Jackknife Inference for Fixed Effects Models

Jackknife Inference for Fixed Effects Models

\xdef\@thefnmark\@footnotetext{$^\dagger$Address: University of Exeter Business School, Rennes Drive, Exeter, Devon, EX4 4PU, United Kingdom. Email: [email removed].}

\xdef\@thefnmark\@footnotetext{$^\ddagger$First Version: February 25, 2026. }

Introduction

Panel models with fixed effects are ubiquitous in empirical research. Such models allow researchers to account for unobserved heterogeneity by introducing unit-specific parameters. For example, individual fixed effects may absorb time-invariant differences across individuals, such as ability, risk preferences, or innate productivity, while time fixed effects may capture aggregate shocks and common trends, such as business cycle fluctuations and policy changes that affect all units. Over time, the fixed effects models researchers wish to estimate have become increasingly complex, including more elaborate fixed effects specifications, such as time-varying group effects and interactive effects, models which feature lagged outcomes and other predetermined regressors, nonlinear and nonstandard models such as binary outcome models and quantile regression, and extensions to panels with three or more dimensions. While it is frequently possible to adapt existing estimation methods to encompass such models, statistical inference has become increasingly challenging, as such estimators typically exhibit highly complex statistical properties.

This notwithstanding, it is often reasonable to expect that such estimators, once suitably centred and scaled, are asymptotically normal with a number of leading order bias terms. Indeed, the accumulated work of several decades has given us a good sense of the likely structure of such results, even when explicit formulae are unavailable. For example, in the context of a generic panel model with one-way fixed effects, we typically expect a standard estimator $\hat{\varphi}$ of a (scalar) common parameter $\varphi$ to be asymptotically distributed

as both dimensions of the panel $N,T \rightarrow \infty$ and $N/T \rightarrow \gamma^2 \in (0, \infty)$. Similarly, for a generic panel model with two-way fixed effects, we typically expect a standard estimator to be asymptotically distributed

under the same asymptotic regime, where we distinguish $\mu_{1}$ and $\mu_{2}$ by their origin in either cross-sectional or time fixed effects. This paper develops a method of inference designed for such situations where, rather than working with model and estimator specific expressions, our method rests upon two high-level assumptions. First, we assume the existence of an estimator of a parameter of interest which, once centred and scaled, is asymptotically normally distributed with a number of unknown bias terms and an unknown variance. Second, we assume the existence of a collection of subsample estimators whose joint distribution (along with the full sample estimator) can be characterised through a bias matrix and a covariance matrix, both of which are implied by the known subsampling design. Under these conditions we derive a jackknife $t$-statistic which provides a general method of inference for fixed effects models which is automatic, computationally inexpensive, tuning parameter-free, and highly model agnostic.

To illustrate the simplicity of our statistics, at a most basic level, we expect for a generic panel model with one-way (cross-sectional) fixed effects that

as $N,T \rightarrow \infty$ and $N/T \rightarrow \gamma^2 \in (0, \infty)$, where $\mathcal{J}$ denotes our jackknife $t$-statistic, $\hat{\varphi}^{(0)}$ denotes an estimator obtained using the full sample, $\hat{\varphi}^{(1)}$ and $\hat{\varphi}^{(2)}$ denote subsample estimators obtained by partitioning the data into the first $1, \dots, T/2$ time periods, and the last $T/2+1, \ldots, T$ time periods, respectively, and $t_1$ denotes the $t$-distribution with one degree of freedom.\footnote{We presume that $T$ is even. The case of time fixed effects is symmetric and we instead partition the cross-section.} Corresponding to (ref), a $( 1 - \alpha )$ two-sided confidence interval for $\varphi$ is obtained as

where $t_{ 1, 1 - \alpha / 2 }$ denotes the $( 1 - \alpha / 2 )$ quantile of the $t$-distribution with one degree of freedom. In parallel, for a generic panel model with two-way (cross-sectional and time) fixed effects we expect that

as $N, T \rightarrow \infty$ and $N / T \rightarrow \gamma^2 \in ( 0, \infty )$, where $\hat{\varphi}^{(0)},\hat{\varphi}^{(1)}$, and $\hat{\varphi}^{(2)}$ are as described previously, and $\hat{\varphi}^{(3)}$, $\hat{\varphi}^{(4)}$ denote subsample estimators obtained by partitioning the data into the first $1, \ldots, N / 2$ cross-sectional units, and the last $N / 2 + 1, \ldots, N$ cross-sectional units, respectively.\footnote{We presume that both $N$ and $T$ are even.} Corresponding to (ref), a $( 1 - \alpha )$ two-sided confidence interval for $\varphi$ is obtained as

