9.5 Weak instruments
6 min read•august 20, 2024
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 and , 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
Top images from around the web for Bias in IV estimators
Frontiers | Reliability Estimation in Multidimensional Scales: Comparing the Bias of Six ... View original
Is this image relevant?
Frontiers | Illustrating Instrumental Variable Regressions Using the Career Adaptability – Job ... View original
Is this image relevant?
Frontiers | Common Methods for Performing Mendelian Randomization View original
Is this image relevant?
Frontiers | Reliability Estimation in Multidimensional Scales: Comparing the Bias of Six ... View original
Is this image relevant?
Frontiers | Illustrating Instrumental Variable Regressions Using the Career Adaptability – Job ... View original
Is this image relevant?
1 of 3
Top images from around the web for Bias in IV estimators
Frontiers | Reliability Estimation in Multidimensional Scales: Comparing the Bias of Six ... View original
Is this image relevant?
Frontiers | Illustrating Instrumental Variable Regressions Using the Career Adaptability – Job ... View original
Is this image relevant?
Frontiers | Common Methods for Performing Mendelian Randomization View original
Is this image relevant?
Frontiers | Reliability Estimation in Multidimensional Scales: Comparing the Bias of Six ... View original
Is this image relevant?
Frontiers | Illustrating Instrumental Variable Regressions Using the Career Adaptability – Job ... View original
Is this image relevant?
1 of 3
- 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 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
- methods aim to provide valid confidence intervals and hypothesis tests even when instruments are weak
- These methods include the , the , and the
- 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 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
- , such as (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
- 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
Key Terms to Review (28)
Anderson-Rubin Test: The Anderson-Rubin test is a statistical method used to assess the validity of instrumental variables in regression analysis, particularly when dealing with weak instruments. This test checks whether the estimated parameters are significantly different from zero under the null hypothesis, which states that the instruments do not affect the endogenous variable. It's particularly useful because it remains valid even when the instruments are weak, providing a more reliable inference in such scenarios.
Asymptotic Bias: Asymptotic bias refers to the difference between the expected value of an estimator and the true parameter value as the sample size approaches infinity. This concept highlights how estimators can behave differently with larger samples, revealing their reliability and consistency. Understanding asymptotic bias is crucial, especially when dealing with weak instruments, as it can lead to misleading conclusions in statistical inference if not properly accounted for.
Bias in estimation: Bias in estimation refers to the systematic error that causes an estimator to consistently overestimate or underestimate the true value of a parameter. This concept is crucial as it affects the accuracy and reliability of statistical inferences made from data. When estimating parameters, such as coefficients in a regression model, bias can arise due to various factors including omitted variable bias, measurement error, or weak instruments, leading to incorrect conclusions about relationships within the data.
Bounds Analysis: Bounds analysis is a technique used in econometrics to provide a range of estimates for causal effects when the model may be influenced by unobserved confounding variables or weak instruments. This method helps to assess the robustness of the estimated effects by establishing lower and upper bounds, allowing researchers to understand the potential variability in their estimates under different assumptions. The focus on bounds is particularly important in the presence of weak instruments, where traditional estimates may be unreliable or biased.
Conditional Likelihood Ratio Test: The conditional likelihood ratio test is a statistical method used to compare the fit of two models when testing a hypothesis about model parameters. It specifically evaluates the likelihood of observing the data under both the null hypothesis and the alternative hypothesis, providing a basis for inference regarding the strength and validity of the model's predictors. This method is particularly important when dealing with weak instruments, as it helps to assess whether the instruments sufficiently explain the variation in the endogenous variable.
Control function approach: The control function approach is a method used in econometrics to address endogeneity issues by introducing an additional variable, called the control function, to account for the correlation between the independent variable and the error term. This technique helps in obtaining consistent estimates of the causal effect of the independent variable on the dependent variable. By incorporating this control function, researchers can better handle situations with weak instruments or when conducting tests such as the Hausman test.
