Generalized Method of Moments (GMM) is a flexible econometric method often used in panel data settings, where multiple entities (N) are observed over multiple periods (T). The effectiveness of GMM depends on the size of both NNN (cross-sectional dimension) and TTT (time dimension).

Sample Size for GMM

Large N, Small T: GMM is most commonly applied in situations with a large NNN (e.g., many individuals, firms, or countries) and a small TTT (e.g., few time periods). This is because GMM exploits the cross-sectional variation more effectively, making it suitable for datasets where NNN is large relative to TTT.

Small N, Small T: When N=15N = 15N=15 and T=5T = 5T=5, both dimensions are small. This can pose challenges because GMM estimators might be biased and inefficient, particularly if TTT is small. The small sample size can lead to weak instruments, undermining the GMM estimates' reliability. This configuration is generally not ideal for GMM.

Large T: As TTT increases, the consistency of GMM estimators improves, but this is often contingent on having a sufficiently large NNN.

Short-Run vs. Long-Run Impacts

• Short-Run Estimates: GMM typically captures the short-run dynamics in the data, especially when used with lagged dependent variables as instruments. The estimates primarily reflect short-run effects.
• Long-Run Estimates: To estimate long-run impacts, you would typically need to consider a model that captures the long-run equilibrium relationships, such as a cointegration framework or adding long-run constraints to your GMM model.

Feasibility of GMM with N=15, T=5

Given N=15N = 15N=15 and T=5T = 5T=5:

• Challenges: GMM might not perform well due to the small sample size, potentially leading to biased estimates and weak instruments.
• Alternative: Depending on the specifics of your data and research question, you might consider other estimation methods better suited to small samples, such as Fixed-Effects or Random-Effects models.

Recommendations

• Use of Estimates: In this scenario, if you proceed with GMM, it’s advisable to interpret the results cautiously and primarily to reflect short-run impacts due to the limited TTT. Estimating long-run effects would require a model explicitly designed for that purpose, potentially with a larger time dimension.

In summary, GMM may not be ideal for N=15N = 15N=15 and T=5T = 5T=5, and the estimates should be interpreted with caution, focusing on short-run impacts. A different approach might be necessary for long-run estimates.

Geetha Baskaran Thank you I really apricate that. In case of increasing N=30 and T=15 , since T>10 are the estimates representing the short run impact as well?

With N=30N = 30N=30 and T=10T = 10T=10 or T=15T = 15T=15, the conditions become more favorable for employing the Generalized Method of Moments (GMM) method.

Feasibility of GMM with N=30 and T=10 or T=15

• Larger N and T: Increasing NNN to 30 and TTT to 10 or 15 provides a more balanced panel dataset. This allows GMM to perform better because:Instrument Strength: A larger NNN improves the strength and reliability of the instruments. Bias Reduction: Increasing TTT reduces the bias typically associated with GMM estimators, making the estimates more reliable. Efficiency: With more time periods, the model can better capture the dynamic aspects of the data, improving the efficiency of the estimators.

Short-Run vs. Long-Run Impacts

• Short-Run Effects: GMM typically captures short-run effects due to its reliance on lagged variables as instruments. Even with T=10T = 10T=10 or T=15T = 15T=15, the estimates from a standard GMM model are generally interpreted as short-run impacts unless the model is explicitly extended to capture long-run relationships.
• Long-Run Effects: To estimate long-run effects, the model would need to incorporate long-run equilibrium constraints or a cointegration framework. Simply increasing TTT doesn’t automatically lead to long-run estimates. However, with a larger TTT, it’s easier to extend the model to explore long-run dynamics if needed.

Recommendations

• GMM Application: With N=30N = 30N=30 and T=10T = 10T=10 or T=15T = 15T=15, GMM becomes a more viable method. The estimators will be more reliable, and the instruments are likely to be stronger.
• Interpretation: The estimates are still generally reflective of short-run effects unless the model is modified to account for long-run relationships.

In summary, with N=30N = 30N=30 and T=10T = 10T=10 or T=15T = 15T=15, GMM is a good choice for capturing the short-run dynamics. If you're interested in long-run effects, you would need to adapt the model accordingly.