Estimation of Panel Data Models with Random Interactive Effects and Multiple Structural Breaks when T is Fixed

Abstract

In this article, we propose a new estimator of panel data models with random interactive effects and multiple structural breaks that is suitable when the number of time periods, T, is fixed and only the number of cross-sectional units, N, is large. This is done by viewing the determination of the breaks as a shrinkage problem, and to estimate both the regression coefficients, and the number of breaks and their locations by applying a version of the Lasso approach. We show that with probability approaching one the approach can correctly determine the number of breaks and the dates of these breaks, and that the estimator of the regime-specific regression coefficients is consistent and asymptotically normal. We also provide Monte Carlo results suggesting that the approach performs very well in small samples, and empirical results suggesting that while the coefficients of the controls are breaking, the coefficients of the main deterrence regressors in a model of crime are not.

Keywords:

• Common factors
• Interactive effects
• Lasso
• Panel data
• Structural change

1 Introduction

Dealing with structural breaks is an important step in most, if not all, empirical economic research. This is particularly true in panel data comprised of many cross-sectional units, such as individuals, firms or countries, which are all affected by major economic events. The worry is that if left unattended, existing breaks will manifest themselves as omitted variables, leading to inconsistent estimates of the slope coefficients of the model. It is therefore, important to know if and when structural breaks have occurred. Of course, such knowledge is rarely available in practice, which means that it has to be inferred from the data. We need to be able to check if there are any breaks present and, if there are, to infer both the break dates and the regime-specific slope coefficients. This should be possible even if the number of time periods, T, is fixed and only the number of cross-sectional units, N, is large, as many economic datasets have this “short” form. The procedure should also be easy to implement, it should not require data to be stationary, and it should be robust to unobserved heterogeneity. This last requirement is potentially very important because unattended heterogeneity can be mistaken for structural breaks. The current article contributes by developing a procedure that meets the above list of demands.

While the literature concerned with structural breaks in time series is huge, the literature concerned with such breaks in panel data is much smaller (see Boldea, Drepper, and Gan 2020, for a recent overview). Yet, panel data are particularly susceptible to structural change. One reason for this is that the sample frequency is usually much lower than in pure series data. Panel datasets therefore, tend to have long time spans, which means that the assumption of constant coefficients is likely to be violated because of major economic events. Another reason is that while T is usually quite small, because the number of cross-sectional units for which time series data is readily available is ever-increasing, N is potentially very large. This is important because the larger is N, the higher the risk that at least some of the cross-sectional units are subject to structural change. A related issue is how many breaks there are. If the literature on structural breaks in panel data is sparse, the part of the literature that deals with an unknown number of breaks is almost nonexistent.¹ The only exceptions known to us are Boldea, Drepper, and Gan (2020), Li, Qian, and Su (2016), and Qian and Su (2016), where the last two studies assume that both N and T are large, which is again something that we would like to avoid in the present article.²

Another prominent feature of the type of disaggregated “micro” panel data that we have in mind, where typically the regressors explain only a small fraction of the variation in the dependent variable, is the presence of unobserved heterogeneity. Studies such as Ahn, Lee, and Schmidt (2013), Bai (2009), Moon and Weidner (2015), Pesaran (2006), Robertson and Sarafidis (2015), and Westerlund, Petrova, and Norkute (2019) allow for unobserved heterogeneity in the form of interactive effects that are handled with by using either some kind of “de-factoring” or generalized method of moments (GMM); however, they do not allow for breaks and many assume that T is large. Li, Qian, and Su (2016) allow for both multiple breaks and interactive effects, but then again in their article T is large. Boldea, Drepper, and Gan (2020) allow for interactive effects without for that matter requiring any correction thereof. This makes their approach very simple, although at a cost in terms of additional restrictive conditions. In particular, it is assumed that the omitted variables bias caused by the omitted interactive effects is time-invariant, up to the breakpoints, which limits the type of effects and regressors that can be permitted.

The proposed methodology builds upon the so-called “adaptive group fused” Lasso approach of Li, Qian, and Su (2016), and Qian and Su (2016), which is suitable when the variation in the slopes has a natural ordering, as when time-stamped like in the current article. However, because in our setup T is fixed, we cannot use principal components as a means to purge the interactive effects as in Li, Qian, and Su (2016). In fact, a major complication when T is fixed is that we cannot easily separate the breaks from the effects. Qian and Su (2016) transform their data by taking first-differences before applying Lasso, which is expected to work also when T is fixed. However, differencing can only handle time-invariant effects. Moreover, while differencing solves the separation problem, it does so in an awkward way, since the (time-varying) slope coefficients in the model for the data in differences are not the same as for the data in levels.

The approach used in the present article can be seen as a reaction to the discussion of the last paragraph. The idea is to apply Lasso to cross-sectionally demeaned data. The demeaning does not affect the slopes and it makes the resulting estimator, henceforth referred to as “post-demeaned Lasso least squares (LS),” “PDL2S” for short, robust to interactive effects, provided that they satisfy a certain random coefficient condition; hence, the term “random interactive effects.”³ Another advantage of the new procedure is that it puts almost no assumptions on the structure of the breaks. In fact, there can be no breaks at all, and if there are breaks present the procedure does not make any assumptions about their number. The procedure is therefore, valid even if some, or indeed all, regimes have a single observation, which is very useful when wanting to detect a break as quickly as possible. Yet another advantage is that the procedure does not place any conditions on the serial correlation properties of the data. Hence, the data can be stationary, as required in the bulk of the previous literature (see Baltagi, Kao, and Liu 2017, for a discussion), but it does not have to be.

The rest of the article is organized as follows. Section 2 describes the model and the PDL2S approach that we will use to estimate it. Section 3 reports our main asymptotic results, whose accuracy in small samples is evaluated by means of Monte Carlo simulation in Section 4. Section 5 presents the results of a small empirical illustration using as an example the economics of crime. Section 6 concludes. All proofs and theoretical results of secondary nature are provided in the supplementary materials.

2 Model and Estimator

Consider a scalar panel data variable $y_{i,t}$, observable across $t=1,\dots ,T$ time periods and $i=1,\dots ,N$ cross-section units. The data generating process of this variable is given by(2.1) $$y_{i,t}=x_{i,t}^{\prime }{\beta }_t+u_{i,t},$$(2.1) (2.2) $$u_{i,t}={\lambda }_i^{\prime }{f}_t+{\epsilon }_{i,t},$$(2.2) where $x_{i,t}$ is a $p\times 1$ vector of known regressors with β_t being a conformable vector of unknown slope coefficients that we allow to change over time, and $u_{i,t}$ is a composite error term that can be both serially and cross-sectionally correlated in a very general fashion. The assumption we make is that $u_{i,t}$ admits to a common factor structure in which f_t and λ_i are $r\times 1$ vectors of unobserved factors and loadings, respectively, and ${\epsilon }_{i,t}$ is a mean zero error term.⁴ The interactive effects are here given by ${\lambda }_i^{\prime }{f}_t$. Any cross-sectional dependence in $u_{i,t}$ is assumed to be captured by these effects, so that the remainder, ${\epsilon }_{i,t}$, is completely idiosyncratic. As usual, r and p are assumed to be fixed numbers.

We will assume that ${\beta }_1,\dots ,{\beta }_T$ takes on m + 1 distinct vectors ${\alpha }_1,\dots ,{\alpha }_{m+1}$, such that(2.3) $${\beta }_t={\alpha }_j$$(2.3) for $t=T_{j-1},\dots ,T_j-1,j=1,\dots ,m+1,m\in [0,T-1],T_0=1$ and $T_{m+1}=T+1$. Hence, in this model, β_t has m + 1 distinct regimes, or m breaks, that occur at time $T_1,\dots ,T_m$. At the one end of the scale, we have m = 0, in which case there is only one regime and ${\beta }_1=\cdots ={\beta }_T={\alpha }_1$, whereas, at the other end, $m=T-1$, which means that there are as many regimes as time periods, and hence ${\beta }_t={\alpha }_t$ for all $t=1,\dots ,T$. It is useful to stack ${\alpha }_1,\dots ,{\alpha }_{m+1}$ and ${\beta }_1,\dots ,{\beta }_T$ into the $(m+1)p\times 1$ and $Tp\times 1$ vectors $A_m=[{\alpha }_1^{\prime },\dots ,{\alpha }_{m+1}^{\prime }]\prime$ and $B_T=[{\beta }_1^{\prime },\dots ,{\beta }_T^{\prime }]\prime$, respectively, and to denote by ${\mathcal{T}}_m=\{T_1,\dots ,T_m\}$ the set of breakpoints when m > 0. If m = 0, then we define ${\mathcal{T}}_m={\mathcal{T}}_0=\varnothing$ (the empty set). It is also useful to note that if $m+1=T$, so that each regime contains only one observation, then the set of breakpoints is given by ${\mathcal{T}}_{T-1}=\{1,\dots ,T\}$. In what follows, we will therefore, use ${\mathcal{T}}_{T-1}$ to denote the full set of time series observations.

Remark 1.

The model in (2.1) supposes that β_t is cross-section invariant, which can be restrictive (see e.g., Baltagi, Griffin, and Xiong 2000). Cross-section invariance is, however, not strictly necessary for our approach to work. Of course, with T fixed, the cross-sectional variation cannot be completely unrestricted. But then this is also not necessary, as in empirical work in economics (and elsewhere) we are typically only interested in the average marginal effect. In the supplementary materials, we therefore, consider a random coefficient extension of (2.1) in which the effect of $x_{i,t}$, denoted ${\beta }_{i,t}$, is allowed to vary over both i and t but where the mean effect is given by $E({\beta }_{i,t})={\beta }_t$. The proposed PDL2S procedure works well even in this case.

