# Influence Analysis with Panel Data111 I thank Marcus Chambers, Marco
Francesconi, Abhimanyu Gupta, Chiara Monfardini, and the participants to the
$10^{th}$ ICEEE, $20^{th}$ German Stata Conference, $28^{th}$ IPDC, 2023
Petralia Workshop, and $29^{th}$ UK Stata Conference for the constructive
suggestions. I am also grateful to Omar Hussein for his insightful comments
and help. This project was supported by the South East Network for Social
Sciences (SeNSS) as part of the Doctoral Training Partnership (award
ES/P00072X/1/2128216). This research was also funded by the UK Economic and
Social Research Council award ES/S012486/1 (MiSoC). The Stata commands for
implementing the methodology presented in the paper are xtlvr2plot for unit-
wise leverage-vs-residual plots, and xtinfluence for the influence analysis
with panel data. The community-contributed are available on
https://github.com/POLSEAN/Influence-Analysis at the time of writing.

Annalivia Polselli Institute of Analytics and Data Science (IADS) and Centre
for Micro-Social Change (MiSoc), University of Essex. Email:
<EMAIL_ADDRESS>
(Current version: )

###### Abstract

The presence of units with extreme values in the dependent and/or independent
variables (i.e., vertical outliers, leveraged data) has the potential to
severely bias regression coefficients and/or standard errors. This is common
with short panel data because the researcher cannot advocate asymptotic
theory. Example include cross-country studies, skill-cell analyses, and
experimental studies. Available diagnostic tools may fail to properly detect
these anomalies, because they are not designed for panel data. In this paper,
we formalise statistical measures for panel data models with fixed effects to
quantify the degree of leverage and outlyingness of units, and the joint and
conditional influences of pairs of units. We first develop a method to
visually detect anomalous units in a panel data set, and identify their type.
Second, we investigate the effect of these units on LS estimates, and on other
units’ influence on the estimated parameters. To illustrate and validate the
proposed method, we use a synthetic data set contaminated with different types
of anomalous units. We also provide an empirical example.

JEL codes: C13, C15, C23.
Keywords: Leverage, outliers, diagnostic measures, Cook’s distance.

##  1 Introduction

Observational data often contain data points that exhibit extreme values in
the response-space (vertical outliers, VO), in the covariate-space (good
leverage points, GL), or in both directions (bad leverage points, BL)
(Rousseeuw and Van Zomeren,, 1990; Silva,, 2001). The presence of these
anomalies creates bias in the least square (LS) estimates (i.e., coefficients
and/or standard errors), and can potentially invalidate the results because
econometric techniques based on the mean are more sensitive to extreme
realisations (Donald and Maddala,, 1993; Bramati and Croux,, 2007; Verardi and
Croux,, 2009). Their impact is more severe with panel data because these
anomalies may be carried over the time series of a unit, exacerbating the bias
(Bramati and Croux,, 2007). This is particularly problematic in short panel
data sets (few cross-sectional units $N$ are observed over multiple time
periods $T$) because the small cross-sectional size does not allow to mobilise
the benefits of the asymptotic theory. These data are common in some fields of
economics due to the structure of the data or the nature of the study. Some
examples are: worldwide country-level macro-data (e.g., 50 United States, 27
European countries); skill-cell data sets (e.g., with education-experience-
state cells as unit of observation); or experimental data, where it is costly
for the researchers to recruit many participants to follow in successive
waves.

Popular tools for the detection of such data points include diagnostic plots
(e.g., leverage-vs-residual plots), and measures of influence (e.g., Cook,’s
(1979) distance). These respectively serve to identify the type of anomaly,
and assess the individual influence on the LS estimates by removing one unit
at a time. Despite the ease of their interpretability and computation, these
measures may fail to detect anomalous units when surrounded by other cases as
they do not assess mutual influence of units. This phenomenon – known as the
_masking effect_ – is documented for Cook-like measures (Atkinson,, 1985;
Chatterjee and Hadi,, 1988; Rousseeuw and Van Zomeren,, 1990; Rousseeuw,,
1991), but can be overcome by measures that delete two cases at a time from
the sample, such as, joint and conditional measures for the influence
(Lawrance,, 1995). A visual inspection of these anomalies may be complicated
with many variables (Rousseeuw,, 1991; Bramati and Croux,, 2007), thus the
need of a method that identifies these anomalies based on statistical
measures.

In this paper, we develop a method to: (i) systematically detect anomalous
units in a panel data set, and identify their type with leverage-vs-residual
plots; and (ii) investigate the effect of these units on LS estimates, and on
other units’ influence on the estimated parameters with influence plots. For
this purpose, we first formalise statistical measures – i.e., individual
leverage and normalised residual squared – that quantify the degree of
leverage and outlyingness of units all over its time series. These are then
used to produce ‘unit-wise’ leverage-vs-residual plots. Second, we build on
Lawrance,’s (1995) pair-wise approach by proposing measures for joint and
conditional influence of units $i$ and $j$ that are designed for panel data
models with fixed effects. We illustrate the method with synthetic data that
have been contaminated with anomalous units. We then apply our method to real
data.

We show that our method detects problematic units that will not otherwise be
detected by the conventional methods. Specifically, leverage-vs-residual plots
provide an initial picture of the anomalies in the data, being more
informative about the existence and type of anomaly. Then, influence plots
complete the analysis by displaying the strength and direction of the
connection between units $i$ and $j$, showing how the influential units affect
the contribution of other units on the LS estimates. The strength of this
method is that a unit, which is not individually influential according to
Cook’s distance, will be detected if it is influential either jointly with
another unit, or in the absence of another highly influential unit. The method
can help the researcher understand how some units may be driving most of the
effect on LS estimates _after_ the main regression analysis, or explore the
data set and identify potential anomalies _before_ any regression analysis.

Once a unit is detected as potentially influential, the researcher should not
proceed with its deletion because it might be a valid observation that should
be properly addressed. For example, robust estimation accounts for anomalies
that affect the estimated coefficients (i.e., VO and BL) (Bramati and Croux,,
2007; Verardi and Croux,, 2009; Aquaro and Čížek,, 2013, 2014; Jiao,, 2022),
and jackknife-type standard errors deal with anomalies that affect the
statistical inference (i.e., GL) (MacKinnon and White,, 1985; Chesher and
Jewitt,, 1987; Davidson et al.,, 1993; MacKinnon,, 2013; Belotti and
Peracchi,, 2020; Polselli,, 2022; MacKinnon et al., 2023a, ; MacKinnon et al.,
2023b, ).222These methods will not be discussed further in this study.

This paper contributes to the literature on diagnostic and influential
measures as follows. While Cook-like distances are still on of the most
popular diagnostic measures for the detection of individually influential
observations,333Several variants of Cook,’s (1979) measure have been proposed
for both cross-sectional and panel data models. In the cross-sectional
framework, Martín and Pardo, (2009) extend the measure to logistic models,
Pinho et al., (2015) to generalized linear mixed models, and Martín, (2015) to
log-linear models. Banerjee and Frees, (1997) are the first to formalise a
leave-one-out diagnostic measure to detect outliers and influential points for
panel data, specifically in linear longitudinal models with random effects.
Belotti and Peracchi, (2020) have extended Banerjee and Frees,’s (1997)
distance to linear panel data models with fixed effects. authors have not yet
addressed how to overcome the _masking effect_. We build on Lawrance,’s (1995)
pair-wise measures to investigate the reciprocal influence of units. We
present an adaptation of the joint and conditional influence measures for
linear panel data models with fixed effects. With this measures, we can detect
units that would have not been detected otherwise by Cook’s distance because
not individually influential. Because the joint and conditional measures
resemble a directed and weighted adjacency list from network analysis,444In a
directed graph, the effect of unit $i$ on unit $j$ differs from $j$ to $i$; in
a weighted graph, the intensity of the effect of each unit differs. we propose
a graphical representation of these measures that plots the influence of unit
$j$ on $i$’s influence such as to show the existence and the strength of the
connections between pairs of units. Using scatter plots or heat plots
facilitates the visual detection and classification of anomalous data points,
and their influence on other units. This representation is innovative in the
literature on diagnostic measures, where Atkinson and Mulira,’s (1993)
stalactite plot pioneers the visual inspection of multivariate outliers in the
data.

This paper is closely related to MacKinnon et al., 2023a ; MacKinnon et al.,
2023b . They derive a measure for the overall leverage of clusters for cross-
sectional data. We follow their derivation to calculate the overall leverage
of a unit all over its history. Because we have repeated measures of the
variables for each unit, our formulation addresses the presence of fixed
effects by transforming the variables with the within-group (WG)
transformation.

The rest of the paper is structured as follows. In Section 2, we start by
defining the three types of anomalies. In Section 3, we first introduce the
panel data models and estimators, and then present the measures of influence
for panel data. Section 4 discusses the proposed method using a synthetic data
set. Section 5 applies the method to an empirical exercise. Section 6
concludes.

##  2 Anomalous Units

Longitudinal data are extremely likely to contain units with values of the
dependent and/or independent variables that follow a different pattern from
the main cloud of the data in the response- and/or covariate-spaces. These are
classifiable as _vertical outliers_ (VO) if the extreme values are in the
outcome variable, _good leverage_ (GL) points if the extreme values are in the
covariates, or _bad leverage_ (BL) points if they are in both directions.

Vertical outliers (unit 30 in Figure 1) are anomalous in the response-factor
space, follow the opposite trend of the main cloud of data points, and lay far
from the regression line as if they were generated from a different process
(Chatterjee and Hadi,, 1986; Greene,, 2012, p.141). They display large squared
normalised residual, and their presence alters the estimated LS intercept
(here, the fitted line shifts upwards), undermining the accuracy of the
estimator but not its precision (Verardi and Croux,, 2009).

