The Stata Journal (yyyy)

vv, Number ii, pp. 1–12

Estimating discrete choice panel data models with interactive fixed effects Markus Eberhardt School of Economics, University of Nottingham Nottingham, United Kingdom [email protected]

Abstract. This article introduces a new Stata command, xtnlmg, which implements (nonlinear) panel time series estimators allowing for heterogeneous slope coefficients across group members and for the presence of strong cross-section dependence: the Boneva and Linton (2017, Journal of Applied Econometrics, forthcoming) Probit Common Correlated Effects (CCE) and Linear Probability Model (LPM) CCE estimators for discrete choice panel estimation. Their implementation builds on the Pesaran (2006, Econometrica 74: 967-1012) linear CCE estimator, which was introduced as part of the Stata command xtmg (Eberhardt 2012, Stata Journal 12: 61-71). Keywords: st0001, xtnlmg, panel econometrics, probit model, parameter heterogeneity, cross-sectional dependence

1

Introduction

For the past decade or so the econometric literature on macro panel data, also known as panel time series econometrics, has focused its attention on addressing strong crosssection dependence in the panel (Chudik et al. 2011). As is still not widely appreciated in the applied empirical literature, strong cross-section dependence is distinct from spatial (or weak) dependence in a simple but profound way: spatial cross-section dependence is subject to distance decay, so that any effect of such dependence is in geographical terms localised and in econometric terms irrelevant for consistent estimation. Strong dependence would arise if we assume away the element of decay, for instance a global shock with heterogeneous impact across locations. If the same global shock drives the evolution of the regressors, as is likely in many empirical applications, the setup induces endogeneity between the observable and unobservable parts of the model. Most progress has been made in the analysis of linear models adopting an unobserved common factor setup (also referred to as ‘interactive fixed effects’ or ‘multifactor error structure’) to model cross-section dependence (seminal contributions include Bai and Ng 2004; Pesaran 2006; Bai 2009) and parts of the literature has now turned to the study of nonlinear models (Chen et al. 2014; Chen 2016), such as logit or probit estimators. These types of models are applied very widely within macroeconomics, most prominently in finance and the empirical analysis of financial crises, which will be the focus of the empirical illustration below, but also in development economics, for instance in the empirical analysis of conflicts. The xtnlmg command introduced in this article c yyyy StataCorp LP

st0001

2

Discrete choice panels with interactive fixed effects

implements a linear probability model as well as a probit estimator recently developed by Boneva and Linton (2017). Their implementation builds on the heterogeneous common correlated effects (CCE) estimator by Pesaran (2006) from the linear regression literature which proxies unobservable common factors using cross-section averages of the model variables. The implementation by Boneva and Linton (2017) shares with this the simple yet powerful setup and is thus easy to use in empirical analysis, especially when panel data are unbalanced with missing observations. The asymptotics these authors develop are in both dimensions of the panel (time series and cross-section) and like the linear routines developed by Pesaran (2006) and Bai (2009) these estimators are most suitable for ‘large N , large T ’ panel datasets. The remainder of this article is organised as follows: Section 2 discusses the empirical model and introduces the implementations by Boneva and Linton (2017). Section 3 provides details on the xtnlmg command, its syntax, options and outputs, Section 4 applies the methods to investigate an early warning system approach for banking crises following the seminal paper by Schularick and Taylor (2012).

2 2.1

Heterogeneous slope estimators for discrete choice panel data models with cross-section dependence Empirical Model

The empirical setup is based on a binary choice model: Yit∗ = α0i dt + βi0 Xit + eit ,