Remark 2.

The fact that the number of time periods within each regime is completely unrestricted is noteworthy because in the existing literature it is standard to assume that the break regimes are expanding with T (see e.g., Baltagi, Feng, and Kao 2016). There is also no need to truncate the sample endpoints, and in this way restrict the breakpoint to the middle of the sample, which is again standard in the literature. This means that breaks can be detected very quickly.

The goal of this article is to infer A_m and ${\mathcal{T}}_m$. Let us therefore, denote by $A_{m^0}^0=[{\alpha }_1^{0\prime },\dots ,{\alpha }_{m^0+1}^{0\prime }]\prime$ the true value of A_m , where m⁰ is the true value of m. The set of true breakpoints is henceforth denoted ${\mathcal{T}}_{m^0}^0=\{T_1^0,\dots ,T_{m^0}^0\}$. It is also useful to introduce $B_T^0=[{\beta }_1^{0\prime },\dots ,{\beta }_T^{0\prime }]\prime$ as the true value of B_T .

Denote by ${\overline{a}}_t=N^{-1}{\sum }_{i=1}^N{a}_{i,t}$ the cross-sectional average of any variable $a_{i,t}$, and let ${\tilde{a}}_{i,t}=a_{i,t}-{\overline{a}}_t$ be the cross-sectionally demeaned version of $a_{i,t}$. In this notation, (2.1) can be written as(2.4) $${\tilde{y}}_{i,t}={\tilde{x}}_{i,t}^{\prime }{\beta }_t+{\tilde{u}}_{i,t}.$$(2.4)

Cross-sectional demeaning is tantamount to demeaning with respect to common time effects. It is important to point out, however, that while we do allow for common time effects, our model does not necessarily include such effects. Hence, unless ${\lambda }_i=\lambda$ for all i, so that ${\lambda }_i^{\prime }{f}_t=\lambda \prime f_t$, the model is misspecified. Demeaning is still key, though, as it enables us to eliminate the mean of λ_i from the regression error in (2.4), which is enough to ensure consistency and asymptotic (mixed) normality as long as the remaining part is uncorrelated with ${\tilde{x}}_{i,t}$.

To estimate $B_T^0$, we propose minimizing the following objective function:(2.5) $${\mathcal{l}}_{\gamma }(B_T)=\frac{1}{N}\sum _{i=1}^N\sum _{t=1}^T{({\tilde{y}}_{i,t}-{\tilde{x}}_{i,t}^{\prime }{\beta }_t)}^2+\gamma ·\sum _{t=2}^T{w}_t\left|\right|{\beta }_t-{\beta }_{t-1}\left|\right|,$$(2.5) where $\gamma =\gamma (N)>0$ is a tuning parameter, w_t is a data-driven weight defined by $w_t=\left|\right|{\dot{\beta }}_t-{\dot{\beta }}_{t-1}|{|}^{-\kappa },\kappa >0$ is a user-specified constant, and ${\dot{\beta }}_t$ is a preliminary estimator of β_t , which is obtained by minimizing the first term in ${\mathcal{l}}_{\gamma }(B_T)$. That is, ${\dot{\beta }}_t$ is simply the period-by-period LS estimator;(2.6) $${\dot{\beta }}_t={\left(\sum _{i=1}^N{\tilde{x}}_{i,t}{\tilde{x}}_{i,t}^{\prime }\right)}^{-1}\sum _{i=1}^N{\tilde{x}}_{i,t}{\tilde{y}}_{i,t}.$$(2.6)

Simple as it may be, it is useful to be able to write this estimator in a more general notation. Let us therefore, introduce(2.7) $$Q_N({\mathcal{T}}_m)=\text{diag}\left(\mathrm{}\frac{1}{N}\sum _{t=T_0}^{T_1-1}\sum _{i=1}^N{\tilde{x}}_{i,t}{\tilde{x}}_{i,t}^{\prime },\dots ,\frac{1}{N}\sum _{t=T_m}^{T_{m+1}-1}\sum _{i=1}^N{\tilde{x}}_{i,t}{\tilde{x}}_{i,t}^{\prime }\mathrm{}\right)\mathrm{},$$(2.7) (2.8) $$R_N({\mathcal{T}}_m)=\left[\begin{array}{c}\frac{1}{N}\sum _{t=T_0}^{T_1-1}\sum _{i=1}^N{\tilde{x}}_{i,t}{\tilde{y}}_{i,t}\\ ⋮\\ \frac{1}{N}\sum _{t=T_m}^{T_{m+1}-1}\sum _{i=1}^N{\tilde{x}}_{i,t}{\tilde{y}}_{i,t}\end{array}\right],$$(2.8) whose dimensions are given by $(m+1)p\times (m+1)p$ and $(m+1)p\times 1$, respectively. These quantities are well defined not only when m > 0 but also when m = 0, in which case $T_{m+1}=T_1=T+1$, and hence $Q_N({\mathcal{T}}_0)=N^{-1}{\sum }_{t=1}^T{\sum }_{i=1}^N{\tilde{x}}_{i,t}{\tilde{x}}_{i,t}^{\prime }$ and $R_N({\mathcal{T}}_0)=N^{-1}{\sum }_{t=1}^T{\sum }_{i=1}^N{\tilde{x}}_{i,t}{\tilde{y}}_{i,t}$. We also note how $Q_N({\mathcal{T}}_{T-1})=\text{diag}(N^{-1}{\sum }_{i=1}^N{\tilde{x}}_{i,1}{\tilde{x}}_{i,1}^{\prime },\dots ,N^{-1}{\sum }_{i=1}^N{\tilde{x}}_{i,T}{\tilde{x}}_{i,T}^{\prime })$ and $R_N({\mathcal{T}}_{T-1})=[N^{-1}{\sum }_{i=1}^N{\tilde{x}}_{i,1}\prime {\tilde{y}}_{i,1},\dots ,N^{-1}{\sum }_{i=1}^N{\tilde{x}}_{i,T}\prime {\tilde{y}}_{i,T}]\prime$. In this notation,(2.9) $${\dot{B}}_T=\left[\begin{array}{c}{\dot{\beta }}_1\\ ⋮\\ {\dot{\beta }}_T\end{array}\right]=\left[\begin{array}{c}{\dot{\beta }}_1({\mathcal{T}}_{T-1})\\ ⋮\\ {\dot{\beta }}_T({\mathcal{T}}_{T-1})\end{array}\right]=Q_N{({\mathcal{T}}_{T-1})}^{-1}{R}_N({\mathcal{T}}_{T-1}).$$(2.9)

The proposed PDL2S estimator of $B_T^0$ is given by(2.10) $${\widehat{B}}_T=\left[\begin{array}{c}{\widehat{\beta }}_1\\ ⋮\\ {\widehat{\beta }}_T\end{array}\right]=\arg \underset{{}_{B_T}}{\min }{\mathcal{l}}_{\gamma }(B_T),$$(2.10) where the dependence on γ here is suppressed for notational simplicity. For a given ${\widehat{B}}_T$, the set of estimated breaks is given by ${\widehat{\mathcal{T}}}_{\widehat{m}}=\{{\widehat{T}}_1,\dots ,{\widehat{T}}_{\widehat{m}}\}$, where ${\widehat{T}}_1<\cdots <{\widehat{T}}_{\widehat{m}}$ for $\widehat{m}>0$ are such that $\left|\right|{\widehat{\beta }}_t-{\widehat{\beta }}_{t-1}\left|\right|\ne 0$ for $t={\widehat{T}}_1,\dots ,{\widehat{T}}_{\widehat{m}}$. If $\left|\right|{\widehat{\beta }}_t-{\widehat{\beta }}_{t-1}\left|\right|=0$ for all $t=1,\dots ,T$, then $\widehat{m}=0$ and ${\widehat{\mathcal{T}}}_{\widehat{m}}={\widehat{\mathcal{T}}}_0=\varnothing$. We also define ${\widehat{T}}_0=1$ and ${\widehat{T}}_{\widehat{m}+1}=T+1$. The set ${\widehat{\mathcal{T}}}_{\widehat{m}}$ divides the sample into $\widehat{m}+1$ regimes such that the parameter estimates remain constant within each regime. The proposed estimator ${\widehat{A}}_{\widehat{m}}$ of $A_{m^0}^0$ is obtained by PDL2S, which is regime-by-regime LS conditional on ${\widehat{\mathcal{T}}}_m$.⁵ In terms of the notation introduced earlier,(2.11) $${\widehat{A}}_{\widehat{m}}=\left[\begin{array}{c}{\widehat{\alpha }}_1\\ ⋮\\ {\widehat{\alpha }}_{\widehat{m}+1}\end{array}\right]=\left[\begin{array}{c}{\widehat{\alpha }}_1({\widehat{\mathcal{T}}}_{\widehat{m}})\\ ⋮\\ {\widehat{\alpha }}_{\widehat{m}+1}({\widehat{\mathcal{T}}}_{\widehat{m}})\end{array}\right]=Q_N{({\widehat{\mathcal{T}}}_{\widehat{m}})}^{-1}{R}_N({\widehat{\mathcal{T}}}_{\widehat{m}}),$$(2.11) where the dependence on ${\widehat{\mathcal{T}}}_{\widehat{m}}$ and γ is again suppressed.

Remark 3.