Leveraged data points appear isolated from the rest of the data but, unlike
VO, follow the same trend of the rest of the data (Chatterjee and Hadi,, 1986;
Greene,, 2012, p.140). Leverage points can be distinguished in ‘good’ or ‘bad’
as follows. The former (unit 20 in Figure 1) exhibit unusually extreme values
in the covariates and lie on the predicted regression line enhancing the
precision of the regression fit; whereas, the latter (unit 10 in Figure 1)
possess extreme values in both input and response direction, lie far from the
plane where the bulk of data points are and are not fitted by the regression
model (Rousseeuw,, 1991; Bramati and Croux,, 2007). While BL realisations
adversely affect the estimated LS coefficients (both the intercept and the
slope of the regression), GL points add variability in the sample allowing for
a better fit of the data at the cost of a deteriorated statistical inference
(MacKinnon and White,, 1985; Chesher and Jewitt,, 1987; Silva,, 2001; Verardi
and Croux,, 2009).

In a panel data sets, anomalous realisations in the time series of a unit may
appear either as isolated cases in the time-series of different units (_cell-
concentrated_ points), or concentrated in the time-series of the same units
(_block-concentrated_ points) (Bramati and Croux,, 2007).

##  3 Measures of Influence in Panel Data Models

Let $\\{y_{it},\mathbf{x}_{it}\\}_{t=1}^{T}$ be repeated measurements of unit
$i\in\mathcal{I}=\\{1,\dots,N\\}$ over $t$ time periods, and that $T_{i}=T$
for all $i$ (i.e., balanced panel).555The data is assumed to be balanced to
simplify the notation, but it can be equally applied to unbalanced panel data
with appropriate change in notation. Consider a linear panel regression model
with unobserved individual heterogeneity as follows

$y_{it}=\mathbf{x}_{it}^{\prime}\bm{\beta}+\alpha_{i}+u_{it},$ (1)

where $y_{it}$ is the response variable for the cross-sectional unit $i$ at
time period $t$; $\mathbf{x}_{it}$ is a $k\times 1$ vector of time-varying
inputs, $\bm{\beta}$ is a $k\times 1$ vector of parameters of interest;
$\alpha_{i}$ is the individual-specific unobserved heterogeneity (or _fixed
effects_); and $u_{it}$ is a stochastic error component.

Stacking observations over time $t$, model (1) becomes

$\mathbf{y}_{i}=\mathbf{X}_{i}\bm{\beta}+\bm{\alpha}_{i}+\mathbf{u}_{i},\,\,\,\text{for
all}\,\,\,i=1,\dots,N,$ (2)

where $\mathbf{y}_{i}$ is a $T\times 1$ vector of the dependent variable;
$\mathbf{X}_{i}$ is a $T\times k$ matrix of time-varying independent
variables; $\bm{\alpha}_{i}=\alpha_{i}\bm{\iota}$ is a $T\times 1$ vector of
unobserved individual fixed effects, and $\bm{\iota}$ is a vector of ones of
order $T$; and $\mathbf{u}_{i}$ is a $T\times 1$ vector of one-way error
component.

Model (2) can be consistently estimated using OLS on the time-demeaned model

$\widetilde{\mathbf{y}}_{i}=\widetilde{\mathbf{X}}_{i}\bm{\beta}+\widetilde{\mathbf{u}}_{i},\,\,\,\text{for
all}\,\,\,i=1,\dots,N,$ (3)

where
$\widetilde{\mathbf{y}}_{i}=(\mathbf{I}_{T}-\,T^{-1}\bm{\iota}\bm{\iota}^{\prime})\mathbf{y}_{i}$
is $T\times 1$;
$\widetilde{\mathbf{X}}_{i}=(\mathbf{I}_{T}-\,T^{-1}\bm{\iota}\bm{\iota}^{\prime})\mathbf{X}_{i}$
is $T\times k$; and
$\widetilde{\mathbf{u}}_{i}=(\mathbf{I}_{T}-\,T^{-1}\bm{\iota}\bm{\iota}^{\prime})\mathbf{u}_{i}$
is $T\times 1$. Note that
$(\mathbf{I}_{T}-\,T^{-1}\bm{\iota}\bm{\iota}^{\prime})\bm{\alpha}_{i}=\bm{0}$
as $T^{-1}\bm{\iota}\bm{\iota}^{\prime}\bm{\alpha}_{i}=\bm{\alpha}_{i}$. The
OLS estimator of (3) is the well-known _within-group_ estimator of the true
population parameter with formula

$\widehat{\bm{\beta}}=\Biggl{(}\sum_{i=1}^{N}\widetilde{\mathbf{X}}_{i}^{\prime}\widetilde{\mathbf{X}}_{i}\Biggr{)}^{-1}\sum_{i=1}^{N}\widetilde{\mathbf{X}}_{i}^{\prime}\widetilde{\mathbf{y}}_{i}.$
(4)

The estimator is unbiased and asymptotically normally distributed under
conventional model assumptions.

The _within-group_ estimator (4) without the whole history of unit $i$ is the
Leave-One-Out (L1O) estimator666The L1O estimator for panel data models is
derived in Banerjee and Frees, (1997) for mixed effects models, and Belotti
and Peracchi, (2020) for fixed effects models. We show the asymptotic
distribution of $\widehat{\bm{\beta}}_{(i)}$ in Appendix C. as follows

$\widehat{\bm{\beta}}_{(i)}=\widehat{\bm{\beta}}-\bigl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\bigr{)}^{-1}\widetilde{\mathbf{X}}_{i}^{\prime}\mathbf{M}_{i}^{-1}\widehat{\mathbf{u}}_{i}$
(5)

where $\mathbf{M}_{i}^{-1}=(\mathbf{I}_{T}-\mathbf{H}_{i})^{-1}$ with
$\mathbf{H}_{i}=\widetilde{\mathbf{X}}_{i}(\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}})^{-1}\widetilde{\mathbf{X}}_{i}^{\prime}$,
and
$\widehat{\mathbf{u}}_{i}=\widetilde{\mathbf{y}}_{i}-\widetilde{\mathbf{X}}_{i}\widehat{\bm{\beta}}$
is the residual term. The L1O estimator estimates the effect of the covariates
on the outcome variable by removing one unit at a time from the sample. This
is a general measure of the influence exerted by that unit on the estimated
coefficients or on the model fit.

The _within-group_ estimator (4) without the full history of pairs of units
$\\{i,j\\}$ is the Leave-Two-Out (L2O) estimator777The derivation of Formula
(6) and its distribution in Appendices B and C. below

$\widehat{\bm{\beta}}_{(i,j)}=\widehat{\bm{\beta}}_{(i)}-\bigl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\bigr{)}^{-1}\bigl{(}\widetilde{\mathbf{X}}_{i}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}+\widetilde{\mathbf{X}}_{j}^{\prime}\bigr{)}\bigl{(}\mathbf{M}_{j}-\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}\bigr{)}^{-1}\bigl{(}\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\widehat{\mathbf{u}}_{i}+\widehat{\mathbf{u}}_{j}\bigr{)}$
(6)

where $\mathbf{M}_{j}=\mathbf{I}_{j}-\mathbf{H}_{j}$ with
$\mathbf{H}_{ij}=\widetilde{\mathbf{X}}_{i}(\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}})^{-1}\widetilde{\mathbf{X}}_{j}^{\prime}$,
and
$\mathbf{H}_{j}=\widetilde{\mathbf{X}}_{j}(\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}})^{-1}\widetilde{\mathbf{X}}_{j}^{\prime}$
noting that $\mathbf{H}_{ji}=\mathbf{H}_{ij}^{\prime}$. The deletion of pairs
of units one at a time from the sample captures the influence exerted by that
pair on the estimated parameters or on the model fit.

###  3.1 Leverage and Normalised Residual Squared

The change in the magnitude of the estimated coefficients or the model fit
after the deletion of a unit from the sample is informative of the influence
of that unit. This can be explained by high residuals, high individual
leverage, or both. In this section, we construct measures for panel data that
assess the individual influence of a unit along these two dimensions. We
define the individual leverage and the normalised residual squared for panel
data, that considers the full time series of a unit (‘unit-wise’ approach).

The leverage of a unit measures the distance of the values of the covariates
of a unit from those of other units in the sample. This information is
provided by the leverage matrix of unit $i$,
$\mathbf{H}_{i}=\widetilde{\mathbf{X}}_{i}\bigl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\bigr{)}^{-1}\widetilde{\mathbf{X}}_{i}^{\prime}$,
which is a $T\times T$ positive-definite and symmetric with diagonal elements
$h_{ii,tt}=\widetilde{\mathbf{x}}_{it}^{\prime}(\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}})^{-1}\widetilde{\mathbf{x}}_{it}$,
and off-diagonal elements
$h_{ii,ts}=\widetilde{\mathbf{x}}_{it}^{\prime}(\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}})^{-1}\widetilde{\mathbf{x}}_{is}$.888With
only one time period (cross-sectional data), the leverage matrix of unit $i$
is the scalar
$h_{i}=\mathbf{x}_{i}^{\prime}(\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}})^{-1}\mathbf{x}_{i}$
with $\mathbf{x}_{i}$ a $k\times 1$ vector of covariates and the constant
term.

