Differentiate between Structural Form Equations and Reduced Form Equations.
Under-Identification vs. Over-Identification Under-Identification: Definition: Under-identification occurs when a statistical model does not have enough information to estimate the parameters of interest uniquely. In other words, the model is underdetermined by the data, leading to multiple possibleRead more
Under-Identification vs. Over-Identification
Under-Identification:
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Definition: Under-identification occurs when a statistical model does not have enough information to estimate the parameters of interest uniquely. In other words, the model is underdetermined by the data, leading to multiple possible parameter estimates that fit the data equally well.
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Consequences:
- Estimates of the parameters may be biased or unreliable.
- Hypothesis tests may be invalid due to the lack of identifying information.
- The model may not provide useful insights or be suitable for making predictions or policy recommendations.
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Example: In a linear regression model with more predictors than observations, the model may be under-identified, as there are infinitely many parameter estimates that can fit the data equally well.
Over-Identification:
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Definition: Over-identification occurs when a statistical model has more identifying information than necessary to estimate the parameters of interest. This situation allows for the model's parameters to be estimated using different sets of identifying restrictions, providing a check on the reliability of the estimates.
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Consequences:
- Provides a means to test the validity of the identifying restrictions.
- Allows for the estimation of more robust and efficient parameter estimates.
- Can lead to more reliable inference and better understanding of the relationships among variables.
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Example: In a simultaneous equations model where each equation is identified by a set of instruments, having more instruments than strictly necessary for identification would lead to over-identification.
Key Differences:
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Nature of the Problem: Under-identification stems from a lack of identifying information, while over-identification arises from an excess of identifying information.
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Consequences: Under-identification leads to unreliable estimates and invalid tests, while over-identification allows for testing the validity of identifying assumptions and potentially improves the reliability of estimates.
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Resolution: Under-identification may require re-specification of the model or additional data, while over-identification can be addressed using statistical tests or by refining the identifying assumptions.
In conclusion, under-identification and over-identification represent two different challenges in statistical modeling, with under-identification leading to unreliable estimates and over-identification providing an opportunity to test the validity of identifying assumptions and potentially improve the reliability of estimates.
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Structural Form Equations vs. Reduced Form Equations Structural Form Equations: Definition: Structural form equations represent the underlying economic relationships between variables in a theoretical model. They express how endogenous variables are determined by exogenous variables and other endogeRead more
Structural Form Equations vs. Reduced Form Equations
Structural Form Equations:
Definition: Structural form equations represent the underlying economic relationships between variables in a theoretical model. They express how endogenous variables are determined by exogenous variables and other endogenous variables in the system.
Characteristics:
Example: In a simple Keynesian model, the consumption function could be a structural form equation expressing how consumption is determined by income and other factors.
Reduced Form Equations:
Definition: Reduced form equations are derived from structural form equations by solving for endogenous variables in terms of exogenous variables. They represent the observed relationships between variables without explicitly modeling the underlying economic mechanisms.
Characteristics:
Example: In the same Keynesian model, the reduced form equation for consumption could express how consumption changes in response to changes in income, without specifying the underlying reasons for this relationship.
Key Differences:
Nature of Relationship: Structural form equations represent causal relationships based on economic theory, while reduced form equations represent statistical relationships observed in the data.
Endogeneity: Structural form equations explicitly model endogeneity, while reduced form equations treat endogenous variables as determined solely by exogenous variables.
Usefulness: Structural form equations are useful for understanding the economic mechanisms at work and for policy analysis, while reduced form equations are useful for empirical estimation and prediction.
Example: In a supply and demand model, the structural form equations would represent how supply and demand are determined by factors such as price and income, while the reduced form equations would show how quantity and price are related without explicitly modeling supply and demand.
In conclusion, structural form equations and reduced form equations represent different ways of modeling relationships between variables, with structural form focusing on underlying causal mechanisms and reduced form focusing on observed statistical associations.
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