JIVE Talking: A Lifeline When Your IVs Are Weak (Or Just Too Many!)
Hi all, this Josh again. I’ve reviewed one of the classics on Instrumental Variables and I thought it would be good to share it with the crowd. If you are interested in this topics I recommend you to also follow
on DiD methods.Jackknife Instrumental Variables Estimation (JIVE) is a clever approach proposed by Joshua Angrist, Guido Imbens, and Alan Krueger in their 1999 Journal of Applied Econometrics paper, "Jackknife Instrumental Variables Estimation."
What’s a Jacknife in Statistics?
The jackknife is repeatedly poking your data by removing one piece at a time and seeing how much your results changes. This change gives you valuable insights into the robustness and accuracy of your findings. It's particularly useful when more complex methods like bootstrapping (another resampling technique) might be too computationally intensive or when you have smaller sample sizes.
JIVE offers a practical alternative that can deliver superior finite-sample properties compared to 2SLS, especially when you're instrument-rich, and performs similarly to Limited Information Maximum Likelihood (LIML) without some of LIML's potential downsides under misspecification.
The Core Problem: Why 2SLS Can Stumble
The bias in 2SLS, especially with many instruments, stems from a subtle but important issue: the instrument used for the i-th observation in the second stage is constructed using information from that same i-th observation in the first stage.
In two-stage least squares (2SLS), we try to solve a problem where one of our main "ingredient" variables (let's call it 'X') is related to the "error" or unobserved factors in our main equation (let's call it 'epsilon'). To fix this, we use a different set of "tool" variables (let's call them 'Z') that are related to 'X' but not directly to our error/epsilon.
The first step in 2SLS is to use our 'tool' variables ('Z') to predict our problematic 'ingredient' variable ('X'). We then use these predicted values of 'X' in our main equation instead of the original 'X'.
However, here's where a problem can pop up, especially when we don't have a huge amount of data. Even though our 'tool' variables ('Z') are good, the predicted values of 'X' from the first step aren't perfectly clean. They still carry a little bit of the 'error' from that first prediction step.
This 'error' in the predicted 'X' values can actually be connected to the 'error' in our main equation ('epsilon').
This connection between the 'error' in our predicted 'X' values and the 'error' in our main equation causes a problem: it makes our 2SLS results a bit "off." This "offness" becomes more noticeable if we use too many 'tool' variables ('Z') compared to the amount of data we have. When this happens, our 2SLS results tend to look more like the results we'd get if we just ignored the problem in the first place (which is what standard OLS regression does).
So, in essence, using too many tools can accidentally bring back some of the original bias we were trying to avoid.The Jackknife Fix: "Leave-One-Out" to the Rescue
JIVE tackles this by borrowing the "leave-one-out" logic from jackknife statistics. The idea is simple yet powerful: when constructing the predicted value (the estimated optimal instrument) for observation i, don't use observation i's data in the first-stage estimation of π.
Imagine we're trying to predict a value for a specific student, but the usual way of doing this in 2SLS (Two-Stage Least Squares) can cause problems because that student's own information is used in the prediction, which can make the prediction "tainted" by unobserved factors affecting that student. The "Jackknife Instrumental Variables Estimators" (JIVE) offer a clever solution.
So, JIVE goes one step further: for each student, it temporarily removes that student's own data from the first-stage calculation. This means that the predicted value for a particular student is made without using any of that student's own "problematic" information. Because of this "leave-one-out" approach, the predicted instrument for a student is no longer correlated with that student's individual unobserved factors, even in smaller datasets.
There are two main versions: JIVE1 and JIVE2.
JIVE1 adjusts the standard 2SLS prediction by taking into account how much influence each observation has in the first stage (called "leverage").
JIVE2 is very similar but makes a slightly different adjustment in its formula.
The great thing about these methods is that even though it sounds like you'd have to do a separate prediction for every single student, there are clever mathematical shortcuts that let you calculate these "leave-one-out" predictions very quickly, without actually running hundreds or thousands of separate regressions.
Both JIVE1 and JIVE2 are considered just as good as 2SLS and another method called LIML in very large datasets, meaning they share the same desirable long-run properties.
