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Simultaneous Equations Models

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Title: Simultaneous Equations Models


1
Chapter 9
  • Simultaneous Equations Models
  • (???????)

2
What is in this Chapter?
  • How do we detect this problem?
  • What are the consequences?
  • What are the solutions?

3
What is in this Chapter?
  • In Chapter 4 we mentioned that one of the
    assumptions in the basic regression model is that
    the explanatory variables are uncorrelated with
    the error term
  • In this chapter we relax that assumption and
    consider the case where several variables are
    jointly determined
  • Predetermined vs. jointly determined
  • Exogenous vs. Endogenous

4
What is in this Chapter?
  • This chapter first discusses the conditions under
    which equations are estimable in the case of
    jointly determined variables (the "identification
    problem") and methods of estimation
  • One major method is that of "instrumental
    variables"
  • Finally, this chapter also discusses causality

5
9.1 Introduction
  • In the usual regression model y is the dependent
    or determined variable and x1, x2, x3... Are the
    independent or determining variables
  • The crucial assumption we make is that the x's
    are independent of the error term u
  • Sometimes, this assumption is violated for
    example, in demand and supply models

6
9.1 Introduction
  • Suppose that we write the demand function as
  • where q is the quantity demanded, p the price,
    and u the disturbance term which denotes random
    shifts in the demand function
  • In Figure 9.1 we see that a shift in the demand
    function produces a change in both price and
    quantity if the supply curve has an upward slope

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9
9.1 Introduction
  • If the supply curve is horizontal (i.e.,
    completely price inelastic), a shift in the
    demand curve produces a change in price only
  • If the supply curve is vertical (infinite price
    elasticity), a shift in the demand curve produces
    a change in quantity only

10
9.1 Introduction
  • Thus in equation (9.1) the error term u is
    correlated with p when the supply curve is upward
    sloping or perfectly horizontal
  • Hence an estimation of the equation by ordinary
    least squares produces inconsistent estimates of
    the parameters

11
9.2 Endogenous and Exogenous Variables
  • In simultaneous equations models variables are
    classified as endogenous and exogenous
  • The traditional definition of these terms is that
    endogenous variables are variables that are
    determined by the economic model and exogenous
    variables are those determined from outside

12
9.2 Endogenous and Exogenous Variables
  • Endogenous variables are also called jointly
    determined and exogenous variables are called
    predetermined. (It is customary to include past
    values of endogenous variables in the
    predetermined group.)
  • Since the exogenous variables are predetermined,
    they are independent of the error terms in the
    model
  • They thus satisfy the assumptions that the x's
    satisfy in the usual regression model of y on x's

13
9.2 Endogenous and Exogenous Variables
  • Consider now the demand and supply mode
  • q a1 b1p c1 y u1 demand function
  • q a2 b2p c2R u2 supply function
    (9.2)
  • q is the quantity, p the price, y the income, R
    the rainfall, and u1 and u2 are the error terms
  • Here p and q are the endogenous variables and y
    and R are the exogenous variables

14
9.2 Endogenous and Exogenous Variables
  • Since the exogenous variables are independent of
    the error terms u1 and u2 and satisfy the usual
    requirements for ordinary least squares
    estimation, we can estimate regressions of p and
    q on y and R by ordinary least squares, although
    we cannot estimate equations (9.2)by ordinary
    least squares
  • We will show presently that from these
    regressions of p and q on y and R we can recover
    the parameters in the original demand and supply
    equations (9.2)

15
9.2 Endogenous and Exogenous Variables
  • This method is called indirect least squaresit
    is indirect because we do not apply least squares
    to equations (9.2)
  • The indirect least squares method does not always
    work, so we will first discuss the conditions
    under which it works and how the method can be
    simplified. To discuss this issue, we first have
    to clarify the concept of identification

16
9.3 The Identification Problem Identification
Through Reduced Form
  • We have argued that the error terms u1 and u2 are
    correlated with p in equations (9.2),and hence if
    we estimate the equation by ordinary least
    squares, the parameter estimates are inconsistent
  • Roughly speaking, the concept of identification
    is related to consistent estimation of the
    parameters
  • Thus if we can somehow obtain consistent
    estimates of the parameters in the demand
    function, we say that the demand function is
    identified