(1)

where Yit∗ is a latent variable relating to the observed response variable, Yit , via the indicator function Yit =1(Yit∗ ). Thus Yit is equal to unity if Yit∗ > 0 and zero otherwise. dt may include observed common factors (e.g. the global oil price) and country fixed effects (when dt = 1 ∀t). So far the setup is identical to that in a standard probit or logit model, except that we do not estimate a single common slope coefficient β for the entire panel but group-specific coefficients βi . Often there will be economic arguments for this crossgroup heterogeneity, while it is also convenient from an econometric theory standpoint: Boneva and Linton (2017) show that the incidental parameter problem arising for fixed effects estimation in discrete choice models (Neyman and Scott 1948) is not present if the coefficients are heterogeneous across groups.1 Furthermore, the na¨ıve pooling of groups to improve efficiency may be counterproductive if predictive power of the model deteriorates in the process due to potential heterogeneity in the determinants across groups (Van den Berg et al. 2008). Alternative nonlinear estimators allowing 1. The incidental parameter problem results from the limited number of observations available to estimate the country-fixed effects, which are ‘nuisance’ parameters in the sense that we are typically not interested in the fixed effects themselves but what they do to the slope coefficients on the variable(s) of interest. When N rises (asymptotically) and T is fixed, the number of these nuisance parameters to be estimated grows as quickly as N , which gives rise to the asymptotic bias.

Eberhardt

3

for common factors (Chen et al. 2014; Chen 2016) assume parameter homogeneity and while conceptually more general than the Probit and LPM CCE are computationally much more demanding. The second innovation in the empirical setup adopted here is the presence of a multifactor error structure: eit = κ0i ft + εit , (2) where ft is a set of common factors with associate group-specific factor loadings κi . In contrast to the related methods for linear models (see Kapetanios et al. 2011) in the discrete choice setup we assume that these factors (as well as the regressors) are stationary processes. The factor setup is referred to as ‘interactive fixed effects’ (IFE, Bai 2009; Chen et al. 2014; Chen 2016) or ‘common correlated effects’ (CCE, Pesaran 2006) in the panel time series literature. On its own the cross-section dependence this setup induces in the model error terms (and if not addressed in the regression residuals) would merely affect the efficiency of the estimator. However, in most empirical applications it is appropriate to assume that the same common factors are not merely correlated with the response variable, but also with the regressors: Xit = A0i dt + Ki0 ft + uit .

(3)

This ubiquity of common factors induces endogeneity between the observed regressors and the error term eit , such that a na¨ıve estimator which ignores the factor structure is asymptotically biased (omitted variable bias).

2.2

Na¨ıve Probit Mean Group model

A first step is to ignore the factor structure and estimate a heterogeneous probit model, which solves the incidental parameter problem for any group-specific effects included in the model. Panel estimates and standard errors can be computed following the Pesaran and Smith (1995) Mean Group approach

2.3

The Boneva and Linton (2017) Probit CCE

The estimators (Probit CCE and LPM CCE) proposed by Boneva and Linton (2017) allow for the presence of unobserved common factors as described above which are proxied by the cross-section averages of the regressors: P r(Yit

= =

0 1 | Xit , dt , ft ) = Φ(α0i dt + βi0 Xit + κ0i ft ) ˜ 0 κi ]dt + βi0 Xit + ψi0 X t ), Φ([αi + K

(4)

˜ 0 κi ] where ψ refers to group-specific coefficients on the cross-section averages and [αi + K highlights that the intercept terms no longer have the same interpretation as in an unaugmented na¨ıve probit model. The empirical setup assumes that the cross-section averages of the regressors Xit and the observed factor(s) dt span the unobserved factors, which means we assume

4

Discrete choice panels with interactive fixed effects

that the regressors are driven by the same common factors as the response variable Yit . Furthermore it is assumed that the number of unobserved common factors does not exceed the number of regressors in the model, and that all of these are stationary. Like in the linear heterogeneous model (Pesaran and Smith 1995) the ‘Mean Group’ Probit-CCE estimator βˆM G of the slope parameters is then obtained as an average of the country coefficients.2 Conveniently, the standard errors for βˆM G can be computed in identical fashion (non-parametrically as the variation of the country estimates around the cross-country mean) to those for the linear estimator. Each estimate βˆi is associated with a marginal effect (βi ), and the xtnlmg routine reports average marginal effects and related standard errors using the above Mean Group principle.