Tibshirani et al. (2005) propose the fused Lasso, which penalizes the ${\mathcal{l}}_1$ norm of both the individual slope coefficients themselves and their differences. Our objective function, which is similar to the adaptive group fused Lasso objective function in Li, Qian, and Su (2016) and Qian and Su (2016), differs from the one used in fused Lasso. The main differences are; (i) the penalization is done by using the Frobenius norm, which enables us to identify breaks in the entire vector β_t , (ii) only the coefficient differences are penalized, which is natural because in the present context there is no reason to shrink the coefficients themselves to zero, and (iii) the coefficient differences are weighted, which is necessary to achieve consistency.

3 Assumptions and Asymptotic Results

3.1 Assumptions

The conditions that we will be working under are given in Assumptions EPS, LAM, Q, MOM, and J. However, before we state these assumptions, we introduce some notations. Specifically, if A is a matrix, ${\lambda }_{\text{min}}(A)$ and ${\lambda }_{\text{max}}(A)$ signify its smallest and largest eigenvalues, respectively, $\text{tr} A$ signifies its trace, and $\left|\right|A\left|\right|=\sqrt{\text{tr} A\prime A}$ signifies its Frobenius norm. If B is also a matrix, then $\text{diag}(A,B)$ denotes the block-diagonal matrix that takes A (B) as the upper left (lower right) block. The symbols ${\to }_d,{\to }_p$ and $MN(·,·)$ signify convergence in distribution, convergence in probability and a mixed normal distribution, respectively. We use w.p.1 (w.p.a.1) to denote with probability (approaching) one. $\mathcal{C}$ denotes the sigma-field generated by $(f_1^{\prime },\dots ,f_T^{\prime })\prime$.

Assumption EPS.

1. ${\epsilon }_{i,t}$ is conditionally independent across i given $\mathcal{C}$ with $\mathbb{E}({\epsilon }_{i,t}|\mathcal{C})=0$ w.p.1;

2. ${\epsilon }_{i,t}$ is independent of $x_{j,s}$ for all i, j, t, and s.

Assumption LAM.

1. ${\lambda }_i=\lambda +{\nu }_i$, where ν_i is conditionally independent across i given $\mathcal{C}$ with $\mathbb{E}({\nu }_i|\mathcal{C})=0_{r\times 1}$ w.p.1;

2. ν_i is independent of $(x_{j,t},{\epsilon }_{j,t})$ for all i, j, and t.

Assumption Q.

1. $\underset{{\mathcal{T}}_m}{\inf }{\lambda }_{\text{min}}[Q_N({\mathcal{T}}_m)]>0$ w.p.1;

2. $Q_N({\mathcal{T}}_m){\to }_p{Q}_0({\mathcal{T}}_m)=\underset{N\to \infty }{\lim }\mathbb{E}[Q_N({\mathcal{T}}_m)|\mathcal{C}]$ as $N\to \infty$, where $0<\underset{{\mathcal{T}}_m}{\inf }{\lambda }_{\text{min}}[Q_0({\mathcal{T}}_m)]<\infty$ w.p.1.

Assumption MOM. $\mathbb{E}\left|\right|{\tilde{x}}_{i,t}|{|}^4<\infty ,|\left|f_t\right||<\infty$ w.p.1, $\mathbb{E}\left|\right|f_t|{|}^4<\infty ,\mathbb{E}|\left|{\nu }_i\right|{|}^4<\infty$ and $\mathbb{E}{\epsilon }_{i,t}^4<\infty$ for all i and t.

Assumption J.

1. (a) $J_{\text{max}}={\max }_{1\le j\le m^0+1}\left|\right|{\alpha }_{j+1}^0-{\alpha }_j^0\left|\right|=O(1)$;

2. $\sqrt{N}\gamma J_{\text{min}}^{-\kappa }\to c_1\in [0,\infty )$ and $\sqrt{N}{J}_{\text{min}}\to c_2\in (0,\infty ]$ as $N\to \infty$, where $J_{\text{min}}={\min }_{1\le j\le m^0+1}$ $\left|\right|{\alpha }_{j+1}^0-{\alpha }_j^0\left|\right|$;

3. $N^{(\kappa +1)/2}\gamma \to \infty$ as $N\to \infty$.

Some comments are in order. Consider Assumption EPS. Many articles in the literature assume that ${\epsilon }_{i,t}$ is (conditionally) independent over i (see e.g., Pesaran 2006; Ahn, Lee, and Schmidt 2013; Moon and Weidner 2015; Robertson and Sarafidis 2015; Westerlund, Petrova, and Norkute 2019), and so do we. Independence is not necessary, though, and can be relaxed to allow for weak cross-sectional dependence at the expense of additional high-level moment conditions (as in Bai 2009). This is demonstrated in Section 4, where we use Monte Carlo simulations to investigate the effect of error cross-section dependence. The heteroscedasticity and serial correlation properties of ${\epsilon }_{i,t}$ are not restricted in any way. As we demonstrate in empirical illustration of Section 5, the assumption that ${\epsilon }_{i,t}$ is independent of $x_{i,t}$, which is the same as in, for example, Bai (2009), Moon and Weidner (2015), Pesaran (2006), and Westerlund, Petrova, and Norkute (2019), is also not necessary but can be relaxed provided that suitable instruments are available. Note also that independence between ${\epsilon }_{i,t}$ and $x_{i,t}$ does not rule out endogeneity, as $x_{i,t}$ can still be correlated with f_t . Lagged dependent variables can be permitted, but then ${\epsilon }_{i,t}$ cannot be serially correlated and we also require ${\lambda }_i=\lambda$ for all i, such that ${\lambda }_i^{\prime }{f}_t=\lambda \prime f_t$ reduces to a common time effect that is eliminated completely by the cross-sectional demeaning. In the supplementary materials, we elaborate on this point.

Assumption LAM is a random coefficient condition that can be seen as the “price” for allowing ${\beta }_1,\dots ,{\beta }_T$ to vary unrestrictedly when T is fixed. It demands that λ_i is randomly distributed with constant mean, and that it is independent of $x_{i,t}$ and ${\epsilon }_{i,t}$. The need for this condition can be explained as follows. As mentioned in Section 2, because of the demeaning, the PDL2S estimator is exactly invariant with respect to ${\lambda }_i^{\prime }{f}_t$ when ${\lambda }_i=\lambda$ for all i. Assumption LAM ensures that the PDL2S estimator is consistent and asymptotically mixed normal even in cases when ${\lambda }_1,\dots ,{\lambda }_N$ are not all equal. Hence, unless ${\lambda }_i=\lambda$ for all i, Assumption LAM binds. This might seem restrictive but actually the evidence that exists suggest that it is not (see e.g., Kapetanios, Serlenga, and Shin 2019; Petrova and Westerlund 2020). The condition is also not very controversial, and has been used extensively in the common correlated effects (CCE) strand of the literature (see Westerlund, Petrova, and Norkute 2019, for an overview). Arguably the study that is closest to ours is that of Boldea, Drepper, and Gan (2020). They assume that $N^{-1}{\sum }_{i=1}^N{x}_{i,t}{\lambda }_i^{\prime }{f}_t{\to }_p{a}_j$ as $N\to \infty$ for all $t=T_{j-1},\dots ,T_j-1$ and $j=1,\dots ,m+1$, so that asymptotically the sample cross-moment of the regressors and the interactive effects is constant within break regimes.⁶ Assumption LAM implies $N^{-1}{\sum }_{i=1}^N{x}_{i,t}{\lambda }_i^{\prime }{f}_t{\to }_p\underset{N\to \infty }{\lim }{N}^{-1}{\sum }_{i=1}^{N}E(x_{i,t}\lambda \prime f_t)$, which may vary freely over t, and is therefore more general. This being said, we want our approach to be as widely applicable as possible. In the supplementary materials, we therefore, discuss ways in which Assumption LAM can be relaxed. One possibility here that we also make use of in the empirical illustration of Section 5 involves transforming all observations on $x_{i,t}$ and $y_{i,t}$ into deviations from their initial values, which is tantamount to allowing for cross-section fixed effects in addition to the random interactive effects allowed under Assumption LAM.

Assumption Q is a noncollinearity condition that rules out cross-section-invariant regressors in $x_{i,t}$. This is the same as the usual time fixed effects-only condition. The simplicity and transparency of this condition is an advantage when compared to studies such as Bai (2009), and Moon and Weidner (2015), where the factors are estimated and the regressors are de-factored, as opposed to just demeaned. As a result, general “low-rank” regressors have to be ruled out in order to ensure that the de-factored regressors have enough variation.⁷ The problem is that the ruled out low-rank regressors depend on λ_i and f_t , which are unknown to the researcher. There is therefore, a risk that the defactoring exhausts too much variation, causing the signal matrix to become (near) singular. This is particularly true in the type of small-T (microeconomic) panels that we have in mind where many regressors have low variation.

Assumption MOM supposes that ${\tilde{x}}_{i,t}$, f_t , ν_i , and ${\epsilon }_{i,t}$ have a certain number of finite moments. Four finite moments are required for ${\tilde{x}}_{i,t}$, which is a standard condition. This condition together with the noncollinearity condition in Assumption Q, and the independence of ${\epsilon }_{i,t}$ and ν_i in Assumptions EPS and LAM are the only conditions placed on the regressors. This is different from the CCE strand of the literature where it is standard to assume that $x_{i,t}$ has a common factor structure that loads on the same factors as $u_{i,t}$ (see Pesaran 2006; Westerlund, Petrova, and Norkute 2019), which is restrictive in itself but also because it rules out models involving, for example, powers or products of the regressors. Boldea, Drepper, and Gan (2020) and Qian and Su (2016) assume that the regressors are independent across the cross-section, which is even more restrictive.⁸ In the present article, $x_{i,t}$ does not have a factor structure, nor does it have to be independent. In fact, $x_{i,t}$ does not even have to be stochastic, but can also contain deterministic terms such as dummy variables. And those regressors that are stochastic can be arbitrarily correlated across both time and cross-section. The same is true for f_t , which is almost completely without restriction. Note in particular how the number of factors, r, is completely unrestricted, provided that it is fixed. This is different from most CCE studies where r is bounded from above by the number of observables, p + 1 (see e.g., Westerlund, Petrova, and Norkute 2019). Moreover, unlike in most GMM- and principal components-based studies, the proposed PDL2S estimator does not depend on the availability of a consistent estimator of r (see e.g., Ahn, Lee, and Schmidt 2013; Bai 2009; Robertson and Sarafidis 2015).