Related literature

The incidental parameter problem (IPP), first articulated by neyman, arises in models where the number of parameters grows with sample size. For panel models with fixed effects this growth is inherent, since the number of fixed effects increases with $N$ and/or $T$.\footnote{For ease of exposition we focus initially on two-dimensional panels. See Section (ref) for discussion of multi-dimensional panels.} The central consequence of the IPP is that estimators of low-dimensional objects of interest, such as slope coefficients and average partial effects, can exhibit nonnegligible bias. Under large $N$, fixed $T$ asymptotics, this bias can be so severe that it results in inconsistency. Under large $N$, large $T$ asymptotics, standard estimators are typically consistent, but their limiting distributions are not centred at zero, referred to as asymptotic bias. As a result, conventional methods of inference built upon known, zero-centred limiting distributions are no longer valid; in particular standard Wald, likelihood ratio, and Lagrange multiplier tests, as well as confidence intervals and $p$-values, will generally be incorrect unless the bias is explicitly accounted for; see, e.g., fernandez2018fixed. This has motivated a large literature which seeks to understand the consequences of the IPP in specific settings and to develop valid methods of inference.

The most standard route to valid inference is to eliminate asymptotic bias so that conventional large sample approximations apply and, in particular, to do so by analytic means. Classically, this is achieved by characterising (i.e., deriving a formula for) the bias, and using this to construct a “bias-corrected” estimator.\footnote{More broadly, related analytic strategies modify the objective function or score; see, e.g., arellano2016likelihood, jochmans2019likelihood, and bonhomme2024neyman.} The main limitation of such approaches is, however, that the appropriate modifications needed to bias-correct, construct a debiased estimator, or otherwise eliminate asymptotic bias are model and estimator-specific, meaning that these need to be derived on a case by case basis. For sufficiently standard problems it is possible to appeal to quite broad results which have been obtained for one-way hahn2004jackknife,hahn2011bias, two-way fernandez2016individual, and three-way fixed effects models czarnowske2025debiased. However, the resulting expressions are very cumbersome, and require a researcher to adapt these to their specific setting, usually by computing layers of derivatives and/or application of the delta method.\footnote{For example, when the parameter of interest is a function of both the fixed effects and common parameters.} Moreover, their implementation is often dependent on tuning parameters to which their performance can be highly sensitive. Beyond these concerns, as fixed effects models have become more complex, results have become increasingly challenging to derive, and, for that matter, increasingly tedious, leading to an absence of results in empirically relevant cases. As a consequence, recent years have seen the widespread adoption of automatic methods of inference for fixed effects models.

The jackknife provides one such approach. hahn2004jackknife proposed leave-one-out jackknife bias correction for static panel models with one-way fixed effects. This has been extended to models with two-way fixed effects in hughes2026jackknife. While such constructions may exhibit favourable higher-order properties (see, e.g. hahn2024efficient), such schemes are not appropriate when data exhibit dependence, which is common in panel data, and, moreover, command a computational cost comparable with the bootstrap due to their large number of auxiliary estimations. As an alternative, dhaene2015split propose a philosophically distinct approach called the split-panel jackknife. Their method is particularly attractive because it is both computationally inexpensive (requiring as little as two auxiliary estimations) and highly model agnostic; in particular, it accommodates dependence. The split-panel jackknife has been extended to two-way fixed effects models in fernandez2016individual and has become the default method of bias correction in many applications. However, jackknife bias correction, by whichever means, does not deliver a complete method of inference on its own: a bias-corrected estimator must be paired with a separate (typically analytic) variance estimator in order to conduct inference. In this sense, existing jackknife approaches provide only a semi-automatic method of inference.

The bootstrap can provide an alternative fully-automatic method of inference for fixed effects models. In particular, if it is possible to devise a bootstrap scheme which correctly replicates the asymptotic distribution of an estimator, including any bias, then inference can be based on the bootstrap distribution without the need to make further adjustments; see, e.g, gonccalves2015bootstrap and higgins2024bootstrap. However, the task of designing such a scheme is nontrivial. For example, gonccalves2015bootstrap show that common bootstrap schemes match the asymptotic variance of the least squares estimator, while failing to replicate its asymptotic bias, in the context of a dynamic panel model with individual fixed effects. Although in this case valid bootstrap schemes can be constructed (see further results in gonccalves2015bootstrap and higgins2025inference), more generally it must be verified that any proposed scheme does indeed yield valid inference for a given application. Moreover, the bootstrap incurs a computational cost which, although sometimes overstated, cannot be dismissed in the context of fixed effects models due to the large sample sizes and the corresponding large number of parameters.