Cragg-Donald Statistic: The Cragg-Donald statistic is a measure used in econometrics to assess the strength of instrumental variables in the context of estimating parameters in a linear regression model. It specifically evaluates whether the instruments are weak, which can lead to biased estimates and unreliable inference if not properly addressed. A higher value of the statistic indicates stronger instruments, making it crucial for determining the validity of the instrumental variable approach.
Exogeneity: Exogeneity refers to a condition where an explanatory variable is not correlated with the error term in a regression model. When a variable is exogenous, it suggests that any changes in this variable do not arise from the model's error, making it crucial for establishing causal relationships and ensuring valid inference in econometric analysis.
F-statistic: The f-statistic is a ratio used in statistical tests to compare the variances of two populations or models, helping to determine if the overall regression model is statistically significant. It plays a vital role in evaluating the goodness of fit of a model, conducting hypothesis tests, and assessing whether a set of independent variables collectively influence a dependent variable.
Finite sample bias: Finite sample bias refers to the discrepancy that can occur between sample estimates and population parameters when a limited number of observations are used for estimation. This bias arises because the estimators may not accurately reflect the true underlying relationships due to sampling variability and can lead to misleading conclusions, especially when using weak instruments in regression analysis.
Inferential Validity: Inferential validity refers to the extent to which the conclusions drawn from a statistical analysis are justified and accurately reflect the true relationships between variables. It focuses on whether the causal inferences made from the data are valid, which is crucial when assessing the effectiveness of instruments used in econometric models. Strong inferential validity ensures that findings can be generalized beyond the specific sample studied, reinforcing the reliability of estimates and predictions.
Instrument Relevance: Instrument relevance refers to the necessity for an instrument variable to be correlated with the endogenous explanatory variable in a regression model. This correlation is crucial because it ensures that the instrument can effectively help identify the causal effect of the explanatory variable on the dependent variable, especially in the presence of omitted variable bias or measurement error.
Instruments must be correlated with the endogenous regressor: In econometrics, the requirement that instruments must be correlated with the endogenous regressor means that valid instrumental variables must have a significant relationship with the variable that is causing endogeneity in the model. This correlation ensures that the instrument can explain some of the variability in the endogenous regressor, helping to isolate causal effects in a regression analysis. If an instrument is not correlated with the endogenous regressor, it fails to fulfill its role and can lead to biased or inconsistent estimates.
Jackknife IV Estimator (JIVE): The Jackknife IV Estimator (JIVE) is a statistical method used to address the issue of weak instruments in instrumental variable estimation. It helps improve the robustness and efficiency of estimates by systematically re-estimating parameters while leaving out one observation at a time. This technique is particularly valuable when traditional IV methods may yield biased or inconsistent estimates due to weak instruments, enhancing the overall reliability of the results.
James Stock: James Stock is a prominent economist known for his work in econometrics, particularly in the area of instrumental variables and the challenges posed by weak instruments. His research highlights the significance of using strong instruments to obtain reliable estimates in econometric models, as weak instruments can lead to biased and inconsistent parameter estimates, impacting the validity of empirical findings.
Kleibergen-Moreira Test: The Kleibergen-Moreira test is a statistical method used to assess the validity of instruments in an econometric model, particularly focusing on weak instruments. It provides a robust framework for testing the null hypothesis that the instruments are valid and sufficiently correlated with the endogenous explanatory variables, while also taking into account the presence of weak instruments which can lead to biased estimates.
Latent variable models: Latent variable models are statistical models that aim to explain observed variables through unobserved, or latent, variables that influence them. These models are particularly useful in econometrics for capturing hidden factors that cannot be directly measured but significantly affect the outcomes of interest. By incorporating latent variables, researchers can better account for measurement error and unobserved heterogeneity in their analyses.
