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Nickell Bias in Panel Local Projection: Financial Crises Are Worse Than You Think$^$
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Key words: Local projection, Nickell bias, impulse response, split-panel jackknife, macro-finance
JEL code: C33, C53, E44, F37, F47
{$^{\dag}$ Ziwei Mei, }[email removed]{. Liugang Sheng, }[email removed]{. Zhentao Shi (Corresponding author),} [email removed]{, 9/F Esther Lee Building, Department of Economics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong SAR, China. Sheng thanks the financial support from the Research Grants Council of the Hong Kong SAR (Project No. 14501821). Shi acknowledges the partial financial support from the National Natural Science Foundation of China (Project No. 72425007). We thank Qiyu Dai, Shu Shen, and Ji Pan for their excellent research assistance. We are also grateful to Geert Dhaene, Raffaella Giacomini, Frank Kleibergen, Byoungchan Lee, {\`O}scar Jord{\`a}, Xun Lu, Ryo Okui, and Liangjun Su for their helpful comments.}
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Financial crises inflict lasting damage on economies, causing output contractions, persistent unemployment, and economic scarring. Ben S. Bernanke, the 2022 Nobel laureate, demonstrated how bank runs played a decisive role in the Great Depression of the 1930s, the worst economic crisis in modern history bernanke1983nonmonetary. The Great Recession following the 2008 financial crisis reignited interest in understanding the relationship between financial shocks and economic recessions. Numerous studies have since demonstrated that financial crises generally produce deeper and more persistent economic losses compared to typical recessions reinhart2009aftermath,laeven2013systemic,schularick2012credit,jorda2013credit.
A central question in this burgeoning literature is to quantify the magnitude and persistence of economic losses caused by financial crises. Accurate assessment of crisis-induced output losses is crucial not only for resolving theoretical debates about crisis persistence, transmission channels, and the speed of economic recovery, but also for guiding policymakers in crafting effective macroeconomic responses, such as monetary easing, fiscal stimulus, and macroprudential regulation, to mitigate economic damage and promote recovery sufi2021financial. As economists employ various measures of financial crises and collect panels of countries with financial shocks, the panel version of jorda2005estimation's local projection (LP) has become widely used by recent empirical analyses of crisis-related economic losses for its simplicity, robustness, and flexible specifications. Moreover, virtually all studies applying panel LP techniques employ fixed-effect (FE) estimators to control for unobserved heterogeneity, cementing FE as the de facto approach in these empirical studies. For example, by applying the FE to various cross-country panel data, romer2017new, baron2021banking, and mian2017household show that financial distress, banking crises, and household debts lead to severe economic contractions, respectively.\footnote{Many other studies, including jorda2013credit, jorda2015leveraged, jorda2016great, zeev2017capital, bhattarai2021local, as well as the recent survey on financial crises conducted by sufi2021financial, also use FE to examine the impacts of financial crises on economic contraction. }
Despite its widespread adoption, the panel LP model is subject to a fundamental limitation: the FE estimator in panel LP inherently incurs the Nickell bias nickell1981biases, even in the absence of lagged dependent variables. To clarify this issue, consider the following illustrative $h$-period-ahead panel LP model:
where $x_{i,t}$ is a measure of financial shock, $y_{i,t+h}$ is the outcome variable, e.g., the logarithm of real GDP, $e_{i,t+h}^{(h)}$ is the error term uncorrelated with $x_{i,t}$, and $\mu_{i}^{(h)y}$ is the individual-specific heterogeneity, or the fixed effect. The IRF $(\beta^{(h)})_{h=0}^{H}$ is of central interest for understanding the dynamic impact of financial crises on economic activities. Despite the absence of the lagged dependent variable in ((ref)), FE for the panel LP model is asymptotically biased when the number of cross-sectional units $N$ and the time periods $T$ are both large; to be precise, the leading case is $(N,T)\to\infty$ and $N/T\to c$ for some constant $c\in(0,\infty)$. It is a type of Nickell bias in the FE estimator for multiple-equation panel vector autoregression (VAR)---the full dynamic model behind the single-equation LP. The Nickell bias has important implications for inference. Standard inference procedures, such as $t$-statistics with critical values based on the standard normal distribution, become unreliable, leading to distorted test sizes, and thus potentially misleading empirical findings. To the best of our knowledge, no systematic study has examined how Nickell bias in panel LP affects estimates of the economic losses associated with financial crises, both qualitatively and quantitatively.
To facilitate the theoretical analysis of the bias, we first present a prototype model where the dependent variable $y_{i,t+1}$ is generated from the following data generating process (DGP)
and the variable of interest $x_{i,t}$ follows a panel autoregression (AR) of order 1:
where $|\rho|<1$ ensures stationarity. This simple model allows us to derive the analytical expression of the bias of the FE estimator of the IRF from this underlying dynamic DGP, which delivers four important implications. Firstly, FE suffers from an intrinsic Nickell bias due to Equation ((ref))'s violation of strict exogeneity.\footnote{ Technically, strict exogeneity is violated if $ \mathbb{E}\left[e_{i,t+h}^{(h)}|({x}_{i,s})_{1\leq s\leq T}\}\right] \neq0 $; see ((ref)) for justifications. } Secondly, FE exhibits an attenuation bias and underestimates the true dynamic effects when $\rho > 0$ in the leading case of economic autoregressive relationships. This result suggests that previous studies on financial crises using FE may have underestimated the impact of financial crises on economic contractions. Third, the Nickell bias tends to rise with the horizon ($h$) and the persistence of the regressor ($\rho$). This implies that FE is more likely to underestimate the long-term impact of financial shocks, and the bias would be larger for more persistent financial shocks. Lastly, the distortion in inference caused by the Nickell bias tends to be more severe for a panel with larger $N$ and smaller $T$.
To eliminate the first-order Nickell bias and restore valid statistical inference in panel LP regressions, we propose the split-panel jackknife (SPJ) estimator Dhaene2015,chudik2018half as a simple, effective, and general solution. It is easy to implement, using the formula: \[ \widehat{\beta}^{(h)\mathrm{spj}}=2\widehat{\beta}^{(h)\mathrm{fe}}-(\widehat{\beta}_{a}^{(h)\mathrm{fe}}+\widehat{\beta}_{b}^{(h)\mathrm{fe}})/2, \] where $\widehat{\beta}^{(h)\mathrm{fe}}$, $\widehat{\beta}_{a}^{(h)\mathrm{fe}}$, and $\widehat{\beta}_{b}^{(h)\mathrm{fe}}$ are the plain FE estimators from all the time periods, the first half ($t\leq T/2$), and the second half ($t>T/2$), respectively. Given that most empirical applications of LP involve at least a moderate $T$, we show that $\widehat{\beta}^{(h)\mathrm{spj}}$ is asymptotically unbiased and follows a zero-mean normal distribution when $N/T^{3}\to0$. Our formal theory of SPJ is developed in a linear model that accommodates extra control variables. The theory can be further extended to entertain other practical specifications in panel data analysis (See Appendix B).
SPJ preserves the key advantages of the LP method. It allows researchers to focus on specifying the main equation of interest linking $y_{i,t+h}$ and $x_{i,t}$ without requiring a full VAR system, and provides a unified regression framework that accommodates a broad range of model specifications, including different lag orders, control variables, and dependent variable forms. It is a practical and theoretically grounded solution for panel data inference.
