A $t$-test for synthetic controls
Victor Chernozhukov Kaspar Wuthrich Yinchu Zhu








A t-test for synthetic controls[This paper was previously circulated as “Inference on average treatment effects in aggregate panel data settings” and “Practical and robust t-test based inference for synthetic control and related methods”.  We are grateful to Jerry Hausman, Guido Imbens, Anna Mikusheva, Yixiao Sun, Stefan Wager, seminar participants at the Federal Reserve Bank of Kansas City, Stanford, UCL, the University of Bern, the University of St. Gallen, and conference participants at the Beyond Big, Corrupted or Missing Data conference at UCSD, the Southern California Winter Econometrics Day 2019 at UCR, the Conference on Synthetic Controls and Related Methods at MIT, and the Econometric Society World Congress 2020 for valuable comments. Wüthrich is also affiliated with the ifo Institute and CESifo. Chernozhukov gratefully acknowledges funding by the National Science Foundation. The usual disclaimer applies.]
    Victor ChernozhukovDepartment of Economics and Center for Statistics and Data Science, Massachusetts Institute of Technology. Email: <vchern@mit.edu>     Kaspar WüthrichDepartment of Economics, University of California, San Diego, 9500 Gilman Dr., La Jolla, CA 92093, USA. Email: <kwuthrich@ucsd.edu>     Yinchu ZhuBrandeis University; 415 South Street, Waltham, MA 02453, USA. Email: <yinchuzhu@brandeis.edu>
    First version on arXiv: December 27, 2018  This version: December 30, 2023
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We propose a practical and robust method for making inferences on average treatment effects estimated by synthetic controls.  We develop a K-fold cross-fitting procedure for bias-correction. To avoid the difficult estimation of the long-run variance, inference is based on a self-normalized t-statistic, which has an asymptotically pivotal t-distribution. Our t-test is easy to implement, provably robust against misspecification, valid with non-stationary data, and demonstrates an excellent small sample performance. Compared to difference-in-differences, our method often yields more than 50% shorter confidence intervals and is robust to violations of parallel trends assumptions. An -package for implementing our methods is available.



Keywords: bias correction, cross-fitting, self-normalization, average treatment effect



A t-test for synthetic controls[This paper was previously circulated as “Inference on average treatment effects in aggregate panel data settings” and “Practical and robust t-test based inference for synthetic control and related methods”.  We are grateful to Jerry Hausman, Guido Imbens, Anna Mikusheva, Yixiao Sun, Stefan Wager, seminar participants at the Federal Reserve Bank of Kansas City, Stanford, UCL, the University of Bern, the University of St. Gallen, and conference participants at the Beyond Big, Corrupted or Missing Data conference at UCSD, the Southern California Winter Econometrics Day 2019 at UCR, the Conference on Synthetic Controls and Related Methods at MIT, and the Econometric Society World Congress 2020 for valuable comments. Wüthrich is also affiliated with the ifo Institute and CESifo. Chernozhukov gratefully acknowledges funding by the National Science Foundation. The usual disclaimer applies.]
    Victor ChernozhukovDepartment of Economics and Center for Statistics and Data Science, Massachusetts Institute of Technology. Email: <vchern@mit.edu>     Kaspar WüthrichDepartment of Economics, University of California, San Diego, 9500 Gilman Dr., La Jolla, CA 92093, USA. Email: <kwuthrich@ucsd.edu>     Yinchu ZhuBrandeis University; 415 South Street, Waltham, MA 02453, USA. Email: <yinchuzhu@brandeis.edu>
    First version on arXiv: December 27, 2018  This version: December 30, 2023
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§ INTRODUCTION


Many existing inference methods for synthetic control (SC) rely on permutation tests and focus on sharp null hypotheses <cit.>. Here we propose a simple and robust t-test for making inferences on the average treatment effect on the treated unit (ATT). Our t-test is very easy to implement. For example, one can obtain (1-α)-confidence intervals as 

    ± (1-α/2) ×.


We consider a synthetic control setup with one treated unit, which is untreated for the first T_0 periods and treated for the remaining T_1 periods, and N untreated control units. Let Y_0t(1) and Y_0t(0) denote the (random) potential outcomes of the treated unit with and without the treatment. The ATT is

    τ = 1/T_1∑_t=T_0+1^T_0+T_1(Y_0t(1)-Y_0t(0)).

The main inferential challenge is that Y_0t(0) is unobserved for t>T_0. SC approximates this unobserved counterfactual using a weighted average of control outcomes, ∑_i=1^Nŵ_iY_it. The weights (ŵ_1,…,ŵ_N) are estimated based on the pre-treatment period and constrained to be positive and add up to one. 

The natural SC estimator of the ATT is

    τ̃= 1/T_1∑_t=T_0+1^T_0+T_1(Y_0t-∑_i=1^Nŵ_iY_it).

In many SC applications, T_0 is smaller than or comparable to N, and the data exhibit persistence and dynamics. In such settings, there are two major challenges when using τ̃ to make inferences about τ. First, τ̃ is biased even under correct specification, and the bias can be substantial under misspecification (Figure <ref>). This bias, which is caused by the error from estimating the high-dimensional (relative to T_0) weights, precludes standard inference procedures. 

Second, even in the ideal case where the true weights are known, estimating standard errors is difficult since they depend on the long-run variance (LRV). It is well-known that classical LRV estimators <cit.> are not accurate enough for reliable inference in small samples. Figure <ref> shows that using the popular Newey-West standard errors yields substantial size distortions.[As pointed out in <cit.>, any consistent estimator of LRV might suffer from robustness issues.] 

We develop an inference method that addresses both challenges. To remove the bias, we propose a K-fold cross-fitting procedure. To avoid the difficult LRV estimation, we propose a self-normalized t-statistic with an asymptotically pivotal t-distribution with K-1 degrees of freedom, which makes our method very easy to implement. Self-normalization further induces higher-order improvements, which lead to excellent small sample properties as confirmed by our simulations (see Appendix <ref>). Figure <ref> shows that the distribution of the bias-corrected estimator is centered at the true value and well-approximated by a normal distribution, even under misspecification. Figure <ref> shows that our t-test exhibits an excellent size accuracy. Our method is implemented in the -package  (<https://github.com/kwuthrich/scinference>). 






The t-test is valid with stationary and non-stationary data. As a result, researchers do not need to pre-test for non-stationarity and choose the inference method according to the pre-test. This is important since non-stationarity is a common feature of SC applications. The existing methods rely on assumptions on the type of non-stationarity (e.g., unit root or deterministic trend) and typically use subsampling procedures for making inferences <cit.>. However, it is well-known that even the slightest misspecification in modeling non-stationarity (e.g., unit root vs. near unit root) yields invalid inferences. Moreover, subsampling may not perform well in small samples. Therefore, we take a different approach and do not make assumptions about the exact nature of the non-stationarity. Instead, we require the control units to be sufficiently similar to the treated unit, which is a crucial contextual requirement in SC studies <cit.>. Avoiding assumptions about the type of non-stationarity is important because one of the main reasons for using SC is that the precise nature of the counterfactual trend is unknown. 

A by-product of our method is improved efficiency over the default alternative method, difference-in-difference (DID). The asymptotic variance of the bias-corrected SC estimator is no larger than that of DID, irrespective of whether the SC model is correctly specified or not. This theoretical finding has important practical implications. In simulations, the proposed confidence intervals are often shorter than those from DID by at least 50%. Moreover, our t-test is valid under violations of the common trends assumption and consequently also more robust than DID.