17
9.3 The Identification Problem Identification
Through Reduced Form
  • Similarly, if we can somehow get consistent
    estimates of the parameters in the supply
    function, we say that the supply function is
    identified
  • Getting consistent estimates is just a necessary
    condition for identification, not a sufficient
    condition, as we show in the next section

18
9.3 The Identification Problem Identification
Through Reduced Form
  • If we solve the two equations in(9.2) for q and p
    in terms of y and R, we get
  • These equations are called the reduced-form
    equations.
  • Equation (9.2) are called the structural
    equations because they describe the structure of
    the economic system.

19
9.3 The Identification Problem Identification
Through Reduced Form
  • We can write equations (9.3) as
  • where v1 and v2 are error terms and

20
9.3 The Identification Problem Identification
Through Reduced Form
  • The ps are called reduced-form parameters.
  • The estimation of the equations (9.4) by ordinary
    least squares gives us consistent estimates of
    the reduced form parameters.
  • From these we have to obtain consistent estimates
    of the parameters in

21
9.3 The Identification Problem Identification
Through Reduced Form
  • Since are all
    single-valued function of the ,they are
    consistent estimates of the corresponding
    structural parameters.
  • As mentioned earlier, this method is known as the
    indirect least squares method.

22
9.3 The Identification Problem Identification
Through Reduced Form
  • It may not be always possible to get estimates of
    the structural coefficients from the estimates of
    the reduced-form coefficients, and sometimes we
    get multiple estimates and we have the problem of
    choosing between them.
  • For example, suppose that the demand and supply
    model is written as

23
9.3 The Identification Problem Identification
Through Reduced Form
  • Then the reduced from is

24
9.3 The Identification Problem Identification
Through Reduced Form
  • or
  • In this case and
    .
  • But these is no way of getting estimates of a1,
    b1, and c1.
  • Thus the supply function is identified but the
    demand function is not.

25
9.3 The Identification Problem Identification
Through Reduced Form
  • On the other hand, suppose that we have the model
  • Now we can check that the demand function is
    identified but the supply function is not.

26
9.3 The Identification Problem Identification
Through Reduced Form
  • Finally, suppose that we have the system

27
9.3 The Identification Problem Identification
Through Reduced Form
  • or
  • Now we get two estimates of b2.
  • One is and the other is
    , and these need not be equal.
  • For each of these we get an estimate of a2, which
    is .

28
9.3 The Identification Problem Identification
Through Reduced Form
  • On the other hand, we get no estimate for the
    parameters a1 , b1, c1, and d1 of the demand
    function.
  • Here we say that the supply function is
    overidentified and the demand function is
    underidentified.
  • When we get unique estimates for the structural
    parameters of an equation fro, the reduced-form
    parameters, we say that the equation is exactly
    identified.

29
9.3 The Identification Problem Identification
Through Reduced Form
  • When we get multiple estimates, we say that the
    equation is overidentified, and when we get no
    estimates, we say that the equation is
    underidentified (or not identified).
  • There is a simple counting rule available in the
    linear systems that we have been considering.
  • This counting rule is also known as the order
    condition for identification.

30
9.3 The Identification Problem Identification
Through Reduced Form
  • This rule is as follows Let g be the number of
    endogenous variables in the system and k the
    total number of variables (endogenous and
    exogenous) missing from the equation under
    consideration.
  • Then

31
9.3 The Identification Problem Identification
Through Reduced Form
  • This condition is only necessary but not
    sufficient.
  • Let us apply this rule to the equation systems we
    are considering.
  • In equations (9.2), g, the number of endogenous
    variable, is 2 and there is only one variable
    missing from each equation (i.e., k1).
  • Both equations are identified exactly.

32
9.3 The Identification Problem Identification
Through Reduced Form
  • In equations (9.5), again g2.
  • There is no variable missing from the first
    equation (i.e., k0) hence it is
    underidentified.
  • There is one variable missing in the second
    equation (i.e., k1) hence it is exactly
    identifies.
  • In equation (9.6)
  • there is no variable missing in the first
    equation hence it is not identified.
  • In the second equation there are two variables
    missing thus kgtg-1 and the equation is
    overidentified.