To illustrate the harm of the Nickell bias and evaluate the effectiveness of the SPJ correction, we revisit four seminal studies examining the macroeconomic consequences of financial crises. These applications are chosen for their prominence in the literature, their diverse yet interconnected economic contexts, and their ability to demonstrate the broad relevance of bias correction methods in empirical macro-finance analysis. Specifically, they cover a comprehensive spectrum of financial disturbances, including general financial distress, banking crises, household debt, and currency collapses.
The first example is romer2017new, which creates a narrative semiannual measure of financial distress index on a 16-bin scale for 24 advanced countries from 1967 to 2012, and shows that the output experiences significant and persistent declines following typical financial distress shocks, highlighting the general macroeconomic vulnerability to financial disruptions. Next, focusing on banking crises, baron2021banking construct a new data set of banking crises based on bank equity crash of more than 30% for 46 advanced and emerging economies over 1870-2016, and find that banking crises cause sharp output contractions and severe credit constraints. Furthermore, mian2017household examine the impact of household debts. They construct a panel of 30 countries from 1960 to 2012, and show how elevated household indebtedness contributes significantly to recessions, particularly evident in the medium-run aftermath of housing booms. The last one is currency crisis, a central concern in international macroeconomics, given its acute and enduring impacts on national economies. cerra2008growth construct a panel of currency crisis episodes across 175 countries from 1965 to 2000, and find that currency crises have persistent negative effects on output.
Our reexamination broadly confirms the original findings from these seminal studies: financial shocks consistently and significantly reduce future economic output. However, compared to the asymptotically unbiased SPJ, FE tends to underestimate the medium- and long-term economic contractions; such underestimation can be substantial in some applications, epitomized by cerra2008growth. The underestimation by FE is consistent with our theoretical finding that the Nickell bias shrinks the FE estimator toward zero; see Remark (ref) for details. Across the four applications we revisited, the relative differences between SPJ and FE at the peak year---when output loss is greatest---range from 16% to 75%. Specifically, SPJ in the romer2017new indicates a peak decline in output of 6.3% approximately three and a half years after a typical financial shock, whereas FE yields a smaller decline of 5.4%. Similarly, for banking crises analyzed by baron2021banking, the FE estimates suggest a cumulative GDP decrease of 3.4% four years after crises, while the SPJ reveals a more pronounced decline of about 4.2%. In mian2017household, a 10 percentage-point increase in the household debt-to-GDP ratio corresponds to a peak GDP contraction of 3.9% using FE estimates, compared to a larger drop of 5.6% estimated with SPJ. Finally, the case of currency crises exhibits the largest discrepancy: SPJ indicates a peak cumulative output loss of approximately 7.9% a decade after the crisis, which is roughly 75% greater than the FE estimates. This substantial discrepancy likely arises due to the large panel of countries ($N=175$) accompanied by a relatively short time length ($T=35$). In summary, our analysis reveals a consistent pattern of underestimation by the FE estimator, particularly for long horizons, when evaluating output losses associated with financial crises. This systematic underestimation mirrors the Nickell bias featured in our prototype model, which biases FE toward zero.
The above empirical findings underscore that the Nickell bias of FE in panel LP is not merely an accidental occurrence but a systematic pathology that must be taken care of by economists who are serious about methodology, and policymakers who are serious about economic consequences. If policymakers rely on the biased empirical estimation method, they are likely to underestimate both the severity and duration of economic downturns, resulting in insufficient or prematurely withdrawn monetary and fiscal interventions. Consequently, stimulus packages, stabilization policies, and financial assistance programs calibrated based on biased estimates will fail to fully address the true magnitude of the crisis. To illustrate this point, we discuss the fiscal policy implications of bias correction in Section (ref). We demonstrate that correcting this bias through SPJ is both essential and economically significant, as it provides policymakers with accurate and unbiased assessments, whereas FE tends to underestimate the required fiscal stimulus to counteract the potential maximum output loss after a typical financial crisis for a given government spending multiplier.
Literature Review and Contributions. Our paper makes econometric contributions to the literature on financial crises by highlighting the Nickell bias of FE in panel LP. The methodology extends beyond the literature on financial crises. The panel LP with fixed effects is also widely used in empirical macroeconomics, for example, to assess the dynamic impacts of political shocks, monetary and fiscal policies, and pandemics acemoglu2019democracy,jorda2022longer,gilchrist2022sovereign. Moreover, its recent adoption in applied microeconomic studies at sectoral, firm, product, or household levels further underscores the broad relevance of addressing Nickell bias, particularly given the large cross-sectional dimensions and short time horizons typical of these datasets bahaj2020home, ottonello2020financial, caldara2022measuring, boehm2023long.\footnote{See jorda2025local for a comprehensive review of the local projection method and its broad applications.} Our SPJ can also be easily applied in those studies to deal with the Nickell bias.
This study is built upon an econometric basis. The finite sample bias of time series AR models is raised by kendall1954. The detrimental effect of such bias is amplified in panel data. nickell1981biases showcases the treacherous nature of panel data: a seemingly innocuous procedure may face unexpected difficulty when a transformation takes care of the individual-specific heterogeneity embodied by the fixed effects; see a recent survey by okui20211. Regarding the estimation of IRFs, the Nickell bias is well-known in dynamic panel VAR literature holtz1988estimating,hahn2002asymptotically,Arellano2003,greenaway2013multistep. With the increasing popularity of single-equation LP for IRF estimation due to its simplicity, the Nickell bias and its implications in the panel LP framework have so far remained unrecognized in the literature. To the best of our knowledge, this paper is the first to point out the omnipresence of Nickell bias even if the lagged dependent variables are absent from the regressors jorda2025local.
For general dynamic panel models, the Nickell bias is usually addressed with the instrumental variable (IV) method Anderson1982,arellano1991some,arellano1995another,blundell1998initial,Hsiao2002 or analytical formulas Kiviet1995,bun2005bias,alvarez2022robust. The IV method is subject to the lack of efficiency and performs poorly with finite samples when the regressor is persistent fernandez2018fixed. The analytical methods require the availability of closed-form formulas. Therefore, case-by-case adjustments and mathematical derivations are necessary for the IV and analytical methods. Dhaene2015 tackle the Nickell bias by SPJ, which is an “automated” estimator that spares applied researchers from the potential weak instrument issue and complex analytical derivations. The application of SPJ goes beyond single-equation linear dynamic panel to multiple-equation models dhaene2016bias and nonlinear models weidner2021bias, due to its simplicity and robustness. We therefore recommend using SPJ in panel LP to preserve the attractive features of LP. Though SPJ is adopted from Dhaene2015, this paper is the first to highlight the necessity of using SPJ specifically in the context of panel LPs to remove the Nickell bias for valid inference.