In this paper we return to the jackknife and develop a complete method of inference underpinned by the “small number of large subsamples” philosophy of the split-panel jackknife. To achieve our purpose, we first construct a minimum-variance unbiased jackknife (MVUJ) estimator of a parameter of interest, obtained as an unbiased linear combination of the full sample and subsample estimators that minimises asymptotic variance. We then use this to construct a jackknife $t$-statistic, which provides a straightforward means to conduct inference, even when the statistical properties of an estimator are highly complex. Beyond our simplest constructions, we develop jackknife $t_q$-statistics with $q > 1$ which yield sharper inference, at the expense of additional complexity, and ultimately derive a jackknife counterpart to Hotelling's $t^2$-statistic hotelling1931generalization which allows us to extend our method of inference to encompass vector parameters and to accommodate joint hypothesis tests. Overall we provide a general framework for the systematic development of jackknife bias-corrected estimators and of jackknife $t$-statistics.

Jackknife Inference

Suppose we observe a panel data set with entries indexed by $i = 1, \ldots, N$ and $t = 1, \ldots, T$, and wish to perform inference on a scalar parameter $\varphi$ in the presence of a large number of fixed effects, which are present in one or more dimensions of the panel. The parameter of interest $\varphi$ might be, for example, an individual slope coefficient, a linear combination of the form $\varphi \coloneqq \ensuremath\boldsymbol{a}^\top \ensuremath\boldsymbol{\beta}$, where $\ensuremath\boldsymbol{\beta}$ is a vector of primitive model parameters, or a more complicated function of primitive parameters (including fixed effects), such as an average partial effect. We make the following assumption.

Assumption (ref) assumes the existence of an estimator $\hat{\varphi}^{(0)}$ that is asymptotically normal, and which may feature one or more bias terms.\footnote{Note that we do not assume $\mu_{r,NT}$ to be bounded nor convergent.} Expansions of the form (ref) have been established for a large range of models, estimators, and parameters of interest. Examples include least squares (LS) and generalized method of moments (GMM) estimators of linear dynamic panel models with individual fixed effects nickell,hahn2002asymptotically,arellano, extremum estimators of nonlinear models with individual fixed effects such as logit and probit models hahn2004jackknife,fernandez2009fixed,hahn2011bias, nonlinear models with individual and time fixed effects fernandez2016individual, and LS and maximum likelihood (ML) estimators of linear and nonlinear models with interactive fixed effects bai2009panel,moon_dynamic_nodate,chen2021nonlinear, amongst many others. When ${\varphi}$ is a common parameter, or a sufficiently smooth function thereof, one typically has $r_{NT} = \sqrt{NT}$. When $\varphi$ is a function of both common parameters and fixed effects, it is possible that the rate may be slower; see fernandez2018fixed for further discussion.

We now introduce some examples to illustrate Assumption (ref) that we shall return to throughout the remainder of the paper.

We now introduce our second assumption which concerns the existence and joint asymptotic distribution of subsample estimators.

Assumption (ref) assumes the existence of subsample estimators $\hat{\varphi}^{(1)}, \ldots, \hat{\varphi}^{(m-1)}$ that exhibit different linear combinations of the same bias terms $\mu_{1,NT}, \ldots, \mu_{R,NT}$. These different linear combinations are represented in the matrix $\ensuremath\boldsymbol{ A }$. The matrix $\ensuremath\boldsymbol{ C }$ encodes the dependence structure amongst the subsamples. Both $\ensuremath\boldsymbol{A}$ and $\ensuremath\boldsymbol{C}$ are treated as known because they are determined by a known, deterministic subsampling scheme, but otherwise Assumption (ref) is deliberately agnostic about underlying model and estimation approach.

To provide some motivation behind Assumption (ref), consider a generic two-way fixed effects model. Suppose an estimator admits an expansion of the form

where $b_1$ and $b_2$ are nonstochastic, and $\nu_{it}$ is mean zero. Consider a subsample which takes $T_1$ time-series observations and $N_1$ cross-sectional observations, and let $\hat{\varphi}^{(1)}$ denote the corresponding estimator computed on this subsample. One typically obtains a corresponding expansion

as $N_1, T_1 \rightarrow \infty$ and $N_1 / T_1 \rightarrow \gamma_1^2 \in ( 0, \infty )$, where $S_1 \coloneqq \{ 1, \ldots, N_1 \} \times \{ 1, \ldots, T_1 \}$.\footnote{For data which exhibit time-series dependence, consecutive observations may be needed in order to preserve the bias.} Assume, for simplicity, that $\nu_{it}$ are identically and independently distributed with $\mathbb{E}[ \nu_{it} ] = 0$, $\mathbb{V}[ \nu_{it} ] = \sigma^2 \in ( 0, \infty )$, and $\mathbb{E}\big[ | \nu_{it} |^{ 2 + \delta } \big] < \infty$ for some $\delta > 0$. Moreover, assume $N_1 / N \rightarrow \gamma_N \in ( 0, 1 ]$ and $T_1 / T \rightarrow \gamma_T \in ( 0, 1 ]$ as $N, T \rightarrow \infty$. Under these conditions,

Define $r_{NT} \coloneqq \sqrt{NT}$,

We then obtain

where

and

We now return to our examples to provide further illustration of Assumption (ref).