33
9.3 The Identification Problem Identification
Through Reduced Form
  • Illustrative Example
  • In Table 9.1 data are presented for demand and
    supply of pork in the United States for 1922-1941

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36
9.3 The Identification Problem Identification
Through Reduced Form
  • Pt, retail price of pork (cents per pound)
  • Qt, consumption of pork (pounds per capita)
  • Yt, disposable personal income (dollars per
    capital)
  • Zt, predetermined elements in pork production.

37
9.3 The Identification Problem Identification
Through Reduced Form
  • The coefficient of Y in the second equation is
    very close to zero and the variable Y can be
    dropped from this equation.
  • This would imply that b20, or supply is not
    responsive to price.
  • In any case, solving from the reduced from to the
    structural from, we get the estimates of the
    structural equation as

38
9.3 The Identification Problem Identification
Through Reduced Form
  • The least squares estimates of the demand
    function are
  • Normalized with respect to Q
  • Normalized with respect to P

39
9.3 The Identification Problem Identification
Through Reduced Form
  • The structural demand function can also be
    written in the two forms
  • Normalized with respect to Q
  • Normalized with respect to P
  • The estimates of the parameters in the demand
    function are almost the same with the direct
    least squares method as with the indirect least
    squares method when the demand function is
    normalized with respect to P.

40
9.3 The Identification Problem Identification
Through Reduced Form
  • Which is the correct normalization?
  • We argued in Section 9.1 that if quantity
    supplied is not responsive to price, the demand
    function should be normalized with respect to P.
  • We saw that fact the coefficient of Y in the
    reduced-form equation for Q was close to zero
    implied that b20 or quantity supplied is not
    responsive to price.

41
9.3 The Identification Problem Identification
Through Reduced Form
  • This is also confirmed by the structural estimate
    of b2, which show a wrong sign for b2 as well but
    a coefficient close to zero.
  • Dropping P from the supply function and using
    OLS, we get the supply function as

42
9.5 Methods of Estimation The Instrumental
Variable Method
  • In previous sections we discussed the indirect
    least squares method
  • However, this method is very cumbersome if there
    are many equations and hence it is not often used
  • Identification problem
  • Here we discuss some methods that are more
    generally applicable
  • The Instrumental Variable Method

43
9.5 Methods of Estimation The Instrumental
Variable Method
  • Broadly speaking, an instrumental variable is a
    variable that is uncorrelated with the error term
    but correlated with the explanatory variables in
    the equation
  • For instance, suppose that we have the equation
  • y ßx u

44
9.5 Methods of Estimation The Instrumental
Variable Method
  • where x is correlated with u
  • Then we cannot estimate this equation by ordinary
    least squares
  • The estimate of ß is inconsistent because of the
    correlation between x and u
  • If we can find a variable z that is uncorrelated
    with u, we can get a consistent estimator for ß
  • We replace the condition cov (z, u) 0 by its
    sample counterpart

45
9.5 Methods of Estimation The Instrumental
Variable Method
  • This gives
  • But can be written as

46
9.5 Methods of Estimation The Instrumental
Variable Method
  • The probability limit of this expression is
  • since cov (z, x) ?0.
  • Hence plim ,thus proving that is a
    consistent estimator for ß.
  • Note that we require z to be correlated with x so
    that cov (z, x) ?0.

47
9.5 Methods of Estimation The Instrumental
Variable Method
  • Now consider the simultaneous equations model
  • where y1, y2 are endogenous variables, z1,
    z2, z3 are exogenous variables, and u1, u2 are
    error term.
  • Since z1 and z2 are independent of u1,
  • cov (z1, u1) 0 , cov (z2, u1) 0
  • However, y2 is not independent of u1
  • cov (y2, u1) ?0.