This paper is closely related to, while different from chudik2018half and herbst2021bias. chudik2018half explore the Nickell bias in a generic linear panel model with weakly exogenous regressors and then extend SPJ to correct it in two-way FE models. Our paper, on the other hand, is specifically motivated by panel LP, and the solutions and empirical examples are tailored for the panel LP. As panel LP involves a series of regressions, the regressor $x_{i,t}$ in ((ref)) violates strict exogeneity when $h\geq 1$. Therefore, weak exogeneity is deduced, rather than assumed, as an intrinsic feature of panel LP in learning the economic IRF. Moreover, we provide an in-depth study of panel VAR($\infty$) about when strict exogeneity is violated in panel LP (see Appendix A.2). herbst2021bias focus on point estimates and investigate the finite sample bias in the time series LP model with and without (in their Appendix A.2) lagged dependent variables and then extend their discussion into panel data; they employ bao2007second for analytical bias correction. Our paper points out that all regressors in panel LP are subject to Nickell bias regardless of the linear specification, and our solution is free of analytical bias calculation. Beyond the impacts on point estimates, we highlight that in panel LP the Nickell bias is not a finite sample issue in statistical inference; it sustains in large samples by shifting the mean of the asymptotic normal distribution of FE, thereby spoiling the theoretical foundation for standard inference using the $t$-statistics. The impact of the bias on statistical inference arises from the asymptotic normal distribution rather than the finite sample standard error.
Data and Software. SPJ is easy to implement and can be extended to many variants of specifications. All empirical applications can be replicated with data and code available at \url{https://metricshilab.github.io/panel-lp-replication/}. This repository also provides links to R and Stata packages to facilitate empirical practices. Given input data, a single-line command of the main function will automatically compute the IRF, standard error, and the corresponding confidence intervals.
Organization. The rest of the paper is organized as follows. In Section (ref), we first use a simple two-equation model to demonstrate the presence of Nickell bias without lagged dependent variables and provide the analytical expression of this bias. We then show that the Nickell bias is a generic phenomenon in panel LP and propose using the SPJ to restore asymptotic normality centered at zero. Moreover, Monte Carlo simulations are carried out to verify the theoretical predictions. Section (ref) estimates the IRFs by FE and SPJ in four empirical examples of macro-finance to evaluate the impact of financial crises. Proofs, theoretical extensions, and additional empirical works are relegated to the Appendix.
The FE estimator is the most popular method that estimates the slope coefficient in the linear panel regression and in the meantime controls the unobservable individual-specific heterogeneity. For simplicity let us consider the regression ((ref)). Let $T_{h}=T-h$ be the effective sample size and $\mathcal{T}^{h}=[T_{h}]=\{1,2,\ldots,T_{h}\}$ be the corresponding index set, where throughout the paper we use $[q]=\{1,2,\cdots,q\}$ to denote the set of positive integers up to a natural number $q$. The FE estimator is
where $\widetilde{y}_{i,t+h}=y_{i,t+h}-T_{h}^{-1}\sum_{t\in\mathcal{T}^{h}}y_{i,t+h}$ is the within-group demeaned dependent variable, and similarly within-group demeaned is the regressor $\widetilde{x}_{i,t}$. To conduct statistical inference about $\beta^{(h)}$, the standard deviation of $\widehat{\beta}^{(h){\rm \mathrm{fe}}}$ is calculated as \[ \widehat{s}^{(h){\rm \mathrm{fe}}}=(\sum_{i\in[N]}\sum_{t,s\in\mathcal{T}^{h}}\widetilde{x}_{i,t}\widetilde{x}_{i,s}\widehat{e}_{i,t+h}^{(h)\mathrm{fe}}\widehat{e}_{i,s+h}^{(h)\mathrm{fe}})^{1/2}/{\sum_{i\in[N]}\sum_{t\in\mathcal{T}^{h}}\widetilde{x}_{i,t}^{2}} \] where $\widehat{e}_{i,t+h}^{(h)\mathrm{fe}}=\widetilde{y}_{i,t+h}-\widetilde{x}_{i,t}\widehat{\beta}^{(h){\rm \mathrm{fe}}}$ is the estimation residual. We then construct the $t$-statistic \[ (\widehat{\beta}^{(h){\rm \mathrm{fe}}}-\beta^{(h)\mathrm{null}})/\widehat{s}^{(h){\rm \mathrm{fe}}}, \] where $\beta^{(h)\mathrm{null}}$ is a hypothesized value under a null of economic interest, and compare the value of the $t$-statistic with a critical value drawn from the standard normal distribution $\mathcal{N}(0,1)$. For example, for a two-sided test with size 5%, we reject the null if the absolute value of the $t$-statistic is larger than 1.96. Does the $t$-statistic based on $\widehat{\beta}^{(h){\rm \mathrm{fe}}}$ provide valid statistical inference for the true IRF $\beta^{(h)}$? This question is complicated by the within-group transformation in the dynamic setting.
The procedure described above is the common practice based on FE. However, there is an intrinsic Nickell bias built into panel LP. jorda2005estimation constructs the time series LP by a VAR system. Here we use ((ref)) and ((ref))---a stylized panel VAR(1)---as the true DGP for demonstration. The dependent variable $y_{i,t+1}$ in ((ref)) is linked to the regressor $x_{i,t+1}$ by the slope parameter $\beta^{(0)}$. In ((ref)) $x_{i,t+1}$ follows one of the simplest time series models---a stationary AR(1) model. Let $\mathbf{x}_{i}^{t}=\left(x_{i,0},x_{i,1},\ldots,x_{i,t}\right)$ be the time series of the regressor from time 0 up to $t$. When assuming $u_{i,t+1}^{x}$ and $u_{i,t+1}^{y}$ independently and identically (i.i.d.) distributed across $i$ and $t$, we have $\mathbb{E}\left[u_{i,t+1}^{y}|\mathbf{x}_{i}^{T}\right]=0$ and thus in ((ref)) strict exogeneity holds. The FE estimator is asymptotically unbiased when $h=0$ in the regression ((ref)).
LP is a series of linear regressions across different horizons. For $h=1$ we substitute ((ref)) into ((ref)) and for $h\geq2$ we repeat the substitution to produce ((ref)) with the following closed-form expressions
Denote $\mathbf{u}_{i}^{x,t}=(u_{i,s}^{x})_{s=1}^{t}$. The composite error term $e_{i,t+h}^{(h)}$ is weakly exogenous\footnote{chudik2018half define weak exogeneity in terms of unconditional moments. We define weak exogeneity in terms of mean independence conditional on the past information, following Mikusheva2023.} in that \[ \mathbb{E}\left[e_{i,t+h}^{(h)}|\mathbf{x}_{i}^{t}\right]=\mathbb{E}\left[\beta^{(0)}(u_{i,t+h}^{x} + \sum_{s=1}^{h-1}\rho^{s}u_{i,t+h-s}^{x})+u_{i,t+h}^{y}\Bigg|x_{i,0},\mathbf{u}_{i}^{x,t}\right]=0. \] However, it is not strictly exogenous as
if $\beta^{(0)}\neq0$.
At first glance, ((ref)) is a seemingly innocuous regression of $y_{i,t+h}$ on another variable $x_{i,t}$. It turns out that ((ref)) and ((ref)) consist of a two-equation panel vector AR model. When the within-group transformation is used to eliminate the fixed effects, the Nickell bias is present even though the lagged $y$ does not explicitly appear on the right-hand side of ((ref)).
The following proposition characterizes the bias in the asymptotic distribution.
