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\textit{Wooldridge, Introductory Econometrics, 4th ed.}

\bigskip \textbf{Chapter 16: Simultaneous equations models}

\bigskip An obvious reason for the endogeneity of explanatory variables in a
regression model is \textbf{simultaneity}:\ that is, one or more of the
``explanatory'' variables are jointly determined with the ``dependent''
variable. Models of this sort are known as \textbf{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
simultaneously determined 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 price-quantity pair that we
observe is that resulting from 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:
\begin{equation}
h_{i}=\beta _{0}+\beta _{1}w_{i}+\beta _{2}z_{1}+u_{i}  \label{labsup}
\end{equation}
where the unit of observation might be the county. This is a \textbf{%
structural equation}, or behavioral equation, relating labor supply to its
causal factors: that is, it reflects 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 difficulty 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 from 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 $z$ variable
expresses wage levels in other areas, which would cet.par. 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 specification of labor demand:
\begin{equation}
h_{i}=\gamma _{0}+\gamma _{1}w_{i}+\gamma _{2}z_{2}+\upsilon _{i}
\label{labdem}
\end{equation}
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 (e.g. the cost
of capital) as this equation's $z$ 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 $\gamma _{2}>0.$ 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 $h_{i}$
and $w_{i}.$ Each of those variables will depend on the exogenous variables
in the entire system--$z_{1}$ and $z_{2}$--as well as the structural errors $%
u_{i}$ and $\upsilon _{i}.$ 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 $z_{1}=z_{2},$ 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 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, cet.par.: more police are available to apprehend murderers, and
perhaps prevent murders, while we might expect that lower-income 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,
reflecting the behavior of city officials), but if we are to consistently
estimate the former equation--the behavioral equation reflecting the
behavior of murderers--we will have to specify the second equation as well,
and collect data for its explanatory factors. 

\textbf{Simultaneity bias in OLS}

What goes wrong if we use OLS\ to estimate a structural equation containing
endogenous explanatory variables? Consider the structural system:
\begin{eqnarray}
y_{1} &=&\alpha _{1}y_{2}+\beta _{1}z_{1}+u_{1}  \label{structsys} \\
y_{2} &=&\alpha _{2}y_{1}+\beta _{2}z_{2}+u_{2}  \nonumber
\end{eqnarray}
in which we are interested in estimating the first equation. Assume that the 
$z$ variables are exogenous, in that each is uncorrelated with each of the
error processes $u.$ What is the correlation between $y_{2}$ and $u_{1}?$ If
we substitute the first equation into the second, we derive:
\begin{eqnarray}
y_{2} &=&\alpha _{2}\left( \alpha _{1}y_{2}+\beta _{1}z_{1}+u_{1}\right)
+\beta _{2}z_{2}+u_{2}  \label{qrf}  \nonumber \\
\left( 1-\alpha _{2}\alpha _{1}\right) y_{2} &=&\alpha _{2}\beta
_{1}z_{1}+\beta _{2}z_{2}+\alpha _{2}u_{1}+u_{2} 
\end{eqnarray}

If we assume that $\alpha _{2}\alpha _{1}\neq 1,$ we can derive the \textbf{%
reduced form} equation for $y_{2}$ as:
\begin{equation}
y_{2}=\pi _{21}z_{1}+\pi _{22}z_{2}+\upsilon _{2}  \label{rf}
\end{equation}
where the reduced form error term $\upsilon _{2}=\alpha _{2}u_{1}+u_{2}.$
Thus $y_{2}$ depends on $u_{1},$ and estimation by OLS\ of the first
equation in (\ref{structsys}) will not yield consistent estimates. We can
consistently estimate the reduced form equation (\ref{rf}) via OLS, and that
in fact is an essential part of the strategy of the 2SLS\ 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 from improperly applying OLS\ to the original structural equation.
The covariance of $y_{2}$ and $u_{1}$ is equal to the covariance of $y_{2}$
and $\upsilon _{2}$:
\begin{eqnarray}
& = & \left[ \alpha _{2}/\left( 1-\alpha _{2}\alpha _{1}\right)
E\left( u_{1}^{2}\right) \right] \nonumber \\ 
& = & \left[ \alpha _{2}/\left( 1-\alpha_{2}\alpha _{1}\right) \right] \sigma _{1}^{2}
\end{eqnarray}
 If we have some priors
about the signs of the $\alpha $ parameters, we may sign the bias.
Generally, it could be either positive or negative; that is, the OLS\
coefficient 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 $%
\alpha _{2}=0:$ that is, if $y_{2}$ was not simultaneously determined with $%
y_{1}.$ But in that case, we do not have a simultaneous system; the model in
that case is said to be a \textbf{recursive} system, which may be
consistently estimated with OLS.