We contribute to the literature on inference methods for SC in settings with one or few treated units. The closest paper to ours is <cit.>, which proposes a subsampling-based approach to inference in SC settings where the number of control units is small. Our t-test does not rely on subsampling, accommodates the empirically relevant case of many controls, and is provably robust against misspecification. 
Our paper is also related to the literature proposing inference procedures based on (penalized) regression estimators and generalizations of SC <cit.>. 
Finally, focusing on average effects, we complement the existing procedures for testing hypotheses about the treatment effect trajectory {Y_0t(1)-Y_0t(0)}_t=T_0+1^T_0+T_1 and making inferences on per-period effects <cit.>.[The time series permutation approach of <cit.> can be extended to test hypotheses about the ATT by collapsing the data across time, provided that T_0≫ T_1.]




  
Notation. Let 1_p denote a p× 1 dimensional vector of ones. For q ≥  1, we denote the ℓ_q-norm of a vector as ·_q.  For a matrix A, we denote A_∞= vec(A)_∞, where vec(A) is the column-wise vectorization of A.  We write a≲ b to
denote a ≤  cb for some constant c > 0 that does not depend on the sample size. We
write a ≍ b to denote a≲ b and b≲ a. For a set A, |A| is the cardinality of A. 



§ A T-TEST FOR SYNTHETIC CONTROLS





 §.§ Setup


Consider a setup with N+1 units and T time periods. Unit i=0 is the treated unit, and units i=1,…,N are the control units. The treated unit is untreated for the first T_0 periods and treated for the remaining T-T_0=T_1 periods. The control units remain untreated for all T periods. Potential outcomes with and without the treatment are denoted as Y_it(1) and Y_it(0). Observed outcomes are given by Y_it=Y_it(0)+α_tD_it, where D_it=1{t>T_0,i=0} and α_t:=Y_1t(1)-Y_0t(0) is the (fixed) treatment effect in period t.

Our goals are (i) to test hypotheses about the ATT, τ=T_1^-1∑_t=T_0+1^Tα_t,

    H_0:τ =τ_0

and (ii) to construct confidence intervals for τ.
The ATT provides an easy-to-interpret summary measure of the overall impact of the treatment and is a standard parameter of interest in DID studies. 



 §.§ Implementing the t-test


To remove the bias of SC, we employ a K-fold cross-fitting procedure, where K is fixed. We choose K consecutive blocks from the pre-treatment period: H_1⋃ H_2⋃⋯⋃ H_K⊆{1,…,T_0}. Define r=min{⌊ T_0/K⌋,T_1 } and let H_k={(k-1)r+1,…,kr} for 1≤ k≤ K.[Here, we choose to use the first K blocks. Other choices such as the last K blocks are also valid.] For simplicity, we assume that T_0/K is an integer. For k=1,…,K, compute

    _k=1/T_1∑_t=T_0+1^T(Y_0t-∑_i=1^N_i,(k)Y_it)-1/|H_k|∑_t∈ H_k(Y_0t-∑_i=1^N_i,(k)Y_it),

where _(k)=(_1,(k),…, _N,(k))' is obtained by applying SC to the data in H_(-k):={1,…,T_0}∖ H_k. This construction ensures that under weak dependence, ŵ_(k) is approximately independent of the data in H_k⋃{T_0+1,…,T}, which allows us to establish the theoretical validity of our procedure under weak conditions. 

The basic idea behind the construction of estimator (<ref>) is as follows. The first component, T_1^-1∑_t=T_0+1^T(Y_0t-∑_i=1^N_i,(k)Y_it), corresponds to the plug-in estimator (<ref>) (with  replaced by _(k)). This estimator is biased (see Figure <ref>). The second component, |H_k|^-1∑_t∈ H_k(Y_0t-∑_i=1^N_i,(k)Y_it), is an estimator of the bias of the plug-in estimator in the pre-treatment period. Under the assumptions specified below, the bias is the same in the pre-treatment and the post-treatment period. As a result, subtracting the second component removes the bias of the plug-in estimator.

For concreteness, we consider the following canonical SC estimator <cit.>:

    _(k)=wmin∑_t∈ H_(-k)(Y_0t-∑_i=1^Nw_iY_it)^2 s.t. w∈𝒲^SC:={w:  w_i≥ 0, ∑_i=1^N w_i=1 }

This estimator is a special case of that in <cit.>. Since our t-test only requires a consistent estimator of the weights, it also works with many other and more general SC estimators.

The final estimator of the ATT is simply the average of _1,…,_K

    τ̂=1/K∑_k=1^K_k.

To avoid the difficult estimation of the LRV, we construct a scale-free test statistic. The idea is to form a ratio in which the numerator and the denominator are both scaled by the long-run standard deviation. Specifically, we construct a t-statistic based on _1,…,_K:

    𝕋_K = √(K)(-τ_0 )/σ̂_,

where 

    σ̂_=√(1+Kr/T_1)√(1/K-1∑_k=1^K (_k -)^2).

Under the null hypothesis (<ref>), 𝕋_K has an asymptotic t-distribution with K-1 degrees of freedom, which suggests the following (1-α) confidence interval for τ:

    CI_K(1-α)= [  -t_K-1(1-α/2)σ̂_/√(K),  +t_K-1(1-α/2) σ̂_/√(K)],

where t_K-1(1-α/2) is the (1-α/2)-quantile of a student t-distribution with K-1 degrees of freedom. 

The test statistic _K and its limiting distribution depend on K. Choosing K seems unavoidable; it is inherent to the cross-fitting procedure required for bias-correction. The choice of K is subject to a trade-off between the expected length[For example, when T_0< T_1 as in our empirical application, the expected length of the confidence interval is  2√(2)√(1+c_0)σ  t_K-1(1-α/2)√(1/K-1)Γ(K/2)/Γ( (K-1)/2).]  of the confidence intervals and their finite sample coverage properties <cit.>. Figure <ref> illustrates this trade-off based on the application-based simulation design of Section <ref>. It shows that choosing a larger K will lead to shorter confidence intervals but may impact the coverage accuracy of our method. When T_0 is rather small as in many SC applications, choosing K=3 yields a good balance between coverage accuracy and length. This choice yields a coverage accuracy that is only slightly worse than that of K=2 while substantially reducing average length relative to K=2. Moreover, it does not yield much wider confidence intervals than K=4 or K=5.





The construction of 𝕋_K is related to <cit.> with two important differences. First, {τ̂_k} naturally arise from our cross-fitting procedure for bias-correction. Second, the τ̂_ks share a common component and, consequently, are not asymptotically independent (Theorem <ref>). Nevertheless, we are able to show that, after scaling the denominator by √(1+K r/T_1), 𝕋_K has an asymptotic t-distribution under the null hypothesis (Theorem <ref>). Our construction of 𝕋_K is also related to the <cit.> variance estimator <cit.>. 





 §.§ Empirical illustration


<cit.> employ SC to estimate the causal effect of the ETA terrorism on per-capita GDP in the Basque Country. They use data on real per-capita GDP (1986 USD, in thousands) of the Basque Country and the other 16 Spanish regions.[The data are available through the -package  <cit.>.] Following <cit.>, our pre-treatment period is 1955–1969 (T_0=15), and the post-treatment period is 1970–1997 (T_1=28). Figure <ref> displays the raw data and shows that there is a pronounced dip in the Basque Country's GDP after 1975. 



We apply our t-test to investigate whether this dip is due to the impact of ETA terrorism. All computations were performed in  <cit.> using our  package. Code to replicate the results is available on GitHub (<https://github.com/kwuthrich/ttest_sc>).

Following the discussion in Section <ref>, we choose K=3. The estimated ATE is τ̂=-0.755, the standard error is σ̂_τ̂/√(K) =0.182, and the t-statistic is 𝕋_K=-4.14. The critical value for α=0.1 is t_2(0.95)=2.92, and the (1-α)-confidence interval is [-1.287, -0.223]. Thus, we reject the null hypothesis that the average effect of terrorism on per-capita GDP is zero at the 10% level. 