We calculate the nuclear (or trace) norm of $\mathbf{H}_{i}$ to summarise the
leverage exerted by each unit over their time series, as in MacKinnon et al.,
2023a ; MacKinnon et al., 2023b  for clustered data.999Similarly, MacKinnon et
al., 2023b  calculate the leverage matrix of cluster $G$ populated with
$N_{g}$ units. They use the nuclear norm operator to provide an overall
assessment of the influence of each group in a way that is computationally
feasible. We face a similar challenge with panel data, where the unit can be
seen as the $N$-th cluster populated with a total of $T$ time realisations.
Therefore, their notion of leverage for cluster-level data is comparable to
some extent to ours for panel data with some adjustments (i.e., the WG
transformation to remove the fixed effects). The overall leverage of unit $i$
can be calculated as
$L_{i}=\mathrm{tr}\big{(}\widetilde{\mathbf{X}}_{i}^{\prime}\widetilde{\mathbf{X}}_{i}\bigl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\bigr{)}^{-1}\big{)}$.
The average measure of the leverage exerted by all units in the sample is
$\frac{1}{N}\sum_{i=1}^{N}\mathrm{tr}(\mathbf{H}_{i})=k/N$ by rearranging
terms in the trace operator, noting that
$\sum_{i=1}^{N}\widetilde{\mathbf{X}}_{i}^{\prime}\widetilde{\mathbf{X}}_{i}=\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}$,
and $k=\mathrm{rank}(\widetilde{\mathbf{X}})$. This result (similar to the
cross-sectional case) can be used to establish the cut-off value for a unit to
be considered as highly influential. For instance, a unit with an individual
leverage that exceeds twice the value of the individual leverage in the
sample, $L_{i}>2k/N$, is a high leverage unit. High leverage units possess
unusually extreme values (too small or too large) in the regressors, and can
be classified as GL unit.

We now define a measure to locate anomalous units in the $y$-space. We define
the normalised residual squared for panel data as
$\widehat{u}^{*}_{it}=\big{(}\widehat{u}_{it}/\sqrt{\sum_{i}\widehat{u}^{2}_{it}}\big{)}^{2}$,
which informs on the outlyingness of unit $i$ at time $t$. We define
$\widehat{\mathbf{u}}^{*}_{i}=[\widehat{u}^{*}_{i1}\,\dots\,\widehat{u}^{*}_{iT}]^{\prime}$
a $T\times 1$ vector that contains the information of the influence exerted by
unit $i$ in each time period $t\in\\{1,\dots,T\\}$ on the least squares
estimates.

We use the Euclidean norm of $\widehat{\mathbf{u}}^{*}_{i}$ to quantify the
influence of unit $i$ in the output space such that
$O_{i}=\sqrt{\sum_{t}\widehat{u}^{*2}_{it}}$. The average normalised residual
squared over the full sample of units is
$N^{-1}\sum_{i=1}^{N}\widehat{u}^{*}_{it}=1/N$ (like in the cross-sectional
case). This value can be used as a reference to compare the levels of
influence of each units in terms of the dependent variables. That is, a unit
is said to be influential in the $y$-space if its value of normalised residual
squared exceeds twice the average value in the sample, i.e., $O_{i}>2/N$.

###  3.2 Measures for Joint and Conditional Influence

Joint influence is the influence exerted by a pair $(i,j)$ on the LS estimates
_jointly_. It is obtained by comparing the estimated coefficients _with_ and
_without_ full history of the pair in the sample. In formulae,

$\mathrm{C}_{ij}(\widehat{\bm{\beta}})=\bigl{(}\widehat{\bm{\beta}}-\widehat{\bm{\beta}}_{(i,j)}\bigr{)}^{\prime}\Bigl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\Bigr{)}\bigl{(}\widehat{\bm{\beta}}-\widehat{\bm{\beta}}_{(i,j)}\bigr{)}(s^{2}K)^{-1}\approx\mathrm{F}(\nu_{1},\nu_{2})$
(7)

where $N$ is the total number of cross-sectional units in the sample;
$\nu_{1}=K$ is the total number of regressors including the constant, and
$\nu_{2}=(N-1)$ is the degrees of freedom at the denominator because of the
clustering;
$s^{2}=\nu_{2}^{-1}\sum_{i=1}^{N}\widehat{\mathbf{u}}_{i}^{\prime}\widehat{\mathbf{u}}_{i}$
is the variance of the fitted model – i.e., residual mean squared error (MSE)
– and is consistent for $\sigma^{2}$. $\mathrm{C}_{ij}(\widehat{\bm{\beta}})$
is symmetric for $i\neq j$. When $i=j$, the measure informs on the individual
influence of unit $i$ on the estimated coefficients for $\beta$. This can be
interpreted as the Cook’s distance for panel data (Banerjee and Frees,, 1997;
Belotti and Peracchi,, 2020). Formula (7) becomes

$\mathrm{C}_{ii}(\widehat{\bm{\beta}})=\bigl{(}\widehat{\bm{\beta}}-\widehat{\bm{\beta}}_{(i)}\bigr{)}^{\prime}\Bigl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\Bigr{)}\bigl{(}\widehat{\bm{\beta}}-\widehat{\bm{\beta}}_{(i)}\bigr{)}(s^{2}K)^{-1}\approx\mathrm{F}(\nu_{1},\nu_{2})$
(8)

$\mathrm{C}_{ii}(\widehat{\bm{\beta}})$ is hence a special case of the more
general measure for the deletion of pairs of units,
$\mathrm{C}_{ij}(\widehat{\bm{\beta}})$.

The joint effect is the ratio

$\mathrm{K}_{j|i}=\frac{\mathrm{C}_{ij}(\widehat{\bm{\beta}})}{\mathrm{C}_{ii}(\widehat{\bm{\beta}})}$
(9)

and informs on unit $i$’s influence within the $(i,j)$ pair. For large values
of this measure, unit $j$ alters the individual effect of unit $i$ on the
estimated coefficients. In other words, the most influential unit in the pair
$j$ contributes to drive most of the effect on LS estimates by _enhancing_ or
_reducing_ the effect of the least influential $i$. The assessment can be done
in conjunction with the conditional effect below.

Because joint measures do not compare individual influences arising _before_
and _after_ the deletion of another unit, the notion of conditional influence
is needed. Conditional influence is the influence exerted by unit $i$ on the
LS coefficients _conditional on_ removing unit $j$ from the sample. It shows
how the absence of unit $j$ alters the influence of unit $i$ on the LS
estimates. In formulae,

$\mathrm{C}_{i(j)}(\widehat{\bm{\beta}})=\bigl{(}\widehat{\bm{\beta}}_{(i,j)}-\widehat{\bm{\beta}}_{(j)}\bigr{)}^{\prime}\Biggl{(}\sum_{i\neq
j}\widetilde{\mathbf{X}}_{i(j)}^{\prime}\widetilde{\mathbf{X}}_{i(j)}\Biggr{)}\bigl{(}\widehat{\bm{\beta}}_{(i,j)}-\widehat{\bm{\beta}}_{(j)}\bigr{)}(s^{2}K)^{-1}\approx\mathrm{F}(\nu_{1},\nu_{2})$
(10)

where $\widetilde{\mathbf{X}}_{i(j)}$ is a $(N-1)\times K$ matrix of time-
demeaned regressors without unit $j$. The value of
$\mathrm{C}_{i(j)}(\widehat{\bm{\beta}})$ is zero for $i=j$, and is not
symmetric for $i\neq j$. The diagnostic measure
$\mathrm{C}_{i(j)}(\widehat{\bm{\beta}})$ is not a test statistics but the
knowledge of its empirical distribution can be used to extrapolate cut-off
values to assess the conditional influence of units.

The conditional effect is the ratio

$\mathrm{M}_{i(j)}=\frac{\mathrm{C}_{i(j)}(\widehat{\bm{\beta}})}{\mathrm{C}_{ii}(\widehat{\bm{\beta}})}$
(11)

quantifying unit $i$’s influence _before_ and _after_ the deletion of unit $j$
from the sample. For $\mathrm{M}_{i(j)}\geq 1$, unit $j$ is said to _mask_ the
influence of unit $i$ because the influence of $i$ increases without $j$ in
the sample. When $\mathrm{M}_{i(j)}<1$, unit $j$ is said to _boost_ the
influence of unit $i$ because the influence of $i$ decreases without $j$. With
this information, we can interpret the results of the joint effect as follows.
When the influence of $i$ is masked (boosted) by $j$, $j$ is reducing
(enhancing) the effect of $i$.

These measures follow a F-distribution with $\nu_{1}$ degrees of freedom at
the numerator and $\nu_{2}$ degrees of freedom at the denominator.101010The
derivation is shown in Appendix C. These two measures are not test statistics
but the information on their empirical distributions can be used to select
distributional cut-off values to identify influential units, as recommended
practice in the cross-sectional framework (Cook,, 1979; Martín and Pardo,,
2009; Martín,, 2015). Cut-off values can be extrapolated from the median of
the inverse cumulative F distribution with $\nu_{1}$ degrees of freedom at the
numerator and $\nu_{2}$ degrees of freedom at the denominator. Alternatively,
non-distributional cut-off values of 1 or $4/N$ can be chosen Bollen and
Jackman, (1985).

The joint and conditional measures are constructed in a way that resembles a
weighted and directed adjacency list from network analysis, which displays the
existence and the strength of the links between pairs of units $(i,j)$. In a
directed graph, the effect of unit $i$ on unit $j$ differs from $j$ to $i$; in
a weighted graph, the intensity of the effect of each unit is different. This
allows us to mobilise graphical tools from network analysis that plot unit
$j$’s influence on $i$’s influence by means of two-way scatter plots, or heat
plots.

##  4 The Method

In this section, we present the method to systematically detect anomalous
units with panel data. Motivated by the fact that visual inspection of the
data becomes complicated with more than two covariates (Rousseeuw,, 1991;
Bramati and Croux,, 2007), we propose a method based on statistical measures
for panel data that facilitates the detection and classification of anomalous
units with many covariates. The method consists of two steps: first, the
detection of anomalous units by type; and, second, the influence analysis to
determine their effect on the LS coefficients, and on other unit’s the
influence on the LS parameters.

This method is mainly designed for short panels because the bias arising from
the presence of anomalous units in the sample is higher as the benefits of the
asymptotic theory cannot be advocated. It is also thought to identify
anomalies within clusters. The method can be employed _before_ the regression
analysis to identify possibly problematic units, or _after_ to understand what
units drive most of the average effect in the LS estimates.

