ch. 14 simultaneous equations

3 reasons for using reduced-form equations

1. do not violate Classical Assumption 3 bc they have no inherent simultaneity, can be estimated with OLS without encountering problems 2. the interpretation of their coefficients as impact multipliers means that they have economic meaning and useful applications of their own 3. reduced-form equations play a crucial role in 2 stage least squares

what is a precondition for the application of 2SLS to equations in simultaneous systems

identification, if equations are not identified in system, SLS cannot be used no matter how large the sample

as fit of the reduced form equation ____, the usefulness of 2sls will increase

increases

does an endogenous variable have to be on the left side at one point in a system of equations?

no, , they are just variables that appear more than once on either left or right side

indirect least squares

regressing the endogenous variable as a function of the instruments. using the coefficient given and setting it equal to B1/(1-B1) to solve for the est coeff -does not give u unbiased coeff, but they are consistent

when there are multiple instruments for a single endogenous varaible, the 2sls estimator is biased in the ___________ direction as the OLS estimator, but it is ____ whereas the OLS estimator is not

same, consistent

test of overidentifying restrictions

test to see whether the different estimates from each instrument in an overidentified system contradict each other aka whether the diff between them are larger than can plausibly be explained by chance -should not contradict, or else at least one of the instruments invalid Ho: the diff estimates of the coeff we get by using diff instruments are not so diff that they could not be explained by chance Ha: the diff coeff produced by diff instruments could not be explained by chance and imply that one or more of of ur instruments is invalid

test of endogeneity (wu-hausman specification test)

testing to see if the right hand side explanatory variable for which you possess an instrument is actually endogenous Ho: variables are exogenous-->no need to do 2SLS bc x is uncorrelated with e Ha: variables are endogenous

an equation is underidentified when

the number of predetermined variables in the system < the number of slope coefficients in the equation in question

an equation is exactly identified when

the number of predetermined variables in the system = the number of slope coefficients in the equation in question

an equation is over identified when

the number of predetermined variables in the system > the number of slope coefficients in the equation in question

impact multipliers

the reduced-form coefficients that measure the impact on the endogenous variable of a 1 unit inc in the value of the predetermined variable, after allowing for the feedback effects from the entire simultaneous system

endogenous variable

variable that are simultaneously/jointly determined in a system of equations (Y's)

instrumental variable ("an instrument")

variable that is 1. highly correlated with the endogenous variable 2. uncorrelated with the error term -used in regression, it avoids the violation of classical assumption 3 by producing predicted values of endogenous variables that can be substituted for the endogenous variables where they appear on the right hand side

exogenous variable

variable that is not simultaneously determined (X's) in a system of equations

simultaneity bias

when applying OLS directly to the structural equations of a simultaneous system, the est. coefficients produced are biased and inconsistent as long as the error term and any of the explanatory variables in equation are correlated, the expected values of the OLS-estimated strutural coefficients are not equal to the true B's

as the sample size gets larger, the 2SLS bias ____, but it is always _____ in a finite sample

falls, nonzero

E. Understand how indirect least squares can produce consistent parameter estimates when there is simultaneity bias.

-ILS does not give us an unbiased estimate of B1 bc B1 is not a linear function of B1/(1-B1) BUT as sample size rises, B1 computed through indirect least squares will converge to the true value of B1

identification

-a structural equation is identified only when the predetermined variables are arranged within the system so as to allow us the use of the observed equilibrium points to distinguish the shape of the equation in question -must have at least one predetermined variable in each equation that is not in the other for equations so that we can identify

C. Be able to explain this graphically in the case of the famous and important example of supply/demand systems.

-in supply and demand, both price and quantity are endogenously determined

B. Understand what the bad consequences of simultaneity bias are for OLS estimates.

-potential bias in all the est. coefficients in a simultaneous equation (since the causation between endogenous variables run both ways, the coeff of an endog variable no longer can be interpreted as its impact on Y1, holding other variables constant and the coeff of a predetermined variable is also impacted bc the endog variable cannot be held constant bc a change in Y affects it as well) -the bias wil have the same sign as the correlation between the error term and the endogenous variable that appers as an explanatory variable in that error term's equation (usually positive/overestimated in economics)

how does two way causation between an X and a Y variable results in the violation of a key assumption of OLS.