48
9.5 Methods of Estimation The Instrumental
Variable Method
  • Since we have three coefficients to estimate, we
    have to find a variable that is independent of
    u1.
  • Fortunately, in this case we have z3 and
    cov(z3,u1)0.
  • z3 is the instrumental variable for y2.
  • Thus, writing the sample counterparts of these
    three covariances, we have three equations

49
9.5 Methods of Estimation The Instrumental
Variable Method
  • The difference between the normal equation for
    the ordinary least squares method and the
    instrumental variable method is only in the last
    equation.

50
9.5 Methods of Estimation The Instrumental
Variable Method
  • Consider the second equation of our model
  • Now we have to find an instrumental variable for
    y1 but we have a choice of z1 and z2
  • This is because this equation is overidentified
    (by the order condition)
  • Note that the order condition (counting rule) is
    related to the question of whether or not we have
    enough exogenous variables elsewhere in the
    system to use as instruments for the endogenous
    variables in the equation with unknown
    coefficients

51
9.5 Methods of Estimation The Instrumental
Variable Method
  • If the equation is underidentified we do not have
    enough instrumental variables
  • If it is exactly identified, we have just enough
    instrumental variables
  • If it is overidentified, we have more than enough
    instrumental variables
  • In this case we have to use weighted averages of
    the instrumental variables available
  • We compute these weighted averages so that we get
    the most efficient (minimum asymptotic variance)
    estimator

52
9.5 Methods of Estimation The Instrumental
Variable Method
  • It has been shown (proving this is beyond the
    scope of this book) that the efficient
    instrumental variables are constructed by
    regressing the endogenous variables on all the
    exogenous variables in the system (i.e.,
    estimating the reduced-form equations).
  • In the case of the model given by equations
    (9.8), we first estimate the reduced-form
    equations by regressing y1 and y2 on z1, z2, z3.
  • We obtain the predicted values
    and use these as instrumental variables.

53
9.5 Methods of Estimation The Instrumental
Variable Method
  • For the estimation of the first equation we use
    , and for the estimation of the second
    equation we use .
  • We can write and as linear function of
    z1, z2, z3.
  • Let us write
  • where the as are obtained from the estimation
    of the reduced-form equations by OLS.

54
9.5 Methods of Estimation The Instrumental
Variable Method
  • In the estimation of the first equation in (9.8)
    we use , z1, z2, and z3 as instruments.
  • This is the same as using z1, z2, z3 as
    instruments because
  • But the first two terms are zero by virtue of the
    first two equations in (9.8).
  • Thus .
    Hence using as an instrumental variable is
    the same as using z3 as an instrumental variable.
  • This is the case with exactly indentified
    equations where there is no choice in the
    instruments.

55
9.5 Methods of Estimation The Instrumental
Variable Method
  • The case with the second equation in (9.8) is
    different.
  • Earlier, we said that we had a choice between z1
    and z2 as instruments for y1.
  • The use of gives the optimum weighting.
  • The normal equations now are
  • since .
  • Thus the optimal weights for z1 and z2 are a11
    and a12.

56
9.5 Methods of Estimation The Instrumental
Variable Method
  • Illustrative Example
  • Table 9.2 provides data on some characteristics
    of the wine industry in Australia for 1955-1956
    to 1974-1975.
  • The demand-supply model for the wine industry

57
9.5 Methods of Estimation The Instrumental
Variable Method
58
9.5 Methods of Estimation The Instrumental
Variable Method
  • where Qt real capital consumption of wine
  • price of wine relative to CPI
  • price of beer relative to CPI
  • Yt real per capital disposable
    income
  • At real per capital advertising
    expenditure
  • St index of storage costs
  • are the endogenous variables
  • The other variable are exogenous.

59
9.5 Methods of Estimation The Instrumental
Variable Method
  • For the estimation of the demand function we have
    only one instrumental variable St.
  • But for the estimation of the supply function we
    have available three instrumental variables
  • The OLS estimation of the demand function gave
    the following results (all variables are in logs
    and figures in parentheses are t-ratios)
  • All the coefficients except that of Y have the
    wrong signs.
  • The coefficient of Pw not only has the wrong sign
    but is also significant.

