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Date Created: 10/03/15
Wooldrtdge Introductory Econometrics 2d ed Chapter 16 Simultaneous equations models An obvious reason for the endogeneity of explanatory variables in a regression model is simultaneity that is one or more of the explanatory variables are jointly determined with the dependent variable Models of this sort are known as simultaneous equations models SEMs and they are widely utilized in both applied microeconomics and macroeconomics Each equation in a SEM should be a behavioral equation which describes how one or more economic agents will react to shocks or shifts in the exogenous explanatory variables ceteris paribus The simultaneouslydetermined variables often have an equilibrium interpretation and we consider that these variables are only observed when the underlying model is in equilibrium For instance a demand curve relating the quantity demanded to the price of a good as well as income the prices of substitute commodities etc conceptually would express that quantity for a range of prices But the only pricequantity pair that we observe is that resulting om market clearing where the quantities supplied and demanded were matched and an equilibrium price was struck In the context of labor supply we might relate aggregate hours to the average wage and additional explanatory factors hi I 30 31 3221 W 1 where the unit of observation might be the county This is l a structural equation or behavioral equation relating labor supply to its causal factors that is it re ects the structure of the supply side of the labor market This equation resembles many that we have considered earlier and we might wonder why there would be any di iculty in estimating it But if the data relate to an aggregate such as the hours worked at the county level in response to the average wage in the county this equation poses problems that would not arise if for instance the unit of observation was the individual derived om a survey Although we can assume that the individual is a price or wage taker we cannot assume that the average level of wages is exogenous to the labor market in Suffolk County Rather we must consider that it is determined within the market affected by broader economic conditions We might consider that the 2 variable expresses wage levels in other areas which would cetpar have an effect on the supply of labor in Suffolk County higher wages in MiddleseX County would lead to a reduction in labor supply in the Suffolk County labor market cet par To complete the model we must add a speci cation of labor demand hi 70 711 7222 11239 2 where we model the quantity demanded of labor as a function of the average wage and additional factors that might shift the demand curve Since the demand for labor is a derived demand dependent on the cost of other factors of production we might include some measure of factor cost eg the cost of capital 2 as this equation s 2 variable In this case we would expect that a higher cost of capital would trigger substitution of labor for capital at every level of the wage so that 72 gt 0 Note that the supply equation represents the behavior of workers in the aggregate while the demand equation represents the behavior of employers in the aggregate In equilibrium we would equate these two equations and expect that at some level of equilibrium labor utilization and average wage that the labor market is equilibrated These two equations then constitute a simultaneous equations model SEM of the labor market Neither of these equations may be consistently estimated via OLS since the wage variable in each equation is correlated with the respective error term How do we know this Because these two equations can be solved and rewritten as two reduced form equations in the endogenous variables h and 10 Each of those variables will depend on the exogenous variables in the entire system 21 and ZQ as well as the structural errors u and 11 In general any shock to either labor demand or supply will affect both the equilibrium quantity and price wage Even if we rewrote one of these equations to place the wage variable on the left hand side this problem would persist both endogenous variables in the system are jointly determined by the exogenous variables and structural shocks Another implication of this structure is that we must have separate explanatory factors in the two equations If 21 22 for instance we would not be able to solve this system and uniquely identify its structural parameters There must be factors that are unique to each structural equation 3 that for instance shift the supply curve without shifting the demand curve The implication here is that even if we only care about one of these structural equations for instance we are tasked with modelling labor supply and have no interest in working with the demand side of the market we must be able to specify the other structural equations of the model We need not estimate them but we must be able to determine what measures they would contain For instance consider estimating the relationship between murder rate number of police and wealth for a number of cities We might expect that both of those factors would reduce the murder rate cetpar more police are available to apprehend murderers and perhaps prevent murders while we might expect that lowerincome cities might have greater unrest and crime But can we reasonably assume that the number of police per capita