Differentiate between Under Identification and Over Identification.

Under-Identification vs. Over-Identification

Under-Identification:

1. 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.

2. 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.

3. 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:

1. 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.

2. 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.

3. 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:

1. Nature of the Problem: Under-identification stems from a lack of identifying information, while over-identification arises from an excess of identifying information.

2. 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.

3. 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.