Let $\mathcal{Z}\sim\mathcal{N}\left(0,\sigma_{xe,h}^{2}/\sigma_{x}^{4}\right)$ denote the zero-mean normal random variable following the limit distribution in ((ref)). Proposition (ref) implies
where “$\stackrel{a}{\sim}$” signifies asymptotic similarity. The bias is of order $1/T$. In terms of point estimates, if $T$ is fixed, $\widehat{\beta}^{(h){\rm \mathrm{fe}}}$ is inconsistent as $N\to\infty$. When $(N,T)\to\infty$ jointly with $N/T\to c \in (0,\infty)$, the FE estimator $\widehat{\beta}^{(h){\rm \mathrm{fe}}}$ is consistent for $\beta^{(h)}$, and hence $\rho$, $\sigma_{u_{x}}^{2}$, $\sigma_{x}^{2}$ and $\sigma_{xe,h}^{2}$ are also consistently estimable. However, when statistical inference is the purpose, we need to examine the bias in the asymptotic distribution. When $N/T\to c \in (0,\infty)$, the bias in ((ref)) satisfying
does not vanish asymptotically and it will distort the test size of the usual inference based on the $t$-statistic.\footnote{ When the lagged dependent variables are present, the well-known explicit Nickell bias in panel LP is pointed out by teulings2014economic. They augment the regression specification as a way to fix the bias, which is valid when the regressors have no dynamics. By contrast, we show that intrinsic Nickell bias occurs even when the lagged dependent variable is not included.} This size distortion is not caused by finite sample standard errors, and it therefore remains even if the asymptotic variances are known. Following hahn2002asymptotically and okui2010asymptotically, one can correct the bias based on ((ref)) for valid asymptotic inference.
In simulation studies and empirical applications, we find that these intriguing phenomena of the FE estimator are common in the simple AR(1) specification as well as more general cases. Although the true DGPs in real data studies are unknown, in the empirical applications in Section (ref) we observe that the bias correction enlarges the magnitude of the IRFs and the biggest discrepancy often occurs on a relatively long horizon.
While we have demonstrated the bias in the FE estimation of ((ref)) from the prototype DGP ((ref)) and ((ref)), the Nickell bias looms in general dynamic systems. In practical use of the panel AR model, researchers may want to include lagged dependent variables as well as other control variables. For example, nickell1981biases considers a panel ARX (in our notation) $y_{i,t+1}=\mu_{i}^{y}+\gamma y_{i,t}+\beta x_{i,t+1}+u_{i,t+1}^{y}$.
Though LP focuses on single-equation regressions instead of simultaneous-equation systems, it is helpful to present the underlying VAR system which implies the series of regressions. In this section, we consider data generated from a panel VAR. Let $\mathbf{x}_{i,t}$ be a $K$-dimensional random vector, and the observed data at time $t$ is combined into a $(K+1)$-vector $\mathbf{w}_{i,t}=(y_{i,t},\mathbf{x}_{i,t}^{\prime})^{\prime}$. We write down a panel structural VAR($p$) model
as the DGP, where $\mathbf{A}_{s}$, $s=0,1,\ldots,p$, are $(1+K)\times(1+K)$ coefficient matrices,\footnote{Without loss of generality, here we write the numbers of the lags of all components in $\mathbf{w}_{i,t}$ being the same for all components. The researcher has the discretion to choose the number of lags for specific variables either by prior knowledge or by some information criterion, and in this case $p$ here is viewed as the largest number of lags with the shorter lags being imputed by known zero coefficients. } and $\boldsymbol{\mu}_{i}^{(0)}$ is the vector of individual-specific fixed effects. The Wold-causal order requests the left-bottom block of $\mathbf{A}_{0}$ to be a $K$-vector of zeros jorda2005estimation, and we standardize its diagonal line as 1. We rewrite ((ref)) as
where the matrices/vectors are partitioned in a compatible manner. The structural form of the first equation is
As derived in Appendix A.1, the predictive equation for $y_{i,t+h}$ is
where $\mathbf{W}_{i,t}=(\mathbf{w}_{i,t}^{\prime},\mathbf{w}_{i,t-1}^{\prime},\ldots,\mathbf{w}_{i,t-p+1}^{\prime})^{\prime}$ is the $p(K+1)$ long vector of regressors, $\boldsymbol{\theta}^{(h)}$ is the corresponding coefficient vector, and the error term \[ e_{i,t+h}^{(h)}=(1,\boldsymbol{0}^{\prime})\sum_{s=0}^{h-1}\mathbf{A}_{0}^{-s}\mathbf{A}_{1}^{s}\mathbf{A}_{0}^{-1}\mathbf{u}_{i,t+h-s}. \] We impose the following assumption to ensure that $e_{i,t+1}^{(h)}$ satisfies weak exogeneity.
Condition (a) rules out cross-sectional dependence for simplicity. The m.d.s. assumption is necessary for the panel LP linear coefficient to be interpreted as IRF. The conditional mean $\mathbb{E}[u_{i,t}^{y}|\mathbf{u}_{i,t}^{x}]=0$ further guarantees that $u_{i,t+1}^{y}$ is mean independent of all the right-hand side regressors in ((ref)). As $e_{i,t+h}^{(h)}$ is a linear combination of $(\mathbf{u}_{i,s})_{s=t+1}^{h}$, it is mean-independent of past information. Condition (b) ensures that the observed variables $\mathbf{w}_{i,t}$ are strictly stationary over time, exhibiting no unit root or explosive behavior. Analyzing panel LP with highly persistent panels is technically involved and beyond the scope of this paper. Our follow-up work \citep*{liao2024nickell} develops a novel method to handle persistent panel data. See Appendix C.5 for details.
When $\mathbf{x}_{i,t}$ is multivariate, without loss of generality we can denote the variable of main economic interest as the first scalar $x_{i,t}^{(1)}$, and the rest of the vector $\mathbf{x}_{i,t}^{(2)}$ as additional control variables to make more plausible the mean independence in ((ref)) and weak exogeneity in ((ref)). In estimation, however, $x_{i,t}^{(1)}$ and $\mathbf{x}_{i,t}^{(2)}$ are symmetric in that they share the same status as regressors accompanying the potential lagged dependent variables in the predictive equation ((ref)). In the panel LP regression, all regressors in $\mathbf{w}_{i,t}$ incur the Nickell bias regardless of the number of lags, which is elaborated in Appendix A.2 for a VAR($\infty$) model.
Closed-form expressions of the bias, as in Proposition (ref) for the prototype model, are intractable in a full-scale dynamic system such as ((ref)). We suggest an automated bias correction method in the next section.
Panel LP requires a correct specification of the main equation of interest, but keeps an agnostic attitude toward the dynamics of $\mathbf{x}_{i,t}$ in the lower block of ((ref)). Unlike the analytic bias correction, Dhaene2015's SPJ is a data-driven method that suits well with the single-equation panel LP.
