Weak instruments can wreak havoc on instrumental variable (IV) estimation, leading to biased and inconsistent results. This topic explores the definition, consequences, and detection of weak instruments, as well as strategies to address this issue in econometric analysis.

Understanding weak instruments is crucial for accurate causal inference. We'll examine methods to detect weak instruments, such as the first-stage F-statistic and Cragg-Donald statistic, and discuss alternative approaches like LIML and JIVE to mitigate their effects on estimation.

Definition of weak instruments

• Weak instruments are instrumental variables that are only weakly correlated with the endogenous explanatory variables in a regression model
• The presence of weak instruments can lead to biased and inconsistent estimates of the causal effect of interest
• Instruments are considered weak when the correlation between the instrument and the endogenous explanatory variable is low relative to the sample size
  • For example, if an instrument is only weakly correlated with the endogenous variable (correlation coefficient of 0.1), it may not provide enough variation to identify the causal effect

Consequences of weak instruments

Bias in IV estimators

• Weak instruments can cause the instrumental variable (IV) estimator to be biased towards the ordinary least squares (OLS) estimator
• The bias of the IV estimator increases as the strength of the instruments decreases
• In the presence of weak instruments, the IV estimator may not be consistent, meaning it does not converge to the true value even as the sample size increases
• The bias can be substantial, especially in small samples or when the endogeneity problem is severe

Misleading inference

• Weak instruments can lead to misleading statistical inference, such as incorrect confidence intervals and hypothesis tests
• The standard errors of the IV estimator may be underestimated, leading to overly narrow confidence intervals and a higher likelihood of Type I errors (rejecting a true null hypothesis)
• Conventional tests, such as the t-test and the Wald test, may have poor size properties and low power when instruments are weak
• Inference based on weak instruments can lead to incorrect conclusions about the significance and magnitude of the causal effect

Detecting weak instruments

First-stage F-statistic

• The first-stage F-statistic is a commonly used diagnostic tool for assessing the strength of instruments
• It tests the joint significance of the excluded instruments in the first-stage regression of the endogenous variable on the instruments and exogenous variables
• A high F-statistic (e.g., greater than 10) suggests that the instruments are strong, while a low F-statistic indicates weak instruments
• However, the F-statistic may not be reliable in the presence of multiple endogenous variables or heteroskedasticity

Cragg-Donald statistic

• The Cragg-Donald statistic is a generalization of the first-stage F-statistic for models with multiple endogenous variables
• It tests the rank condition for identification and provides a measure of the strength of the instruments
• A higher Cragg-Donald statistic indicates stronger instruments, while a low value suggests weak instruments
• The Cragg-Donald statistic is often compared to critical values derived by Stock and Yogo (2005) to assess instrument strength

Stock-Yogo critical values

• Stock and Yogo (2005) provide critical values for the Cragg-Donald statistic to test for weak instruments
• The critical values are based on the maximum acceptable bias of the IV estimator relative to the OLS estimator (e.g., 5%, 10%, 20%, 30%)
• If the Cragg-Donald statistic exceeds the relevant critical value, the instruments are considered strong enough to limit the bias of the IV estimator
• The Stock-Yogo critical values provide a formal test for weak instruments and help researchers determine the reliability of their IV estimates

Dealing with weak instruments

Selecting stronger instruments

• One approach to dealing with weak instruments is to carefully select instruments that are more strongly correlated with the endogenous explanatory variables
• Researchers can draw on economic theory, institutional knowledge, or prior empirical evidence to identify potential instruments
• Stronger instruments can help reduce the bias and improve the precision of the IV estimator
• However, finding suitable instruments that satisfy the exclusion restriction and relevance condition can be challenging in practice

Limited information maximum likelihood (LIML)

• LIML is an alternative estimator that is more robust to weak instruments compared to the standard IV estimator
• It is a maximum likelihood estimator that accounts for the presence of weak instruments and provides more reliable estimates
• LIML has better finite sample properties than the IV estimator and is less biased in the presence of weak instruments
• However, LIML may have higher variance than the IV estimator and can be sensitive to the specification of the model

Jackknife IV estimator (JIVE)

• The jackknife IV estimator is another alternative estimator designed to mitigate the bias caused by weak instruments
• JIVE removes the correlation between the instruments and the error term by leaving out each observation when estimating the fitted values of the endogenous variable
• This jackknife procedure reduces the bias of the IV estimator, especially in small samples
• JIVE can provide more reliable estimates than the standard IV estimator in the presence of weak instruments
• However, JIVE may have higher variance than the IV estimator and can be computationally intensive

Weak-instrument robust inference

• Weak-instrument robust inference methods aim to provide valid confidence intervals and hypothesis tests even when instruments are weak
• These methods include the Anderson-Rubin test, the Kleibergen-Moreira test, and the conditional likelihood ratio test
• These tests are robust to weak instruments and maintain the correct size even when the instruments are weak
• Weak-instrument robust inference can help researchers draw reliable conclusions about the causal effect of interest
• However, these methods may have lower power compared to conventional tests when the instruments are strong

Weak instruments in practice

Examples of weak instruments

• Weak instruments can arise in various empirical settings, such as:
  • Using lagged variables as instruments (e.g., lagged GDP growth as an instrument for current GDP growth)
  • Employing geographical or historical variables as instruments (e.g., distance to a port as an instrument for trade)
  • Using institutional or policy changes as instruments (e.g., changes in compulsory schooling laws as an instrument for education)
• In these cases, the instruments may only weakly correlate with the endogenous explanatory variable, leading to weak instrument problems

Empirical studies with weak instruments

• Many empirical studies in economics and other social sciences have encountered weak instrument issues
• Examples include studies on the returns to education, the impact of foreign aid on economic growth, and the effect of institutions on development
• Researchers have used various methods, such as LIML, JIVE, and weak-instrument robust inference, to address the weak instrument problem
• Careful examination of the first-stage results and diagnostic tests is crucial to assess the strength of instruments and the reliability of the IV estimates

Alternatives to instrumental variables

Control function approach

• The control function approach is an alternative method for addressing endogeneity when suitable instruments are not available
• It involves estimating the endogenous explanatory variable as a function of the exogenous variables and the residuals from this first-stage regression
• The estimated residuals are then included as an additional regressor in the main equation to control for endogeneity
• The control function approach can provide consistent estimates of the causal effect, even in the absence of valid instruments
• However, the control function approach relies on distributional assumptions and may be sensitive to misspecification

Latent variable models

• Latent variable models, such as structural equation models (SEM), can be used to estimate causal effects when the endogenous variable is unobserved or measured with error
• These models specify the relationships between the observed variables and the latent variables using a system of equations
• Latent variable models can account for measurement error and provide estimates of the causal effect based on the estimated latent variables
• However, latent variable models rely on distributional assumptions and may be sensitive to model misspecification

Bounds analysis

• Bounds analysis is a non-parametric approach that provides bounds on the causal effect when instruments are not available or are weak
• It relies on weaker assumptions than the IV approach and does not require point identification of the causal effect
• Bounds analysis uses the observed data to construct upper and lower bounds on the causal effect, allowing for partial identification
• The width of the bounds depends on the strength of the assumptions made and the quality of the data
• Bounds analysis can provide informative results, even when point identification is not possible, but the bounds may be wide if the assumptions are weak or the data are limited