\textbf{Identifying and estimating a structural equation}

The tool that we will apply to consistently estimate structural equations
such as (\ref{structsys}) is one that we have seen before: two-stage least
squares (2SLS). The application of 2SLS\ in a structural system is more
straightforward than the general application of instrumental variables
estimators, since the specification of the system makes clear what variables
are available as instruments. Let us first consider a slightly different
two-equation structural system:
\begin{eqnarray}
q &=&\alpha _{1}p+\beta _{1}z_{1}+u_{1}  \label{ss2} \\
q &=&\alpha _{2}p+u_{2}  \nonumber
\end{eqnarray}
We presume these equations describe the workings of a market, and that the
equilibrium condition of market clearing has been imposed. Let $q$ be per
capita milk consumption at the county level, $p$ be the average price of a
gallon of milk in that county, and let $z_{1}$ be the price of cattle feed.
The first structural equation is thus the supply equation, with $\alpha
_{1}>0$ and $\beta _{1}<0$: 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 $\alpha _{2}<0,
$ reflecting the slope of the demand curve in the $\{p,q\}$ plane. Given a
random sample on $\{p,q,z_{1}\},$ what can we achieve? The demand equation
is said to be identified--in fact, \textbf{exactly identified}--since one
instrument is needed, and precisely one is available. $z_{1}$ 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 $z_{1}$ available
to identify the demand curve. Intuitively, we can think of variations in $%
z_{1}$ 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 \textbf{unidentified}. 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 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 flakes, orange juice
might be a healthy substitute for quenching one's thirst? Then the supply
curve would be identified--exactly identified--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 2SLS\ would be used to ``boil them
down'' into the single instrument needed. In that case, we would say that
the supply curve would be \textbf{overidentified}.

The identification status of each structural equation thus hinges upon 
\textbf{exclusion restrictions}: our \textit{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:
\begin{eqnarray}
hours &=&f_{1}\left( \log (wage),educ,age,kl6,wifeY\right) \nonumber 
\label{labmkt} \\
\log (wage) &=&f_{2}\left( hours,educ,xper,xper^{2}\right)  
\end{eqnarray}
The first 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 non-wage 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; \texttt{age, kl6} and \texttt{wifeY}
are available to identify the supply equation, while  \texttt{xper} and 
\texttt{xper}$^{2}$ are available to identify the demand equation. This is
the\textbf{\ order condition} for identfication, essentially counting
instruments and variables to be instrumented; each equation is
overidentified. But the order condition is only necessary; the sufficient
condition is the \textbf{rank condition}, which essentially states that in
the reduced-form equation:
\begin{equation}
\log (wage)=g\left( educ,age,kl6,wifeY,xper,xper^{2}\right) 
\label{rf3}
\end{equation}

at least one of the population coefficients on $\{xper,xper^{2}\}$ must be
nonzero. But since we can consistently estimate this equation with OLS, we
may generate sample estimates of those coefficients, and test the joint null
that both coefficients are zero. If that null is rejected, then we satisfy
the rank condition for the first equation, and we may proceed to estimate it
via 2SLS. The equivalent condition for the demand equation is that at least
one of the population coefficients $\{age,kl6,wifeY\}$ in the
regression of $hours$ on the system's exogenous variables is nonzero. If any
of those variables are significant in the equivalent reduced-form equation,
it may be used as an instrument to estimate the demand equation via 2SLS. 

The application of two-stage least squares (via Stata's \texttt{ivregress 2sls}
command) involves identifying the endogenous explanatory variable(s), 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 explanatory 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 defined
by an equation, and we can proceed to evaluate the identification status of
each equation in turn, given the appropriate exclusion restrictions. Note
that if an equation is \textbf{unidentified}, 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. 

\pagebreak
\textbf{Simultaneous equations models with time series}

One of the most common applications of 2SLS\ in applied work is the
estimation of structural time series models. For instance, consider a simple
macro model:
\begin{eqnarray}
C_{t} &=&\beta _{0}+\beta _{1}\left( Y_{t}-T_{t}\right) +\beta
_{2}r_{t}+u_{1t}  \label{macro}  \nonumber \\
I_{t} &=&\gamma _{0}+\gamma _{1}r_{t}+u_{2t}  \nonumber \\
Y_{t} &=&C_{t}+I_{t}+G_{t} 
\end{eqnarray}

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 suffice, and consumption will be exactly identified. 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, $Y_{t-1}$ (firms make
investment spending plans based on the demand for their product). This would
allow $Y_{t-1},$ a \textbf{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 series data, it should be
clear that those models will generally require instrumental variables
techniques such as 2SLS\ for the consistent estimation of their component
relationships.


\end{document}