Denote the FE estimator of ((ref)) as $\widehat{\boldsymbol{\theta}}^{(h)\mathrm{fe}}$, and let $\widehat{\boldsymbol{\theta}}_{a}^{(h)\mathrm{fe}}$ and $\widehat{\boldsymbol{\theta}}_{b}^{(h)\mathrm{fe}}$ be the FE estimators using the first half of observations in the set $\mathcal{T}_{a}^{h}:=\left\{ t\leq T_{h}/2\right\} $ and the second half of observations in the set $\mathcal{T}_{b}^{h}:=\left\{ T_{h}/2<t\leq T_{h}\right\} $, respectively. The SPJ estimator is defined as
To establish the asymptotic properties of the SPJ estimator, we impose the following Assumption (ref) that consists of high-level conditions of the law of large numbers and central limit theorem commonly seen in the literature of panel data, say bai2009panel. Define \[ \widehat{\mathbf{Q}}_{k}:=\frac{1}{NT_{h}/2}\sum_{i\in[N]}\sum_{t\in\mathcal{T}_{k}^{h}}\widetilde{\mathbf{W}}_{i,t}\widetilde{\textbf{\ensuremath{\mathbf{W}}}}_{i,t}^{\top}, \] for $k\in\{a,b\}$ associated with the two halves of the data over the time dimension, where $\widetilde{\textbf{\ensuremath{\mathbf{W}}}}_{i,t}=\textbf{\ensuremath{\mathbf{W}}}_{i,t}-T_{h}^{-1}\sum_{t\in\mathcal{T}^{h}}\textbf{\ensuremath{\mathbf{W}}}_{i,t}$. Let $\widehat{\mathbf{Q}}:=(\widehat{\mathbf{Q}}_{a}+\widehat{\mathbf{Q}}_{b})/2$. We use $\boldsymbol{1}\left\{ \cdot\right\} $ to denote the indicator function.
The SPJ estimator is asymptotically normal without a bias term if $T$ is non-trivial relative to $N$ in that $N/T^{3}\to0$.
The asymptotic distribution centered around zero allows us to invoke the common procedure for inference. To construct a feasible $t$-statistic, we compute the \[ \widehat{\mathbf{R}}=(NT_{h})^{-1}\sum_{i\in[N]}\sum_{t,s\in\mathcal{T}^{h}}\mathbf{d}_{i,t}^{*}\mathbf{d}_{is}^{*\top}\widehat{e}_{i,t+h}^{(h){\rm spj}}\widehat{e}_{i,s+h}^{(h){\rm spj}}, \] where $\widehat{e}_{i,t+h}^{(h){\rm spj}}=\widetilde{y}_{i,t+h}-\widetilde{\textbf{\ensuremath{\mathbf{W}}}}_{i,t}^{\top}\widehat{\boldsymbol{\theta}}_{h}^{{\rm spj}}$ is the SPJ's estimated residual. Given some standard assumptions, $\widehat{\mathbf{R}}$ consistently estimates the individual-clustered variance $\mathbf{R}$. The standard practice of statistical inference based on the $t$-statistic is asymptotically valid.
The condition $N/T^3 \to 0$ for SPJ accommodates the leading case of $N/T \to c \in (0, \infty)$ in macro-finance applications. While not designed for large-$N$-small-$T$ panels, SPJ effectively eliminates bias and ensures valid inference under moderate $N/T$ ratios, as evidenced by our numerical and empirical results. Real data has a finite number of observations. As a rule of thumb for empirical applications, we recommend our method for panels with $T\geq 30$ and $N/T \leq 10$.
To illustrate the presence of Nickell bias and the finite sample performance of SPJ, we conduct Monte Carlo simulation exercises based on the prototype model of the simple form as in Equations ((ref)) and ((ref)). We generate the innovations $\varepsilon_{i,t}$ and $u_{i,t}$ from independent $\mathcal{N}(0,1)$, and the fixed effect is generated by $\mu_{i}^{(0)y}=0.2\sqrt{T}\bar{x}_{i}+\xi_i $ where $\xi_i\sim\mathcal{N}(0,1)$ is independent of all other variables. For the contemporaneous connection between $y_{i,t+1}$ and $x_{i,t+1}$, we specify $\beta^{(0)}= -0.6$, where the negative relationship is motivated by the impact of financial crises on the real economy. The AR(1) coefficient $\rho\in\{0,0.2,0.5,0.8\}$ varies the persistence of the predictor. We consider the sample sizes $(N,T)\in\{(30,60),(30,120),(50,120)\}$ in line with the empirical examples. All simulation results are produced by 1000 independent replications.
Proposition (ref) offers the closed-form formula of the bias for the AR(1) specification of $x$. It provides an “oracle” debiased (DB) estimator (See Appendix A.4 for details) to take advantage of the closed-form formula, which is based on the oracle of correct specification of AR(1). It will be compared with the plain FE and the SPJ estimators that keep an agnostic attitude about the dynamics of $x$.
Figure (ref) reports the average of estimated IRFs over the 1000 replications and compares them with the true IRFs. With $\beta^{(0)}<0$, the true IRFs are negative and decay to zero exponentially fast. The FE estimator exhibits an upward bias, indicating a positive Nickell bias with the opposite sign of the true IRFs. In other words, the FE estimator underestimates the negative effects of $x_{i,t}$ on $y_{i,t+h}$. Moreover, the bias worsens as the horizon $h$ increases. This finding is predicted by Proposition (ref) and discussed in Remark (ref).
Regarding the influence of $\rho$, FE's bias is relatively small when $\rho=0$ or 0.2. As $\rho$ increases to 0.5 and 0.8, the bias becomes more pronounced. The bias is mitigated when $T$ grows from $60$ to $120$. When $N$ grows from $30$ to $50$, the bias becomes larger. In comparison, the oracle DB estimator, as well as our recommended SPJ estimator, is very close to the true IRF.
When it comes to the estimation errors measured by root-mean-square error (RMSE), Figure (ref) shows that the FE estimator produces the largest error among the three methods in all cases, while the oracle DB estimator yields the smallest RMSE. RMSEs of all the three estimators get larger as $\rho$ gets closer to 1. The recommended method, SPJ, again exhibits performance similar to that of the oracle DB estimator in terms of RMSEs.
Figure (ref) plots the empirical coverage probabilities of the confidence intervals (CIs) by inverting the $t$-statistic. The nominal 95% probability is marked by the horizontal dashed line. For each $(N,T)$ and $\rho$, the coverage rate of CIs from the FE estimator is close to the nominal probability when $h=0$, while the bias emerges when $h>0$ and the coverage rate falls short of 0.95 as $h$ grows. Such Nickell bias is present even though $\rho=0$ when the IRF equals zero as $h>0$, and the performance further deteriorates as $x_{i,t}$ gets more persistent under a larger $\rho$. When $N=30,$ the Nickell bias is alleviated as $T$ grows from 60 to 120, while it materializes again as $N$ increases to $50.$ These numerical findings echo the closed-form expression of the bias in Proposition (ref). The CIs from the DB estimator, as an oracle estimator when the closed-form expression of the Nickell bias is known, have coverage probabilities very close to 95% in all cases. The SPJ estimator closely resembles the oracle DB estimator, except for the slight deviation in the case of a small sample size $(N,T)=(30,60)$ and $\rho=0.8$; here SPJ still substantially mitigates the severe bias from the FE. To summarize, the simulation results provide a clear picture of the bias issue for the FE estimator in panel LP, and support the practical values of the SPJ estimator in bias correction.
If statistical inference is of key research interest, Figure (ref) shows that the Nickell bias in FE causes substantial under-coverage of the confidence interval (or equivalently, excessively high rejection rate in the $t$-test). Such distortion violates the principle of statistical inference and results in misleading empirical findings. Therefore, the Nickell bias must be removed in statistical inference, and SPJ is preferred to FE in this sense.