Minimum-Variance Unbiased Jackknife Estimator

We now discuss how to combine a set of subsample estimators and the full sample estimator to produce a minimum-variance unbiased jackknife (MVUJ) estimator. Consider a vector $\ensuremath\boldsymbol{v} \in \mathbb{R}^{m}$ such that

Then under Assumption (ref),

that is, $\ensuremath\boldsymbol{v}^\top \ensuremath\boldsymbol{\hat{\varphi}}$ is asymptotically unbiased. In general, the restrictions (ref) - (ref) do not identify a unique $\ensuremath\boldsymbol{v}$. Indeed, define

so that (ref) and (ref) can be restated as $\ensuremath\boldsymbol{D}^\top \ensuremath\boldsymbol{v} = \ensuremath\boldsymbol{d}$. Since $\textup{rank}( \ensuremath\boldsymbol{A} ) = R$ and $\ensuremath\boldsymbol{\iota}_m \notin \textup{col}( \ensuremath\boldsymbol{A} )$, we have $\textup{rank}( \ensuremath\boldsymbol{D} ) = R + 1$, and hence the feasible set

is nonempty whenever $m \geq R + 1$, and has infinitely many elements when $m \geq R + 2$. One can obtain an asymptotically unbiased jackknife estimator by solving the linear system $\ensuremath\boldsymbol{D}^\top \ensuremath\boldsymbol{v} = \ensuremath\boldsymbol{d}$ without reference to $\ensuremath\boldsymbol{C}$. For example, the minimum Euclidean norm solution is given by

Yet, since $\ensuremath\boldsymbol{C}$ is known, we can go further and construct an unbiased estimator with minimal asymptotic variance. In particular, an MVUJ weights vector is obtained by solving \footnotetext{Since $\textup{rank}( \ensuremath\boldsymbol{D} ) = R + 1$ (because $\textup{rank}( \ensuremath\boldsymbol{A} ) = R$ and $\ensuremath\boldsymbol{\iota}_m \notin \textup{col}( \ensuremath\boldsymbol{A} )$), $\ensuremath\boldsymbol{D}$ has full column rank, hence $\ensuremath\boldsymbol{D}^\top \ensuremath\boldsymbol{D}$ is invertible.}

Using a solution to (ref), we define an MVUJ estimator as

When $\ensuremath\boldsymbol{C}$ is positive definite, the programme (ref) has a unique solution which is given by

If $\ensuremath\boldsymbol{C}$ is not positive definite, the minimiser of (ref) need not be unique. In this case, one may compute an MVUJ weights vector by solving the system

where $\ensuremath\boldsymbol{\pi}$ is an $(R + 1) \times 1$ vector of Lagrange multipliers.\footnote{A necessary condition for $\ensuremath\boldsymbol{v}$ to be a solution to (ref) is that there exists a vector of Lagrange multipliers $\ensuremath\boldsymbol{\pi}$ such that $

\binom{\ensuremath\boldsymbol{v}}{\ensuremath\boldsymbol{\pi}} = \binom{\ensuremath\boldsymbol{0}_m}{\ensuremath\boldsymbol{d}}$; see, e.g., \cite{nocedal2006numerical} Lemma 16.1 and Theorem 16.2. When $\ensuremath\boldsymbol{C} \succ \ensuremath\boldsymbol{0}$, this system has a unique solution $( \ensuremath\boldsymbol{v}^\ast, \ensuremath\boldsymbol{\pi}^\ast )$. Eliminating $\ensuremath\boldsymbol{\pi}^\ast$ yields (ref).} We now return to our examples.

Jackknife $t$-Statistic

We now construct a jackknife $t$-statistic that delivers inference without knowledge of $\sigma^2$ (nor indeed $\ensuremath\boldsymbol{\mu}_{NT}$). Consider a vector $\ensuremath\boldsymbol{u}^\ast$ such that

where $\ensuremath\boldsymbol{A}$ and $\ensuremath\boldsymbol{C}$ are as in Assumption (ref). We establish the existence of such a vector in the following result.

Now that we have established the existence of vectors $\ensuremath\boldsymbol{v}^\ast$ and $\ensuremath\boldsymbol{u}^\ast$ which satisfy (ref), (ref), and (ref) - (ref), we use these to obtain the following result.