Assumption J imposes some conditions on the tuning parameter γ and the size of the breaks, and is easy to justify. For example, if we assume that all the breaks are bounded away from zero and infinity, then Assumption J requires that $\gamma =O(N^{-(1+\delta )/2})$ with $\delta \in [0,\kappa )$. One way to satisfy Assumption J is therefore, to set γ proportional to .. (as in, e.g., Belloni et al. 2016; Hansen and Liao 2019). We also note that the breaks do not have to be bounded away from zero and hence that some, or indeed all, breaks may be shrinking to zero. The breaks therefore, do not have to be “large” for our procedure to be able to detect them, which is reassuring.

3.2 Asymptotic Results

Our first main result characterizes the limit of ${\widehat{\beta }}_t$.

Theorem 1.

Suppose that Assumptions EPS, LAM, Q, MOM, and J hold. Then, uniformly in $t\in {\mathcal{T}}_{T-1}$,$$\left|\right|{\widehat{\beta }}_t-{\beta }_t^0\left|\right|=O_p(N^{-1/2}).$$

Theorem 1 establishes that the PDL2S estimator is consistent and that the rate of convergence is given by $N^{-1/2}$, which is the highest possible rate for the type of parametric fixed-T panel data models that we consider. In Lemma A.1 of the supplementary materials, we show that the preliminary period-by-period LS estimator, ${\dot{\beta }}_t$, is consistent at the same rate, which is just as expected because T is fixed. Hence, from a rate of convergence point of view, nothing is gained by using PDL2S. However, the preliminary estimator does not account for the fact that the slopes are constant within break regimes. It is therefore, not as efficient as PDL2S. It is also completely uninformative regarding the number of breaks and their location. This brings us to our second main result.

Theorem 2.

Suppose that Assumptions EPS, LAM, Q, MOM, and J hold. Then, as $N\to \infty$,$$\mathbb{P}(||{\widehat{\beta }}_t-{\widehat{\beta }}_{t-1}||=0 \text{for}\mathrm{ }\text{all} t\in {\mathcal{T}}_{m^0}^{0c}={\mathcal{T}}_{T-1}\setminus {\mathcal{T}}_{m^0}^0)\to 1.$$

The set ${\mathcal{T}}_{m^0}^{0c}$ is the complement of ${\mathcal{T}}_{m^0}^0$. Hence, since .. is constant within break regimes, we have that ${\beta }_t^0={\beta }_{t-1}^0$ for all $t\in {\mathcal{T}}_{m^0}^{0c}$. Theorem 2 states that ${\widehat{\beta }}_t-{\widehat{\beta }}_{t-1}$ is strongly consistent for ${\beta }_t^0-{\beta }_{t-1}^0$ when $t\in {\mathcal{T}}_{m^0}^{0c}$, which is a reflection of the usual sparsity result in the variable selection literature (see e.g., Fan and Li 2006). But from Theorem 1, we know that ${\widehat{\beta }}_t-{\widehat{\beta }}_{t-1}$ is consistent for all t, including $t\in {\mathcal{T}}_{m^0}^0$. This means that PDL2S is able to identify the true model in (2.1) with the correct number of breaks and break dates. The following corollary to Theorems 1 and 2 formalizes this.

Corollary 1.

Suppose that Assumptions EPS, LAM, Q, MOM, and J hold. Then, as $N\to \infty$,

(a) $\mathbb{P}(\widehat{m}=m^0)\to 1$;

(b) $\mathbb{P}({\widehat{\mathcal{T}}}_{\widehat{m}}={\mathcal{T}}_{m^0}^0|\widehat{m}=m^0)\to 1$.

Theorem 3 reports the asymptotic distribution of the PDL2S estimator, and it does so conditional on the high probability even that $\widehat{m}=m^0$.

Theorem 3.

Suppose that Assumptions EPS, LAM, Q, MOM, and J hold, and that $\widehat{m}=m^0$. Then, as $N\to \infty$,$$\sqrt{N}({\widehat{A}}_{\widehat{m}}-A_{m^0}^0){\to }_{d}MN(0_{(m^0+1)p\times 1},Q_0^{-1}{\Omega }_0{Q}_0^{-1}),$$where$$\begin{array}{c}{Q}_0=Q_0({\mathcal{T}}_{m^0}^0),\\ {\Omega }_0=\underset{N\to \infty }{\lim }\frac{1}{N}\sum _{i=1}^N\mathbb{E}(u_i{u}_i^{\prime }|\mathcal{C}),\\ u_i=u_i({\mathcal{T}}_{m^0}^0)=\left[\begin{array}{c}\sum _{t=T_0^0}^{T_1^0-1}{\tilde{x}}_{i,t}{\tilde{u}}_{i,t}\\ ⋮\\ \sum _{t=T_{m^0}^0}^{T_{m^0+1}^0-1}{\tilde{x}}_{i,t}{\tilde{u}}_{i,t}\end{array}\right].\end{array}$$

The definitions of Q₀ and ${\Omega }_0$ in Theorem 3 reveal that the PDL2S estimator is asymptotically equivalent to the infeasible LS estimator of (2.4) that takes all the breaks as known. In this sense, PDL2S is “oracle efficient.” That being said, ${\Omega }_0$ does depend on ${\tilde{u}}_{i,t}$, which is a function of ${\tilde{\nu }}_i^{\prime }{f}_t$. Hence, while oracle efficient in the sense that it is asymptotically equivalent to the known break LS estimator, the PDL2S estimator is not asymptotically equivalent to the LS estimator that takes both the breaks and the factors as known. As pointed out in Section 2, the demeaning removes the mean of λ_i , and this is enough to ensure $\sqrt{N}$-consistency and asymptotic (mixed) normality as long as ${\epsilon }_{i,t}$ and ν_i are uncorrelated with ${\tilde{x}}_{i,t}$. However, this does not mean that the PDL2S estimator is asymptotically invariant with respect to ${\lambda }_i^{\prime }{f}_t$, and Theorem 3 confirms this. In Section 4, we use Monte Carlo simulations as a means to investigate how the variance of the PDL2S estimator is affected by the interactive effects.

The asymptotic distribution of $\sqrt{N}({\widehat{A}}_{\widehat{m}}-A_{m^0}^0)$ is normal conditional on $\mathcal{C}$, which means that unconditionally it is mixed normal (see Andrews 2005, for a discussion). The asymptotic distribution therefore, supports standard normal and chi-squared inference. Of course, for such standard inference to be possible, we need a consistent estimator of $Q_0^{-1}{\Omega }_0{Q}_0^{-1}$. Let us therefore, define ${\widehat{u}}_{i,t}={\tilde{y}}_{i,t}-{\tilde{x}}_{i,t}^{\prime }{\widehat{\alpha }}_j$, where $t={\widehat{T}}_{j-1},\dots ,{\widehat{T}}_j-1$ with $j=1,\dots ,\widehat{m}+1$. A natural estimator of $Q_0^{-1}{\Omega }_0{Q}_0^{-1}$ given by(3.1) $$Q_N{({\widehat{\mathcal{T}}}_{\widehat{m}})}^{-1}\widehat{\Omega }{Q}_N{({\widehat{\mathcal{T}}}_{\widehat{m}})}^{-1}$$(3.1) where $Q_N({\widehat{\mathcal{T}}}_{\widehat{m}})$ is as before and the $(\widehat{m}+1)p\times (\widehat{m}+1)p$ matrix $\widehat{\Omega }$ is given by(3.2) $$\widehat{\Omega }=\frac{1}{N}\sum _{i=1}^N{\widehat{u}}_i{\widehat{u}}_i^{\prime },$$(3.2) where(3.3) $${\widehat{u}}_i={\widehat{u}}_i({\widehat{\mathcal{T}}}_{\widehat{m}})=\left[\begin{array}{c}\sum _{t={\widehat{T}}_0}^{{\widehat{T}}_1-1}{\tilde{x}}_{i,t}{\widehat{u}}_{i,t}\\ ⋮\\ \sum _{t={\widehat{T}}_m}^{{\widehat{T}}_{m+1}-1}{\tilde{x}}_{i,t}{\widehat{u}}_{i,t}\end{array}\right].$$(3.3)

The consistency of this estimator is a direct consequence of the consistency of ${\widehat{A}}_{\widehat{m}},\widehat{m}$, and ${\widehat{\mathcal{T}}}_{\widehat{m}}$.

Corollary 2.

Suppose that Assumptions EPS, LAM, Q, MOM, and J hold. Then, as $N\to \infty$,$$Q_N{({\widehat{\mathcal{T}}}_{\widehat{m}})}^{-1}\widehat{\Omega }{Q}_N{({\widehat{\mathcal{T}}}_{\widehat{m}})}^{-1}{\to }_p{Q}_0^{-1}{\Omega }_0{Q}_0^{-1}.$$

Remark 4.