In this section, we apply our method to the four renowned empirical studies on macro-finance linkage as mentioned in the Introduction. They all center on the aftermath of financial crises on output, but each of them places a distinct emphasis on different financial shocks. romer2017new design a new financial crisis chronology to discuss the responses of output and unemployment following the overall financial distress. baron2021banking explore the role of banking crises in the credit crunch and output reduction diamond1983bank,bernanke2018real. The third one and its related studies mian2017household,mian2010household highlight the rising household debts in depressing output in the mid-run. The fourth study cerra2008growth examines the persistent damage of currency crises on national economies. These studies employ cross-country panel data and adopt the panel LP regression Eq.((ref)): $y_{i,t+h}$ denotes some measure of economic output, and the regressor vector $\mathbf{W}_{i,t}$ includes a key variable of interest $x_{i,t}^{(1)}$---a certain measure of financial shocks.\footnote{The first three studies adopt the panel LP method, and the fourth study cerra2008growth adopts the dynamic panel with distributed lag model, which also suffers the Nickell bias. The FE estimates are closely aligned with their estimation results for the currency crises, which will be discussed later in detail.} We revisit these works and contribute a methodological edge to the literature on financial crises.
Crisis chronology can be traced back to the records of caprio1996bank about bank crisis events in the 1990s. Thereafter, reinhart2009aftermath, schularick2012credit, laeven2013systemic, and jorda2013credit have adopted more precise subjective standards and quantitative data to measure financial crises and evaluate the associated economic losses. In a comprehensive study, romer2017new (RR, henceforth) create a narrative semiannual measure of financial distress for 24 advanced countries from 1967 to 2012, based on the contemporaneous narrative accounts of country conditions on the disruptions to credit supply in the OECD Economic Outlook. To capture the variation in financial disruption across countries and time periods, they classify financial distress into five categories: credit disruption, minor crisis, moderate crisis, big crisis, and extreme crisis, and each category is further broken into minus, regular, and plus. Thus, their new index of financial distress has a scale of 16, where 0 represents no financial distress, and a higher value indicates a more severe financial crisis. Based on this novel measure of general financial distress, they use FE to estimate panel LP and find that the average decline in output after a financial crisis is moderate in size, though persistent and statistically significant.
We revisit RR's open-access semi-annual dataset, an unbalanced panel of 24 OECD countries. RR construct the financial distress index from 1967 to 2012, but between 1967 and 1979 only Germany had a “Credit disruption (regular)” in 1974. Our analysis restricts the time period from 1980 to 2012; this sample adjustment barely changes their estimation results.
RR's panel LP specification is
for $h=0,1,...,10$,\footnote{ The only difference from Eq.((ref)) is the presence of the time fixed-effect $g_t^{(h)y}$; See Sections B.2.3 and C.3 for such an extension.} where the dependent variable $y_{i,t+h}$ is the logarithmic real GDP or the unemployment rate, and $x_{i,t}^{{\rm FD}}$ is their new index on financial distress. $\mu_{i}^{(h)y}$ and $g_{t}^{(h)}$ indicate country and time fixed effects for each horizon $h$. Moreover, four lags of the dependent variables and the financial distress index are included as control variables. The linear coefficient $\beta^{(h)}$ is the IRF of the economic activities in period $h$ ahead of financial shocks occurring at time $t$, which corresponds to a minor level shift in financial distress as RR defines. To obtain a typical shock in financial distress, RR multiplies the estimates of $\beta^{(h)}$ by 7, which corresponds to a “Moderate crisis (minus)” financial crisis.
Figure (ref)'s panels (A) and (B) show the IRFs of the real GDP and the unemployment rate to a typical financial shock, respectively, estimated by RR's FE and our recommended SPJ. SPJ confirms RR's finding that output falls and the unemployment rate rises significantly and persistently following the financial crisis, but its impulse responses exhibit considerable differences from those of the FE estimates, particularly for the long horizons.
Panel (A) shows that the immediate aftermath of a moderate crisis is a fall in GDP of 2.2% based on the SPJ estimate, which is close to the FE estimate. However, the differences between the SPJ and the FE estimates become more pronounced as the horizon $h$ rises. According to SPJ, the decline in output grows substantially 3.5 years after the financial shock and peaks at 6.3%, which is about 16% higher than the FE estimates. The differences between the two estimates are also persistent. After 5 years ($h=10$), the decline in output based on SPJ returns to about 5%, which remains 28% larger in magnitude than the corresponding FE.
Consistent with the responses of real GDP, Panel (B) suggests that the FE estimator also tends to underestimate the aftermath of a moderate financial crisis on the unemployment rate. A financial shock raises the unemployment rate steadily. As with GDP, the unemployment rate continued to increase through 3.5 years following the impulse, peaking at 2.6 percentage points based on the SPJ estimate (24% higher than the FE estimate), and after that, the differences between the two estimates remain.
Overall, FE tends to underestimate the IRFs, which is likely caused in part by the persistence of financial distress. To check the persistence of the financial distress index, we adopt a simple panel AR(1) model: $x_{i,t}^{{\rm FD}}=\mu_{i}+g_{t}+\rho x_{i,t-1}^{{\rm FD}}+e_{i,t},$ where $x_{i,t}^{{\rm FD}}$ is the financial distress index, and $\mu_{i}$ and $g_{t}$ denote country and year fixed effects. This regression yields the autoregressive coefficient $\widehat{\rho}= 0.836,$ $(s.e.=0.031, R^2= 0.8177)$, comparable to $\rho = 0.8$ in our simulation. Thus, the financial distress index is persistent, which may play a role in the differences between the FE and SPJ estimates. Though each of these two estimators falls into the 95% confidence interval of the other estimator, it is still important to correct the Nickell bias in FE. As demonstrated by the simulation results in Section (ref), the coverage of confidence intervals of the FE estimates can be severely distorted (Figure (ref)) in the presence of Nickell bias (Figure (ref)), and the conventional hypothesis testing for the null hypothesis $\mathbb{H}_0:\beta^{(h)}=0$ using the $t$ statistics by FE is invalid.
The banking system is vital to the modern financial system, and a vast literature has studied the financial intermediary of banks and how banking crises led to a sustained decline in economic activities diamond1983bank,calomiris2003fundamentals, gertler2010financial, brunnermeier2014macroeconomic, rampini2019financial. Ben Bernake argues that bank runs played a critical role in the Great Depression of the 1930s, and the collapse of Lehman Brothers triggered global financial panics, which eventually turned into a worldwide financial crisis during 2007--2008 and the prolonged Great Recession bernanke1983nonmonetary, bernanke2018real. The recent 2023 meltdown of Silicon Valley Bank, the second largest failure of a financial institution in U.S. history, again spurred worries about the global financial markets.
baron2021banking (BVX, henceforth) study the aggregate output loss associated with bank crises. Different from the traditional approach that takes banking panics as banking crises, BVX argue that a large decline in bank equity return is the prerequisite for banking crises.\footnote{Many scholars have emphasized the criticality of bank equity. gertler2010financial introduce financial intermediaries into their model, showing that disruptions in financial intermediation depress aggregate economic activity. he2013intermediary find the asymmetry of capital market risk premiums during banking crises, that is, when bank equity is low, bank losses have a significant impact on the risk premium, but when bank equity is high, it has little effect. brunnermeier2014macroeconomic suggest that a decline in bank assets through the price channel creates greater risk as the economy moves away from a steady state. rampini2019financial provide a dynamic model including bank net assets and find that low liquidity in banks will lead to more severe and prolonged recessions during banking crises.} Specifically, they construct a new data set of bank equity return for 46 advanced and emerging economies over 1870--2016, and define a bank crisis as a bank equity crash of more than 30%. Their FE estimates find that bank equity crashes predict a sizable and long-lasting decline in future real GDP.