It goes beyond the scope of this paper to examine how anomalous units should
be treated once they are detected with the proposed method. The deletion of
any anomalous unit from the sample is not always the best option because the
econometric literature offers techniques to handle these anomalies properly. A
strand of the literature focuses on developing robust estimators to
consistently estimate the parameter of interest by handling VO and BL points
(Bramati and Croux,, 2007; Verardi and Croux,, 2009; Aquaro and Čížek,, 2013,
2014; Jiao,, 2022). Another focuses on the downward bias of the statistical
inference in the presence of GL points, recommending the use of jackknife-type
standard errors (MacKinnon and White,, 1985; Chesher and Jewitt,, 1987;
Davidson et al.,, 1993; MacKinnon,, 2013; Belotti and Peracchi,, 2020;
Polselli,, 2022; MacKinnon et al., 2023b, ).

###  4.1 Detection of Anomalous Units

The first step consists in detecting the presence of anomalous units, and
identifying their type from their location in the leverage-residual space. A
graph that plots the individual leverage over the normalised residual squared
(i.e., leverage-versus-residual plot) informs on the presence and the type of
anomalous unit in the sample based on their location in the plane.

Figure 2 plots the individual leverage over the normalised squared residual.
The solid red lines mark the average values for individual leverage and
normalised residuals squared. Cut-off values for high leverage and high
residuals can be arbitrarily set as twice the average values. With this
respect, plausible GL units possess high leverage but low squared residual and
are located in the top-left quadrant; possible VO have high residual squared
but low leverage and are located in the bottom-right quadrant; possible BL
units display both high leverage and residual squared are in the top-right
quadrant. Non-influential units are grouped in the cloud of points in the
bottom-left quadrant or around the intersection of the two lines as they
display low leverage and residual.

We illustrate the method to detect anomalous units in a leverage-residual plot
with synthetic data that has been previously contaminated with anomalous
units: unit 10 is a BL, unit 20 a GL, and unit 30 a VO. From the graph, the
type of three units is correctly detected based on their leverage and
residuals, and easily identifiable from their location on the plane. The rest
of the units are not influential, as per data generating process, being
confined to the bottom-left quadrant.

###  4.2 Influence Analysis

Once anomalous units are detected and classified with the leverage-vs-
residuals plot, the second part of the influence analysis shows how anomalies
affect the LS estimates, and other units’ influence on the estimated
coefficients in terms of joint and conditional influence. Because the output
resembles a weighted and directed adjacency matrix, we can mobilise graphical
tools from network analysis (e.g., heat plots and scatter plots) that plot the
measure of pair-wise influence of unit $j$ on unit $i$. In this example, we
visualise the links and strength between pairs of units using heat plots.

Figure 3 shows four plots with values of the joint influence, joint effects,
conditional influence, and conditional effects (respectively from top to
bottom, left to right). The colour scale from dark blue to red shows the
degree of influence/effect from the smallest to the largest value. The
influence analysis consists of four steps as follows.

1\. The joint influence plot displays the joint influence of pairs of units
when $i\neq j$, and the individual influence when $i\neq j$. The diagonal
elements represent Cook’s distance for panel data. Units 10 and 20 – generated
as BL and GL units, respectively – have high individual influence, largely
exceeding the distributional cut-off 0.694 and the conventional Cook’s
Distance cutoff of 1. These units reciprocally have high joint influence in
correspondence of another anomaly of the same type (e.g., with unit 30 as
being a VO) and also with the rest of or almost all the units.

2\. The joint effect plot shows joint effects $\mathrm{K}_{j|i}$ of pairs of
units. The values of this measure are extremely high for pairs of units
${i,j}$ equal to $\\{17,10\\}$, $\\{17,20\\}$. The $j$-th units that exert
most of the influence in the pair are the BL and GL units that was already
detected with the leverage-vs-residual plot and with the joint influence plot
whereas the $i$-th unit was not. Detected units $j=\\{10,20\\}$ _alter_ the
individual effect of units $i=17$ by either _reducing_ or _enhancing_ its
effect in the LS estimation. The direction of the effect (upwards or
downwards) can be inferred from the outcome of the conditional effect in step
4.

3\. The conditional influence plot highlights the influence of unit $i$
_conditional on_ removing $j$ from the sample. The conditional influence is
quite high for already individually influential units (i.e., 10, and 20), by
construction; these are the BL and GL units. This measures are informative for
the construction of the conditional effects.

4\. The conditional effect plot plots the measure $\mathrm{M}_{i(j)}$ for the
pair $\\{i,j\\}$. Units $j=\\{10,20\\}$ are _masking_ the influence of unit
$i=17$ as $\mathrm{M}_{i(j)}\gg 1$; no unit is _boosting_ the effect of any
other. This information is useful to conclude that the $j$-th units detected
in step 2 are _reducing_ the effect of unit 17.

##  5 Empirical example

In this section, we illustrate an application of the method for the detection
of anomalous units in panel data sets. We use the data and the regression
equation corresponding to Column (2a) of Table 4 in Berka et al.,
(2018).111111With this exercise, we do not intent to support or invalidate the
original results. The sample consists of 9 countries observed over the time
period 1995–2007. The illustrative example is relevant to our case because of
the small cross-sectional sample size, and the ease of interpretability of the
regression output.

Berka et al., (2018) study the relationship between real exchange rate and
sectoral productivity in nine Eurozone countries finding a strong correlation
between productivity and real exchange rates among high-income countries with
floating nominal exchange rates. The estimating regression equation is as
follows

$RER_{it}=\beta
TFP_{it}+\mathbf{x}_{it}^{\prime}\bm{\gamma}+\alpha_{i}+u_{it},$ (12)

where $RER_{it}$ is the log real exchange rate (expenditure-weighted)
expressed as EU15 average relative to country $i$ (an increase is a
depreciation) in period $t$; $TFP_{it}$ log of TFP level of traded relative to
nontraded sector in EU12 relative to country i at time t; $\mathbf{x}_{it}$
are other covariates; $\alpha_{i}$ is the country fixed effects; and $u_{it}$
is the error term. The coefficients in Equation (12) are estimated using OLS
after the _within-group_ transformation.

We are interested in identifying anomalous countries that may drive most of
the effect on the LS estimate of $TFP$ based on Equation (12). The leverage-
vs-residual plot (Figure 4) highlights a group of six countries with
individual leverage above the mean value – where Italy may be a potential
leverage units – and only one country (Ireland) with the potential to be a BL
or VO.

Figure 5 shows the joint and conditional influence of pairs of countries. The
labels refer to 1-Austria, 2-Belgium, 3-Finland, 4-France, 5-Germany,
6-Ireland, 7-Italy, 8-Netherlands, 9-Spain. On the main diagonal, individually
influential countries correspond to 6 (Ireland) and 7 (Italy). These exert
high joint influence (red and pink squares) with other non-influential
countries – i.e., Ireland on Netherlands (unit 8), and Italy on Austria (unit
1) and Belgium (unit 2). Looking at the joint effects, units 1, 3, 6, 7, and 8
_alter_ the effect of unit 9 (Spain) on the LS estimates. The direction of the
effect (upwards or downwards) can be inferred from the output of the
conditional effect. Countries 6, 7, and 8 seems to be _masking_ the effect of
country 9 whereas countries 1 and 3 are _boosting_ its contribution to the
$TFP$ coefficient. Therefore, we can conclude by looking again at the joint
effects plot that the former group of countries is _reducing_ the effect of
Spain while the latter is _enhancing_ its influence.

##  6 Conclusion

The presence of outliers and leveraged data points has the potential to bias
regression coefficients and standard errors, which necessitates appropriate
methods to detect and identify them. Available measures – such as, leverage-
vs-residual plots and Cook-like measures – may fail to correctly detect and
classify them in panel data set. The method we propose overcomes this
limitation by taking into account the panel structure of the data. We
formalised the average leverage and average normalised residuals in a panel
data setting to produce unit-wise leverage-residual plots. Then, we developed
two diagnostic measures for panel data models – based on Lawrance,’s (1995)
cross-sectional measures – and showed their statistical distributions.

Overall, the method detected those units that exert more influence in the data
set by taking into account the full history of a unit. The analysis of the
joint and conditional effects showed how their presence alters the influence
of other units in the sample and, hence, how their presence is driving the
average effect on the LS estimates. The strength of this method is that a unit
that is not individually influential according to Cook’s distance will always
be detected if turns up to be influential _jointly with_ or _conditional on_
another highly influential unit. This will help the researcher understand how
anomalous units drive their LS estimates. As a novelty, the individual and
bilateral (joint and conditional) influence exerted by units in the sample are
displayed with a network representation (e.g., heat plots and scatter plots).

The method can be used before the actual regression analysis to investigate
the presence of anomalies in the sample, or after the regression analysis to
understand what units are driving the estimated coefficients or standard
errors.

Once anomalous units are properly detected and identified, the researcher
should deal with their presence following the recommendations in the
econometric literature. For example, the literature recommends the use of
robust estimators of the median (e.g., M-estimators, S-estimators, etc.) in
the presence of outliers (Bramati and Croux,, 2007; Verardi and Croux,, 2009;
Aquaro and Čížek,, 2013, 2014; Jiao,, 2022), whereas heteroskedasticity-
consistent standard errors based on jackknife methods in the presence of
leveraged data (Hinkley,, 1977; MacKinnon and White,, 1985; Chesher and
Jewitt,, 1987; Davidson et al.,, 1993; MacKinnon,, 2013; Belotti and
Peracchi,, 2020; Polselli,, 2022; MacKinnon et al., 2023a, ; MacKinnon et al.,
2023b, ).