§ THEORETICAL PROPERTIES WITH STATIONARY DATA


Here, we show that our t-test is valid with stationary data, robust to misspecification, and more efficient than DID. We consider inference in a repeated sampling framework under asymptotics where T_0,T_1→∞, and our results also allow for N→∞.



 §.§ Validity under correct specification



We start by establishing the properties of our procedure under correct specification. Assuming correct specification allows use to explain the econometric intuition and the main theoretical arguments in a simple and transparent way. In Section <ref>, we show that our method remains valid under misspecification.

Following the SC literature, we predict Y_0t(0) for t>T_0 using a linear combination of (Y_1t(0),…,Y_Nt(0)). Defining Y_t(0):=Y_0t(0), Y_t:=Y_0t, and X_t:=(Y_1t(0),…,Y_Nt(0))', the linear prediction model can be written as

    Y_t(0) = X_t'w + u_t,    w∈𝒲^SC,    E ( X_t u_t )= 0,    E(u_t)=0[This zero-mean assumption is made for expositional convenience. Our theoretical results can accommodate E(u_t) 0 (see Section <ref>).],  for 1≤ t≤ T.

Importantly, we interpret model (<ref>) as a statistical or predictive model and not as a structural model.[See, for example, <cit.> and <cit.> for similar interpretations.] In this framework, correct specification means that there exists a time-invariant predictive relationship with weights that satisfy the SC constraints. In other words, correct specification requires that the linear projection coefficient satisfies the SC constraints, i.e., [E(X_t X_t')]^-1E(X_t Y_t(0))∈𝒲^SC. This can be viewed as a minimal requirement for canonical SC to “work”. Linear predictive models like (<ref>) can be justified based on factor models <cit.> and vector autoregressive models <cit.> for the potential outcomes. In Section <ref>, we show that our t-test remains valid under misspecification, i.e., when [E(X_t X_t')]^-1E(X_t Y_t(0))∉𝒲^SC. Our approach can alternatively be justified based on restrictions on selection into treatment such as unconfoundedness-type assumptions.

We make the following assumption. 

E(X_t) does not depend on t and {u_t}_t=1^T is covariance-stationary. 


The assumption of stationary prediction errors is weak since it is not imposed on the observed variables. The condition that E (X_t) does not depend on t can be viewed as a weaker version of covariance-stationarity.  In Section <ref>, we establish the properties of our method with non-stationary data.

By simple algebra, Assumption <ref> gives us the following observation.

 Suppose that the model is (<ref>). Let Assumption <ref> hold.
Then, for 1≤ k≤ K,

    τ̂_k-τ=(1/T_1∑_t=T_0+1^Tu_t-1/|H_k|∑_t∈ H_ku_t)+(1/|H_k|∑_t∈ H_k_t-1/T_1∑_t=T_0+1^T_t)'Δ_(k),

where X̃_t=X_t-E(X_t) 
and Δ_(k)=_(k)-w. 

To establish the asymptotic properties of our method, we need to show that the second term in Lemma <ref> is negligible. For that, we impose two additional assumptions.


First, we impose ℓ_2-consistency of the SC estimator ŵ_(k).
 max_1≤ k≤ K_(k)-w_2=o_P(1). 

Consistency of the SC estimator follows from standard arguments and can be established when N is large relative to T_0. The following lemma provides simple sufficient conditions, allowing the number of control units N to be large.

 Suppose that the model is (<ref>). Suppose further that the following conditions hold for 1≤ k≤ K.
 
    
  * There exists a constant κ>0 such that min_1≤ k≤ Kλ_min(Σ_(-k))≥κ, where Σ_(-k)=|H_(-k)|^-1∑_t∈ H_(-k)E(X_tX_t').
    
  * |H_(-k)|^-1∑_t∈ H_(-k)(X_tX_t'-E(X_tX_t'))_∞=o_P(1) and |H_(-k)|^-1∑_t∈ H_(-k)X_tu_t_∞=o_P(1)

 Then Assumption <ref> holds.
 
The first condition in Lemma <ref> requires the eigenvalues of Σ_(-k) to be bounded away from zero to achieve identification of w. The second condition holds under weak serial dependence, mild conditions on the tail of the distribution of the variables, and conditions on N. For example, when the entries of X_t and u_t are sub-Gaussian, we can allow for log N=o(√(T_0)); when the entries of X_t and u_t have bounded rth moment for r>2, then we can typically allow for N=o(T_0^r/4). This feature is essential because N and T_0 have a similar order of magnitude in many SC applications. Finally, we emphasize that Lemma <ref> does not impose any sparsity assumptions on the weights.

Second, we impose weak dependence assumptions on the data. 

Suppose the following conditions hold.
  

  * There exists a constant κ_1>0 such that for any A⊆{1,…,T},
the largest eigenvalue of E[|A|^-1(∑_t∈ AX̃_t)(∑_t∈ AX̃_t)']
is bounded above by κ_1. 

  * There exists a sequence ρ_T>0 such that P(max_1≤ t≤ TX̃_t_∞≤ρ_T)→1. 

  * The data {(X_t,u_t)}_t=1^T are β-mixing with coefficient
satisfying (γ_T)→0 for some sequence γ_T
satisfying 0<γ_T<r/2 and ρ_Tγ_T=o_P(min{√(T_0),√(T_1)}).

  * {u_t}_t=1^T satisfies max_1≤ t≤ TE|u_t|^q=O(1) and (i)≲ i^-η for some constants q> 2 and η>q/(q-2).



The weak dependence is stated in terms of β-mixing, which holds for a large class of stochastic processes. Under Assumption <ref>, Δ_(k) is approximately independent of the data in H_k⋃{T_0+1,…,T} such that the ℓ_2-consistency of ŵ_(k) can be used to bound the second term in Lemma <ref>.

The following theorem establishes the asymptotic distribution of the component estimators. Let g_c_0,K=K { c_0<1}+(K/c_0) {1≤ c_0≤ K } +  { c_0>K}.

 Suppose that the model is (<ref>). Let Assumptions  <ref>, <ref>, and <ref> hold. Suppose that T_0,T_1→∞ and that T_0/T_1→ c_0 for some c_0∈[0,∞]. Then, 

    √(min{T_0,T_1})[ _1-τ;    ⋮; _K-τ ]d→[ √(min{c_0,1})ξ_0-√(g_c_0,K)ξ_1;                              ⋮; √(min{c_0,1})ξ_0-√(g_c_0,K)ξ_K ]σ,

where σ^2=lim_T→∞E(T^-1/2∑_t=1^Tu_t)^2 and ξ_0,…,ξ_K are independent N(0,1) random
variables.


Theorem <ref> imposes no restrictions on the relative magnitude of T_0 and T_1, accommodating a wide range of applications. This feature adds useful robustness since researchers do not have to choose between different asymptotic approximations and inference procedures depending on the relative magnitude of T_0 and T_1. 

The next theorem establishes the asymptotic distribution of the ATT estimator and t-statistic.
 
Let the assumptions in Theorem <ref> hold. Then (i)

    √(min{T_0,T_1})(-τ) d→ N(0,(min{c_0,1} + g_c_0,KK^-1 )σ^2)

and (ii), under the null hypothesis (<ref>), 𝕋_Kd→ t_K-1, where the random variable t_K-1 has a standard t-distribution with K-1 degrees of freedom.

Part (i) of Theorem <ref> establishes the asymptotic normality of the ATT estimator. Making inferences directly based on this result requires estimating the LRV σ^2, which is difficult in small sample settings. We therefore use the self-normalized test statistic 𝕋_K, which allows us to avoid estimating the LRV. Part (ii) demonstrates that 𝕋_K has an asymptotically pivotal student t-distribution with K-1 degrees of freedom. This result is very useful from a practical perspective as one does not have to simulate non-standard critical values, nor rely on subsampling or permutation distributions. 