The baseline LP specification in BVX is
for $h=1,2,...,6$, where the dependent variable $\Delta_{h}y_{i,t+h}$ is the change in the logarithm of real GDP from year $t$ to $t+h$; to be consistent with the other two empirical studies in this section, we multiply the change by 100 to represent percentages. The regressor $x_{i,t}^{{\rm B}}=\mathbf{1}\{r_{i,t}^{{\rm B}}\leq-30\%\}$ is a dummy variable for the bank equity crash when the bank equity index dropped more than 30%, and similarly $x_{i,t}^{{\rm N}}=\mathbf{1}\{r_{i,t}^{{\rm N}}\leq-30\%\}$ is a dummy for the nonfinancial equity crash. The control variables include three lags of bank and nonfinancial equity crash dummies, and the contemporaneous and up to three-year lagged real GDP growth and change in credit-to-GDP.
The dependent variable here is the $h$-order difference $\Delta_{h}y_{i,t+h}$, and therefore the slope coefficients $\beta^{(h)\mathrm{B}}$ and $\beta^{(h)\mathrm{N}}$ represent cumulative responses. As detailed in Appendix B.1, FE under this specification also suffers from the Nickell bias and underestimates the magnitude of the IRFs, and the bias can be corrected by SPJ. Figure (ref)'s Panel (A) shows the output declines significantly and persistently after a banking crisis. Compared with SPJ, FE tends to underestimate output loss, and the gap widens over time. The SPJ suggests that the cumulative decline in GDP peaks four years after the bank crisis at 4.2%, while the corresponding output gap is 3.4% by FE. Panel (B) reiterates the real output losses following a crash in nonfinancial equity. In the first three years after the nonfinancial stock crash, the IRF by SPJ is very close to the one by FE, whereas the differences in the two IRFs become visible afterward. FE again tends to underestimate.
In the baseline specification, we do not control for time fixed effects as they may absorb the global impact of some big banking crises. BVX's FE estimates find that the inclusion of time fixed effects in the panel LP mitigates the output loss. We find similar results by SPJ, and thus the difference between the two estimates slightly declines. On the other hand, the gap between the two estimated IRFs of nonfinancial firms' equity crashes becomes more visible.
Prior to 2007, the subprime mortgage market in the United States developed rapidly due to the continued prosperity of the U.S. housing market and the low interest rates. Household debt levels were on the rise, and mortgages were inversely proportional to household income mian2009consequences. The 2007--2008 global financial crisis reminds us of another important source of financial crisis: household debt. Expansion of household debt boosted consumer demand mian2011house, fueled housing speculation mian2022credit, and raised house prices justiniano2019credit. A bubble cannot last forever. The cooling of the U.S. housing market, especially the hike in short-term interest rates, overloaded repayment burdens upon home buyers, leading to large-scale defaults that eventually triggered the subprime mortgage crisis.\footnote{In addition, the expansion of household debt in most developed economies such as the United States mainly affects the economy through the channel of consumption demand mian2018finance,mian2020does. The higher the debt leverage, the greater the responsiveness of household consumption propensity to changes in housing wealth mian2013household.} The global financial crisis highlights that household debts would not only affect economic fluctuations in the short term but also hamper economic growth in the medium and long term.
In an influential study, mian2017household (MSV, henceforth) explore the dynamics between household debt, nonfinancial firms debt, and economic fluctuations, with a cross-country panel via both VAR and LP. They find that the household debt shock leads to a boom-recession cycle, that is, GDP first increases for 2-3 years but then decreases substantially. More precisely, they find that a rise in the household debt to GDP ratio from four years ago to last year predicts a substantial decline in subsequent real GDP growth from the current year onward. By contrast, the output declines immediately following firm debt shocks, whereas recovers gradually afterward. We will compare the baseline panel LP results using the FE estimation in MSV's Figure 2 (p.1770) with the results based on SPJ.
Collecting an unbalanced panel of 30 countries from 1960 to 2012, MSV specify the LP regression as
for $h=1,2,...,10$, where $y_{i,t+h}$ is the logarithmic real GDP. The key explanatory variables of interest are the household debt to GDP ratio $x_{i,t}^{{\rm HH}}$ and nonfinancial firm debt to GDP ratio $x_{i,t}^{{\rm F}}$. Control variables include four lags of household debt to GDP ratio and nonfinancial firm debt to GDP ratio, and contemporaneous and up to four-year lagged logarithmic real GDP.
Our empirical analysis supports and reinforces MSV's main findings that household debt booms predict a sustained decline in output after a temporary rise. SPJ's IRFs in Figure (ref) maintain the boom-recession cycle, with long-term real GDP declines greater than MSV's FE. In Panel (A) SPJ resembles FE in the first to three years, indicating a temporary rise in GDP due to the housing market boom. The output has started to decline persistently since the fourth year. The trough is observed in the seventh year, where FE estimates a ten percentage point shock in the household debt to GDP ratio is associated with a 3.9% drop, in contrast with SPJ's 5.6% drop. This is consistent with the literature that finds debt booms may distort resource allocation and human capital accumulation, and thus reduce long-run growth gopinath2017capital, charles2018housing, borio2016labour. As a comparison, Panel (B) shows that the IRFs of output to firm debt shocks are close.
In our final application, we examine the impact of currency crises on output loss, a significant concern in international economics due to their severe and enduring effects on national economic output burnside2001prospective, hong2005recovery, cerra2008growth. These crises, marked by sharp currency depreciations, can trigger widespread economic disruptions and persistent contractions. For instance, the Asian Financial Crisis of 1997-1998 led to substantial economic contractions in Southeast Asia, notably in Indonesia, Thailand, and South Korea. Similarly, the Argentine Crisis of 2001-2002 resulted in a severe economic depression and rising unemployment.
We used the data set from cerra2008growth, which includes panel data on currency crises for 175 countries from 1965 to 2000. An exchange market pressure index (EMPI) is constructed as the sum of the percentage depreciation in the exchange rate and the percentage loss in foreign exchange reserves, allowing for cross-country comparability. A dummy variable for a currency crisis is assigned to a specific year and country if the EMPI is in the upper quartile of all observations within the panel. Our panel local projection (LP) specification is as follows:
for $h=1,2,...,10$. Here, the dependent variable $\Delta_{h}y_{i,t+h}$ represents the change in the logarithm of real GDP from year $t-1$ to $t+h$. The regressor $x_{i,t}$ is a dummy variable indicating a currency crisis. Control variables include four lags of the currency crisis dummies and real GDP growth.
Figure (ref) illustrates the cumulative changes in real GDP following currency crises, using estimates from both FE and SPJ. This comparison highlights significant differences in how each method captures the economic impact of such crises.
The FE estimator shows a decline in real GDP immediately after a currency crisis. This decline persists over several years, indicating that currency crises have a lasting negative effect on economic output. However, the FE estimates suggest a moderate initial drop in GDP, which stabilizes relatively quickly. This might provide a somewhat reassuring picture to policymakers, suggesting that while the impact is negative, it is not overwhelmingly severe.