2.4

Boneva and Linton (2017) LPM CCE

The linear probability model version of the estimator is produced in an identical fashion to the Probit CCE model, except that the implementation is a simple ordinary least squares regression at the group level. The LPM can clearly be motivated from the linear common correlated effects estimator (Pesaran 2006).

3

The xtnlmg command

3.1

Syntax

xtnlmg depvar



indepvars



if



in



, cce lpm trend robust caall nomarg  noconstant level(#) res(varname) pred(varname) obspred(varname)

3.2

Options

A na¨ıve Probit Mean Group estimator is set as the default. cce implements the Boneva and Linton (2017) Probit CCE estimator. The regression output by default computes the average marginal effects – see option nomarg below for the probit output. By default the empirical results are based on adopting the cross-section averages computed from the variables of those groups for which a Probit CCE can be computed – see option caall for alternatives. lpm implements a linear probability model based on the Pesaran (2006) Common Correlated Effects estimator. Since the underlying model is linear the regression results presented are (by construction) average marginal effects averaged across panel groups. trend specifies each group-specific regression to be augmented with a linear trend term. 2. As an option the xtnlmg routine allows for the use of robust regression to compute the means, which is the standard practice in the application of linear CCE models.

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robust specifies the use of the rreg command to construct the coefficient averages across N panel members reported (see Hamilton 1992, for details). This puts less emphasis on outliers in the computation of the average coefficient. The default is unweighted averages. caall Individual groups are dropped from the panel when (a) their dependent variable is always 0 or always 1, or (b) when the regressor(s) predict the outcome perfectly in the group regression. By default the cross-section averages included in the Probit CCE are only computed from those groups which are not dropped from the analysis. Since the cross-section averages capture unobservable factors it may be preferable to construct these averages from all the available data instead, which can be enabled with this option. nomarg By default average marginal effects are presented as the results for the na¨ıve Probit MG and Probit CCE estimators. With this option the results presented are the original probit estimates. This option does not work with the LPM CCE. noconstant suppresses the constant term. This is generally not recommended. level(#) specifies the confidence level, as a percentage, for confidence intervals. The default is level(95) or as set by set level; see [U] 23.5 Specifying the width of confidence intervals. res(varname) provides Pearson residuals for the model which are stored in varname. These can then be subjected to diagnostic tests, including testing for cross-section dependence (see xtcd if installed), which in the present case of a nonlinear model is discussed in Hsiao et al. (2012). Note that the residual series constructed are not based on the linear prediction of the averaged MG estimates but are derived from the group-specific regressions. This is similar to the post-estimation command predict with the option group(varname) in the Random Coefficient Model estimator xtrc, although in the latter this only allows predicted values but not residuals to be computed. pred(varname) provides predicted values which are stored in varname. These series are again based on the linear prediction of the group-specific regressions. These predicted values should be used to investigate the predictive power of the model in form of the AUROC (area under the receiver operating characteristic curve) statistic – roctab and roccomp. obspred(varname) provides predicted values for a Probit CCE or LP¡ CCE model where the predictions are only based on the observable variables (note that the group-specific constant is also not included). This allows for a comparison of the AUROC statistic for the full model with that for the observed variables only. The option only works for the Probit CCE (option cce) and LPM CCE (option lpm) with resulting predictions stored in varname.

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Discrete choice panels with interactive fixed effects

3.3

Saved results

The xtnlmg routine environment saves the following information to e(): Scalars e(N)

number of observations

e(g min)

e(N g)

number of groups

e(g max)

e(sigma)

mean squared error

e(g avg)

e(df m)

model degrees of freedom

e(chi2)

lowest number of observations in an included group highest number of observations in an included group average number of observations in an included group Wald chi-squared statistic

Macros e(cmd) e(depvar) e(title2)

xtnlmg dependent variable estimator selected

e(ivar) e(tvar)

group (panel) variable time variable

Matrices e(b)

coefficient vector

e(V)

group-specific regression

e(deviance)

variance–covariance matrix of the estimators deviance statistic for each group perfectly identified panel members dropped (vector)

e(betas)

unidentified panel members dropped (vector) e(includedid) vector indicating panel members included e(unident)

