot be correlated with the error term ɛ1 i 2. ˆXi is a good proxy for X i if the predicted value of X and X itself are highly correlated. In other words, the first stage regression should have a high R- 5

7 square; the three exogenous variables,p, Q and R, must explain a large part of the variation of X i. 3. ˆXi can identify the first structural equation if it is not a perfect linear function of the included exogenous variables in that equation. The first equation has two exogenous variables included, P i and Q i. The instrument is a perfect linear function of three exogenous variables,p i, Q i and R i. The additional variable R i allows us to identify the equation. Question: Why is R i special in the identification problem? R i is an exogenous variable in the system that can be excluded from the structural equation. Suppose R i was not included in the first stage. What would this do to the second stage? (Hint: Multicollinearity) 2.2 The Order Condition for Identification A structural equation can be identified if for every endogenous variable it contains in the right hand side, there exists at least one exogenous variable in the system that can be excluded from this equation. In our example, the first equation had one endogenous variable in the right hand side X, and there was one exogenous variable in the system R that was excluded from that equation. Therefore the first equation of the system is Exactly Identified. The second equation in the model has one endogenous variable in the right hand side Y and there are two exogenous variables in the system Q and R that are excluded from that equation. Therefore the second equation of the system is Overidentified Moving on to our system of defense and health expenditure equations, neither equation is identified. Question: Why? Apply the Order Condition. Question: In order to identify the HE equation, we need to include an exogenous variable that influences DE but not HE. In order to identify the DE equation, we need to include an exogenous variable that influences HE but not DE. Why would this help? Can you think of such variables to include? Can you specify a complete 2SLS model that would help us to establish a causal relationship from DE to HE? 6