A major point about Corollary 2 is that the asymptotic covariance matrix of the PDL2S estimator is very easily estimable. This stands in sharp contrast to the large-T framework that typically involves some kind of heteroscedasticity and autocorrelation consistent (HAC) correction (see e.g., Bai 2009; Pesaran 2006), which is not only difficult to implement but also known to lead to poor small-sample properties.

Consider testing the null hypothesis of $H_0:R{A}_{m^0}^0=r$, where R is a $q\times (m^0+1)p$ matrix of rank $q\le (m^0+1)p$ and r is a $q\times 1$ vector. Again, conditional on the high probability event that $\widehat{m}=m^0$, the relevant Wald test statistic is given by(3.4) $$W=N(R{\widehat{A}}_{\widehat{m}}-r)\prime {[R{Q}_N{({\widehat{\mathcal{T}}}_{\widehat{m}})}^{-1}\widehat{\Omega }{Q}_N{({\widehat{\mathcal{T}}}_{\widehat{m}})}^{-1}R\prime ]}^{-1}(R{\widehat{A}}_{\widehat{m}}-r).$$(3.4)

Suppose that H₀ is true. Then, because of Theorem 3 and Corollary 2,(3.5) $$\begin{array}{c}W=\sqrt{N}(R{\widehat{A}}_{\widehat{m}}-r)\prime {(R{Q}_0^{-1}{\Omega }_0{Q}_0^{-1}R\prime )}^{-1}\sqrt{N}(R{\widehat{A}}_{\widehat{m}}-r)\\ \hspace{1em}+o_p(1){\to }_d{\chi }^2(q)\end{array}$$(3.5)

as $N\to \infty$. Similarly, if q = 1, then the t-statistic(3.6) $$t=\frac{\sqrt{N}(R{\widehat{A}}_{\widehat{m}}-r)}{\sqrt{R{Q}_N{({\widehat{\mathcal{T}}}_{\widehat{m}})}^{-1}\widehat{\Omega }{Q}_N{({\widehat{\mathcal{T}}}_{\widehat{m}})}^{-1}R\prime }}$$(3.6) has a limiting N(0, 1) distribution under H₀.

All the results reported so far are conditional on the tuning parameter γ. While in theory any choice satisfying Assumption J will do, as with most other tuning parameters, in practice the results can be sensitive to different specifications of γ. It might therefore, be preferable to set this parameter in a data-driven fashion. In this article, we follow Li, Qian, and Su (2016), and Qian and Su (2016), and set γ by minimizing an information criterion;(3.7) $$\widehat{\gamma }=\arg \underset{{}_{\gamma }}{\min }\text{IC}(\gamma ),$$(3.7) with(3.8) $$\text{IC}(\gamma )={\widehat{\sigma }}^2({\widehat{\mathcal{T}}}_{\widehat{m}}(\gamma ))+\varphi ·p[\widehat{m}(\gamma )+1],$$(3.8) where ${\widehat{\mathcal{T}}}_{\widehat{m}}(\gamma )$ and $\widehat{m}(\gamma )$ are ${\widehat{\mathcal{T}}}_{\widehat{m}}$ and $\widehat{m}$, respectively, when treated as functions of γ, $\varphi =\varphi (N)>0$ is a penalty, and(3.9) $${\widehat{\sigma }}^2({\mathcal{T}}_m)=\frac{1}{NT}\sum _{i=1}^N\sum _{j=1}^{m+1}\sum _{t=T_{j-1}}^{T_j-1}{({\tilde{y}}_{i,t}-{\tilde{x}}_{i,t}^{\prime }{\widehat{\alpha }}_j)}^2.$$(3.9)

Theorem 4.

Suppose that Assumptions EPS, LAM, Q, MOM, and J hold, that $\varphi \to 0$ and that $N\varphi \to \infty$. Then, as $N\to \infty$,$$\mathbb{P}[\widehat{m}(\widehat{\gamma })=m^0]\to 1.$$

As usual, the penalty $\varphi$ is not unique and has to be set by the researcher. Analogous to Qian and Su (2016), in this article we set $\varphi =(\hspace{0.17em}\text{ln}\hspace{0.17em}N)/N$, which makes $\text{IC}(\gamma )$ similar to the conventional Schwarz Bayesian information criterion (BIC).

4 Monte Carlo Simulations

4.1 Setup

In this section, we use Monte Carlo simulations as a means to evaluate the finite sample properties of the proposed PDL2S approach. The data generating process used for this purpose is given by a restricted version of (2.1) and (2.2) that sets p = 4 and r = 5. Similarly to Qian and Su (2016), we consider $m^0\in \{0,1,2\}$ with ${\beta }_t=0_{p\times 1}$ when $m^0=0,{\beta }_t=1_{p\times 1}·1(T/2\le t\le T)$ when $m^0=1$ and ${\beta }_t=1_{p\times 1}·[1(\lfloor T/3\rfloor \le t<\lfloor 2T/3\rfloor )+2·1(\lfloor 2T/3\rfloor \le t\le T)]$ when $m^0=2$, where $1(·)$ and $\lfloor ·\rfloor$ are the indicator and integer part functions, respectively, and $0_{p\times 1}$ ($1_{p\times 1}$) is a $p\times 1$ vector of zeroes (ones).⁹

If $u_{i,t}$ and $x_{i,t}$ are independent, the interactive effects in (2.1) can be ignored without risking the consistency of the regular post-Lasso LS estimator based on raw (nondemeaned) data. Hence, in order to make $u_{i,t}$ and $x_{i,t}$ dependent, we allow $x_{i,t}$ to load on the same set of factors as $u_{i,t}$, which as pointed out in Section 3.1 is a requirement in CCE. Specifically, $x_{i,t}$ is generated according to the following factor model:(4.1) $$x_{i,t}={\Gamma }_i{f}_t+{\nu }_{i,t},$$(4.1) where(4.2) $$f_t=(1-\phi )+\phi f_{t-1}+{\eta }_t,$$(4.2) with $f_0=0_{r\times 1},\phi \in \{0.8,1\}$ and ${\eta }_t\sim N(0_{r\times 1},I_r)$. Hence, while stationary (although highly persistent) when $\phi =0.8$, when $\phi =1$, f_t is unit root nonstationary. The data generating process considered for ${\nu }_{i,t}$ is also very general and is the same as in, for example, Petrova and Westerlund (2020). It is given by(4.3) $${\nu }_{i,t}=\pi {\nu }_{i,t-1}+e_{i,t}+\sum _{j=1}^K\pi (e_{i-j,t}+e_{i+j,t}),$$(4.3) where ${\nu }_{1,0}=\cdots ={\nu }_{N,0}=0_{p\times 1},\pi \in \{0.4,0.8\}$, K = 10 and $e_{i,t}\sim N(0_{p\times 1},I_p)$. This means that ${\nu }_{i,t}$ is weakly correlated over time as well as with 2K of its neighboring cross-sectional units.¹⁰ If $\pi =0.4$, we say that the error dependence is “low,” whereas if $\pi =0.8$ the error dependence is said to be “high.” A similar process is used for generating ${\epsilon }_{i,t}$;(4.4) $${\epsilon }_{i,t}=\pi {\epsilon }_{i,t-1}+{\xi }_{i,t}+\sum _{j=1}^K\pi ({\xi }_{i-j,t}+{\xi }_{i+j,t}),$$(4.4) where ${\epsilon }_{1,0}=\cdots ={\epsilon }_{N,0}=0$ and ${\xi }_{i,t}\sim N(0,{\sigma }_i^2)$ with ${\sigma }_i^2\sim U(0.5,1)$. Hence, ${\epsilon }_{i,t}$ is not only weakly serially and cross-sectionally correlated but also heteroscedastic. Finally, to ensure that Assumption LAM is met the loadings in Γ_i and λ_i are drawn independently from N(2, 1).

As for the sample size, we consider all combinations of $N\in \{25,50,100,200,600\}$ and $T\in \{5,10,20\}$, where the values considered for T are intentionally smaller than those considered for N.

We report four performance measures; the frequency of false detection of the estimated number of breaks, the frequency of false detection of the estimated breakpoints given that the number of breaks is selected correctly, the average number of estimated breaks, and the mean squared error (MSE) of the PDL2S estimator (times 100), computed as the average $\left|\right|{\widehat{A}}_{\widehat{m}}-A_{m^0}^0\left|\right|/(\widehat{m}+1)p$ across the Monte Carlo replications, whose number is here set to 1000.

The estimation code was written in Python, which is one of the most common programming languages in applications of the Lasso. Following the previous literature on the adaptive Lasso (see Qian and Su 2016), we set κ = 2. For a given value γ, we optimize (2.5) using the convex optimization package CVXPY. We then determine the most appropriate value of γ by minimizing the information criterion in (3.8). To accomplish this, we need to choose a suitable grid containing values of γ that yield the true breaks. One way to do so is to first select an interval $[{\gamma }_{\text{max}},{\gamma }_{\text{min}}]$, where ${\gamma }_{\text{min}}$ (${\gamma }_{\text{max}}$) is chosen so that the number of estimated breaks is zero (“many”) (see Qian and Su 2016). We then slice $[{\gamma }_{\text{max}},{\gamma }_{\text{min}}]$ into 50 evenly sized intervals on a log-scale, optimize (2.5) at each value and select as $\widehat{\gamma }$ the value that minimizes the information criterion in (3.8).

