Chapter 14: Simultaneous Equations Systems

Chapter 14: Simultaneous Equations Systems

14-2 Simultaneous Equations

• Many important economic and business models involve simultaneous equations
  • Examples:
  • Supply and demand
  • Keynesian aggregate demand
  • Rational expectations schemes
  • Price of housing
• Classical Assumption III (the assumption that explanatory variables are uncorrelated with the error term) is often violated in simultaneous systems.
• This violation leads to biased OLS coefficient estimates.

14-3 The Nature of Simultaneous Equations Systems

• The Chicken or the Egg Dilemma
  • The relationship between variables is joint and bidirectional (two-way causal effects).
• In a standard econometric equation of the form: $Y<em>t = b</em>0 + b<em>1X</em>{1t} + b<em>2X</em>{2t} + e_t$
  • Both Y affects Xs and Xs affect Y.

14-4 Endogenous vs. Exogenous Variables

• Endogenous variables: Determined within the system.
• Exogenous variables: Not determined within the system.
• Example of simultaneous equations:
  • $Y<em>{1t} = a</em>0 + a<em>1Y</em>{2t} + a<em>2X</em>{1t} + a<em>3X</em>{2t} + e_{1t}$
  • $Y<em>{2t} = b</em>0 + b<em>1Y</em>{1t} + b<em>2X</em>{3t} + b<em>3X</em>{2t} + e_{2t}$
• The equations are known as structural equations, with coefficients expressing economic relationships.

14-5 Structural Equations

• Significance of structural equations lies in the characterization of underlying economic theory by relating endogenous variables to both endogenous and exogenous variables.
• Not all variables are inherently exogenous or endogenous; context matters for their classification.
  • Lagged endogenous variables are included to aid in analysis but are not simultaneously determined.

14-6 Simultaneous Systems Example: Soft-Drink Industry

• Supply and demand equations:
  • $Q<em>{Dt} = a</em>0 + a<em>1P</em>t + a<em>2X</em>{1t} + a<em>3X</em>{2t} + e_{Dt}$
  • $Q<em>{St} = b</em>0 + b<em>1P</em>t + b<em>2X</em>{3t} + e_{St}$
  • Where:
  • $Q_{Dt}$ = quantity demanded
  • $Q_{St}$ = quantity supplied
  • $P_t$ = price
• Both price and quantity are determined simultaneously.

14-10 Violation of Classical Assumption III

• In simultaneous systems, an increase in the error term (e.g., $e<em>{1}$) affects the endogenous variable (e.g., $Y</em>{1}$).
  • Leads to a loop effect where increases propagate through the system.

14-12 Reduced-Form Equations

• These equations simplify simultaneous equations by expressing endogenous variables in terms of fixed variables and error terms without violating Classical Assumption III.
• Example equations:
  • $Y<em>{1t} = p</em>0 + p<em>1X</em>{1t} + p<em>2X</em>{2t} + p<em>3X</em>{3t} + v_{1t}$
  • $Y<em>{2t} = p</em>4 + p<em>5X</em>{1t} + p<em>6X</em>{2t} + p<em>7X</em>{3t} + v_{2t}$

14-16 Understanding Simultaneity Bias

• Simultaneity bias arises from correlation within a simultaneous system, causing OLS coefficients to be biased.
• Example of bias illustrated through concurrent variations in the variables, impacting OLS estimates.

14-20 Two-Stage Least Squares (2SLS)

• 2SLS is an effective method to correct for simultaneity bias.
  • It substitutes endogenous variables with estimates from regressions of reduced-form equations.
• Instrumental variables are required for this technique; they must be correlated with the endogenous variables but not with the error term.

14-31 Identification Problem

• An equation can only be subjected to 2SLS if it is identified, meaning sufficient exogenous variables exist to distinguish it from others within the system.
• An example of simultaneous supply and demand equations reinforces the need for distinguishable variables to perform identification for appropriate estimation.

14-36 The Order Condition of Identification

• For an equation to be identified, the number of predetermined variables (exogenous + lagged endogenous) must be greater than or equal to the number of slope coefficients of interest.
• Illustrated through supply and demand scenarios and the Keynesian model, showing the identification conditions clearly with respect to their variable structures.