Theorem <ref> only requires ℓ_2-consistency _(k). As a result, our method is generic and can be used in conjunction with the many other ℓ_2-consistent estimators (e.g., Lasso). This is similar to the double machine learning literature <cit.>, where the first-order asymptotics of the final estimator depend on the influence function in the de-biasing process, but not on the properties of the estimators of the nuisance parameters, beyond a rate requirement.  

 The results in <cit.> show that t-tests remain valid when the variances of the component estimators are heterogenous. This suggests that our method remains valid under certain forms of non-stationarity in {u_t}; for example, when the variance of {u_t} varies across blocks. 


The alternative fixed-b approach <cit.> does not naturally arise from the cross-fitting procedure and encounters extra technical difficulty due to the error from estimating the high-dimensional weights. The usual justification for the fixed-b approach is its higher-order improvement in Gaussian location models. Our procedure achieves the same property in this setting; see Appendix <ref>.






 §.§ Validity under misspecification

In this section, we show that our procedure is robust against model misspecification. Towards this end, let w_*:=min_v∈𝒲^SC E(Y_t(0)-X_t'v)^2 denote the pseudo-true parameter value and define the pseudo-true residuals as u_*,t:=Y_t(0)-X_t'w_*.

Misspecification arises due to 𝒲^SC being “too small” such that [E(X_t X_t')]^-1E(X_t Y_t(0))∉𝒲^SC and E(X_t u_*,t) 0 in general. Misspecification affects our analysis in two important ways. First, _(k) is inconsistent for w. To overcome this challenge, we develop novel results on the pseudo-true ℓ_2-consistency of _(k). We show that under weak conditions, _(k)-w_*_2=o_P(1). Based on this result, we prove that, for 1≤ k≤ K, 

    τ̂_k-τ=1/T_1∑_t=T_0+1^Tu_*,t
    
    -1/|H_k|∑_t∈ H_ku_*,t+o_P(1/√(min{T_0,T_1})),

Second, there is generally no guarantee that E(u_*,t)=0.  However, as long as {u_*,t}_t=1^T is stationary, T_1^-1∑_t=T_0+1^Tu_*,t-|H_k|^-1∑_t∈ H_ku_*,t is mean-zero and asymptotically normal under suitable conditions.

The following assumption provides sufficient conditions for the pseudo-true consistency. 

	  Suppose that {(Y_t(0),X_t)}_t=1^T is covariance-stationary and satisfies the following conditions.
 

  * (_(-k)-μ)-(_(-k)-Σ) w_*_∞=o_P(1), where μ=EX_tY_t(0), Σ=EX_tX_t', _(-k)=|H_(-k)|^-1∑_t∈ H_(-k)X_tY_t(0) and _(-k)=|H_(-k)|^-1∑_t∈ H_(-k)X_tX_t'. 

  * _(-k)-Σ_∞=o_P(1) and λ_min(Σ)≥ c. 
	
	


Assumption <ref> essentially requires a law of large numbers uniformly across entries of X_t Y_t(0) and X_t X_t'. The following lemma establishes consistency under misspecification.


	Under Assumption <ref>, _(k)-w_*_2=o_P(1). In particular,
	
    _(k)-w_*_2^2≤4_(-k)-Σ_∞+2(_(-k)-_(-k) w_*)-(μ-Σ w_*)_∞/c.



Because the previous theoretical results still hold with u_t replaced by _t:=u_*,t-E(u_*,t), we have the following result.


	 Let Assumptions <ref> with u_t replaced by _t and <ref> hold. Suppose further that T_0,T_1→∞ and that T_0/T_1→ c_0
	for some c_0∈[0,∞]. Then, (i)
	
    √(min{T_0,T_1})(-τ) d→ N(0,(min{c_0,1} + g_c_0,KK^-1 )σ_∗^2),
where σ_*^2=lim_T→∞E(T^-1/2∑_t=1^T_t)^2and (ii), under the null hypothesis (<ref>), 𝕋_Kd→ t_K-1.
	
	


Theorem <ref> shows that misspecification does not affect the validity of the t-test with stationary data. However, given that the variance of the estimator depends on the LRV of the pseudo-true residuals σ_∗^2, misspecification will affect the efficiency our our method.



 §.§ De-biased SC is more efficient than DID




Here we show that bias-corrected SC is more efficient than the default alternative DID, irrespective of whether SC is correctly specified or not.[The recent work by <cit.> complements our results by studying efficiency of SC methods under design-based uncertainty.] Such efficiency considerations are important since SC applications often feature small sample sizes.

Suppose that T_0<T_1 as in our empirical application. The DID estimator of the ATT  can be written as <cit.>

    τ̂^DID=1/T_1∑_t=T_0+1^T(Y_t-X_t'w_DID)-1/T_0∑_t=1^T_0(Y_t-X_t'w_DID),

where w_DID:=(1/N,…,1/N)'. For simplicity, suppose that E(Y_t(0))=0 and E(X_t)=0 and assume that the data are iid. Then a CLT gives 

    √(T_0)(τ̂^DID -τ)d→ N(0,(c_0+1)σ_DID^2),  σ^2_DID=E(Y_t(0)-X_t'w_DID)^2.

For SC, the previous results imply

    √(T_0)(τ̂-τ)d→ N(0,(c_0+1)σ_∗^2),  σ^2_∗=E(Y_t(0)-X_t'w_*)^2,

where w_*=min_v∈𝒲^SC E(Y_t(0)-X_t'v)^2. To show that the SC estimator is more efficient than DID, it suffices to note that 
    σ_*^2=E(Y_t(0)-X_t'w_*)^2=min_v∈𝒲^SCE(Y_t(0)-X_t'v)^2≤ E(Y_t(0)-X_t'w_DID)^2=σ_DID^2.

This theoretical result is confirmed by the simulations in Section <ref>, where SC often yields much shorter confidence intervals than DID.



§ THEORETICAL PROPERTIES WITH NON-STATIONARY DATA


In many applications the data are non-stationary. An important rationale for applying SC is that the precise structure of the non-stationarity is unknown such that one has to rely on controls to identify counterfactual trends. Therefore, unlike the existing literature <cit.>, we do not make assumptions on the exact form of non-stationarity and instead restrict the heterogeneity in the non-stationarity across control units. In essence, we require the control units to be sufficiently similar to the treated unit, which is an important contextual requirement in SC studies <cit.>. 



 §.§ Unrestricted common non-stationarity


We start by establishing the validity of our procedure under the following general class of non-stationary processes. 

  Assume that X_t=1_Nθ_t+Z_t, 1≤ t≤ T, where θ_t∈ℝ is an unrestricted stochastic process and E(Z_t) does not depend on t. Moreover, suppose that {u_t}_t=1^T is covariance-stationary and E(Z_tu_t)=0.

Assumption <ref> requires that X_t can be decomposed into a common non-stationary component θ_t and a stationary component Z_t. Importantly, {θ_t} can be an arbitrary stochastic process (e.g., a deterministic trend, random walk, or general ARIMA process) such that researchers are not required to impose further restrictions on the non-stationarity as long as it is shared among units. This feature brings in very useful robustness. Even for low-dimensional problems it is well-known that the slightest misspecification in modeling non-stationarity (e.g., unit root vs. near unit root) can yield invalid inferences.[For example, there is a large literature on making inference on the scalar parameter β in Y_t=X_t β +U_t, where X_t=ρ X_t-1+V_t and ρ=1+c/T. The key difficulty is that the asymptotic distribution of the usual t-statistic for β depends on  c, but c cannot be consistently estimated <cit.>. Hence, using a unit root process (c=0) for the dynamics of x_t is not robust to barely undetectable misspecification (e.g., local-to-unity c<0).] 


The next theorem establishes the validity of our procedure.