Functions e(sample)

4

e(pident)

marks estimation sample

Empirical Example: early warning system for systemic banking crises

The 2007/8 Global Financial Crisis (GFC) triggered a renewed debate on the relationship between banks and their lending on the one hand and banking crises on the other. Economic historians have contributed significantly to this debate given the nature of banking crises in advanced economies as ‘rare events.’ The construction of long panel datasets (Reinhart and Rogoff 2009; Jord`a et al. 2017) allowed for the empirical analysis of the role played by credit booms over a long time horion, returning to earlier theories by Minsky and Kindleberger. The article by Schularick and Taylor (2012, henceforth ST) is a seminal contribution to this vibrant literature, which cemented the “new (near) consensus” (Bordo and Meissner 2016, 386) of financial crises as ‘credit booms gone bust.’ Their empirical analysis is based on a panel of 14 countries over the 1870-2010 period, excluding the years of and immediately after the first and second world wars as well as observations for ‘ongoing crises’ (years in crisis after the crisis start year). This literature employs ‘early warning systems’ where dummies for the start year of a banking crisis are regressed on lags of financial and monetary characteristics. In this empirical illustration I focus on the analysis of real credit growth (loans by banks to

Eberhardt

7

private non-financial firms), which enters the empirical models as a lag polynomial from t − 1 to t − 5. The dataset must be tsset before use. The first two columns of Table 1 replicate the results from pooled logit models with and without country fixed effects in ST (Table 3, columns [4] and [5]). The upper panel shows the raw logit output, in a lower panel I provide the sum of the five lag coefficients along with average marginal effects. ST put a lot of emphasis on the predictive power of their empirical model, which they capture in form of the AUROC statistic (area under the receiver operating characteristic curve) – a value of 0.5 would make the predictive power as sound as that of a coin flip, statistically significant higher values (with a maximum of unity) indicate improved predictive power. The logit fixed effects model in column [2] achieves an AUROC value of 0.717. ST adopt this specification as their benchmark model and show in additional analysis that introducing additional variables does not improve this power of prediction. Columns [3] to [6] report results from estimators implement using xtnlmg: a na¨ıve probit mean group model in [3], a CCE linear probability model in [4] and Probit-CCE models in [5] and [6], where the latter differ by the sample of countries used in computing the cross-section averages (ten in the former, fourteen in the latter) – in all cases the results reported are robust means across countries. The code for these models is as follows (output is only presented for the Probit-CCE model to save space): . global credit "l1_dlloansr l2_dlloansr l3_dlloansr l4_dlloansr l5_dlloansr" . tsset ccode year . xtnlmg crisisST $credit, robust pred(pMG) [... Output omitted ...] . xtnlmg crisisST $credit, robust lpm pred(pLPM) obspred(obspLPM) [... Output omitted ...] . xtnlmg crisisST $credit, robust cce pred(pCCE) obspred(obspCCE) Boneva & Linton (2017) Discrete Choice Common Correlated Effects estimator (based on GLM-Probit model) All coefficients presented are average margins across groups Averages computed as outlier-robust means (using rreg) Mean Group type estimation Group variable: ccode

Number of obs Number of groups

= =

915 10

min = avg = max =

76 91.5 108

= =

. .

Obs per group:

chi2(.) Prob > chi2

--------------------------------------------------------------------------------crisisST | Coef. Std. Err. z P>|z| [95% Conf. Interval] ----------------+----------------------------------------------------------------