The simulations are too time consuming for a personal computer. We used the UPPMAX (Uppsala Multidisciplinary Center for Advanced Computational Science) cluster Rackham, which is accessible via the SNIC (Swedish National Infrastructure for Computing). Rackham consists of 486 nodes, each containing two 10-core Intel Xeon V4 central processing units.

4.2 Results

Tables 1–4 contain the results, which are reported for different constellations of $\phi$ and π. We consider four cases; (i) stationary factors ($\phi =0.8$) and low error dependence ($\pi =0.4$), (ii) nonstationary factors ($\phi =1$) and low error dependence ($\pi =0.4$), (iii) stationary factors ($\phi =0.8$) and high error dependence ($\pi =0.8$), and (iv) nonstationary factors ($\phi =1$) and high error dependence ($\pi =0.8$).

Table 1 Simulation results for the case with stationary factors ($\varphi =0.8$) and low error dependence ($\pi =0.4$).

	T = 5					T = 10					T = 20
N	25	50	100	200	600	25	50	100	200	600	25	50	100	200	600
$m^0=0$
NB	0.000	0.000	0.000	0.000	0.000	0.001	0.000	0.000	0.000	0.000	0.001	0.000	0.000	0.000	0.000
BP	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	0.000	0.000	0.000	0.000	0.000	0.001	0.000	0.000	0.000	0.000	0.001	0.000	0.000	0.000	0.000
MSE	0.086	0.077	0.041	0.044	0.018	0.083	0.038	0.033	0.011	0.010	0.035	0.026	0.009	0.012	0.001
$m^0=1$
NB	0.019	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
BP	0.001	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	0.983	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
MSE	0.387	0.120	0.106	0.062	0.041	0.142	0.090	0.061	0.019	0.018	0.105	0.040	0.025	0.020	0.011
$m^0=2$
NB	0.151	0.045	0.005	0.000	0.000	0.003	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
BP	0.000	0.000	0.000	0.000	0.000	0.001	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	1.840	1.955	1.995	2.000	2.000	1.997	2.000	2.000	2.000	2.000	2.000	2.000	2.000	2.000	2.000
MSE	1.571	0.519	0.144	0.053	0.036	0.188	0.075	0.048	0.044	0.016	0.106	0.068	0.031	0.016	0.011

Table 2 Simulation results for the case with nonstationary factors ($\varphi =1$) and low error dependence ($\pi =0.4$).

	T = 5					T = 10					T = 20
N	25	50	100	200	600	25	50	100	200	600	25	50	100	200	600
$m^0=0$
NB	0.001	0.000	0.000	0.000	0.000	0.004	0.000	0.000	0.000	0.000	0.038	0.002	0.000	0.000	0.000
BP	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	0.001	0.000	0.000	0.000	0.000	0.004	0.000	0.000	0.000	0.000	0.038	0.002	0.000	0.000	0.000
MSE	0.119	0.108	0.035	0.020	0.022	0.056	0.060	0.039	0.027	0.016	0.077	0.052	0.031	0.012	0.011
$m^0=1$
NB	0.015	0.001	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	3.000	0.000	0.000	0.000	0.000
BP	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	0.985	0.990	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.003	1.000	1.000	1.000	1.000
MSE	0.351	0.096	0.076	0.081	0.030	0.131	0.043	0.039	0.041	0.021	0.050	0.053	0.0139	0.024	0.014
$m^0=2$
NB	0.153	0.043	0.012	0.000	0.000	0.002	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
BP	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	1.847	1.957	1.988	2.000	2.000	1.998	2.000	2.000	2.000	2.000	2.000	2.000	2.000	2.000	2.000
MSE	1.596	0.457	0.191	0.039	0.029	0.124	0.073	0.068	0.024	0.016	0.099	0.058	0.027	0.026	0.000

Table 3 Simulation results for the case with stationary factors ($\varphi =0.8$) and high error dependence ($\pi =0.8$).

	T = 5					T = 10					T = 20
N	25	50	100	200	600	25	50	100	200	600	25	50	100	200	600
$m^0=0$
NB	0.011	0.050	0.062	0.000	0.001	0.017	0.047	0.060	0.000	0.001	0.014	0.057	0.043	0.014	0.001
BP	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	0.011	0.001	0.068	0.000	0.001	0.018	0.051	0.000	0.000	0.001	0.014	0.069	0.047	0.015	0.001
MSE	0.128	0.102	0.132	0.023	0.044	0.076	0.074	0.071	0.035	0.025	0.049	0.038	0.011	0.028	0.000
$m^0=1$
NB	0.011	0.024	0.030	0.007	0.001	0.003	0.018	0.010	0.002	0.000	0.015	0.019	0.004	0.003	0.000
BP	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	1.005	1.024	1.033	1.007	1.001	1.003	1.020	1.010	1.002	1.000	1.015	1.023	1.005	1.003	1.000
MSE	0.313	0.329	0.177	0.123	0.043	0.173	0.144	0.133	0.051	0.041	0.054	0.120	0.019	0.031	0.015
$m^0=2$
NB	0.098	0.037	0.008	0.003	0.000	0.003	0.013	0.004	0.002	0.000	0.005	0.013	0.006	0.000	0.000
BP	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	1.909	1.992	2.007	2.003	2.000	2.003	2.012	2.004	2.002	2.000	2.005	2.013	2.006	2.000	2.000
MSE	1.182	0.485	0.238	0.113	0.057	0.208	0.181	0.143	0.062	0.023	0.148	0.127	0.076	0.050	0.020

Table 4 Simulation results for the case with nonstationary factors ($\varphi =1$) and high error dependence ($\pi =0.8$).

	T = 5					T = 10					T = 20
N	25	50	100	200	600	25	50	100	200	600	25	50	100	200	600
$m^0=0$
NB	0.015	0.049	0.048	0.019	0.000	0.028	0.057	0.051	0.020	0.001	0.057	0.068	0.034	0.007	0.003
BP	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	0.017	0.051	0.052	0.02	0.000	0.029	0.068	0.056	0.02	0.001	0.061	0.075	0.037	0.007	0.005
MSE	0.179	0.069	0.074	0.073	0.040	0.104	0.049	0.071	0.041	0.025	0.038	0.034	0.032	0.014	0.012
$m^0=1$
NB	0.003	0.021	0.021	0.006	0.000	0.002	0.012	0.009	0.001	0.000	0.005	0.016	0.007	0.002	0.000
BP	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	1.003	1.024	1.023	1.006	1.000	1.002	1.015	1.009	1.001	1.000	1.005	1.016	1.007	1.002	1.000
MSE	0.270	0.238	0.150	0.061	0.056	0.170	0.151	0.051	0.063	0.053	0.102	0.059	0.049	0.035	0.013
$m^0=2$
NB	0.103	0.026	0.008	0.001	0.000	0.002	0.005	0.004	0.000	0.000	0.003	0.003	0.006	0.000	0.000
BP	0.000	0.000	0.000	0.000	0.000	0.000	0.001	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	1.898	1.998	2.007	2.001	2.000	2.002	2.005	2.004	2.003	2.000	2.003	2.004	2.006	2.000	2.000
MSE	1.208	0.412	0.238	0.130	0.048	0.115	0.121	0.143	0.062	0.032	0.100	0.065	0.076	0.024	0.015

NOTE: “NB”, “BP”, “AVE” and “MSE” refer to the frequency of false detection of the estimated number of breaks, the frequency of false detection of the estimated breakpoints given that the number of breaks is selected correctly, the average number of estimated breaks, and the MSE of PDL2S estimator times 100, computed as the average $100·\left|\right|{\widehat{\mathit{A}}}_{\widehat{m}}-{\mathit{A}}_{m^0}^0\left|\right|/(\widehat{m}+1)$ p across the Monte Carlo replications, respectively. m⁰, $\varphi$, and π refer to the number of breaks, the persistence of the factors, and the serial and cross-sectional correlation of the errors in the equations for $y_{i,t}$ and $x_{i,t}$, respectively.

We begin by considering the results reported in Table 1 for the case with stationary factors and low error dependence. The first thing to note is that PDL2S does very well even when N and T are as small as N = 25 and T = 5. In fact, in this case the breaks are estimated perfectly. The only exception is when $m^0=2$, in which case the number of breaks is not always estimated correctly. However, this is only a small-sample effect that goes away with increasing values of N and T. The breakpoints are always estimated correctly. Hence, as expected given Corollary 1, the PDL2S approach is robust to the number of breaks, and works very well even if there are no breaks at all. The MSE decreases with increasing values of N and T. The effect of N is anticipated and is a reflection of the consistency of the PDL2S estimator (Theorem 1). The improvement that comes from increasing T cannot be explained by our theoretical results, which are silent about the effect of T. It suggests that T does not have to be “small” but that the estimator works well also when T is relatively large.

As expected given the unrestricted specification of the factors, increasing their persistence from $\phi =0.8$ to $\phi =1$ has no effect on the results. This is clear from comparing the results reported in Table 1 with those reported in Table 2. In the literature it is common to assume that the factors are stationary (see e.g., Bai 2009; Pesaran 2006), which rules out factors that are, for example, breaking or trending. This may be justified in some applications, but certainly not in general. The fact that the performance of the PDL2S estimator is unaffected by the specification of the factors is therefore, a great advantage.

High error dependence generally leads to worse performance than if the dependence is low, as is evident by comparing the results reported in Tables 1 and 3. Note in particular how the gain in performance that comes from increasing N is relatively slow when the dependence is high, which is partly expected given the high level of error cross-section correlation in this case. The effect is not detrimental, though, and so performance is still acceptable. Hence, as discussed in Section 3, error cross-section independence is not necessary. This is true not only when the factors are stationary, but also when they are nonstationary, as they are in Table 4.