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##  7 Figures

Figure 1: Example of anomalous units

_Note: The graphs show the relationship between the outcome variable and the
only regressor. The original data is contaminated with three anomalous units:
10 is a bad leverage unit, 20 is a good leverage unit, and 30 is a vertical
outlier. From left to right: scatter plot of original data, scatter plot data
after the within-group transformation. The red dotted line is fitted using
uncontaminated units only; the dashed line using uncontaminated and good
leverage points; the dash-dot line using uncontaminated and bad leverage
points; the solid line uncontaminated units and vertical outliers._

Figure 2: Comparison of leverage-versus-residual scatter plots

_Note: The graph plots the leverage against normalised residual squared,
evaluating the influence of each unit when its full time series is considered.
The horizontal and vertical red lines mark the average values for high
leverage and normalized residual squared, respectively. A unit can be
considered anomalous if it exceeds twice the value of the individual leverage
and/or the individual normalised residuals squared. The data is contaminated
with three anomalous units: 10 is a bad leverage unit, 20 is a good leverage
unit, and 30 is a vertical outlier._

Figure 3: Influence analysis, heat plot

_Note: The graphs plot the influence measures/effects for units $(i,j)$. The
graph on the top-left displays the joint influence (or individual influence
when $i=j$); the top-right graphs displays the joint effect; the bottom-left
graph the conditional influence; the bottom-right graph the conditional
effect. The data is contaminated with three anomalous units: 10 is a bad
leverage unit, 20 is a good leverage unit, and 30 is a vertical outlier._

Figure 4: Leverage-vs-residual plots, Berka et al., (2018)

_Note: Data and regression equation corresponds to Column (2a) of Table 4 in
Berka et al., (2018). From the graph, Italy can be a potential leverage point
while Ireland a potential outlier._

Figure 5: Influence Analysis, Berka et al., (2018)

_Note: Data and regression equation corresponds to Column (2a) of Table 4 in
Berka et al., (2018). The labels refer to 1-Austria, 2-Belgium, 3-Finland,
4-France, 5-Germany, 6-Ireland, 7-Italy, 8-The Netherlands, 9-Spain. From the
graph, the influence of Spain is masked and reduced by Ireland, Italy, and The
Netherlands while is boosted and enhanced by Austria and Finland._

##  Appendix A Data Generating Process

We use a synthetic data set to explain each step. The data generating process
mainly follows Bramati and Croux,’s (2007) simulation design with some
differences. We generate a regression model as follows

$\displaystyle
y_{it}=\beta_{0}+\beta_{1}x_{it}+\alpha_{i}+\epsilon_{it},\text{ for all
}i=1,\dots,N\text{ and }t=1,\dots,T$ (13) $\displaystyle
x_{it}\mathbin{\overset{}{\kern
0.0pt\sim}}\mathcal{N}(0,1);\,\,\alpha_{i}\mathbin{\overset{}{\kern
0.0pt\sim}}\mathcal{U}(0,20);\,\,\epsilon_{it}\mathbin{\overset{}{\kern
0.0pt\sim}}\mathcal{N}\bigl{(}0,1\bigr{)},$ (14)

where $\beta_{0}=1$ and $\beta_{1}=0.5$. We set $N=100$ and $T=20$. Equation
(13) is estimated using the _within-group_ approach.

Because we are interested in the simultaneous influence exerted by multiple
atypical observations in the sample, we generate a synthetic data set with two
anomalies per type – i.e., VO, GL and BL units. VO (units 10 and 40) are
obtained by adding values drawn from $\mathcal{N}(50,1)$ to the original
dependent variable. GL units (units 20 and 50) are generated by replacing the
original value of the regressor with values drawn from $\mathcal{N}(15,1)$. BL
units (units 30 and 60) are created by adding $\mathcal{N}(50,1)$ to the
original dependent variable, and replacing the original values of the
independent variable with $\mathcal{N}(10,1)$. We generate _block-
concentrated_ anomalies by contaminating $t\leq 10$ in the time series of
units 10, 20, and 30, and $t\leq 5$ in the time series of units 40, 50, and
60.

##  Appendix B Derivation of L2O Estimator

Suppose that the panel is balanced with $T_{i}=T_{j}=T$.121212This assumption
simplifies the notation. However, the L2O estimator can also be estimated with
unbalanced panel data, provided thatthe formula is modified to account for
$T_{i}\neq T_{j}$. Let the FE estimator $\bm{\beta}$ without units $i$ and $j$
be

$\displaystyle\widehat{\bm{\beta}}_{(i,j)}$
$\displaystyle=\biggl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}-\widetilde{\mathbf{X}}_{i}^{\prime}\widetilde{\mathbf{X}}_{i}-\widetilde{\mathbf{X}}_{j}^{\prime}\widetilde{\mathbf{X}}_{j}\biggr{)}^{-1}\biggl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{Y}}-\widetilde{\mathbf{X}}_{i}^{\prime}\widetilde{\mathbf{y}}_{i}-\widetilde{\mathbf{X}}_{j}^{\prime}\widetilde{\mathbf{y}}_{j}\biggr{)}$
$\displaystyle=\biggl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}-\widetilde{\mathbf{X}}_{ij}^{\prime}\widetilde{\mathbf{X}}_{ij}\biggr{)}^{-1}\biggl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{Y}}-\widetilde{\mathbf{X}}_{ij}^{\prime}\widetilde{\mathbf{y}}_{ij}\biggr{)},$
(15)

where $\widetilde{\mathbf{X}}$ is a ${NT\times k}$ matrix,
$\widetilde{\mathbf{X}}_{l}$ for $l\in\\{i,j\\}$ is a $T\times k$ matrix, and
$\widetilde{\mathbf{X}}_{ij}=\bigl{[}\widetilde{\mathbf{X}}_{i}\hskip
8.53581pt\widetilde{\mathbf{X}}_{j}\bigr{]}^{\prime}$ a ${2T\times k}$ matrix;
$\widetilde{\mathbf{Y}}$ is a ${NT\times 1}$ vector,
$\widetilde{\mathbf{y}}_{l}$ for $l\in\\{i,j\\}$ is a ${T\times 1}$ vector,
$\widetilde{\mathbf{y}}_{ij}=\bigl{[}\widetilde{\mathbf{y}}_{i}\hskip
8.53581pt\widetilde{\mathbf{y}}_{j}\bigr{]}^{\prime}$ is a ${2T\times 1}$
vector.

Apply the Woodbury’s formula131313The standard Woodbury’s formula calculates
the inverse of the sum of matrices as follows
$(\mathrm{A}+\mathrm{B}\mathrm{D}\mathrm{C})^{-1}=\mathrm{A}^{-1}-\mathrm{A}^{-1}\mathrm{B}\,(\mathrm{D}^{-1}+\mathrm{C}\mathrm{A}^{-1}\mathrm{B})^{-1}\mathrm{C}\mathrm{A}^{-1}$.
In this specific case, we have
$\mathrm{A}=\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}$,
$\mathrm{B}=-\widetilde{\mathbf{X}}_{ij}^{\prime}$,
$\mathrm{C}=\widetilde{\mathbf{X}}_{ij}$, and
$\mathrm{D}=\mathbf{I}_{i}\oplus\mathbf{I}_{j}\equiv\mathbf{I}_{ij}$ which is
a $2T\times 2T$ block diagonal matrix with identity matrices $\mathbf{I}_{i}$
and $\mathbf{I}_{j}$ on the main diagonal, and off-diagonal block matrices of
zeros of dimension $T\times T$. to Equation (B) such that the L2O can be
expressed as

$\displaystyle\widehat{\bm{\beta}}_{(i,j)}$
$\displaystyle=\Bigr{\\{}\bigl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\bigr{)}^{-1}+\bigl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\bigr{)}^{-1}\widetilde{\mathbf{X}}_{ij}^{\prime}\bigl{(}\mathbf{I}_{ij}-\mathbf{\overline{H}}_{ij}\bigr{)}^{-1}\widetilde{\mathbf{X}}_{ij}\bigl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\bigr{)}^{-1}\Bigr{\\}}\biggl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{Y}}-\widetilde{\mathbf{X}}_{ij}^{\prime}\widetilde{\mathbf{y}}_{ij}\biggr{)}$
$\displaystyle=\widehat{\bm{\beta}}-\bigl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\bigr{)}^{-1}\widetilde{\mathbf{X}}_{ij}^{\prime}\mathbf{\overline{M}}_{ij}^{-1}\widehat{\mathbf{u}}_{ij},$
(16)

where
$\mathbf{\overline{H}}_{ij}=\widetilde{\mathbf{X}}_{ij}\bigl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\bigr{)}^{-1}\widetilde{\mathbf{X}}_{ij}^{\prime}=\begin{bmatrix}\mathbf{H}_{i}&\mathbf{H}_{ij}\\\
\mathbf{H}_{ij}^{\prime}&\mathbf{H}_{j}\end{bmatrix}=\begin{bmatrix}\widetilde{\mathbf{X}}_{i}(\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}})^{-1}\widetilde{\mathbf{X}}_{i}^{\prime}&\widetilde{\mathbf{X}}_{i}(\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}})^{-1}\widetilde{\mathbf{X}}_{j}^{\prime}\\\
\widetilde{\mathbf{X}}_{j}(\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}})^{-1}\widetilde{\mathbf{X}}_{i}^{\prime}&\widetilde{\mathbf{X}}_{j}(\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}})^{-1}\widetilde{\mathbf{X}}_{j}^{\prime}\end{bmatrix}$,
noting that $\mathbf{H}_{ji}=\mathbf{H}_{ij}^{\prime}$;
$\mathbf{\overline{M}}_{ij}=(\mathbf{I}_{ij}-\mathbf{\overline{H}}_{ij})=\begin{bmatrix}\mathbf{M}_{i}&-\mathbf{H}_{ij}\\\
-\mathbf{H}_{ij}^{\prime}&\mathbf{M}_{j}\end{bmatrix}=\begin{bmatrix}\mathbf{I}_{i}-\mathbf{H}_{i}&-\mathbf{H}_{ij}\\\
-\mathbf{H}_{ij}^{\prime}&\mathbf{I}_{j}-\mathbf{H}_{j}\end{bmatrix}$; and
$\widehat{\mathbf{u}}_{ij}\big{[}\widehat{\mathbf{u}}_{i}\,\,\widehat{\mathbf{u}}_{j}\big{]}$
is a $2t\times 1$ vector.