8

Discrete choice panels with interactive fixed effects l1_dlloansr | .1512436 .1360916 1.11 0.266 -.115491 .4179782 l2_dlloansr | .5131053 .2112007 2.43 0.015 .0991596 .927051 l3_dlloansr | -.1320237 .1537542 -0.86 0.391 -.4333765 .169329 l4_dlloansr | .1751943 .1260038 1.39 0.164 -.0717686 .4221572 l5_dlloansr | .2136913 .0790898 2.70 0.007 .0586781 .3687045 l1_dlloansr_avg | .4647465 .3575735 1.30 0.194 -.2360848 1.165578 l2_dlloansr_avg | -.5477375 .3681889 -1.49 0.137 -1.269375 .1738996 l3_dlloansr_avg | .9976566 .4050769 2.46 0.014 .2037205 1.791593 l4_dlloansr_avg | -.7083833 .4888262 -1.45 0.147 -1.666465 .2496983 l5_dlloansr_avg | .0085689 .3736268 0.02 0.982 -.7237262 .740864 --------------------------------------------------------------------------------Cross-section averaged regressors in the CCE model marked by the suffix avg. Linear predictions are stored in variable: pCCE Linear predictions for observed variables only are stored in variable: obspCCE Omitted panel members: If the dependent variable does not change over time, then no relationship can be identified. This is the case for 0 panel member(s), which is/are dropped from the sample: none omitted We also drop panel members when outcomes are perfectly predicted, which is the case in 4 panel member(s): 1 2 3 11 The sample size is thus N=10. IDs for these are contained in e(includedid). The estimated deviance for N=14 countries is provided in e(deviance). Collinear variables: These (e.g. time-invariant) covariates are automatically dropped in the groupspecific regressions. The number of groups identified for each covariate is: l1_dlloansr 10 out of a possible 10 groups. l2_dlloansr 10 out of a possible 10 groups. l3_dlloansr 10 out of a possible 10 groups. l4_dlloansr 10 out of a possible 10 groups. l5_dlloansr 10 out of a possible 10 groups.

The regression output contains a lot of additional information: if the dependent variable does not vary over time in group j, this group is dropped from the analysis. Similarly, if the model for group j is perfectly predicted, this group is dropped. In both cases the routine reports the groups which have been dropped, using the group code used for setting the cross-section dimension of the panel with tsset. Finally, if variable k turned out to be time-invariant for group j, this variable is dropped from the model as indicated at the very end of the above output. Focusing on the sum of average marginal effects (MFX) it appears that these are very much higher than in the pooled logit models, in excess of three times higher in the Probit-CCE models. The Probit-CCE model in [5] can only be implemented for ten of the fourteen sample countries since outcomes are perfectly predicted for four countries.3 Naturally, it is conceivable that the substantial rise in average marginal effects is due to this differential sample. I therefore re-estimate all other models with N = 10 and report the sum of marginal effects in a separate row (results for [5] and [6] are naturally unchanged): although the LPM-CCE shows a significant increase in average marginal effects, the two pooled model results are almost unchanged, such that the substantially 3. These countries are Australia (122 observations, 2 crises), Canada (121, 3), Switzerland (120, 4) and the Netherlands (76, 2).

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9

higher average marginal effects in heterogeneous CCE models is not driven by sample selection. The AUROC statistics further highlight a substantial improvement in predictive power moving from pooled to heterogeneous models, rising from around 0.7 to 0.9. I find that these AUROC statistics are statistically significantly different from the preferred ST model in [2]. Figure 1 plots the ROC for models in [2]-[5] along with statistical tests. . roccomp crisisST temp2 pMG . roccomp crisisST temp2 pLPM . roccomp crisisST temp2 pCCE . roccurve crisisST temp2 pMG pLPM pCCE, title("") lw(medthick) lcolor(black gs3 /// gs9 gs13) lpattern(shortdash solid solid solid) legend(title("") /// order(1 "Logit FE (ST), AUROC = 0.717" 2 "Naive Heterog. Probit, AUROC = 0.842" /// 3 "LPM CCE, AUROC = 0.895" 4 "Probit CCE, AUROC = 0.914") ring(0) rows(4) pos(5) /// region(fcolor(none) lcolor(none))) xlabel(#5, tposition(inside) nogrid) /// ylabel(#5, tposition(inside)) xsize(5) ysize(5) scheme(s1color) /// xmtick(, tposition(inside)) ymtick(, tposition(inside)) /// xtitle("False Positive Rate", size(3) justification(center) margin(b=0)) /// ytitle("True Positive Rate (% predicted)", size(3) justification(center) /// orientation(vertical)) xline(0(.2)1,lstyle(grid) lw(.02) lcolor(white)) /// yline(0(.2)1,lstyle(grid) lw(.02) lcolor(white))