Suppose that the model is (<ref>). Let Assumptions <ref> with X_t replaced by Z_t and <ref> hold. Let the conditions of Lemma <ref> hold with X_t replaced by Z_t. Suppose that T_0,T_1→∞ and that T_0/T_1→ c_0 for some c_0∈[0,∞]. Then (i)

    √(min{T_0,T_1})(-τ) d→ N(0,(min{c_0,1} + g_c_0,KK^-1 )σ^2)

and (ii), under the null hypothesis (<ref>), 𝕋_Kd→ t_K-1.


Theorem <ref> exploits the specific structure of SC estimators. It relies on two insights. First, under Assumption <ref>, 1_N'ŵ=1_N'w=1 implies that X_t'(ŵ-w)=Z_t'(ŵ-w). As a result, we have a version of Lemma <ref> with X̃_t replaced by Z̃_t, which is stationary.  Second, _(k) is consistent and is only a function of {Z_t}_t∈ H_(-k) under Assumption <ref>. Thus, under weak dependence of {Z_t}, _(k)-w is approximately independent of {Z_t}_t∈ H_k⋃{T_0+1,…,T}, and the result follows from similar arguments as before. 







 §.§ Sparse deviations from common non-stationarity



Here we study a more general setting that allows for deviations from the common non-stationarity. Consider the following assumption.


[]
	Assume that 	X_t=_Nθ_t+Z_t+βξ_t, 1≤ t≤ T,
	where θ_t is an unrestricted stochastic process, E(Z_t)=0, ξ_t is a non-stationary process, β∈^N is a
	vector of zeros except that one entry is one. Assume that {θ_t}_t=1^T,
	{Z_t}_t=1^T, and {ξ_t}_t=1^T are mutually independent.
	Moreover, Y_t(0)=X_t'w+u_t, 1≤ t≤ T,
	where E(u_t)=0, w∈𝒲^SC, and (β)⋂(w)=∅.

Assumption <ref> considers sparse deviations
from Assumption <ref>. All control units share a common non-stationary component
θ_t, except one unit for which the non-stationary component is θ_t+ξ_t.
We do not need to know which unit has a different trend but assume that the treated unit does not have a different trend. 
We conjecture that the results be extended to allow for deviations by more than one unit; we leave this extension for future research.



The following weak assumption restricts the size of the deviation ξ_t. It can be verified for many non-stationary processes including unit roots and
polynomial trends; see Appendix <ref>.

	Assume T_1≪ T_0/log N.
	Suppose that 
	
    ∑_t=T_0+1^Tξ_t/√(∑_t∈ H_(-k)ξ_t^2)=O_P(T_1T^-1/2)    and    ∑_t=T_0+1^Tξ_t/∑_t∈ H_(-k)ξ_t^2=O_P(T_1T^-3/2).

	Moreover, the same condition holds with ∑_t=T_0+1^Tξ_t
	in the numerator replaced by ∑_t∈ H_kξ_t.


Assumptions <ref> and <ref> allow for general non-stationary trends and deviations without imposing specific structures on the dynamics. This is important in practice because procedures that rely on specific assumptions on the non-stationarity are typically not robust against deviations from these assumptions, and even the slightest misspecification can yield invalid inferences. 

 DID is generally inconsistent under sparse deviations from a common non-stationarity. For example, suppose that there is a linear trend, i.e., ξ_t=t. Then DID is inconsistent unless T/N→ 0. 


Finally, we impose conditions on the stationary component Z_t.

	Let Assumption   <ref>  hold with X_t replaced
	by Z_t. In addition, suppose that the following conditions hold.
	 
	
  * ∑_t∈ H_(-k)Z_tu_t_∞=O_P(√(T_0log N))
	
  * ∑_t∈ H_(-k)Z_tξ_t_∞/√(∑_t∈ H_(-k)ξ_t^2)=O_P(√(log N))
	
  * (∑_t∈ H_(-k)u_tξ_t)/√(∑_t∈ H_(-k)ξ_t^2)=O_P(1)
	
  * ∑_t∈ H_(-k)(Z_tZ_t'-EZ_tZ_t')_∞=O_P(√(T_0log N))
	


The next theorem establishes the asymptotic validity of the proposed inference method when there are sparse deviations.

	Let  Assumptions <ref>,
	 <ref>, and <ref>  hold. Then, for 1≤ k≤ K,
	
    √(T_1)(_k-τ)=1/√(T_1)∑_t=T_0+1^Tu_t-1/√(|H_k|)∑_t∈ H_ku_t+o_P(1).

	Moreover, under the null hypothesis (<ref>), 𝕋_Kd→t_K-1.


To our knowledge, Theorem <ref> guarantees the inference validity of SC under the most general available conditions without imposing a specific structure on the dynamics of the non-stationarity. 




§ APPLICATION-BASED SIMULATIONS



It is important to understand the advantages and limitations of the t-test in practice. Here we present simulation evidence demonstrating when it works well and when it does not. We calibrate the simulations to the application in  <cit.> and set T_0=15, T_1=28, and N=16. Because the results in Section <ref> require T_0 to be larger than T_1, we also analyze a setting where T_0=150. We generate the treated outcome as

    Y_t(0)=X_t'w+u_t,  u_t=ρ_uu_t-1+v_t,  v_tiid∼ N(0,σ^2_v),  1≤ t≤ T,

and set α_t=τ=0. The parameters σ^2_v and ρ_u are obtained by fitting an AR(1) model to the empirical SC residuals. The degree of persistence in u_t is high: ρ_u=0.61. We compare our method to DID with K-fold cross-fitting and K-fold cross-fitting based on the true w (Oracle). We show results for K∈{2,3}. The nominal coverage is 1-α=0.9.

We consider three stationary and six non-stationary DGPs. We first make the data stationary by subtracting the contemporaneous control average as in <cit.> and fit a factor model with four factors to the stationary data. The control outcomes are 

    Y_it(0)=λ_i'f_t+η_it,   1≤ i≤ N,  1≤ t≤ T,

where f_t=(f_1t,…, f_4t)', f_stiid∼ N(0,σ^2_f_s), η_it=ρ_iη_it-1+ϵ_it, and ϵ_it∼ N(0,σ_ϵ_i^2). 
(λ_1,…,λ_N), (σ^2_f_1,…,σ^2_f_4), (ρ_1,…,ρ_N),  and (σ^2_ϵ_1,…,σ^2_ϵ_N) are obtained from and estimated based on the factor model fitted to the data. The degree of persistence is high: med(ρ_1,…,ρ_N)=0.75. Let ŵ^SC be the SC estimate of w based on the stationary data. 

We specify the three stationary DGPs as follows:
  

  1. Stationary, SC. w=ŵ^SC

  2. Stationary, DID. w=(1/N,…,1/N)'

  3. Stationary, misspecification. w=-(1/N,2/N,…,N/N)'


For the non-stationary DGPs, we augment (<ref>) with a unit-specific non-stationary component {θ_it}

    Y_it(0)=θ_it+λ_i'f_t+η_it,   1≤ i≤ N,  1≤ t≤ T.

We specify the six non-stationary DGPs as follows:


  4. Common trend. θ_it=t, w=ŵ^SC.

  5.  Common random walk. θ_it=θ_t, where θ_t=θ_t-1+ξ_t, ξ_tiid∼ N(0,1), w=ŵ^SC

  6.  Common trend + sparse deviation. θ_it=t+ 1{i=1}· t, w=ŵ^SC

  7.  Common random walk + sparse deviation. θ_it=θ_1t+1{i=1}·θ_2t, where θ_1t=θ_1t-1+ξ_1t, ξ_1tiid∼ N(0,1) and θ_2t=θ_2t-1+ξ_2t, ξ_2tiid∼ N(0,1), w=ŵ^SC

  8. Heterogeneous trends. θ_it=i+i· t, w=ŵ^SC

  9. Random walks with heterogeneous drifts. θ_it=i+θ_it-1+ξ_it, ξ_itiid∼ N(0,1), w=ŵ^SC

DGP4 and DGP5 satisfy the assumptions in Section <ref>, DGP6 and DGP7 satisfy the assumptions in Section <ref>, and DGP8 and DGP9 allow for studying the performance of the t-test when our theoretical assumptions are violated.