-simultaneous equations violates the assumption of independence between the error term and the explanatory variables (classical assumption 3) -when there is endogenous variables that are jointly determined in a system of simultaneous equations, a change in the error term will work its way through the system and eventually affect an explanatory variable in the same equation, violating the assumption ex: demand curve shifting up from an increase in the error term in demand equation--> BOTH price and quantity will increase , and the increase in the error term is correlated with an inc in an IV

properties of 2SLS(3)

1. 2SLS estimates are still biased (2SLS just minimizes bias) 2. if the fit of the reduced-form equation is poor (aka the instrumental variable isnt highly correlated with the original endogenous variable), then 2SLS will not rid the equation of bias and is ineffective, best way to judge fit of reduced form is with F-statistic 3. 2SLS estimates have increased variances and Standard errors than OLS estimates

H. Understand the importance of an F statistic computed from the first stage regression to determine whether the instrumental variables are sufficiently correlated with the endogenous variable to permit us to proceed, the correct F statistic to compute in various cases, and the rule of thumb for an acceptable first stage F value.

1. if there is a single endogenous variable and none of the exogenous variables in first stage regression appear among the explanatory variables in the equations -->how strong the correlation is between the endogenous variable and the instruments is a function of the size of the F-statistic in the first stage of 2sls's output. -if the F-statistic in the first stage regression > 10, our instrumental variable will suffice 2. if at least one of the exogenous and predetermined variables in first stage regression appears among the explanatory variables in the equations you are estimating --> we do an F test on all the coefficients of variables not used as explanatory variables (aka the instruments) H0= B2= B3= B4, Ha: not true and if the F-statistic from our test is > 10, the instruments have a sufficiently strong correlation with the endogenous variable

Understand what is done in the first and second stage of 2SLS when those stages are performed individually, and not as a part of a packaged 2SLS command.

1st stage: Run OLS for each of the endogenous variables that appear as explanatory variables in the structural equations in the system as a function of all the predetermined variables in the reduced form equation --> since the predetermined variables are uncorrelated with the reduced form error term, the OLS est of the reduced form coeff are unbiased and can be used to calculate est. of endgoenous variables 2nd stage: Substitute the reduced form Y's for the Y's that appear on the RIGHT SIDE ONLY aka endogenous independent variables of the structural equations, and then estimate these revised equations

simultaneous equations violate which classical assumption?

3: the error term and each explanatory variable must be uncorrelated with each other

predetermined variable

all exogenous and lagged endogenous variables (not simultaneously determined in the current time period, so not included in endogenous variables) -are determined outside of the system of specified equations or prior to the current period

how many structural equations must there be in a system of equation

as many equations as endogenous variables

F. Understand the advantage of 2SLS over indirect least squares when you have more valid instruments than endogenous variables.

When our equation is overidentified, we have more than 1 instrumental variable to estimate only one parameter aka we get 3 diff B values. 2SLS is superior because it combines these valid estimates of B1

Two-stage least squares

a method of avoiding simultaneity bias by systematically creating variables to replace the endogenous variables where they appear as explanatory variables in a simultaneous equations system

order condition for identification

a necessary condition for a structural equation to be identified -an equation meets the order condition of identification if the number of predetermined variables in the entire simultaneous system must be >= the number of slope coefficients in the equation of interest

how to obtain reduced form equations

by plugging in the systems of structural equations into each other , regrouping terms, and solving for Y-->gets rid of inherent simultaniety

structural equation

characterize the underlying economic theory behind each endogenous variable by expressing it in terms of both endogenous and exogenous variables ex: Y1 = a0 + aX + aY + Ax2 Y2= B0

if exogenous variables _________, we can use them as instruments for price in the supply curve

do not appear in the supply curve

if you do not get an F statistic > 10 after 1st stage regression, what is a possible remedy?

drop the weaker instruments (not stat sig/small t) and see if it will boost the first stage F statistic

each reduced form equation includes only one ______ and that each equation has exactly the same set of _______.

endogenous variable (DV) predetermined variables

reduced form equations

equations that expresses a particular endogenous variable solely in terms of an error term and all the predetermined (exogenous + lagged endogenous) variables in the simultaneous system