Simultaneous Equations

1
Simultaneous Equations

• Estimation Techniques

2
Indirect Least Squares

• Estimate the reduced form equations
• If structural equation is exactly identified,
  there will be a unique way to calculate the
  parameters.
• Estimates of reduced form parameters are
  unbiased, but estimates of the structural
  parameters will not be. Both are consistent.

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Example
4
Instrumental Variables

• Find a variable Z that is uncorrelated with the
  error term and highly correlated with the
  right-hand side endogenous variable- Y2
• Use Z as a substitute for Y2
• Estimates will be consistent

5
Two Stage Least Squares

• Estimate the reduced form for all the endogenous
  variables that appear on the right hand side.
  Save the predicted values
• Estimate the structural equation but use as
  instruments the predicted endogenous variables
  obtained in the first stage
• Estimates are consistent Shown to be superior
  to many other estimators

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Example
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Limitations of Two Stage Least Squares

• Equations can not be underidentified
• Need good instruments
• Estimates of coefficients are still biased, but
  they are now consistent
• The stronger the correlation among the
  pre-determined variables, the less precise 2SLS.

8
Regulation in the Contact Lens Industry

• Hass-Wilson, Deborah, Tying Requirements in
  Markets with Many Sellers the Contact Lens
  Industry, Review of Economics and Statistics 69
  (February 1987) 170-175.

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Hypothesis

• Price of contact lens will be higher in states
  that prohibit the fitting of contact lenses by
  independent opticians and thereby tie the sale
  of contact lenses to the services of
  ophthalmologists and optometrists.

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The Model

• Oligopoly model
• Firms choose price and quality to maximize
  profits
• Price will depend on marginal cost, market
  elasticity of demand and sellers elasticity of
  demand
• Marginal cost depends on quality

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Two Equations

• Ph(QUALW, SOFT, FITOPH, FITOPTOM, EXOPH, Y,
  INPUT, R-FIT, LIC, R-AD, REG)
• QUALWg(P, FITOPH, FITOPOTOM, SEX, AGE, FAIL,
  WEARTIME, HOURS, DIRT, DAMAGE, WARP, SOFT, R-FIT,
  R-AD, LIC, REG)

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DATA

• Cross sectional data collected by the FTC from
  consumers in 18 urban areas -354 observations
• Price is measured as the adjusted package price
  of the eye examination, keratometric examination
  and contact lenses.
• Quality is measured as an index of the wearers
  eye health or the relative presence of 7
  potentially pathological eye conditions caused by
  poorly fitted contact lenses.

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Predetermined Variables

• SOFT 1 for soft contact lenses, 0 otherwise
• FITOPH1 if fitting was done by ophthalmologist
• FITOPTOM1 if fitting was done by optometrist
• EXOPH 1 if examined by an ophthalmologist
• YIncome
• INPUT - price of inputs
• R-FIT1 if the consumers state has restrictions
  on fitting by opticians
• R-AD 1 if the consumers state restricts
  advertising
• LIC1 if the state requires licenses for opticians

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Predetermined Variables, cont.

• REGIndex of other commercial practice
  restrictions
• SEX1 for male
• Age Age of consumer
• FAIL1 if consumer was unsuccessful in wearing
  contact lenses before
• WEARTIME Wear time prior to the exam
• HOURS Average hours worn per day
• DIRT 1 if lenses were dirty
• DAMAGE 1 if lenses were damaged
• WARP1 if lenses were warped.

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Consequences of Ignoring Simultaneity

• Biased coefficients -- positive bias on estimated
  coefficient of QUALW
• Potential problem with estimating effect of R-FIT
  and REG on price if R-FIT and REG affect QUALW
• Estimator not consistent

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Identification

• Price equation has 11 slope coefficients. There
  are 18 predetermined variables so the price
  equation is over-identified
• Quality equation has 16 slope coefficients so it
  is also over-identified
• Critical to have at least one pre-determined
  variable that explains quality but not price and
  at least one that explains price not quality.

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2SLS Results for Price
Standard errors in parentheses
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2SLS Results for Quality
Standard errors in parentheses
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Findings

• Contact lens prices are approximately 17.30 or
  8 higher in states with restrictions
• Other restrictions also increase prices
• No effect on quality of tying requirements
• No difference in quality between optometrists,
  ophthalmologists, and opticians

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Using Stata

• . regress y1 y2 x (x z)