We refer to $\mathcal{J}$ as the jackknife $t$-statistic, and to $\tilde{\sigma}^2$ and $\tilde{\sigma}$ as the jackknife variance and jackknife standard error, respectively. The jackknife $t$-statistic provides a straightforward way to conduct inference on $\varphi$. In particular, by constructing $\ensuremath\boldsymbol{\hat{\varphi}}$ and then calculating $\ensuremath\boldsymbol{u}^\ast$ and $\ensuremath\boldsymbol{v}^\ast$ we can readily obtain hypothesis tests, $p$-values, and confidence intervals, as discussed below.\\

Hypothesis tests. To test

we form the jackknife $t$-statistic

A level-$\alpha$ two-sided jackknife $t$-test rejects $H_0$ whenever $| \mathcal{J} | > t_{1,1-\alpha/2}$, where $t_{1,1-\alpha/2}$ is the $( 1 - \alpha/2 )$ quantile of a $t$-distribution with $1$ degree of freedom. One-sided tests are obtained similarly. \\

$p$-values. For the two-sided test of $H_0: \varphi = \varphi_0$ against $H_A: \varphi \neq \varphi_0$, the $p$-value associated with the test statistic (ref) is

where $F( \cdot )$ denotes the CDF of the $t$-distribution with $1$ degree of freedom. For one-sided alternatives $H_A: \varphi > \varphi_0$ and $H_A: \varphi < \varphi_0$, the corresponding $p$-values are

Confidence intervals. A $(1-\alpha)$ confidence interval for $\varphi$ is given by

where $t_{1,1-\alpha/2}$ is the $( 1 - \alpha/2 )$ quantile of the $t$-distribution with $1$ degree of freedom. One-sided confidence intervals can be obtained similarly. We now return to our examples to illustrate our method of inference.

Jackknife $t_q$-Statistic

Section (ref) describes the simplest construction, in which we find only a single variance weights vector $\ensuremath\boldsymbol{u}^\ast$ and obtain a jackknife $t$-statistic which is asymptotically $t$-distributed with one degree of freedom. However, in many settings it is possible to find multiple variance weights vectors $\ensuremath\boldsymbol{u}_1^\ast, \ldots, \ensuremath\boldsymbol{u}_q^\ast$ satisfying (ref) - (ref) through the generation of further subsamples. These can be utilised to construct a statistic which is $t_q$-distributed with $q > 1$. This has the advantage that the critical values $t_{q,1-\alpha/2}$ are smaller than $t_{1,1-\alpha/2}$ and therefore produce shorter confidence intervals, for example. We now formalise this. Suppose there exist $q$ vectors $\ensuremath\boldsymbol{u}_1^\ast, \ldots, \ensuremath\boldsymbol{u}_q^\ast$ such that, writing

the following conditions hold:

These conditions are natural extensions of (ref) - (ref). Using $\ensuremath\boldsymbol{v}^\ast$ and $\ensuremath\boldsymbol{U}^\ast$ we obtain the following result.

We now return to our examples to illustrate the construction of a jackknife $t_q$-statistic with $q > 1$.

Jackknife $t^2_q$ Statistic

In this section we present a statistic which nests those presented in Section (ref) and (ref) as special cases. In particular, we show how it is possible to construct a jackknife counterpart to Hotelling's $t^2$-statistic hotelling1931generalization. This allows us to extend our method of inference to encompass vector parameters and to accommodate joint hypothesis tests. We begin by expanding Assumptions (ref) and (ref).

Assumption (ref) is a natural generalisation of Assumption (ref). Examples of such results may be found in the references cited in the discussion that surrounds that assumption.

Now, consider a vector-valued function $\ensuremath\boldsymbol{g}( \ensuremath\boldsymbol{\varphi} ) \in \mathbb{R}^{k \times 1}$ which is continuously differentiable in a neighbourhood of $\ensuremath\boldsymbol{\varphi}$ with derivative $\ensuremath\boldsymbol{G}( \ensuremath\boldsymbol{\varphi} ) \in \mathbb{R}^{k \times p}$. Let

be a $mk \times 1$ vector. Under Assumption (ref) and (ref), using $\ensuremath\boldsymbol{A}$ and $\ensuremath\boldsymbol{C}$ it is possible to construct an MVUJ weights vector $\ensuremath\boldsymbol{v}^{\ast}$, as well as variance weights vectors $\ensuremath\boldsymbol{u}_{1}^{\ast}, \ldots, \ensuremath\boldsymbol{u}_{q}^{\ast}$, in the same manner as described in Sections (ref) and (ref), respectively. Indeed, we will presume that it is possible to construct $q \geq k$ such variance weights vectors. Define the $k \times k$ matrix

which generalises the jackknife variance described in Theorems (ref) and (ref). Using this we form the following statistic

We refer to (ref) as the jackknife $t^2_q$-statistic. Theorem (ref) below characterises the asymptotic distribution of this statistic.