Expanding last block of Equation (16) with the formula for the inverse of
partitioned matrices141414The formula is  $\begin{bmatrix}A&B\\\
C&D\end{bmatrix}^{-1}=\begin{bmatrix}A^{-1}+A^{-1}B(D-CA^{-1}B)^{-1}CA^{-1}&-A^{-1}B(D-CA^{-1}B)^{-1}\\\
-(D-CA^{-1}B)^{-1}CA^{-1}&(D-CA^{-1}B)^{-1}\end{bmatrix}$ provided that
$A^{-1}$ exists and $(D-CA^{-1}B)$ is invertible., provided that
$\mathbf{M}_{i}$ and
$\bigl{(}\mathbf{M}_{j}-\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}\bigr{)}$
are non-singular to be invertible Equation (16) becomes

$\displaystyle\widehat{\bm{\beta}}_{(i,j)}$
$\displaystyle=\widehat{\bm{\beta}}-\bigl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\bigr{)}^{-1}\Bigl{(}\widetilde{\mathbf{X}}_{i}^{\prime}\mathbf{M}_{i}^{-1}\widehat{\mathbf{u}}_{i}+\bigl{(}\widetilde{\mathbf{X}}_{i}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}+\widetilde{\mathbf{X}}_{j}^{\prime}\bigr{)}\bigl{(}\mathbf{M}_{j}-\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}\bigr{)}^{-1}\bigl{(}\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\widehat{\mathbf{u}}_{i}+\widehat{\mathbf{u}}_{j}\bigr{)}\Bigr{)}$
$\displaystyle=\widehat{\bm{\beta}}_{(i)}-\bigl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\bigr{)}^{-1}\bigl{(}\widetilde{\mathbf{X}}_{i}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}+\widetilde{\mathbf{X}}_{j}^{\prime}\bigr{)}\bigl{(}\mathbf{M}_{j}-\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}\bigr{)}^{-1}\bigl{(}\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\widehat{\mathbf{u}}_{i}+\widehat{\mathbf{u}}_{j}\bigr{)}$
(17)

where
$\widehat{\bm{\beta}}_{(i)}=\widehat{\bm{\beta}}-\bigl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\bigr{)}^{-1}\widetilde{\mathbf{X}}_{i}^{\prime}\mathbf{M}_{i}^{-1}\widehat{\mathbf{u}}_{i}$
is the L1O estimator.

##  Appendix C Distribution of Influence Measures

###  C.1 Model assumptions for the convergence of the estimators

Let $\\{\mathbf{y}_{i},\mathbf{X}_{i}\\}_{t=1}^{T}$ be a sequence of
independent and identically distributed (_iid_) sequence of random variables,
and $\\{\mathbf{u}_{i}\\}_{t=1}^{T}$ be an independent but not identically
distributed (_inid_) sequence of error terms for all $i=1,\dots,N$. Under this
notation $T$ is the full set of time information, and the total number of
observations in the sample is given by $n=N\cdot T$ with balanced data sets.
The length of the time period is fixed and smaller than the number of units
$T\ll N$ to guarantee the conventional $N\to\infty$ asymptotics.

Define
$\widetilde{\mathbf{w}}_{i}=(\mathbf{I}_{T}-\,T^{-1}\bm{\iota}\bm{\iota}^{\prime})\mathbf{w}_{i}$
the variable after the within-group (mean-centered) transformation. Then, we
rely on the assumptions below for the convergence of the estimators

  1. asm.1

_Strict exogeneity_ :
$\mathbb{E}\bigl{(}\widetilde{\mathbf{u}}_{i}|\widetilde{\mathbf{X}}_{i}\bigr{)}=\mathbb{E}(\widetilde{u}_{it}|\widetilde{\mathbf{x}}_{it})=0$,
for all $i=1,\dots,N$ and $t=1,\dots,T$;

  2. asm.2

_Rank condition_ : $\mathbf{S}_{N}\equiv
N^{-1}\sum_{i=1}^{N}\widetilde{\mathbf{X}}_{i}^{\prime}\widetilde{\mathbf{X}}_{i}$
is a finite symmetric matrix with full column rank $k$.

  3. asm.3

_Moment conditions_ :

    1. i.

$\mathbb{E}\big{\|}\widetilde{\mathbf{X}}_{i}^{\prime}\widetilde{\mathbf{X}}_{i}\big{\|}<\infty$
for
$\widetilde{\mathbf{X}}_{i}^{\prime}\widetilde{\mathbf{X}}_{i}\in\mathbb{R}^{k\times
k}$;

    2. ii.

$\mathrm{sup}_{i}\,\mathbb{E}\big{\|}\widetilde{\mathbf{X}}_{i}^{\prime}\widetilde{\mathbf{u}}_{i}\big{\|}^{2+\delta}<\infty$
for some $\delta>0$, $\forall i$ and
$\widetilde{\mathbf{X}}_{i}^{\prime}\widetilde{\mathbf{u}}_{i}\in\mathbb{R}^{k}$,

where $\|\cdot\|$ denotes the Euclidean norm.

  4. asm.4

_Heteroskedasticity_ :
$\overline{\bm{\Sigma}}_{N}=N^{-1}\sum_{i=1}^{N}\bm{\Sigma}_{i}\to\bm{\Sigma}$,
where the matrix of the heteroskedastic disturbances
$\bm{\Sigma}_{i}=\mathbb{E}\bigl{(}\widetilde{\mathbf{u}}_{i}\widetilde{\mathbf{u}}_{i}^{\prime}|\widetilde{\mathbf{X}}_{i}\bigr{)}=\mathrm{diag}\bigl{\\{}\sigma_{it}^{2}\bigr{\\}}$
is symmetric of dimension $T$, finite, positive definite, and diagonal.

  5. asm.5

_Convergence of variance-covariance matrix_ :
$\overline{\mathbf{V}}_{N}=N^{-1}\sum_{i=1}^{N}\,\mathbf{V}_{i}\to\mathbf{V}$,
where
$\mathbf{V}_{i}=\mathbb{E}\bigl{(}\widetilde{\mathbf{X}}_{i}^{\prime}\bm{\Sigma}_{i}\widetilde{\mathbf{X}}_{i}\bigr{)}$
and $\mathbf{V}$ is a finite positive definite $k\times k$ matrix.

###  C.2 Joint Influence

We first establish the asymptotic properties of $\widehat{\bm{\beta}}_{(i)}$
and show its equivalence to $\widehat{\bm{\beta}}$ in the sense that

$\sqrt{N}\bigl{(}\widehat{\bm{\beta}}_{(i)}-\bm{\beta}\bigr{)}=\sqrt{N}\bigl{(}\widehat{\bm{\beta}}-\bm{\beta}\bigr{)}+o_{p}(1)$

implying that $\widehat{\bm{\beta}}_{(i)}$ and $\widehat{\bm{\beta}}$ share
the same asymptotic distribution. Note that $\widehat{\bm{\beta}}$ is a
consistent estimator of $\bm{\beta}$ under the least square assumptions in
this paper. We first show the consistency of $\widehat{\bm{\beta}}_{(i)}$.
Using the L1O formula, we know that

$\displaystyle\widehat{\bm{\beta}}_{(i)}-\bm{\beta}$
$\displaystyle=(\widehat{\bm{\beta}}-\bm{\beta})-\bigl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\bigr{)}^{-1}\widetilde{\mathbf{X}}_{i}^{\prime}(\mathbf{I}_{T}-\mathbf{H}_{i})^{-1}\bigl{(}\widetilde{\mathbf{u}}_{i}-\widetilde{\mathbf{X}}_{i}(\widehat{\bm{\beta}}-\bm{\beta})\bigr{)}$
(18)

Using the Triangle Inequality,

$\small\big{\|}(\widehat{\bm{\beta}}_{(i)}-\bm{\beta})\big{\|}\leq\big{\|}(\widehat{\bm{\beta}}-\bm{\beta})\big{\|}+\bigg{\|}\biggl{(}\frac{1}{N}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\biggr{)}^{-1}\bigg{\|}\,\frac{1}{N}\big{\|}\widetilde{\mathbf{X}}_{i}\big{\|}\big{\|}(\mathbf{I}_{T}-\mathbf{H}_{i})^{-1}\big{\|}\bigl{(}\big{\|}\widetilde{\mathbf{u}}_{i}\big{\|}+\big{\|}\widetilde{\mathbf{X}}_{i}\big{\|}\big{\|}\widehat{\bm{\beta}}-\bm{\beta}\big{\|}\bigr{)}$
(19)

is $o_{p}(1)$ because $\big{\|}\widehat{\bm{\beta}}-\bm{\beta}\big{\|}$ is
$o_{p}(1)$;
$\bigl{(}N^{-1}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\bigr{)}^{-1}=\mathbf{S}_{XX}^{-1}+o_{p}(1)$
by asm.2, _WLLN_ and _Slutsky’s theorem_ ; $\widetilde{\mathbf{X}}_{i}$ and
$\widetilde{\mathbf{u}}_{i}$ are $O_{p}(1)$ because random variables with
finite moments by asm.3 and, therefore,
$N^{-1}\big{\|}\widetilde{\mathbf{X}}_{i}\big{\|}=O_{p}(N^{-1})$; and

$\big{\|}(\mathbf{I}_{T}-\mathbf{H}_{i})^{-1}\big{\|}\leq\big{\|}\mathbf{I}_{T}\big{\|}+\big{\|}\widetilde{\mathbf{X}}_{i}\big{\|}^{2}\big{\|}\bigl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}-\widetilde{\mathbf{X}}_{i}^{\prime}\widetilde{\mathbf{X}}_{i}\bigr{)}^{-1}\big{\|}\\\
$ (20)

is O(1) because the first term on the right-hand-side, $\sqrt{T}$, is O(1)
without a remainder term, and the second component is bounded above by
$o_{p}(1)$ random variable.