is exogenous to the murder rate Probably not in the sense that cities striving to reduce crime will spend more on police Thus we might consider a second structural equation that expressed the number of police per capita as a function of a number of factors We may have no interest in estimating this equation which is behavioral re ecting the behavior of city of cials but if we are to consistently estimate the former equation the behavioral equation re ecting the behavior of murderers we will have to specify the second equation as well and collect data for its explanatory factors Simultaneity bias in OLS What goes wrong if we use OLS to estimate a struc 4 tural equation containing endogeneous explanatory variables Consider the structural system 21 06122 3121 U1 3 92 06231 3222 U2 in which we are interested in estimating the rst equation Assume that the 2 variables are exogenous in that each is uncorrelated with each of the error processes u What is the correlation between yg and U1 If we substitute the rst equation into the second we derive 92 062 041212 3121 1 3222 U2 4 lt1 05201 yg 0523121 1 3222 O gul U2 5 If we assume that 052051 7 1 we can derive the reduced form equation for yg as 32 772121 772222 139 U2 6 where the reduced form error term 112 ozgul U2 Thus yg depends on U1 and estimation by OLS of the rst equation in 3 will not yield consistent estimates We can consistently estimate the reduced form equation 6 via OLS and that in fact is an essential part of the strategy of the ZSLS estimator But the parameters of the structural equation are nonlinear transformations of the reduced form parameters so being able to estimate the reduced form parameters does not achieve the goal of providing us with point and interval estimates of the structural equation In this special case we can evaluate the simultaneity bias that would result om improperly applying OLS to the original structural equation The covariance of y2 and U1 is equal to the covariance of y2 and v2 Cglt1 052051 E 052 l 052051 0 If we have some priors about the signs of the oz parameters we may sign the bias Generally it could be either positive or negative that is the OLS coe icient estimate could be larger or smaller than the correct estimate but will not be equal to the population parameter in an expected sense unless the bracketed expression is zero Note that this would happen if 052 0 that is if yg was not simultaneously determined with yl But in that case we do not have a simultaneous system the model in that case is said to be a recursive system which may be consistently estimated with OLS Identifying and estimating a structural equation The tool that we will apply to consistently estimate structural equations such as 3 is one that we have seen before twostage least squares ZSLS The application of ZSLS in a structural system is more straightforward than the general application of instrumental variables estimators since the speci cation of the system makes clear what variables are available as instruments Let us rst consider a slightly different twoequation structural system q mp 3121 m 7 052 NZ 6 Q We presume these equations describe the workings of a market and that the equilibrium condition of market clearing has been imposed Let g be per capita milk consumption at the county level p be the average price of a gallon of milk in that county and let 21 be the price of cattle feed The rst structural equation is thus the supply equation with 11 gt 0 and l lt 0 that is a higher cost of production will generally reduce the quantity supplied at the same price per gallon The second equation is the demand equation where we presume that 052 lt 0 re ecting the slope of the demand curve in the 19 q plane Given a random sample on 19 q 21 what can we achieve The demand equation is said to be identi ed in fact exactly identi ed since one instrument is needed and precisely one is available 21 is available because the demand for milk does not depend on the price of cattle feed so we take advantage of an exclusion restriction that makes 21 available to identify the demand curve Intuitively we can think of variations in 21 shifting the supply curve up and down tracing out the demand curve in doing so it makes it possible for us to estimate the structural parameters of the demand curve What about the supply curve It also has a problem of simultaneity bias but it turns out that the supply equation is unidenti ed Given the model as we have laid it out there is no variable available to serve as an instrument for p that is we need a variable that affects demand and shifts the demand curve but does not directly affect supply In this case no such variable is available and we cannot apply the instrumental variables 7 technique without an instrument What if we went back to the drawing board and realized that the price of orange juice should enter the demand equation although it tastes terrible on corn akes orange juice might be a healthy substitute for quenching one s thirst Then the supply curve would be identi ed exactly identi ed since we now would have a single instrument that served to shift demand but did not enter the supply relation What if we also considered the price of beer as an additional demand factor Then we would have two available instruments presuming that each is appropriately correlated and ZSLS would be used to boil them down into the single instrument needed In that case we would say that the supply curve