Using the result in Theorem (ref), one can readily conduct hypothesis tests, obtain associated $p$-values, and construct confidence ellipsoids for the parameter of interest. Compared to Theorem (ref) and (ref), Theorem (ref) imposes a stronger condition on the bias term since it assumes $\| ( \ensuremath\boldsymbol{A} \otimes \ensuremath\boldsymbol{I}_{p} ) \ensuremath\boldsymbol{\mu}_{NT} \|_2 = \mathcal{O}( 1 )$. This assumption can be dispensed with when $\ensuremath\boldsymbol{g}( \cdot )$ is linear.

Further Matter

Multi-dimensional Fixed Effects

We have focused thus far on two-dimensional panels since these remain the most commonly encountered in practice. However, panels with three or more dimensions are increasingly available, for example exporter-importer-year data and employer-employee-year data. Our framework extends naturally to multi-dimensional panels and provides a systematic way to construct bias-corrected jackknife estimators and jackknife $t$-statistics.

Consider a multi-dimensional panel with dimension sizes $N_1,\ldots,N_K$ and let $N \coloneqq N_1 \times \ldots \times N_{K}$. We generalise Assumptions (ref) and (ref).

Results of the form (ref) have been established for Poisson models with three-way fixed effects in weidner2021bias, and for a broader class of nonlinear models with three-way fixed effects in czarnowske2025debiased. Moreover, there are other three-way models, and $K$-way models more generally, for which results of the form (ref) could reasonably be supposed to hold.

The leading order bias of estimators of fixed effects models often scales in a predictable way with sample size, such that the matrix $\ensuremath\boldsymbol{A}$ can be determined from knowledge only of the order of the bias. In the context of a two-way fixed effects model, we usually expect the (leading) order bias of an estimator to be of the form

Heuristically, the rate can be understood as being determined by the relative order of the standard error and the dimension of the fixed effects. For example, with fixed effects $\ensuremath\boldsymbol{\lambda}^{(1)}$ and $\ensuremath\boldsymbol{\lambda}^{(2)}$, where $\textup{dim}( \ensuremath\boldsymbol{\lambda}^{(1)} ) = N_1$ and $\textup{dim}( \ensuremath\boldsymbol{\lambda}^{(2)} ) = N_2$, and with the order of the standard error as $\sqrt{N}$,

As another example, consider $K = 3$. With fixed effects $\ensuremath\boldsymbol{\lambda}^{(1)}$, $\ensuremath\boldsymbol{\lambda}^{(2)}$, and $\ensuremath\boldsymbol{\lambda}^{(3)}$, where $\textup{dim}\big( \ensuremath\boldsymbol{\lambda}^{(1)} \big) = N_1$, $\textup{dim}\big( \ensuremath\boldsymbol{\lambda}^{(2)} \big) = N_2$, and $\textup{dim}\big( \ensuremath\boldsymbol{\lambda}^{(3)} \big) = N_3$, and with the order of the standard error as $\sqrt{ N }$,

Or indeed, the fixed effects may be present in multiple dimensions simultaneously, for example, $\ensuremath\boldsymbol{\lambda}^{(1,2)}$, $\ensuremath\boldsymbol{\lambda}^{(2,3)}$, and $\ensuremath\boldsymbol{\lambda}^{(1,3)}$, where $\textup{dim}( \ensuremath\boldsymbol{\lambda}^{(1,2)} ) = N_1 N_2$, $\textup{dim}( \ensuremath\boldsymbol{\lambda}^{(2,3)} ) = N_2 N_3$, and $\textup{dim}( \ensuremath\boldsymbol{\lambda}^{(1,3)} ) = N_1 N_3$, and with the order of the standard error as $\sqrt{N}$,

Applying this same heuristic we are able to determine the (likely) structure of the matrix $\ensuremath\boldsymbol{A}$ in multi-dimensional fixed effects models.

The following example illustrates the construction of an MVUJ estimator and jackknife $t$-statistic in a three-way fixed effects model.

Higher-order Bias Corrections

In this section we discuss how higher-order bias can be embedded in our framework, thereby allowing us to construct higher-order MVUJ estimators. To fix ideas, suppose an estimator $\hat{\varphi}^{(0)}$ of a parameter of interest in a two-way fixed effects model admits an $\ell$-th order expansion

as $N, T \to \infty$ and $N / T \rightarrow \gamma^2 \in ( 0, \infty )$, where $z_{NT} \xrightarrow{d} \mathcal{N}( 0, \sigma^2 )$. Assume $T = ( \ell + 1 ) T_{\ell+1} $ where $T_{\ell+1}$ is an integer. Partition the time-dimension into $\ell + 1$ disjoint blocks of equal length with

For $j = 1, \ldots, \ell$ define subsamples

Analogously, assume $N = ( \ell + 1 ) N_{\ell+1}$ where $N_{\ell+1}$ is an integer, and partition the cross-section into $\ell + 1$ disjoint blocks of equal length with

For $j = \ell + 1, \ldots, 2 \ell$ define the subsamples

Figure (ref) below illustrates the subsampling scheme with $\ell = 2$.