Why Should You Care? The JIVE Advantage
The Monte Carlo simulations in the original paper are quite revealing:
Reduced Bias & Better Coverage with Many Instruments: In models with many instruments (some potentially weak), JIVE1 and JIVE2 consistently show less median bias than 2SLS and their confidence interval coverage rates are much closer to the nominal level, similar to LIML. 2SLS, in contrast, can be badly biased with poor coverage in these scenarios.
Performance under Misspecification: This is where JIVE can really shine, sometimes even over LIML. In one of their simulations (Model 5), an instrument Zi2 directly affects Yi but not Xi, and is incorrectly omitted from the main equation. In this case, 2SLS, JIVE1, and JIVE2 remain consistent because the estimated instrument doesn't depend on Zi2. LIML, however, is affected by this direct effect and becomes inconsistent. The JIVE estimators demonstrated superiority in this setup.
Robustness in Non-Linear/Heteroscedastic First Stages: In a model with a non-linear and heteroscedastic first stage (Model 3), JIVE2 performed particularly well, showing the best median-bias and median absolute error, even outperforming JIVE1 and LIML.
Signaling Non-Identification: When instruments are truly irrelevant (Model 4, all πj=0), JIVE and LIML estimates become much more dispersed than OLS or 2SLS. This is a good thing, as it signals to the researcher that the instruments aren't providing reliable information, whereas 2SLS might still give deceptively precise (but biased) estimates.
Getting Practical: JIVE in Stata (Conceptually)
While there isn't a one-click jive command in Stata (though user-written versions might exist!), implementing it based on the formulas is straightforward. Here's the conceptual Stata workflow for JIVE1 (JIVE2 is a minor modification):
Stata
// Assume:
// y: dependent variable
// x: endogenous regressor (for simplicity, one x)
// z1 z2 ... zk: your instruments
// w1 w2 ... : exogenous controls (if any)
// 0. Preliminaries
global N = _N // Store sample size for JIVE2 if needed
// 1. First Stage Regression (Regress X on Zs and other exogenous controls)
regress x z* w*
predict x_hat // This is Z_i * pi_hat (including contribution from w*)
predict h, leverage // This is h_i, the leverage for observation i
// 2. Construct the JIVE instrument for x
// For JIVE1:
gen x_jive1 = (x_hat - h * x) / (1 - h)
// For JIVE2 (if you wanted it):
// gen x_jive2 = (x_hat - h * x) / (1 - 1/$N)
// 3. Second Stage: Run IV using the JIVE instrument
// Using JIVE1 instrument:
ivregress 2sls y w* (x = x_jive1)
// Note: The standard errors from this direct ivregress might need adjustment
// as they don't account for the first-stage estimation of the instrument.
// However, the paper reports coverage rates using asymptotic standard errors
// for a just-identified IV estimator using the JIVE-constructed instruments,
// and finds them to be remarkably accurate for JIVE and LIML[cite: 68, 69, 71].
Important Considerations for Stata Users:
The leverage hi is from the regression of X on Z (and any other exogenous regressors W that are also in the first stage for X).
The paper's authors note that for JIVE, reporting coverage rates based on asymptotic standard errors for a just-identified IV estimator using the constructed JIVE instrument works well.
The empirical example using Angrist and Krueger (1991) data showed that with 180 instruments, JIVE and LIML produced notably higher (and potentially more reliable) estimates of the returns to schooling than 2SLS. The JIVE standard errors were also considerably lower than those for the USSIV estimators discussed in Angrist and Krueger (1995).
When to Reach for JIVE?
Consider using JIVE when:
You have a large number of instruments relative to your sample size.
You suspect your instruments might be weak.
You're concerned about the finite-sample bias of 2SLS.
You want a robustness check against LIML, especially if you suspect certain types of model misspecification where LIML might struggle.
Reference:
Angrist, J. D., Imbens, G. W., & Krueger, A. B. (1999). Jackknife Instrumental Variables Estimation. Journal of Applied Econometrics, 14(1), 57–67.