would be overidenti ed The identi cation status of each structural equation thus hinges upon exclusion restrictions our a priori statements that certain variables do not appear in certain structural equations If they do not appear in a structural equation they may be used as instruments to assist in identifying the parameters of that equation For these variables to successfully identify the parameters they must have nonzero population parameters in the equation in which they are included Consider an example hours f1 lt10gltwage educ age kidsltfi nwz39fez39nCQS logltwagegt f2 hours educ Eper Eper2 The rst equation is a labor supply relation expressing the number of hours worked by a married woman as a function of her wage education age the number of preschool children and 8 nonwage income including spouses s earnings The second equation is a labor demand equation expressing the wage to be paid as a function of hours worked the employee s education and a polynomial in her work experience The exclusion restructions indicate that the demand for labor does not depend on the worker s age nor should it the presence of preschool kids or other resources available to the worker Likewise we assume that the woman s willingness to participate in the market does not depend on her labor market experience One instrument is needed to identify each equation age kidslt6 and nwifenc are available to identify the supply equation while Xper and Xper2 are available to identify the demand equation This is the order condition for ident cation essentially counting instruments and variables to be instrumented each equation is overidenti ed But the order condition is only necessary the suf cient condition is the rank condition which essentially states that in the reducedform equation logltwagegt g educ age kidsltf nwz39fez39nc Eper1per2 9 at least one of the population coef cients on p8T Eperg must be nonzero But since we can consistently estimate this equation with OLS we may generate sample estimates of those coef cients and test the joint null that both coef cients are zero If that null is rejected then we satisfy the rank condition for the rst equation and we may proceed to estimate it via ZSLS The equivalent condition for the demand equation is that at least 9 one of the population coef cients age kidsltf mud f einc in the regression of hours on the system s exogenous variables is nonzero If any of those variables are signi cant in the equivalent reducedform equation it may be used as an instrument to estimate the demand equation via ZSLS The application of twostage least squares via Stata s ivreg command involves identifying the endogenous explana tory variables the exogenous variables that are included in each equation and the instruments that are excluded from each equation To satisfy the order condition the list of excluded instruments must be at least as long as the list of endogenous ex planatory variables This logic carries over to structural equation systems with more than two endogenous variables equations a structural model may have any number of endogenous variables each de ned by an equation and we can proceed to evalu ate the identi cation status of each equation in turn given the appropriate exclusion restrictions Note that if an equation is unidenti ed due to the lack of appropriate instruments then no econometric technique may be used to estimate its parameters In that case we do not have knowledge that would allow us to trace out that equation s slope while we move along it Simultaneous equations models with time series One of the most common applications of ZSLS in applied work is the estimation of structural time series models For instance consider a simple macro model Gt 2 o 1ltYt Ti grtult 10 10 It 70 71th U2t Y1 Ct It Gt In this system aggregate consumption each quarter is determined jointly with disposable income Even if we assume that taxes are exogenous and in fact they are responsive to income the consumption function cannot be consistently estimated via OLS If the interest rate is taken as exogenous set for instance by monetary policy makers then the investment equation may be consistently estimated via OLS The third equation is an identity it need not be estimated and holds without error but its presence makes explicit the simultaneous nature of the model If r is exogenous then we need one instrument to estimate the consumption function government spending will suf ce and consumption will be exactly identi ed If r is to be taken as endogenous we would have to add at least one equation to the model to express how monetary policy reacts to economic conditions We might also make the investment function more realistic by including dynamics that investment depends on lagged income for instance Yl1 rms make investment spending plans based on the demand for their product This would allow Yl39g1 a predetermined variable to be used as an additional instrument in estimation of the consumption function We may also use lags of exogenous variables for instance lagged taxes or government spending as instruments in this context Although this only scratches the surface of a broad set of issues relating to the estimation of structural models with time 11 series data it should be clear that those models Will generally require instrumental variables techniques such as ZSLS for the consistent estimation of their component relationships 12
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