One important contribution of the nonlinear CCE estimators is the ability to capture unobserved time-varying heterogeneity. In a lower panel of Table 1 I produce AUROC statistics based on linear predictions from the respective model results but limited to observable variables (i.e. excluding estimates on intercepts and on cross-section averages). In order to compare like with like I report the AUROC based on linear predictions from a model without country fixed effects for the ST benchmark model in [2]. Note that based on these statistics the CCE models, with the exception of the variant in [6], still perform better (in terms of predictive power) than the pooled logit models.

5

Acknowledgements

This routine acts as a wrapper for the generalised linear model (glm) routine. It builds on my earlier linear mean group routine (xtmg). Any remaining errors are my own.

6

References

Bai, J. 2009. Panel Data Models with Interactive Fixed Effects. Econometrica 77(4): 1229–1279. Bai, J., and S. Ng. 2004. A PANIC attack on unit roots and cointegration. Econometrica 72(4): 191–221. Van den Berg, J., B. Candelon, and J.-P. Urbain. 2008. A cautious note on the use of panel models to predict financial crises. Economics Letters 101(1): 80–83. Boneva, L., and O. Linton. 2017. A discrete-choice model for large heterogeneous panels with interactive fixed effects with an application to the determi-

10

Discrete choice panels with interactive fixed effects Table 1: Banking crisis prediction [1] Logit None None n/a Pooled

[2] Logit FE Country None n/a Pooled

[3] Probit Country None n/a Heterog

[4] LPM-CCE Country Included 14 Heterog

[5] Probit-CCE Country Included 10 Heterog

[6] Probit-CCE Country Included 14 Heterog

1st Lag

-0.257 (2.511)

-0.398 (2.568)

0.478 (1.529)

-0.077 (0.135)

2.014 (2.040)

3.698 (2.439)

2nd Lag

6.956 (2.309)***

7.138 (2.569)***

5.593 (1.438)***

0.446 (0.107)***

5.476 (2.652)**

7.664 (3.180)**

3rd Lag

1.079 (3.175)

0.888 (3.326)

0.955 (1.623)

-0.077 (0.083)

-1.756 (2.458)

-2.771 (2.566)

4th Lag

0.290 (1.370)

0.203 (1.446)

0.667 (1.633)

0.070 (0.067)

2.839 (2.386)

3.546 (1.861)*

5th Lag

2.035 (0.767)***

1.867 (0.805)**

2.640 (1.171)**

0.156 (0.067)**

3.368 (1.256)***

6.077 (1.997)***

error

10.103 (3.117)***

9.697 (3.476)***

10.333 (3.328)***

0.518 (0.214)**

11.942 (4.951)**

18.214 (5.486)**

error

0.348 (0.080)***

0.301 (0.096)***

0.735 (0.224)***

0.518 (0.214)**

0.921 (0.330)***

1.109 (0.356)***

(N = 10)\ error

0.374 (0.113)***

0.365 (0.126)***

0.842 (0.270)***

0.729 (0.283)***

0.921 (0.330)***

1.109 (0.356)***

0.673 0.036 0.103

0.717 0.035 n/a

0.842 0.024 0.000

0.895 0.020 0.000

0.914 0.016 0.000

0.911 0.015 0.000

0.752 0.036 0.045

0.745 0.036 0.062

0.709 0.037 0.714

1,272 14 90.9 53

915 10 91.5 46

915 10 91.5 46

Estimator Fixed Effects Cross-Section Averages N for CA Specification ∆ln(Real Credit)

P

lags Standard P MFX] Standard P MFX] Standard

AUROC Standard error Comparison‡ (p) AUROC (observables) Standard error Comparison‡ (p) Observations (n) Countries (N ) Average T Number of Crises