Let $\hat{\varphi}^{(j)}$ be the estimator computed on subsample $S_j$ for $j = 1, \ldots, m$ where $m \coloneqq 2\ell$. Suppose each subsample estimator admits the expansion

for $j = 1, \ldots, \ell$, and

for $j = \ell + 1, \ldots, m$. Now, define $\ensuremath\boldsymbol{\hat{\varphi}} \coloneqq ( \hat{\varphi}^{(0)}, \hat{\varphi}^{(1)}, \ldots, \hat{\varphi}^{(m)} )^\top$, $r_{NT}\coloneqq \sqrt{NT}$, $R \coloneqq 2\ell$, and

with

Then

with

and

where $\kappa_j = \frac{ j }{ \ell + 1 }$. Finally, if, for example,

with $\nu_{it} \sim \textup{iid}( 0, \sigma^2 )$ and, for $j = 1, \ldots, \ell$,

then $\ensuremath\boldsymbol{z}_{NT} \xrightarrow{d} \mathcal{N}( \ensuremath\boldsymbol{0}, \sigma^2 \ensuremath\boldsymbol{C} )$ with

for $j_1, j_2 = 1, \ldots, \ell$. With $\ensuremath\boldsymbol{A}$ and $\ensuremath\boldsymbol{C}$ determined, one may construct an $\ell$-th order corrected MVUJ estimator and the corresponding $\ell$-th order jackknife $t$-statistic as previously.

We now illustrate the construction of a higher-order MVUJ estimator and associated jackknife $t$-statistic.

Numerical Examples

In this section we showcase both the performance and the versatility of our method of inference through two numerical examples. The first considers inference on a slope coefficient in a linear regression model with predetermined regressors and individual fixed effects. The second considers inference on a slope coefficient in a panel probit model with interactive fixed effects.

Linear Regression with Individual Effects

In our first example the data are generated according to

where $\varphi=0.5$, $\lambda_i \sim \textup{iid}\ \mathcal{N}(0,1)$, and $\varepsilon_{it} \sim \textup{iid}\ \mathcal{N}(0,1)$. We study the LS estimator of $\varphi$, denoted by $\hat{\varphi}^{(0)}$. Assume $T = 2T_2$, $N = 2N_2$, and $N = 5N_5$ where $T_2$, $N_2$ and $N_5$ are integers. Consider the following subsamples

Let $\hat{{\varphi}}^{(j)}$ denote the LS estimator computed on subsample $S_j$. We compare three different constructions. The first, JK(a), utilises subsamples $S_1$ and $S_2$, alongside $S_0$. With $\ensuremath\boldsymbol{\hat{{\varphi}}}_{(a)} \coloneqq ( \hat{{\varphi}}^{(0)}, \hat{{\varphi}}^{(1)}, \hat{{\varphi}}^{(2)} )$, the MVUJ estimator is

There is one admissible variance weights vector

yielding the jackknife $t$-statistic

The second, JK(b), uses additional cross-sectional partitions. In particular $\ensuremath\boldsymbol{\hat{{\varphi}}}_{(b)} \coloneqq (\hat{{\varphi}}^{(0)}, \hat{{\varphi}}^{(1)},\hat{{\varphi}}^{(2)},\hat{{\varphi}}^{(3)},\hat{{\varphi}}^{(4)})^\top$. The MVUJ estimator is not unique, so we use a convenient construction

Two admissible variance weights vectors are

yielding

Finally, JK(c) uses additional cross-sectional partitions. In particular,

where $\hat{{\varphi}}^{(4+g)}$ is the LS estimator on the $g$-th cross-sectional fifth. The MVUJ estimator is not unique so we again use a convenient construction

Five admissible variance weights vectors are

yielding

Table {(ref)} below reports empirical coverage probabilities and average length of a 95% two-sided confidence interval for ${\varphi}$ constructed by inverting the different jackknife $t$-statistics outlined above. We also report empirical bias and standard error of the different estimators. We compare our jackknife confidence intervals with two alternative constructions. The first utilises the bias correction provided by hahn2011bias which is based on an analytical characterisation. The second uses the moving block bootstrap scheme described in higgins2025inference.

Overall jackknife bias correction (all schemes) outperforms both analytical bias correction and bootstrap bias correction in terms of bias reduction, with the latter two being quite sensitive to the choice of tuning parameters (bandwidth, block size), especially in small samples. On the other hand, a small inflation of the standard error is observed for the jackknife bias-corrected estimator when compared to the uncorrected LS estimator. This, however, quickly diminishes as sample size increases. In terms of coverage, those confidence intervals obtained by inverting a jackknife $t$-statistic considerably outperform their analytical and bootstrap counterparts, both in their nominal coverage, and in their reliability, with bootstrap confidence intervals exhibiting particular sensitivity to block length. The observed length of each jackknife confidence interval is as expected, with the use of additional variance weights vectors producing shorter confidence intervals. This is illustrative of the trade-off between the most simple constructions, which produce wider confidence intervals, and those which are more complex, but yield shorter confidence intervals. Nonetheless, it should be emphasised that even the most complex construction, JK(c), which produces a length comparable to other methods, is still computationally trivial in comparison to the bootstrap, does not require the use of cumbersome analytic formulae, and does not utilise any tuning parameters.