All-in-all, we find that the PDL2S estimator performs very well in the type of small-T panels considered, and that it does so under a wide range of empirically relevant scenarios. It should therefore, be an attractive alternative to the already existing menu of estimators of panel regression models with possible interactive effects and breaks.

5 Empirical Illustration

5.1 Motivation

There is a large and growing empirical literature concerned with the socioeconomic determinants of crime. Usual suspects include deterrence variables capturing the probabilities of apprehension and punishment, and variables that control for the relative rate of return of legal opportunities. One of the main conclusions from this literature is that aggregate data do not provide much support of the deterrence idea that policy can reduce crime by raising expected costs (see e.g., Dills, Miron, and Summers 2010).

One of the most widely held explanations for this lack of empirical support is the presence of unobserved heterogeneity, which, unless appropriately accounted for, may well render the LS estimator biased and inconsistent. Cornwell and Trumbull (1994) were among the first to make this point. According to them, the issue of unobserved heterogeneity cannot be ignored, and there are by now plenty of research that confirms this (see e.g., Bushway, Brame and Paternoster 1999; Cherry and List 2002; Worrall and Pratt 2004). Another explanation is that while most theories of crime are about the behavior of individuals, many studies use aggregated data, usually at the state or country level, even though there is by now plenty of evidence to suggest that individual (crime) behavior is not well preserved under aggregation (see Lott and Mustard 1997). Yet another explanation for the lack of support of the deterrence idea is the presence of structural breaks. According to McDowall and Loftin (2005, p. 359), “[c]onventional explanations of crime rate trends assume that changes in the rates follow a process that is linear and constant […] Questioning the conventional assumptions, an emerging class of historical contingency theories stresses variation in the crime-generating mechanism. According to contingency explanations, the process underlying the rates […] has a structure that shifts over time.” The concern is that failure to control for such shifts is likely to result in inconsistent estimates of the model parameters.

Although the presence of unobserved heterogeneity and structural breaks have been more or less ignored in most studies, some attempts have been made to obtain at least a partial solution. Cornwell and Trumbull (1994) use data on 90 counties in North Carolina between 1981 and 1987. The fact that their dataset has a panel structure makes it possible to control for certain types of unobserved heterogeneity while at the same time maintaining a relatively low level of aggregation. By contrast, studies such as Batton and Jensen (2002) and Carlson and Michalowski (1997) employ aggregate time series data that they split into subperiods based on major events in order to account for structural change.

Of course, while potentially quite useful by themselves, these solutions are bound to be inadequate in any application that is characterized by both unobserved heterogeneity and structural breaks. One possibility is to use panel data to account for unobserved heterogeneity and to slice up the sample period to account for breaks. But then this means that the breaks are treated as known, which is risky, as misplaced breaks are just as problematic as omitted breaks. This is important, as there is usually great uncertainty over both the number of breaks and their location. As an example, Batton and Jensen (2002) used the Chow test to test for the presence of breaks at given dates. They urged caution in interpreting their test results, since almost every potential breakpoint was found to be significant.

The discussion of the last paragraph suggests that there is a need for an approach that is general enough to accommodate not only unobserved heterogeneity but also structural breaks. The PDL2S estimator fits this bill and we will therefore, use it in this empirical illustration.

5.2 Main Results

The data that we will use are the same as in Cornwell and Trumbull (1994) (see also Baltagi 2006; Baltagi and Liu 2009).¹¹ Hence, in this illustration, N = 90 and T = 7, which means that it is important to use techniques that do not require T to be large. This is another reason for considering PDL2S.

The included deterrence variables, which are standard in the literature, are the probability of arrest (PRBARR), the probability of conviction given arrest (PRBCONV), the probability of a prison sentence given a conviction (PRBPRIS), the average prison sentence in days (AVGSEN), and the number of police per capita (POLPC). In addition to the deterrence variables, the dataset contains a number of controls. There are wage variables that are intended to capture opportunities in the legal sector. These are the average weekly wages in construction (WCON), transportation, utilities and communication (WTUC), wholesale and retail trade (WTRD), finance, insurance and real estate (WFIR), services (WSER), manufacturing (WMFG), federal government (WFED), state government (WSTA) and local government (WLOC). Population density (DENSITY), and percent young male (PCTYMLE) are also included, as crime tends to depend on these. All-in-all there are p = 16 regressors, which are again similar to those considered previously in the literature (see e.g., Ghasemi 2017, and the references provided therein). All regressors are transformed into logs, as is the crime rate (CRMRTE).¹²

The PDL2S estimator is implemented as described in Section 4, except that we now take deviations from initial values, as mentioned in Section 3.1, to control for county fixed effects. This means that the first time observation, 1981, is lost and hence that the effective number of time observations is given by $T-1=6$. We estimate $\widehat{m}(\widehat{\gamma })=2$ breaks in 1985 and 1986, which means that according to our break detection procedure there are three regimes, 1982–1984, 1985 and 1986–1987. The resulting PDL2S results are reported in the top panel of Table 5, which also contains PDL2S, and two-way fixed effects LS results for a model without breaks.

Table 5 Empirical estimation results.

	Breaks allowed (PDL2S)			No breaks
Regressor	1982–1984	1985	1986–1987	PDL2S	FE
Controls included in break estimation
PRBARR	–0.459***	–0.613***	–0.301***	–0.414***	–0.354***
PRBCONV	–0.271***	–0.422***	–0.222***	–0.297***	–0.282***
PRBPRIS	–0.196***	–0.196	–0.128	–0.168***	–0.173***
AVGSEN	0.057	–0.084	–0.134**	–0.022	–0.003
POLPC	0.483***	0.548***	0.414***	0.456***	0.413**
DENSITY	–0.005	–0.149	0.293	–0.568	0.414
WCON	0.017	0.490**	0.358**	–0.004	–0.038
WTUC	0.026	0.778*	0.084	0.056**	0.046***
WTRD	0.040	–0.103	–0.038	0.055	–0.021
WFIR	–0.002	0.080	0.025	–0.023	–0.004
WSER	–0.182	0.584	–0.217	0.010	0.009
WMGF	–0.047	–0.080	–0.080	–0.166	–0.360***
WFED	–0.120	–0.124	0.023	–0.155	–0.309*
WSTA	–0.030	–0.738	0.293	–0.038	0.053
WLOC	–1.705***	–1.989**	–0.271	0.147	0.182*
PCTYMLE	2.504***	1.608	1.217**	1.408***	0.627*
Controls projected out prior to break estimation
PRBARR	–0.444***			–	–0.720***
PRBCONV	–0.307***			–	–0.546***
PRBPRIS	–0.118**			–	0.248***
AVGSEN	–0.017			–	–0.087
POLPC	0.449***			–	0.366***

NOTE: The dependent variable is the crime rate (CRMRTE). The deterrence regressors are cases the probability of arrest (PRBARR), the probability of conviction given arrest (PRBCONV), and the probability of a prison sentence given a conviction (PRBPRIS). The included control variables are average prison sentence in days (AVGSEN), the number of police per capita (POLPC), population density (DENSITY), percent young male (PCTYMLE), and the average weekly wage by industry, where the included industries are construction (WCON), transportation, utilities and communication (WTUC), wholesale and retail trade (WTRD), finance, insurance and real estate (WFIR), services (WSER), manufacturing (WMFG), federal government (WFED), state government (WSTA), and local government (WLOC). All variables are expressed in logs. “FE” refers to the usual two-way fixed effects LS estimator. The PDL2S estimator allows for county fixed effects using the initial value adjustment discussed in Section 3.1. When no breaks are allowed PDL2S is simply LS with initial value adjustment and cross-county demeaning. The symbols *, ** and *** denote statistical significance at the 10%, 5% and 1% levels, respectively. In the top panel the controls are included in the PDL2S procedure, whereas in the bottom panel they are projected out prior to applying PDL2S, which means that they do not affect the break estimation procedure. When the controls are included we estimate two breaks, while when they are projected out we estimate no breaks.

We begin by considering the results for the deterrence variables. The first thing to note is that while there is some variation in the results, the estimated effects of the deterrence variables appear to be quite stable over time. We also see that the signs of the estimated deterrence effects are largely as expected based on economic theory. PRBARR and PRBCONV are two of the variables that have attracted most interest in the previous literature (see e.g., Cherry and List 2002; Cornwell and Trumbull 1994; Ghasemi 2017). Their effects are estimated to be significantly negative, which is evidence in support of the deterrence idea. The estimated effects of PRBPRIS are also negative but they are generally smaller (in absolute value) than those of PRBARR and PRBCONV, and they are not always significant. The same is true for AVGSEN. Together with the significantly negative effect of PRBARR and PRBCONVI, the insignificance of PRBPRIS and AVGSEN suggests that imprisoning more criminals, or imprisoning them for longer, is not as effective as increasing the risk of apprehension or conviction once arrested. This is consistent with the idea that the consequences of being arrested and found guilty of a crime do not stop with the punishment of the criminal justice system, but that they also include other “costs” to the individual, such as the social stigmatization that come with a conviction (see Bun et al. 2020). POLPC enters significantly but with an unexpected positive sign, a finding that is consistent with the results of Cornwell and Trumbull (1994), Baltagi (2006), and Baltagi and Liu (2009).