As a result, $\widehat{\bm{\beta}}_{(i)}$ is a consistent estimator of the
true value of the parameter $\bm{\beta}$ as
$\widehat{\bm{\beta}}_{(i)}=\bm{\beta}+o_{p}(1)$ from (19). Therefore,
removing one unit does not have an impact on the estimates of the true value
of the parameter as the cross-sectional units increase to infinity.

We now show that the estimators $\widehat{\bm{\beta}}_{(i)}$ and
$\widehat{\bm{\beta}}$ have the same asymptotic distribution by applying the
Reverse Triangle Inequality to
$\sqrt{N}\bigl{(}\widehat{\bm{\beta}}-\bm{\beta}\bigr{)}-\sqrt{N}\bigl{(}\widehat{\bm{\beta}}_{(i)}-\bm{\beta}\bigr{)}$
such that

$\big{\|}\sqrt{N}\bigl{(}\widehat{\bm{\beta}}-\bm{\beta}\bigr{)}-\sqrt{N}\bigl{(}\widehat{\bm{\beta}}_{(i)}-\bm{\beta}\bigr{)}\big{\|}\leq\biggl{\|}\biggl{(}\frac{1}{N}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\biggr{)}^{-1}\biggr{\|}\frac{1}{\sqrt{N}}\,\Big{\|}\widetilde{\mathbf{X}}_{i}\Big{\|}\big{\|}(\mathbf{I}_{T}-\mathbf{H}_{i})^{-1}\big{\|}\bigl{(}\big{\|}\widetilde{\mathbf{u}}_{i}\big{\|}+\big{\|}\widetilde{\mathbf{X}}_{i}\big{\|}\big{\|}\widehat{\bm{\beta}}-\bm{\beta}\big{\|}\bigr{)}\\\
$ (21)

is bounded above by a $o_{p}(1)$ random variable because the first component
of (21) is $\big{(}\mathbf{S}_{XX}^{-1}+o_{p}(1)\big{)}$ as
$N^{-1}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\overset{p}{\to}\mathbb{E}(\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}})=\mathbf{S}_{XX}>0$
by the Central Limit Theorem; the second term is
$,O_{p}(N^{1/r-1/2})=O_{p}(1)$ for $r\geq 2$ under asm.3.asm.3i.; the third
component is $O(1)$ by (20); and the last quantity in parenthesis is
$O_{p}(1)$.

We can conclude that $\widehat{\bm{\beta}}_{(i)}$ and $\widehat{\bm{\beta}}$
share the same asymptotic distribution as

$\sqrt{N}\bigl{(}\widehat{\bm{\beta}}_{(i)}-\bm{\beta}\bigr{)}=\sqrt{N}\bigl{(}\widehat{\bm{\beta}}-\bm{\beta}\bigr{)}+o_{p}(1)$
(22)

where
$\sqrt{N}\bigl{(}\widehat{\bm{\beta}}-\bm{\beta}\bigr{)}\overset{d}{\rightarrow}\mathcal{N}\bigl{(}\bm{0},\mathbf{S}_{XX}^{-1}\bm{\Sigma}\mathbf{S}_{XX}^{-1}\bigr{)}$.
This result holds for all $i=1,\dots,N$.

Now, we find the distribution of
$\widehat{\bm{\beta}}-\widehat{\bm{\beta}}_{(i)}$ using Shwarz Inequality and
Triangle Inequality

$\displaystyle\big{\|}\widehat{\bm{\beta}}-\widehat{\bm{\beta}}_{(i)}\big{\|}$
$\displaystyle=\big{\|}\bigl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\bigr{)}^{-1}\widetilde{\mathbf{X}}_{i}^{\prime}(\mathbf{I}_{T}-\mathbf{H}_{i})^{-1}\bigl{(}\widetilde{\mathbf{u}}_{i}-\widetilde{\mathbf{X}}_{i}(\widehat{\bm{\beta}}-\bm{\beta})\bigr{)}\big{\|}$
$\displaystyle\leq\big{\|}N\bigl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\bigr{)}^{-1}\big{\|}N^{-1}\big{\|}\widetilde{\mathbf{X}}_{i}\big{\|}\big{\|}(\mathbf{I}_{T}-\mathbf{H}_{i})^{-1}\big{\|}\bigl{(}\big{\|}\widetilde{\mathbf{u}}_{i}\big{\|}+\big{\|}\widetilde{\mathbf{X}}_{i}\big{\|}\big{\|}\widehat{\bm{\beta}}-\bm{\beta}\big{\|}\bigr{)}$
(23)

is $o_{p}(1)$ because
$\big{\|}\widehat{\bm{\beta}}-\bm{\beta}\big{\|}\overset{p}{\to}0$;
$\bigl{(}N^{-1}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\bigr{)}^{-1}=\mathbf{S}_{XX}^{-1}+o_{p}(1)$
by asm.2, _WLLN_ and _Slutsky’s theorem_ ; $\widetilde{\mathbf{X}}_{i}$ and
$\widetilde{\mathbf{u}}_{i}$ are $O_{p}(1)$ because random variables with
finite moments by asm.3 and, therefore,
$N^{-1}\big{\|}\widetilde{\mathbf{X}}_{i}\big{\|}=O_{p}(N^{-1})$ for $r\geq
2$; and $\big{\|}(\mathbf{I}_{T}-\mathbf{H}_{i})^{-1}\big{\|}$ is bounded
above by $o_{p}(1)$ random variable as in (20).

We follow the same reasoning as above to derive the distribution of
$\widehat{\bm{\beta}}_{(i,j)}$. We re-write (17) by adding and subtracting
$\bm{\beta}$ as follows

$\widehat{\bm{\beta}}_{(i,j)}-\bm{\beta}=\bigl{(}\widehat{\bm{\beta}}_{(i)}-\bm{\beta}\bigr{)}-\bigl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\bigr{)}^{-1}\bigl{(}\widetilde{\mathbf{X}}_{i}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}+\widetilde{\mathbf{X}}_{j}^{\prime}\bigr{)}\bigl{(}\mathbf{M}_{j}-\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}\bigr{)}^{-1}\bigl{(}\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\widehat{\mathbf{u}}_{i}+\widehat{\mathbf{u}}_{j}\bigr{)}$
(24)

Using Triangle Inequality, we have that

$\displaystyle\small\big{\|}\widehat{\bm{\beta}}_{(i,j)}-\bm{\beta}\big{\|}$
$\displaystyle\leq\big{\|}\widehat{\bm{\beta}}_{(i)}-\bm{\beta}\big{\|}$ (25)
$\displaystyle+\bigg{\|}\biggl{(}\frac{1}{N}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\biggr{)}^{-1}\bigg{\|}$
(26) $\displaystyle\hskip
14.22636pt\biggl{(}\frac{1}{N}\big{\|}\widetilde{\mathbf{X}}_{i}\big{\|}\big{\|}\mathbf{M}_{i}^{-1}\big{\|}\big{\|}\mathbf{H}_{ij}\big{\|}+\frac{1}{N}\big{\|}\widetilde{\mathbf{X}}_{j}\big{\|}\biggr{)}$
(27) $\displaystyle\hskip
14.22636pt\bigg{\|}\bigl{(}\mathbf{M}_{j}-\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}\bigr{)}^{-1}\bigg{\|}$
(28) $\displaystyle\hskip
14.22636pt\bigl{(}\big{\|}\mathbf{H}_{ij}\big{\|}\big{\|}\mathbf{M}_{i}^{-1}\big{\|}\big{\|}\widehat{\mathbf{u}}_{i}\big{\|}+\big{\|}\widehat{\mathbf{u}}_{j}\big{\|}\bigr{)}$
(29)

is $o_{p}91)$ because the term on the right-hand-side of (25) is $o_{p}(1)$ by
result (19), (26) is $\mathbf{S}_{XX}^{-1}+o_{p}(1)$ as $N\to\infty$ and $T$
fixed, and the rest is $O_{p}(1)$. Specifically, (27) is $O_{p}(1)$ because of
$N^{-1}\big{\|}\widetilde{\mathbf{X}}_{l}\big{\|}=O_{p}(N^{-1})$ for
$l\in\\{i,j\\}$, $\big{\|}\mathbf{M}_{i}^{-1}\big{\|}=O(1)$ by (20), and
$\big{\|}\mathbf{H}_{ij}\big{\|}=N^{-1/2}\big{\|}\widetilde{\mathbf{X}}_{i}\big{\|}\big{\|}(N^{-1}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}})^{-1}\big{\|}N^{-1/2}\big{\|}\widetilde{\mathbf{X}}_{j}\big{\|}=o_{p}(1)$;
component (28) is $O_{p}(1)$ because
$\bigl{(}\mathbf{M}_{j}-\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}\bigr{)}^{-1}=\i_{T}+o_{p}(1)$
since
$\mathbf{M}_{j}=\mathbf{I}_{T}-N^{-1/2}\widetilde{\mathbf{X}}_{j}(N^{-1}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}})^{-1}N^{-1/2}\widetilde{\mathbf{X}}_{j}^{\prime}=\mathbf{I}_{T}+o_{p}(1)$;
and, (29) is $O_{p}(1)$ because
$\big{\|}\widehat{\mathbf{u}}_{l}\big{\|}=\big{\|}\widetilde{\mathbf{u}}_{l}\big{\|}+\big{\|}\widetilde{\mathbf{X}}_{l}\big{\|}\big{\|}(\widehat{\bm{\beta}}-\bm{\beta})\big{\|}=O_{p}(1)$
by asm.3.