Limited Information Maximum Likelihood (LIML): Limited Information Maximum Likelihood (LIML) is an estimation technique used in econometrics that focuses on estimating the parameters of a specific equation within a larger system of equations while treating the other equations as fixed. This method is particularly useful in situations where there are concerns about the validity and strength of instruments, which can affect the reliability of estimates. By using LIML, researchers can obtain more consistent estimates even when instruments are weak or invalid.
Mark Watson: Mark Watson is a notable economist who has contributed significantly to the fields of time series analysis and econometrics. His work often focuses on issues such as weak instruments in regression analysis, where he emphasizes the consequences of using instruments that do not sufficiently correlate with the endogenous explanatory variables, leading to biased estimates and unreliable inference.
Overidentification: Overidentification occurs when there are more instruments available than the number of endogenous variables in a model. This situation allows for the possibility of testing the validity of the instruments, which can lead to better model estimates. Additionally, overidentification can be critical for assessing joint hypotheses about parameters, ensuring that the model does not suffer from weak instruments that could bias the results.
Simultaneous equations models: Simultaneous equations models are statistical models that involve multiple equations where the dependent variables are interrelated and can influence each other simultaneously. These models are essential for capturing the complexity of economic relationships, where changes in one variable can affect others at the same time, which is particularly important when addressing issues like endogeneity and structural relationships.
Stock-Yogo Critical Values: Stock-Yogo critical values are specific threshold values used in econometrics to assess the strength of instruments in instrumental variable estimation. These values help determine whether an instrument is considered weak or strong, impacting the validity of the estimated parameters and the reliability of inference. A weak instrument can lead to biased estimates, making it essential to compare the test statistic to these critical values to ensure robust results.
Structural Equation Models: Structural equation models (SEMs) are a set of statistical techniques that allow researchers to analyze complex relationships among variables by specifying a series of equations. These models can incorporate both direct and indirect effects, providing a comprehensive understanding of the relationships within a theoretical framework. SEMs are particularly useful in exploring causal relationships and are often used in social sciences to test hypotheses about variable interdependencies.
Two-stage least squares (2sls): Two-stage least squares (2SLS) is an estimation technique used to provide consistent estimates of parameters in a regression model when there is endogeneity or correlation between the independent variables and the error term. This method employs instrumental variables to remove bias by first predicting the values of the endogenous variables using instruments and then substituting those predicted values back into the original equation for final estimation. Its effectiveness hinges on the validity of the instruments used, addressing issues related to weak instruments and allowing for diagnostic tests like the Hausman test.
Underidentification: Underidentification occurs when a model has fewer valid instruments than the number of endogenous variables it aims to estimate. This situation creates ambiguity in estimating causal relationships because there isn't enough information to uniquely determine the effects of the endogenous variables. It often leads to biased estimates and makes it challenging to draw reliable conclusions from the model.
Weak instrument bias: Weak instrument bias refers to the distortion in the estimation of causal relationships that occurs when an instrumental variable is not strongly correlated with the endogenous explanatory variable. This situation can lead to unreliable and inconsistent parameter estimates, ultimately compromising the validity of causal inference in econometric models. The implications of weak instruments are critical in understanding the limits of instrumental variable approaches, especially when addressing endogeneity issues.
Weak instrument test: The weak instrument test is a statistical procedure used to evaluate the strength of instrumental variables in regression analysis. It determines whether an instrument is correlated with the endogenous explanatory variable but weakly related, which can lead to biased and inconsistent estimates in the presence of endogeneity. Understanding this test is crucial for ensuring that valid inferences can be made from econometric models that rely on instrumental variables.
Weak-instrument robust inference: Weak-instrument robust inference refers to methods used in econometrics that provide reliable statistical conclusions even when the instruments used for estimating causal relationships are weak. Weak instruments can lead to biased or inconsistent estimates, making it crucial to have robust techniques that account for this uncertainty. This concept is important because it ensures that researchers can still make valid inferences from their models, despite the limitations posed by weak instruments.