Consider next the results for the controls. The first thing to note is that most variables are not significant, as expected given the results reported in the previous literature (see e.g., Cornwell and Trumbull 1994; Baltagi 2006; Baltagi and Liu 2009). We also note that unlike for the deterrence variables, for the controls there are some marked jumps in the results over time, which consistent with the many major events of the period being considered (see e.g., Carlson and Michalowski 1997).¹³ To take an extreme example, consider WTUC. The estimated effect of this variable goes from 0.026 and insignificant in the first regime to 0.778 and highly significant in the second, which represents an increase of almost 3000%, only to go down to 0.084 and insignificant again in the third regime. This last observation is important because it suggests that the detected breaks might not be due to the deterrence variables but that they might be due to the other, economically and statistically relatively unimportant, control variables. The controls might therefore, do more damage than good here.

In order to be able to test for the presence of breaks in the deterrence variables only, instead of including the controls in the estimated equation we project them out prior to applying PDL2S. Specifically, a two-stage procedure, in which each deterrence variable and CRMRTE are first regressed onto all the controls, is applied. This is done point-wise in time, which means that we allow the coefficients of the controls to break every year. In the second stage, we then take the residuals of the first-stage CRMRTE and deterrence variable regressions, and feed them to the PDL2S procedure. According to the results reported in the bottom panel of Table 5, there are no breaks in the second-stage regression, which supports the idea that conditional on the controls there are no breaks. We also see that the estimated deterrence effects are almost identical to the ones obtained when the controls are included but no breaks are allowed, suggesting again that the controls are unimportant. This gives credence to studies such as Ghasemi (2017) where only deterrence variables are used.

As pointed out in Section 5.1, while the literature generally agrees that the presence of breaks is a concern, the role of these breaks in the crime generating process is still an unsettled issue. The following quotation from McDowall and Loftin (2005, p. 361) captures the sentiment in the literature: “contingency theories do not disagree with the conventional approach about the variables that produce the rate changes. Instead, they add a new layer of complexity to allow for the context within which the rate-generating process operates. If these theories are correct, they could significantly improve knowledge about how crime rates change over time. If they are incorrect, they might needlessly complicate attempts to refine the standard approach.” The results reported here suggest that while the crime process is breaking, the effect of the deterrence variables has been stable over time.

5.3 Robustness

In order to get a feeling for the validity of the random interactive effects assumption (extended to allow for county fixed effects), we computed the average correlation coefficient of the PDL2S residuals for all pairs of counties, and the CD test of Pesaran (2021), which tests the null hypothesis of no cross-sectional correlation. If the random interactive effects assumption is correct, the regression errors should be cross-county uncorrelated, whereas if the assumption is incorrect there should be some remaining cross-county correlation. Hence, only if the residuals are cross-county uncorrelated can we conclude in favor of the random interactive effects assumption. The average correlation coefficient is –0.01 and the CD statistic is –1.56, which is insignificant even at the liberal 10% level.¹⁴ We take these results to suggest that there are no major violations of the random interactive effects assumption.¹⁵

Cornwell and Trumbull (1994) argue that PRBARR and POLPC may be endogenous. As pointed out in Section 2, we do not require strict exogeneity but only exogeneity conditional on the factors. In order to assess the validity of this condition, we employed a post-demeaned version of the Lasso GMM of Qian and Su (2016). The instruments used are the same as in Cornwell and Trumbull (1994). They are the fraction of crimes that involve face-to-face contact, and per capita tax revenue. While the first instrument is likely to be correlated with PRBARR, as face-to-face contact makes it possible for victim to identify the offender, the second is likely to correlate with POLPC, as counties with preferences for law enforcement will vote for higher taxes to fund a larger police force.¹⁶ The results, available upon request, are very similar to those reported in Table 5. The main difference is that the standard errors are much larger in the GMM specification, which is consistent with the results of Baltagi (2006), Baltagi and Liu (2009), Bun et al. (2020), and Cornwell and Trumbull (1994). We interpret these results as providing evidence in favor of the PDL2S results reported in Table 5. The logic goes as follows: While LS and GMM are both consistent under exogeneity (conditional on the factors), the GMM instruments are not as informative as the regressors that they replace, leading to variance inflation. This is consistent with Bun et al. (2020), and Cornwell and Trumbull (1994), who on efficiency grounds prefer LS over GMM.

6 Conclusion

The present article considers what we believe to be an empirically very relevant scenario, namely, a researcher faced with the task of estimating a panel data model with unobserved heterogeneity and slope coefficients that may be subject to multiple structural breaks. The researcher wants to be able to estimate not only the slope coefficients within each regime, but also the unknown breakpoints and their number. Moreover, because the panel dataset is short, estimation must be possible even if the number of time periods, T, is fixed and only the number of cross-sectional units, N, is large. The current article contributes by developing a Lasso-based approach that meets this list of demands.

Our asymptotic results show that with probability approaching one the new approach correctly determines the number of breaks and their locations, and that the estimator of the regime-specific regression coefficients is consistent and asymptotically normal. Simulation results are also provided to suggest that the asymptotic predictions are borne out well in small samples.

Acknowledgments

The authors would like to thank Ivan Canay (Coeditor), an Associate Editor and two anonymous referees for many valuable comments and suggestions.

Supplementary Materials

The supplement provides (i) proofs of the asymptotic results provided in Section 3.2 of the main article, (ii) details of the extensions mentioned in Sections 2 and 3.1 of the same article, and (iii) some Monte Carlo results pertaining to these extensions.

Additional information

Funding

Westerlund would like to thank the Knut and Alice Wallenberg Foundation for financial support through a Wallenberg Academy Fellowship.

Notes

1 An incomplete list of studies dealing with a single structural break in panel data include Antoch et al. (2019), Baltagi, Feng, and Kao (2016), Baltagi, Kao, and Liu (2017), Hidalgo and Schafgans (2017), Karavias, Narayan, and Westerlund (2022), and Zhu, Sarafidis, and Silvapulle (2020).

2 Qian and Su (2016) recognize the importance of allowing T to be finite and discuss likely implications for theory, but they do not provide any formal results for the fixed-T case. Similarly, while in Baltagi, Feng, and Kao (2016) there is a discussion of how to proceed in the presence of multiple breaks, their theory supposes that there is just one break.

3 This condition can be restrictive but it is need for the proofs; see Section 3.1 for a discussion.

4 As we explain later in Section 3, the type of factors that can be permitted under our assumptions is very broad. This suggests that there is no need to discriminate between known and unknown factors, but that one can just as well treat them all as unknown. This is the main rationale for writing (2.2) in terms of (the unknown) f_t only.

5 One can also use the regular Lasso estimator of $A_{m^0}^0$ , as given by $[{\widehat{\beta }}_{{\widehat{T}}_0}\prime ,\dots ,{\widehat{\beta }}_{{\widehat{T}}_{\widehat{m}}}\prime ]\prime$. However, as is well known in the literature, post-Lasso typically outperforms regular Lasso, and our (unreported) Monte Carlo results confirm this. In this article, we therefore, focus on post-Lasso LS.

6 The need for this condition is partly expected given the discussion in Section 1 on the difficulty of separating the breaks from the interactive effects. Boldea, Drepper, and Gan (2020) do not do anything to control for the interactive effects but apply LS as if there were no effects present at all. This means that they have to put enough structure on the effects so as to ensure that they do not interfere with their break estimation procedure. One of the terms in the resulting omitted interactive effects bias of the LS estimator is given by $N^{-1}{\sum }_{i=1}^N{x}_{i,t}{\lambda }_i^{\prime }{f}_t$. If this is not constant within break regimes, the interactive effects will be mistaken for structural breaks.

7 Certain low-rank regressors can be permitted but they then require special treatment (see Bai 2009).

8 The condition that the regressors are identically distributed can be relaxed (see Boldea, Drepper, and Gan 2020). However, it is still necessary that the sample second moment matrix of the regressors is asymptotically time-invariant (within break regimes).

9 We experimented with differently spaced breaks. The results, available upon request, suggests that the conclusions are unaffected by the spacing of the breaks and that the PDL2S approach works well even if there are regimes that contains only one time period, which corroborates our asymptotic results.

10 The cross-sectional sum in ${\nu }_{i,t}$ is truncated at beginning and end when not enough cross-sections are available. For example, when generating ${\nu }_{1,t}$ , the sum only includes $e_{2,t},\dots ,e_{11,t}$.

11 The data can be downloaded online from the Journal of Applied Econometrics data archive, available at http://qed.econ.queensu.ca/jae/.

12 We refer to Cornwell and Trumbull (1994) for a more detailed description of the data.

13 Most important, there was (i) the election of Ronald Reagan in 1980 and the political-economic reorganization that followed, (ii) a displacement of nonwhite inner-city males from the regular labor force to the criminogenic informal drug economy, and (iii) a steep increase in juvenile violent crime.

14 Juodis and Reese (2022) argue that the CD test can be misleading when applied to cross-sectionally demeaned data in that it will tend to reject too often. We are unable to reject, suggesting that this tendency to over-reject is not an issue.

15 Unreported results confirm that the regressors are not uncorrelated across counties. This means that misspecification of the regression function, such as when breaks are omitted or misplaced, should manifest itself as cross-correlated residuals, which is not what we find.

16 Hence, there are two instruments, one for each of the two endogenous regressors. This means that the model is just identified. We experimented with using the one-year lagged values of PRBARR and POLPC as additional instruments. Because the resulting model is overidentified, we can apply the overidentifying restrictions J-statistic to assess the validity of the instruments. The instruments passed the test. The problem is that the lags do not appear to be very relevant, in that PRBARR and POLPC are basically serially uncorrelated, which casts doubt on the results based on the larger instrument set. For this reason, we follow the previous literature and focus on the just-identified model specification. All other regressors are treated as exogenous and are therefore, included in the set of instruments.