Table <ref> presents the results. Under stationarity, our t-test exhibits an excellent performance despite the strong serial correlation, even when T_0=15. Under correct specification, coverage and average length of the confidence intervals are similar to the Oracle. Consistent with our theory, misspecification does not affect coverage accuracy but yields wider confidence intervals. 

With non-stationary data and T_0=15, the t-test performs well when there is a common non-stationarity but may exhibit some undercoverage under deviations from the common non-stationarity. When T_0=150, the t-test works well under all DGPs, even under DGP8 and DGP9 featuring non-sparse deviations not covered by our theoretical results. Finally, a comparison of SC and DID shows that SC is more robust and often yields more than 50% shorter confidence intervals. 


1.25





§ RECOMMENDATIONS FOR PRACTICE



We conclude with some guidance for empirical practice. First, in small samples, relying on classical estimators of the LRV yields substantial size distortions. By contrast, we demonstrate that using a carefully constructed self-normalized test statistic leads to an excellent performance, even when the degree of persistence is high. Given that the number of pre-treatment periods is small in typical SC applications, we recommend using the proposed self-normalized t-statistic and avoiding the difficult estimation of the LRV. Second, with non-stationary data, the robustness of the t-test improves substantially as T_0 increases. Therefore, we recommend collecting enough pre-treatment data in such settings.   Third, we find that our method is more robust and efficient than DID. Therefore, we recommend it whenever the DID assumptions are questionable. 




apalike






 



Appendix (for online publication)






§ HIGHER-ORDER IMPROVEMENTS


Suppose that we observe {y_t}_t=1^T from the following
Gaussian location model: 

    y_t=β+u_t,

where u_t is a mean zero covariance-stationary Gaussian process
with ∑_h=-∞^∞h^2γ(h)<∞ and γ(h)=Eu_tu_t-h.
This model has been considered for example by <cit.> and <cit.>. <cit.> show that the usual consistent estimators using Bartlett
kernel for the LRV (e.g., Newey-West estimator) would generate a test that has size
distortion at least O(T^-1/2). They also show that the fixed-b
approach would have size distortion of order O(T^-1). These
results are the typical explanation for why the fixed-b approach
is more accurate than the classical LRV estimation approach. We now show that in this setting our self-normalization
approach also enjoys the higher-order improvement as it has size distortion
of order O(T^-1).

We first outline the inference procedure in the Gaussian location
model in (<ref>). Let K≥2 be a fixed integer
and define G=T/K. For simplicity, we assume that G is an integer.
Then we construct blocks {H_k}_k=1^K with H_k={(k-1)G+1,…,kG}.
Let _k=G^-1∑_t∈ H_ky_t for 1≤ k≤ K
and =K^-1∑_k=1^K_k. For testing H_0: β=0,
we define the test statistic

    𝐓_K=√(K)/√((K-1)^-1∑_k=1^K(_k-)^2).

The critical value is t_K-1(1-α/2). 

	Consider the model in (<ref>).
	Suppose that H_0: β=0 holds. Then 
	
    |P(|𝐓_K|>t_K-1(1-α/2))-α|=O(T^-1).



By Theorem <ref>, the “cross-fitted” self-normalized t-test also has size distortion O(T^-1) in the Gaussian location model (<ref>). Although it is quite difficult to derive the higher-oder asymptotics outside the Gaussian location model, Theorem <ref> suggests that, for size considerations, the cross-fitted self-normalized test is expected to have similar properties as fixed-b methods.
However, in high-dimensional settings, we are not aware of any existing results that would allow for establishing the validity of the fixed-b approach in our context.







§ VERIFICATION OF ASSUMPTION <REF>



Here we illustrate the verification of Assumption <ref> based on two leading examples.

[Unit root]
	Assume that ξ_t=∑_i=1^tv_t with a mean-zero stationary
	process v_t. Then by the functional central limit theorem, we
	can show that T^-1/2ξ_[rT] tends to B(r), where B(·)
	is a standard Brownian motion. Thus, the continuous mapping theorem
	and the condition of T_1≪ T_0 imply T_1^-1T^-1/2∑_t=T_0+1^Tξ_td→ B(1)
	and T^-2∑_t∈ H_(-k)ξ_t^2=T^-2∑_t=1^T_0ξ_t^2-T^-2∑_t∈ H_(k)ξ_t^2d→∫_0^1B^2(s)ds.
	Therefore, Assumption <ref> holds. 

[Polynomial trend]
	Assume that ξ_t=∑_k=1^Mc_kt^k with constants c_1,…,c_M,
	where M≥1 is fixed. [M=1 corresponds to a linear trend and
	M=2 represents a quadratic trend.] Recall the elementary relation
	of (∑_i=1^ni^γ)/(n^1+γ/(1+γ))→1
	as n→∞ for any fixed γ>0. If T_1≪ T_0,
	then Assumption <ref> holds:
	
    ∑_t=T_0+1^Tξ_t/√(∑_t∈ H_(-k)ξ_t^2)   =   (1+o(1))·(1+M)^-1[(1+T_1/T_0)^M+1-1]T_0^M+1/√((1+2M)^-1T_0^2M+1)
       =   O(T_1T^-1/2)

	and, similarly,
	
    ∑_t=T_0+1^Tξ_t/∑_t∈ H_(-k)ξ_t^2   =   (1+o(1))·(1+M)^-1[(1+T_1/T_0)^M+1-1]T_0^M+1/(1+2M)^-1T_0^2M+1
       =   O(T_1T^-M-1)
       =   o(T_1T^-3/2).
 





§ PROOFS




 §.§ Lemmas




  §.§.§ Proof of Lemma <ref>


Let μ=E(X_t). Notice that for T_0+1≤ t≤ T, 

    Y_t-X_t'_(k)=α_t+u_t-X_t'Δ_(k)=α_t+u_t-μ'Δ_(k)-_t'Δ_(k)

and for t∈ H_k

    Y_t-X_t'_(k)=u_t-X_t'Δ_(k)=u_t-μ'Δ_(k)-_t'Δ_(k).


Therefore, 

    _k-τ    =T_1^-1∑_t=T_0+1^T(Y_t-X_t'_(k))-1/|H_k|∑_t∈ H_k(Y_t-X_t'ŵ_(k))-τ
        =T_1^-1∑_t=T_0+1^T(α_t+u_t-μ'Δ_(k)-_t'Δ_(k))-1/|H_k|∑_t∈ H_k(u_t-μ'Δ_(k)-_t'Δ_(k))-τ
        =T_1^-1∑_t=T_0+1^Tu_t-1/|H_k|∑_t∈ H_ku_t+1/|H_k|∑_t∈ H_k_t'Δ_(k)-T_1^-1∑_t=T_0+1^T_t'Δ_(k).


The proof is complete. 






  §.§.§ Proof of Lemma <ref>


We prove the results for the following more general estimator:

    ŵ_(k)=wmin∑_t∈ H_(-k)(Y_t-X_t'w)^2  s.t.   w∈𝒲,

where 𝒲 is a subset of {w:w_1≤ Q} with Q=O(1). For SC, we see that 𝒲={w=(w_1,…,w_N)': w_i≥ 0, ∑_i=1^N w_i=1 } and thus Q=1. 






We use the notation E_T,(-k) for |H_(-k)|^-1∑_t∈ H_(-k). To simplify notations, we use δ=ŵ_(k)-w instead of  Δ_(k)=ŵ_(k)-w. Let η_1=E_T,(-k)(X_tX_t'-E(X_tX_t'))_∞
and η_2=E_T,(-k)X_tu_t_∞. 