In contrast, the SPJ estimates reveal a more severe and protracted decline in GDP, suggesting that traditional FE estimates may significantly underestimate the true economic cost of currency crises. Quantitatively, the FE estimates show an initial GDP decline of 3% following a currency crisis, stabilizing around a 4.5% decrease after five years.\footnote{This quantitative result closely aligns with Figure 3 in cerra2008growth, who employ a dynamic panel with distributed lags model, which also suffers from the Nickell bias.} The SPJ estimator, however, indicates a more substantial and persistent output drop, exceeding 7% after five years. This finding corroborates hong2005recovery, who argue that GDP levels remain permanently below their initial trend following currency crises. Furthermore, while the FE estimate shows a peak cumulative output loss of approximately 5.1% in the sixth year post-crisis, the SPJ estimate suggests a 7.1% output decline over the same horizon---about 40% higher than FE. Moreover, SPJ suggests that the peak effect is about 7.9% cumulative output loss about a decade after the crisis, which is about 75% higher than that from FE.
A notable aspect of Figure (ref) is that the FE estimators fall outside the 95% confidence intervals of the SPJ estimates for long horizons ($h>6$). This statistically significant difference between the SPJ and FE estimators underscores the substantial bias in the FE estimates, emphasizing the importance of robust estimation techniques in capturing the full extent of economic disruptions caused by currency crises. Our econometric theory posits that Nickell bias is more severe in panels with a larger number of cross-sectional units $N$ and a smaller number of time periods $T$, which is the case in this example.
Our empirical analysis demonstrates that financial shocks induce substantial output losses, corroborating the consensus that recessions triggered by financial crises are both deeper and more protracted than typical downturns. Moreover, our comparison of FE with SPJ reveals a systematic downward bias in the former, despite the distinct nature of financial crises, as well as variations in measurement and sample coverage. This bias leads to an underestimation of financial crisis costs, usually with large magnitudes in the long horizons. These findings underscore that the Nickell bias is pervasive, and it requires correction for valid inference.
Those case studies are in line with several key findings in our prototype model and simulation exercises. First, in all four cases, FE exhibits attenuation relative to SPJ, consistent with Remark (ref) for the prototype model and Remark B1 in Appendix B.1 for the differenced dependent variable. This result indicates that previous studies on financial crises using FE may underestimate the impact of financial crises. Second, the gaps between SPJ and FE tend to be larger for long horizons than short horizons, indicating that the FE estimator is more likely to underestimate the long-term impact of financial crises. Lastly, among all applications, the discrepancies between FE and SPJ are the most statistically significant in cerra2008growth, where the relative magnitude of the sample sizes $N/T$ is the largest. This result echoes the bias term in ((ref)) of Proposition (ref) that a larger ratio $N/T$ undermines the statistical inference for FE.
Next, we discuss the magnitudes of the Nickell bias in the four cases and its importance for policy design. Table (ref) presents the absolute relative differences between the two estimates at the peak period of the output declines following financial shocks. Column (6) shows that the FE estimates of economic contraction following financial distress, banking crises, household debts, and currency crises as key results in those four studies suffer from considerable underestimation, ranging from 16% to 75%. Moreover, our results also indicate that the economic recovery from financial crises is slower than the FE estimates suggest. This helps partially explain why the recovery from the 2008 global financial crisis was weak and slow imf2018world.
Note that the estimated quantitative effects of financial crises on output vary widely in the literature for various reasons, such as the measurement of financial crisis, the time periods, the sample of countries, and the measurement issues of GDP sufi2021financial. Our study showcases the important estimation bias arising from the common econometric method used in the literature, and provides better econometric tools for practitioners to achieve more accurate estimates of financial crises on output loss.
Lastly, although the numerical differences between the SPJ and FE estimates are visually small in some empirical applications, they remain highly relevant for policymakers designing effective responses to financial crises. Fiscal stimulus policies are commonly employed to mitigate economic damage following severe financial crises. For example, in response to the subprime mortgage crisis of 2007-2008, the U.S. government under President Obama enacted the American Recovery and Reinvestment Act (ARRA) in 2009, which provided approximately \$831 billion, around 5.8% of GDP, to avoid further economic deterioration.
Assuming a government spending multiplier of 1.0, a relatively high value according to ramey2019ten and ramey2018government, the choice of estimation methods between FE and SPJ influences the magnitude of a stimulus package designed to counteract the maximum output loss following a financial crisis.\footnote{ramey2019ten conducted a recent survey of fiscal multipliers, finding that government purchase multipliers typically range between 0.6 and 1. A smaller multiplier implies the need for a larger stimulus package to offset the same amount of economic contraction. We select a multiplier of 1.0 for straightforward comparison. However, even under alternative fiscal multipliers, our qualitative conclusion---that the fiscal stimulus is likely insufficient based on FE estimates---remains unchanged.} Take romer2017new as an example. Following a typical financial crisis, the FE estimates would require a fiscal stimulus of approximately 5.43% of GDP, which equals the estimated output loss at the peak in Column (5) of Table (ref) divided by the fiscal multiplier, while the SPJ estimates recommend a larger stimulus of about 6.29% of GDP. Consequently, failing to account for Nickell bias---using FE rather than SPJ---would lead policymakers to underestimate the necessary stimulus by roughly 0.85% of GDP. Our simple calculation indicates that the size of ARRA may be smaller than the desired level, and thus a larger stimulus could have accelerated recovery from the Great Recession.
This simple policy calibration applies to the other three cases. To save space, we next focus on the currency crisis, which witnesses the largest discrepancy between SPJ and FE estimates. Based on the results in Table (ref), following a typical currency crisis, FE would call for a fiscal stimulus of approximately 4.51% of GDP, in comparison to that of 7.89% of GDP according to SPJ. There is a 3.38% gap to eliminate the potential loss of production at the peak. Developing countries experiencing a currency crisis are usually required to consolidate their fiscal expenditure. Our analysis implies that those fiscal consolidation measures may make their economies more difficult to recover from the currency crisis.
As a summary, the Nickell bias of FE estimates for financial crises is not merely a statistical concern; it also has significant implications for policymaking and real-world economic outcomes. Ignoring this bias may lead to suboptimal policy decisions that prolong economic hardship and delay recovery from severe crises.
The Nickell bias for FE is well-known in linear dynamic panel data models where lagged dependent variables serve as regressors. Yet, it has not been studied in empirical macroeconomic applications invoking the panel LP.\footnote{ In panel LP, chong2012harrod, choi2018oil and hobijn2021using are aware of the explicit Nickell bias and they suggested or used lagged dependent variables as IVs.} We show that an asymptotic bias emerges in the FE estimator for panel LP even if no lagged dependent variables are present, which is the case in many empirical applications, for example, chodorow2019macro, ottonello2020financial, and bahaj2022employment, to name a few. Furthermore, in the general specification, we illustrate that all predetermined regressors incur Nickell bias. It is imperative to correct this bias for valid asymptotic inference.
The SPJ method is capable of correcting the Nickell bias for general panel LP models. After bias correction, the standard statistical inference based on the $t$-statistic remains valid. By applying our bias correction procedure to four empirical studies on assessing the economic losses associated with financial crises, we show that FE tends to underestimate the economic contraction following financial crises in the mid- and long- runs. Our method extends beyond the literature on financial crises, and can be easily applied in other studies using panel LP.
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