We show that the estimators $\widehat{\bm{\beta}}_{(i,j)}$ and
$\widehat{\bm{\beta}}$ have the same asymptotic distribution. Apply the
Reverse Triangle Inequality to
$\sqrt{N}\bigl{(}\widehat{\bm{\beta}}-\bm{\beta}\bigr{)}-\sqrt{N}\bigl{(}\widehat{\bm{\beta}}_{(i,j)}-\bm{\beta}\bigr{)}$
such that we have

$\displaystyle\big{\|}\sqrt{N}\bigl{(}\widehat{\bm{\beta}}-\bm{\beta}\bigr{)}$
$\displaystyle-\sqrt{N}\bigl{(}\widehat{\bm{\beta}}_{(i,j)}-\bm{\beta}\bigr{)}\big{\|}$
$\displaystyle\leq\biggl{\|}\biggl{(}\frac{1}{N}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\biggr{)}^{-1}\biggr{\|}\biggl{\|}\frac{1}{\sqrt{N}}\,\widetilde{\mathbf{X}}_{i}^{\prime}\mathbf{M}_{i}^{-1}\widehat{\mathbf{u}}_{i}\biggr{\|}$
(30)
$\displaystyle+\bigg{\|}\biggl{(}\frac{1}{N}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\biggr{)}^{-1}\bigg{\|}\bigg{\|}\frac{1}{\sqrt{N}}\biggl{(}\widetilde{\mathbf{X}}_{i}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}+\widetilde{\mathbf{X}}_{j}^{\prime}\biggr{)}\bigg{\|}$
(31) $\displaystyle\hskip
14.22636pt\bigg{\|}\bigl{(}\mathbf{M}_{j}-\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}\bigr{)}^{-1}\bigg{\|}$
(32) $\displaystyle\hskip
14.22636pt\bigg{\|}\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\widehat{\mathbf{u}}_{i}+\widehat{\mathbf{u}}_{j}\bigg{\|}$
(33) $\displaystyle=o_{p}(1)$

is $o_{p}(1)$ because (30) is $o_{p}(1)$ by result (21), and the overall
quantity in (31) is $o_{p}(1)$, since the first component is
$\mathbf{S}_{XX}^{-1}+o_{p}(1)$ and multiplies a quantity that is $O_{p}(1)$
as in (26), noting that
$N^{-1/2}\big{\|}\widetilde{\mathbf{X}}_{i}\big{\|}=O_{p}(N^{1/r-1/2})$ and,
hence, $O_{p}(1)$ with $r\geq 2$; (32) is $O_{p}(1)$ as in (19); (33) is
$O_{p}(1)$ as in (29). Therefore, $\widehat{\bm{\beta}}_{(i,j)}$ and
$\widehat{\bm{\beta}}$ share the same asymptotic distribution.

Last step consists in deriving the joint distribution of
$\widehat{\bm{\beta}}$ and $\widehat{\bm{\beta}}_{(i,j)}$. Rearranging
Equation (17), we obtain

$\widehat{\bm{\beta}}-\widehat{\bm{\beta}}_{(i,j)}=\bigl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\bigr{)}^{-1}\widetilde{\mathbf{X}}_{i}^{\prime}\mathbf{M}_{i}^{-1}\widehat{\mathbf{u}}_{i}\\\
+\bigl{(}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\bigr{)}^{-1}\bigl{(}\widetilde{\mathbf{X}}_{i}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}+\widetilde{\mathbf{X}}_{j}^{\prime}\bigr{)}\\\
\bigl{(}\mathbf{M}_{j}-\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}\bigr{)}^{-1}\bigl{(}\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\widehat{\mathbf{u}}_{i}+\widehat{\mathbf{u}}_{j}\bigr{)}$
(34)

Using Shwarz Inequality and Triangle Inequality

$\displaystyle\big{\|}\widehat{\bm{\beta}}-\widehat{\bm{\beta}}_{(i,j)}\big{\|}$
$\displaystyle\leq\biggl{\|}\biggl{(}\frac{1}{N}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\biggr{)}^{-1}\biggr{\|}\biggl{\|}\frac{1}{N}\,\widetilde{\mathbf{X}}_{i}^{\prime}\mathbf{M}_{i}^{-1}\widehat{\mathbf{u}}_{i}\biggr{\|}$
(35)
$\displaystyle+\bigg{\|}\biggl{(}\frac{1}{N}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\biggr{)}^{-1}\bigg{\|}\bigg{\|}\frac{1}{N}\biggl{(}\widetilde{\mathbf{X}}_{i}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}+\widetilde{\mathbf{X}}_{j}^{\prime}\biggr{)}\bigg{\|}$
(36) $\displaystyle\hskip
14.22636pt\bigg{\|}\bigl{(}\mathbf{M}_{j}-\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}\bigr{)}^{-1}\bigg{\|}$
(37) $\displaystyle\hskip
14.22636pt\bigg{\|}\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\widehat{\mathbf{u}}_{i}+\widehat{\mathbf{u}}_{j}\bigg{\|}$
(38)

where the left-hand-side is $o_{p}(1)$ following the reasoning used in
(25)-(29). Thus, the difference
$\widehat{\bm{\beta}}-\widehat{\bm{\beta}}_{(i,j)}\overset{p}{\to}\bm{0}$ as
$N\to\infty$ and $T$ fixed.

###  C.3 Conditional Influence

Rearrange (24)

$\begin{split}\small\sqrt{N}&\bigl{(}\widehat{\bm{\beta}}_{(i,j)}-\bm{\beta}\bigr{)}-\sqrt{N}\bigl{(}\widehat{\bm{\beta}}-\bm{\beta}\bigr{)}=\\\
&-\biggl{(}\frac{1}{N}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\biggr{)}^{-1}\biggl{\\{}\frac{1}{\sqrt{N}}\,\widetilde{\mathbf{X}}_{i}^{\prime}\mathbf{M}_{i}^{-1}\widehat{\mathbf{u}}_{i}+\frac{1}{\sqrt{N}}\bigl{(}\widetilde{\mathbf{X}}_{i}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}+\widetilde{\mathbf{X}}_{j}^{\prime}\bigr{)}\bigl{(}\mathbf{M}_{j}-\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}\bigr{)}^{-1}\bigl{(}\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\widehat{\mathbf{u}}_{i}+\widehat{\mathbf{u}}_{j}\bigr{)}\biggr{\\}}\end{split}$
(39)

and (18) replacing $i$ with $j$

$\sqrt{N}\bigl{(}\widehat{\bm{\beta}}_{(j)}-\bm{\beta}\bigr{)}-\sqrt{N}\bigl{(}\widehat{\bm{\beta}}-\bm{\beta}\bigr{)}=-\biggl{(}\frac{1}{N}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\biggr{)}^{-1}\frac{1}{\sqrt{N}}\,\widetilde{\mathbf{X}}_{j}^{\prime}\mathbf{M}_{j}^{-1}\widehat{\mathbf{u}}_{j}.$
(40)

The difference of (39) and (40) is as follows

$\sqrt{N}(\widehat{\bm{\beta}}_{(i,j)}-\bm{\beta})-\sqrt{N}(\widehat{\bm{\beta}}_{(j)}-\bm{\beta})=\\\
\biggl{(}\frac{1}{N}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\biggr{)}^{-1}\biggl{\\{}\frac{1}{\sqrt{N}}\,\widetilde{\mathbf{X}}_{j}^{\prime}\mathbf{M}_{j}^{-1}\widehat{\mathbf{u}}_{j}-\frac{1}{\sqrt{N}}\,\widetilde{\mathbf{X}}_{i}^{\prime}\mathbf{M}_{i}^{-1}\widehat{\mathbf{u}}_{i}\\\
-\frac{1}{\sqrt{N}}\bigl{(}\widetilde{\mathbf{X}}_{i}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}+\widetilde{\mathbf{X}}_{j}^{\prime}\bigr{)}\bigl{(}\mathbf{M}_{j}-\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}\bigr{)}^{-1}\bigl{(}\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\widehat{\mathbf{u}}_{i}+\widehat{\mathbf{u}}_{j}\bigr{)}\biggr{\\}}$
(41)

Then, by the Reverse Triangle Inequality

$\displaystyle\big{\|}\sqrt{N}$
$\displaystyle(\widehat{\bm{\beta}}_{(i,j)}-\bm{\beta})-\sqrt{N}(\widehat{\bm{\beta}}_{(j)}-\bm{\beta})\big{\|}$
$\displaystyle\leq\biggl{\|}\biggl{(}\frac{1}{N}\widetilde{\mathbf{X}}^{\prime}\widetilde{\mathbf{X}}\biggr{)}^{-1}\biggl{\|}\biggl{\\{}\biggl{\|}\frac{1}{\sqrt{N}}\,\widetilde{\mathbf{X}}_{j}^{\prime}\mathbf{M}_{j}^{-1}\widehat{\mathbf{u}}_{j}\biggl{\|}+\biggl{\|}\frac{1}{\sqrt{N}}\,\widetilde{\mathbf{X}}_{i}^{\prime}\mathbf{M}_{i}^{-1}\widehat{\mathbf{u}}_{i}\biggl{\|}$
(42)
$\displaystyle+\bigg{\|}\frac{1}{\sqrt{N}}\bigl{(}\widetilde{\mathbf{X}}_{i}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}+\widetilde{\mathbf{X}}_{j}^{\prime}\bigr{)}\bigg{\|}\bigg{\|}\bigl{(}\mathbf{M}_{j}-\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\mathbf{H}_{ij}\bigr{)}^{-1}\bigg{\|}\bigg{\|}\bigl{(}\mathbf{H}_{ij}^{\prime}\mathbf{M}_{i}^{-1}\widehat{\mathbf{u}}_{i}+\widehat{\mathbf{u}}_{j}\bigr{)}\biggl{\|}\biggr{\\}}$
(43)

is $o_{p}(1)$ by same arguments used for (21) and (30)-(33). Therefore,

$\sqrt{N}\big{(}\widehat{\bm{\beta}}_{(i,j)}-\bm{\beta}\big{)}=\sqrt{N}\big{(}\widehat{\bm{\beta}}_{(j)}-\bm{\beta}\big{)}+o_{p}(1)$
(44)

which is asymptotically equivalent to the FE estimator $\widehat{\bm{\beta}}$
by result (22) relative to unit $j$.