Notice that E_T,(-k)(u_t-X_t'δ)^2≤ E_T,(-k)u_t^2.
Therefore, 

    E_T,(-k)(X_t'δ)^2≤2(E_T,(-k)X_tu_t)'δ≤2η_2δ_1.


Notice that 

    |δ'(E_T,(-k)X_tX_t'-Σ_(-k))δ|≤δ_1^2‖ E_T,(-k)X_tX_t'-Σ_(-k)‖ _∞=η_1δ_1^2.


This means that 

    E_T,(-k)(X_t'δ)^2≥δ'Σ_(-k)δ-η_1δ_1^2≥κδ_2^2-η_1δ_1^2.


It follows that 

    κδ_2^2-η_1δ_1^2≤2η_2δ_1.


Since δ_1≤ŵ_(k)_1+w_1≤2Q,
we obtain the result by δ_2^2≤(η_1δ_1^2+2η_2δ_1)/κ.  





  §.§.§ Proof of Lemma <ref>

	
We prove the result for the more general estimator (<ref>), where 𝒲 is assumed to be a convex subset of {w:w_1≤ Q}, Q=O(1). This estimator encompasses SC.
	
	For simplicity, we write =ŵ_(k), =_(-k)
	and =_(-k). Let ξ=(- w_*)-(μ-Σ w_*). By assumption, ξ_∞=o_P(1). 
	
	We rewrite 
	
    w_* =min_v v'Σ v-2μ'v     s.t.     v∈𝒲

	and 
	
    =min_v v' v-2'v     s.t.     v∈𝒲

	
	Let Δ=-w_*. 	For any λ∈[0,1], define w_λ=w_*+λΔ.
	Then by the definition of w_*, we have that w_λ'Σ w_λ-2μ' w_λ≥ w_*'Σ w_*-2μ' w_*,
	which means 
	
    λ^2Δ'ΣΔ≥2λ(μ-Σ w_*)'Δ.

	
	Thus, for any λ∈(0,1), we have that λΔ'ΣΔ≥ 2(μ-Σ w_*)'Δ.
	Since this holds for any λ∈(0,1), we have that 
	
    (μ-Σ w_*)'Δ≤0.

	
	Now by definition, we have that 
	
    '-2'≤ w_*' w_*-2' w_*.

	
	It follows that 
	
    Δ'Δ≤2(- w_*)'Δ.

	
	Notice that 
	
    (- w_*)'Δ =(μ-Σ w_*)'Δ+ξ'Δ(i)≤ξ'Δ≤ξ_∞Δ_1≤2ξ_∞Q,

	where (i) holds by (<ref>). The above two displays imply
	that 
	
    Δ'Δ≤2ξ_∞Q.

	
	Therefore, 
	
    2ξ_∞Q ≥Δ'ΣΔ+Δ'(-Σ)Δ   ≥   Δ'ΣΔ--Σ_∞Δ_1^2
       ≥    cΔ_2^2-4-Σ_∞Q^2.

	
	The desired result follows. 





 §.§ Main results






  §.§.§ Proof of Theorem <ref>




	By Lemma <ref>, we have that 
	
    |τ̂_k-τ-(1/T_1∑_t=T_0+1^Tu_t-1/|H_k|∑_t∈ H_ku_t)|
    	≤| 1/|H_k|∑_t∈ H_k_t'Δ_(k)|+ |1/T_1∑_t=T_0+1^T_t'Δ_(k)|,

	where X̃_t=X_t-E(X_t) 
	and Δ_(k)=_(k)-w.[We note that working with X̃_t instead of X_t is key for our theoretical argument since E(X̃_t)=0.] We now bound the two terms on the right-hand side of (<ref>). 
	
Fix k∈{1,…,K}. Define B_k to be the “two-sided buffer”,
i.e., the set that contains the smallest γ_T numbers and
the largest γ_T numbers in  H_k={(k-1)r+1,…,kr} for 1≤ k≤ K. Also define A_k=H_k\ B_k,
i.e., A_k={t: (k-1)r+1+γ_T+1≤ t≤ kr-γ_T}.
Thus, 

    ∑_t∈ H_kX̃_t'Δ_(k)=∑_t∈ A_kX̃_t'Δ_(k)+∑_t∈ B_kX̃_t'Δ_(k).


The second term can be bounded by 

    |∑_t∈ B_kX̃_t'Δ_(k)|≤max_1≤ t≤ T_0X̃_t_∞Δ_(k)_1|B_k|=2γ_Tmax_1≤ t≤ T_0X̃_t_∞Δ_(k)_1.


Thus, 

    P(|∑_t∈ B_kX̃_t'Δ_(k)|≤2ρ_Tγ_TΔ_(k)_1)→1.


On the other hand, we use Berbee's coupling to bound ∑_t∈ A_kX̃_t'Δ_(k).
By Theorem 16.2.1 of <cit.>, on an enlarged probability
space, there exist random variables {X̅_t}_t∈ A_k
such that (1) {X̅_t}_t∈ A_k and {X̃_t}_t∈ A_k
have the same distribution, (2) {X̅_t}_t∈ A_k is
independent of data in {1,...,T_0}\ H_k and (3)
P({X̅_t}_t∈ A_k≠{X̃_t}_t∈ A_k)≤(γ_T).
Notice that Δ_(k) is independent of {X̅_t}_t∈ A_k.
Hence, 

    E[(∑_t∈ A_kX̅_t'Δ_(k))^2|Δ_(k)]   =   Δ_(k)'E[(∑_t∈ A_kX̅_t)(∑_t∈ A_kX̅_t)']Δ_(k)
       (i)≤   |A_k|κ_1Δ_(k)_2^2,

where (i) follows by Assumption <ref> and the
fact that {X̅_t}_t∈ A_k and {X̃_t}_t∈ A_k
have the same distribution. Thus, ∑_t∈ A_kX̅_t'Δ_(k)=O_P(√(|A_k|)Δ_(k)_2).
Since P({X̅_t}_t∈ A_k≠{X̃_t}_t∈ A_k)≤(γ_T)=o(1),
it follows that 

    ∑_t∈ A_kX̃_t'Δ_(k)=O_P(√(|A_k|)Δ_(k)_2).


Now by (<ref>) and (<ref>),

    ∑_t∈ H_kX̃_t'Δ_(k)   =   O_P(ρ_Tγ_TΔ_(k)_1+√(|A_k|)Δ_(k)_2)
       (i)=   O_P(ρ_Tγ_TΔ_(k)_1+√(r)Δ_(k)_2),

where (i) follows by the assumption that γ_T=o(T_0) and |A_k|<|H_k|=r.
Similarly, we can show

    T_1^-1∑_t=T_0+1^TX̃_t'Δ=O_P(T_1^-1ρ_Tγ_TΔ_(k)_1+T_1^-1/2Δ_(k)_2).

Since r≍min{T_0,T_1}, the above two displays and (<ref>) imply 
	
    |τ̂_k-τ-(1/T_1∑_t=T_0+1^Tu_t-1/|H_k|∑_t∈ H_ku_t)|=O_P(ρ_Tγ_TΔ_(k)_1/min{T_0,T_1}+ Δ_(k)_2/√(min{T_0,T_1})).


By   Assumptions <ref> and <ref>, we have
	
    √(min{T_0,T_1})|τ̂_k-τ-(1/T_1∑_t=T_0+1^Tu_t-1/|H_k|∑_t∈ H_ku_t)|=o_P(1 ).

The desired result in Theorem <ref> follows from Assumption <ref> and the usual CLT for dependent processes, e.g., Theorem 5.20 of <cit.>. 






  §.§.§ Proof of Theorem <ref>

Part (i) is a direct consequence of Theorem <ref>. For Part (ii), recall that 

    𝕋_K=√(K)(-τ_0)/√(1+Kr/T_1)×√((K-1)^-1∑_k=1^K(_(k)-)^2).