60
9.5 Methods of Estimation The Instrumental
Variable Method
  • Treating Pw as endogenous and using S as an
    instrument, we get following results
  • The coefficient of Pw still has a wrong sign but
    it is at least not significant.
  • In any case the conclusion we arrive at is that
    the quantity demanded is not responsive to prices
    and advertising expenditures but is responsive to
    income.
  • The income elasticity of demand for wine is about
    4.0 (significantly greater than unity).

61
9.6 Methods of Estimation The Two-Stage Least
Squares Method
  • The 2SLS method differs the IV method described
    in Section 9.5 in that the s are used as
    regressors rather than as instruments, but the
    two methods give identical estimates.
  • Consider the equation to be estimated
  • The other exogenous variables in the system are
    z2, z3, and z4.

62
9.6 Methods of Estimation The Two-Stage Least
Squares Method
  • Let be the predicted value of y2 from, a
    regression on y2 on z1, z2, z3, and z4 (the
    reduces-form equation).
  • Then where v2, the
    residual, is uncorrelated with each of the
    regressors, z1, z2, z3, and z4 and hence with
    as well. (This is the property of least squares
    regression that we discussed in Chapter 4.)

63
9.6 Methods of Estimation The Two-Stage Least
Squares Method
  • The normal equations for the efficient IV method
    are
  • Substituting we get

64
9.6 Methods of Estimation The Two-Stage Least
Squares Method
  • But these are the normal equations if we replace
    y2 by in (9.9) and estimate the equation by
    OLS.
  • This method of replacing the endogenous variables
    on the right-hand side by their predicted values
    from the reduced form and estimating the equation
    by OLS is called the two-stage least squares
    (2SLS) method.

65
9.6 Methods of Estimation The Two-Stage Least
Squares Method
  • The name arises from the fact that OLS is used in
    two stages
  • Stage 1. Estimate the reduced-form equations
    by
  • OLS and obtain the predicted
    s.
  • Stage 2.Replace the right-hand side endogenous
  • variables by s and estimate
    the
  • equation by OLS.

66
9.6 Methods of Estimation The Two-Stage Least
Squares Method
  • Note that the estimates do not change even if we
    replace y1 by in equation (9.9).
  • Take the normal equations (9.12).
  • Now substitute in equations
    (9.12).
  • We get

67
9.6 Methods of Estimation The Two-Stage Least
Squares Method
  • The last terms of these two equations are zero
    and the equations that remain are the normal
    equations from the OLS estimation of the equation
  • Thus in stage 2 of the 2SLS method we can replace
    all the endogenous variables in the equation by
    their predicted values from the reduced forms and
    then estimate the equation by OLS.

68
9.10 Granger Causality
  • Granger starts from the premise that the future
    cannot cause the present or the past.
  • If event A occurs after event B, we know that A
    cannot cause B.
  • At the same time, if A occurs before B, it does
    not necessarily imply that A causes B.
  • For instance, the weatherman's prediction occurs
    before the rain. This does not mean that the
    weatherman causes the rain.

69
9.10 Granger Causality
  • In practice, we observe A and B as time series
    and we would like to know whether A precedes B,
    or B precedes A, or they are contemporaneous
  • For instance, do movements in prices precede
    movements in interest rates, or is it the
    opposite, or are the movements contemporaneous?
  • This is the purpose of Granger causality
  • It is not causality as it is usually understood

70
9.10 Granger Causality
  • Granger devised some tests for causality (in the
    limited sense discussed above) which proceed as
    follows.
  • Consider two time series, yt and xt.
  • The series xt fails to Granger cause yt if in a
    regression of yt on lagged ys and lagged xs,
    the coefficients of the latter are zero.
  • That is, consider
  • Then if ßi0 (i1,2,....,k), xt fails to cause
    yt.
  • The lag length k is, to some extent, arbitrary.

71
9.10 Granger Causality
  • Learner suggests using the simple word
    "precedence" instead of the complicated words
    Granger causality since all we are testing is
    whether a certain variable precedes another and
    we are not testing causality as it is usually
    understood
  • However, it is too late to complain about the
    term since it has already been well established
    in the econometrics literature. Hence it is
    important to understand what it means
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