Probit Model with Interactive Effects

In our second experiment we consider a panel probit model with interactive fixed effects and either exogenous or predetermined covariates. The data are generated according to

where $\varphi = 1$, and $\lambda_i \sim \textup{iid}\ \mathcal{N}( 0, 1 )$, $\gamma_t \sim \textup{iid}\ \mathcal{N}\big( 0, 1 \big)$, and $\varepsilon_{it} \sim \textup{iid}\ \mathcal{N}( 0, 1 )$ are mutually independent. We consider two designs for $x_{it}$

where $\eta_{it} \sim \textup{iid}\ \mathcal{N} ( 0 , 1 )$ and is independent of $\lambda_{i}$, $\gamma_{t}$, and $\varepsilon_{it}$. Our first jackknife estimator is based on the subsamples

where we assume $N = 2 {N_{2}}$ and $T = 2 {T_2}$ where ${N_2}$ and ${T_2}$ are integers. Let $\ensuremath\boldsymbol{ {\hat{\varphi}} }_{(a)} \coloneqq ( {\hat{\varphi}}^{(0)}, {\hat{\varphi}}^{(1)}, {\hat{\varphi}}^{(2)}, {\hat{\varphi}}^{(3)}, {\hat{\varphi}}^{(4)} )^{ \top }$. We construct an MVUJ estimator

Two admissible variance weights vectors are

with which we construct two jackknife $t$-statistics

and

Our second set of subsamples utilise $S_{ 1 }$, $S_{ 2 }$, and

where we assume $N = 5 N_{5}$ where $N_5$ is an integer. Let

where ${\hat{\varphi}}^{(4+g)}$ is computed on the $g$-th cross-sectional fifth $S_{4+g}$. We construct an MVUJ estimator

together with the jackknife standard error

where

This yields a jackknife $t$-statistic

Table {(ref)} below reports empirical coverage probabilities and length of a 95% two-sided confidence interval for ${\varphi}$ constructed by inverting $\mathcal{J}{(a)}$, $\mathcal{J}{(b)}$, and $\mathcal{J}{(c)}$. We also report empirical bias and standard error of the bias-corrected estimates.

Both jackknife bias-corrected estimators achieve substantial bias reduction. When the covariate is exogenous there is very little inflation of the standard error when compared to the uncorrected ML estimator. When the covariate is predetermined, inflation of the standard error is observed. This occurs due to moderate dependence between partitions of the data in finite samples which diminishes as sample size (in particular the time-dimension) increases. Overall coverage is consistently close to its nominal value, particularly in the design where the covariate is exogenous. As expected, the use of additional variance weights vectors to produce $t_q$-statistics with $q > 1$ significantly reduces the length of confidence intervals.

Conclusion

This paper develops a straightforward and computationally inexpensive method of inference for fixed effects models. More broadly, this paper provides a general framework for the systematic development of jackknife bias-corrected estimators and jackknife $t$-statistics, that accommodates a wide range of models and estimation approaches, multi-dimensional panels, and higher-order bias corrections. Thus far, we have largely proceeded from the point at which a suitable collection of subsamples is obtained. Ongoing work develops general principles for constructing such subsamples, including optimality. Results in this regard will appear in subsequent revisions.

Beyond fixed effects models, there has been increased interest in automatic inference procedures in other settings where estimators exhibit asymptotic bias. Much of this literature emphasises bootstrap or analytic bias correction (see, e.g., cavaliere2024bootstrap). Our method provides an alternative to the bootstrap, one which both exhibits a substantially reduced computational burden and enjoys a high degree of model agnosticism. As such, our method may be usefully employed elsewhere.

One promising line of enquiry is two-stage estimators, where the final stage estimator inherits bias from the first stage. Examples of this include models with many first-stage covariates cattaneo2019two and factor-augmented regressions with estimated factors goncalves2014estimating. In such instances, we may derive jackknife bias-corrected estimators and jackknife $t$-statistics through the generation of subsamples. Another promising line of enquiry is semiparametric and nonparametric estimators where asymptotic bias can arise as a consequence of smoothing calonico2014robust,calonico2018effect. In such instances, we may manipulate bandwidth in order to produce a collection of estimators which exhibit different linear combinations of the same bias terms. Ongoing work explores the applications of our method in these settings.