0.673 0.036 n/a 1,272 14 90.9 53

1,272 14 90.9 53

1,092 12 91.0 50

Notes: Standard errors in parentheses – logit ones are heteroskedasticity-robust, those in [3] to [6] are based on Boneva and Linton (2017). Statistical significance at the 10%, 5% and 1% level indicated using *, ** and ***, respectively. ] Marginal effects are evaluated at the mean. Results presented in columns [1] and [2] are replications of ST (Table 3, Columns [4] and [5]). \ Since the Probit-CCE model is only estimable for ten of the fourteen sample countries we re-estimate the sum of marginal effects for all other models using the same ten-country sample only. ‡ ‘Comparison’ reports the p-value for a test of common AUROC with the respective model and that of a pooled logit model in [2] – the focus is on the full model and separately on observed variables (‘observables’: lagged credit growth) as indicated. Rejection of the test implies that these models have statistically significantly higher predictive power than the model preferred by ST in [2].

Eberhardt

11 1

True Positive Rate (% predicted)

.8

.6

.4

Logit FE (ST), AUROC = 0.717

.2

Naive Heterog. Probit, AUROC = 0.842 LPM CCE, AUROC = 0.895 Probit CCE, AUROC = 0.914

0 0

.2

.4

.6

.8

1

False Positive Rate

Figure 1: ROC plots for selected models nants of corporate bond issuance. Journal of Applied Econometrics forthcoming. http://dx.doi.org/10.1002/jae.2568. Bordo, M. D., and C. M. Meissner. 2016. Fiscal and financial crises. Handbook of Macroeconomics 2: 355–412. Chen, M. 2016. Estimation of nonlinear panel models with multiple unobserved effects. Chen, M., I. Fernandez-Val, and M. Weidner. 2014. Nonlinear panel models with interactive effects. arXiv preprint arXiv:1412.5647 . Chudik, A., M. H. Pesaran, and E. Tosetti. 2011. Weak and Strong Cross Section Dependence and Estimation of Large Panels. Econometrics Journal 14(1): C45–C90. Eberhardt, M. 2012. Estimating panel time-series models with heterogeneous slopes. Stata Journal 12(1): 61. Hamilton, L. C. 1992. How Robust is Robust Regression? Stata Technical Bulletin 1(2).

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Discrete choice panels with interactive fixed effects

Hsiao, C., M. H. Pesaran, and A. Pick. 2012. Diagnostic Tests of Cross-section Independence for Limited Dependent Variable Panel Data Models. Oxford Bulletin of Economics and Statistics 74(2): 253–277. ` M. Schularick, and A. M. Taylor. 2017. Macrofinancial history and the new Jord` a, O., business cycle facts. NBER Macroeconomics Annual 31(1): 213–263. Kapetanios, G., M. H. Pesaran, and T. Yamagata. 2011. Panels with Nonstationary Multifactor Error Structures. Journal of Econometrics 160(2): 326–348. Neyman, J., and E. L. Scott. 1948. Consistent estimates based on partially consistent observations. Econometrica: Journal of the Econometric Society 1–32. Pesaran, M. H. 2006. Estimation and inference in large heterogeneous panels with a multifactor error structure. Econometrica 74(4): 967–1012. Pesaran, M. H., and R. P. Smith. 1995. Estimating long-run relationships from dynamic heterogeneous panels. Journal of Econometrics 68(1): 79–113. Reinhart, C. M., and K. S. Rogoff. 2009. This Time is Different: Eight Centuries of Financial Folly. Princeton University Press. Schularick, M., and A. M. Taylor. 2012. Credit booms gone bust: monetary policy, leverage cycles, and financial crises, 1870–2008. The American Economic Review 102(2): 1029–1061. About the authors Markus Eberhardt is an associate professor in the School of Economics, Univerity of Nottingham, a research affiliate at the Centre for Economic Policy Research (CEPR), and a research associate at the Centre for the Study of African Economies (CSAE), Department of Economics, University of Oxford.