Unreliable Residuals In Ivreghdfe With Clustered Errors?

Have you ever encountered inconsistencies when working with instrumental variable regressions, particularly with the ivreghdfe command in Stata? It's a common challenge, especially when dealing with clustered standard errors. Let's dive into a discussion about the reliability of residuals from ivreghdfe when you specify clustered standard errors, and how this can impact your analysis. This article aims to explain the issue, provide a reproducible example, and discuss the implications for your research.

The Core Issue: Understanding Residual Inconsistencies

When using the ivreghdfe command, specifying clustered standard errors can sometimes lead to unreliable residuals. By unreliable, we mean that the residuals obtained differ significantly from those derived using default homoskedastic or robust standard errors. This discrepancy can be quite concerning because, under normal circumstances, residuals from models with different error structures should be highly correlated. If you find that your residuals are almost uncorrelated under clustered standard errors compared to homoskedastic or robust errors, it’s a sign that something isn’t quite right.

To put it simply, residuals are the differences between the observed values and the values predicted by your regression model. They play a crucial role in various diagnostic tests and can even be used directly in further analysis. When these residuals become unreliable, it raises questions about the validity of your model and the conclusions you can draw from it. It's like having a faulty speedometer in your car – it might still give you a reading, but can you trust it?

The core of the issue lies in how ivreghdfe handles the clustering of standard errors. Clustering adjusts the standard errors to account for correlation within groups, which is a common practice in econometrics when dealing with panel data or other grouped structures. However, the process of adjusting standard errors might inadvertently affect the calculation of residuals, leading to the observed inconsistencies. Understanding this potential pitfall is the first step in ensuring the robustness of your results. This issue highlights the importance of careful model diagnostics and the need to validate your results using multiple approaches.

A Reproducible Example: Seeing the Problem in Action

To illustrate this issue, let's walk through a reproducible example using Stata. This example will help you see firsthand the discrepancies in residuals when specifying clustered standard errors with ivreghdfe.

First, we'll use the sysuse auto command to load Stata's built-in auto dataset. This dataset contains information about various car models, including their miles per gallon (mpg), weight, price, and other characteristics. We will use this dataset to run a series of instrumental variable regressions using ivreghdfe.

sysuse auto

Next, we'll run three different ivreghdfe regressions. In each regression, we'll model mpg as a function of weight, using price as an endogenous variable instrumented by headroom. We'll also absorb the fixed effects for i.foreign to control for differences between domestic and foreign cars.

The first regression will use the default homoskedastic standard errors. We'll save the residuals from this regression as resid_1:

ivreghdfe mpg weight (price = headroom), absorb(i.foreign) resid(resid_1)

The second regression will use robust standard errors, which are less sensitive to violations of homoskedasticity. We'll save these residuals as resid_2:

ivreghdfe mpg weight (price = headroom), absorb(i.foreign) resid(resid_2) vce(robust)

Finally, the third regression will use clustered standard errors. To create a clustering variable, we'll generate a variable group_id that divides the dataset into two groups. We'll then specify vce(cluster group_id) to cluster the standard errors at the group_id level. The residuals from this regression will be saved as resid_3:

gen group_id = (_n <= 35) + 1
ivreghdfe mpg weight (price = headroom), absorb(i.foreign) resid(resid_3) vce(cluster group_id)

Now, let's compare the residuals from these three regressions. We'll use the corr command to calculate the correlations between resid_1, resid_2, and resid_3:

corr resid_*

You should observe that resid_1 and resid_2 are perfectly correlated, as expected. However, resid_3 (the residuals from the clustered regression) will likely have a much lower correlation with resid_1 and resid_2. This discrepancy highlights the issue of unreliable residuals when using clustered standard errors with ivreghdfe.

This example clearly demonstrates the problem. By running these commands, you can see the stark differences in the residuals and understand why this issue is a significant concern. The next step is to explore the implications of these unreliable residuals and how they might affect your research.

Implications and Deeper Issues: What Does This Mean for Your Research?

The presence of unreliable residuals can have significant implications for your research. Residuals are not just byproducts of a regression; they are essential for various diagnostic tests and further analyses. If the residuals are flawed, any subsequent analysis that relies on them could also be compromised.

One of the primary concerns is the validity of diagnostic tests. Many tests, such as those for heteroskedasticity or autocorrelation, rely on the properties of the residuals. If the residuals are inconsistent, these tests may yield misleading results. For instance, you might incorrectly conclude that your model suffers from heteroskedasticity when the issue is simply the unreliable residuals from the clustered standard errors.

Furthermore, if you are using the residuals directly in further analysis, such as in a two-stage least squares (2SLS) procedure or in a control function approach, the results could be biased. These methods often assume that the residuals are well-behaved and uncorrelated with the regressors, assumptions that may not hold if the residuals are unreliable. This can lead to incorrect inferences and potentially flawed conclusions.

Another critical question is whether the coefficient vector and covariance matrix are still reliable when you specify clustering. While the residuals may be problematic, it is possible that the coefficient estimates themselves are still consistent and efficient. However, this is not guaranteed. The issue of unreliable residuals raises concerns about the overall validity of the model, and it is crucial to investigate further whether the clustering is affecting the coefficient estimates as well. If the coefficients and covariance matrix are also affected, the entire analysis could be at risk.

Therefore, even if the issue is