We notice that r/T_1→min{c_0/K,1}. Then Theorem <ref> and the continuous mapping theorem imply that   𝕋_Kd→𝒯_K, where 

    𝒯_K    =√(K)(√(min{c_0,1})ξ_0-√(g_c_0,K))/√(1+min{c_0,K})×√((K-1)^-1∑_k=1^Kg_c_0,K(ξ_k-)^2)
        =√(K/g_c_0,K(1+min{c_0,K}))(√(min{c_0,1})ξ_0-√(g_c_0,K))/√((K-1)^-1∑_k=1^K(ξ_k-)^2),

where  =K^-1∑_k=1^Kξ_k.

Notice that  is independent of ∑_k=1^K(ξ_k-)^2.
Therefore, the numerator is independent of the denominator. The variance
of the numerator is 

    K/g_c_0,K(1+min{c_0,K})×(min{c_0,1} +g_c_0,KK^-1)=
    1     if 1≤ c_0 ≤ K
    
    1     if c_0<1
    
    1     if c_0>K.


Hence, the numerator has a standard normal distribution. It follows
that _K∼ t_K-1. The proof is complete. 



  §.§.§ Proof of Theorem <ref>


Note that all the results in Lemma <ref> and Theorem <ref> still hold with u_t replaced by u_*,t. Therefore, the result follows by Lemma <ref> and similar arguments as in Theorems <ref>–<ref>.








  §.§.§ Proof of Theorem <ref>


We start by deriving two auxiliary results. 


Suppose the model is (<ref>). Let Assumption <ref> hold. 

Then, for 1≤ k≤ K,

    τ̂_k-τ=(1/T_1∑_t=T_0+1^Tu_t-1/|H_k|∑_t∈ H_ku_t)+(1/|H_k|∑_t∈ H_k_t-1/T_1∑_t=T_0+1^T_t)'Δ_(k),

where _t=Z_t-E(Z_t) and Δ_(k)=_(k)-w. 



Let μ=E(Z_t). Notice that, because 1_N'w=1_N'_(k)=1, for T_0+1≤ t≤ T, 

    Y_t-X_t'_(k)=α_t+u_t-Z_t'Δ_(k)=α_t+u_t-μ'Δ_(k)-_t'Δ_(k)

and for t∈ H_k

    Y_t-X_t'_(k)=u_t-Z_t'Δ_(k)=u_t-μ'Δ_(k)-_t'Δ_(k).


Therefore, 

    _k-τ    =T_1^-1∑_t=T_0+1^T(Y_t-X_t'_(k))-1/|H_k|∑_t∈ H_k(Y_t-X_t'ŵ_(k))-τ
        =T_1^-1∑_t=T_0+1^T(α_t+u_t-μ'Δ_(k)-_t'Δ_(k))-1/|H_k|∑_t∈ H_k(u_t-μ'Δ_(k)-_t'Δ_(k))-τ
        =T_1^-1∑_t=T_0+1^Tu_t-1/|H_k|∑_t∈ H_ku_t+1/|H_k|∑_t∈ H_k_t'Δ_(k)-T_1^-1∑_t=T_0+1^T_t'Δ_(k).


The proof is complete.





Suppose that the model is (<ref>). Let Assumptions <ref> and  conditions of Lemma <ref> hold with X_t replaced by Z_t hold. Suppose that ŵ_(k) is obtained from (<ref>), where 𝒲⊆{w:w_1≤ Q,1_N'w=1}, Q=O(1) (this estimator nests SC). Then ŵ_(k)-w_2=o_P(1). In particular, 

    ‖ŵ_(k)-w‖ _2^2   ≤   4|H_(-k)|^-1∑_t∈ H_(-k)(Z_tZ_t'-E(Z_tZ_t'))_∞Q^2/κ
          +4|H_(-k)|^-1∑_t∈ H_(-k)Z_tu_t_∞Q/κ.


 Note that, under Assumption <ref>, we have that X_t'Δ_(k)=Z_t'Δ_(k), because 1_N'w=1_N'ŵ_(k)=1. The result now follows from similar arguments as in Lemma <ref> with X_t replaced by Z_t. 



 Assumption  <ref> and 1_N'w=1 imply that

    Y_t=X_t'w+u_t=θ_t+Z_t'w+u_t

Thus, since 𝒲^SC⊂{v:1_N'v=1},

    ŵ_(k)   =   v∈𝒲^SCmin∑_t∈ H_(-k)(Y_t-X_t'v)^2
       =   v∈𝒲^SCmin∑_t∈ H_(-k)(θ_t+Z_t'w+u_t-θ_t-Z_t'v)^2
       =   v∈𝒲^SCmin∑_t∈ H_(-k)(Z_t'w+u_t-Z_t'v)^2

This shows that _(k) only depends on {Z_t}_t∈ H_(-k) and, thus, is approximately independent of {Z_t}_t∈ H_(k)⋃{T_0+1,...,T}. Therefore, using Lemma <ref> and the same arguments as in Theorem <ref>, one can show that
	
    |τ̂_k-τ-(1/T_1∑_t=T_0+1^Tu_t-1/|H_k|∑_t∈ H_ku_t)|=O_P(ρ_Tγ_TΔ_(k)_1/min{T_0,T_1}+ Δ_(k)_2/√(min{T_0,T_1})).


The result now follows by Lemma <ref> and the same arguments as in Theorems <ref>–<ref>. 




  §.§.§ Proof of Theorem <ref>




We first derive a result on the error bounds for SC. Then we prove Theorem <ref>. 

	Assume that Y=Xw+u
	and X=θ_N'+Z+ξβ', where u∈^T, w∈^N
	satisfies _N'w=1 and w≥0 and β=(0,...,0,1,0,...,0)'∈^N
	has jth component taking value 1 and satisfies (w)⋂(β)=∅.
	Consider the estimator 
	
    =min_v∈^N Y-Xv_2^2     s.t.     v≥0, _N'v=1.

	
	Suppose that the eigenvalues of EZ'Z/T are between c_0^-1
	and c_0. Then on the event _1, 
	
    -w_2≤√(c_0/T[8λ_1^2+2c_1^2+4λ_2+4λ_3]),

	where 
	
    _1={Z'ξ_∞≤λ_1ξ_2}⋂{ |u'ξ|≤ c_1ξ_2}⋂{Z'u_∞≤λ_2}⋂{Z'Z-EZ'Z_∞≤λ_3} .

	
	Moreover, on the event _1⋂_2, we have 
	
    |β'(-w)|≤ c_2^-1ξ_2^-2[2λ_1ξ_2+4λ_3+c_0^3/2T^1/2(4λ_1+2c_1+3√(λ_2)+3√(λ_3))],

	where 
	
    _2={min_1≤ j≤ N(ξ+Z_(N))'(ξ+Z_(N)-Z_(j))≥ c_2ξ_2^2}⋂{u_2≤ c_3√(T)}⋂{max_1≤ j≤ NZ_(j)_2≤ c_3√(T)}
    ⋂{ c_2ξ_2≥3(λ_1+c_1)}⋂{ c_2ξ_2^2≥6c_3√(T)(√(Tc_0)+√(λ_3)+c_3√(T))}

	and Z_(j)∈^T denotes the jth column of Z∈^T× N.


	Without loss of generality, we assume that β=(0,...,0,1)'∈^N.
	Then (w)⊆{1,...,N-1}. Let Δ=-w and Σ_Z=EZ'Z/T.
	We proceed in three steps.
	
	Step 1: bound Δ_2.
	
	Recall that Y=Xw+u. By definition, we have Y-X_2^2≤Y-Xw_2^2.
	This means that 
	
    XΔ_2^2≤2u'XΔ.

	We notice that _N'Δ=0, X=θ_N'+Z+ξβ'
	and thus
	
    XΔ=ZΔ+ξβ'Δ.

	
	Therefore, we have 
	
    ZΔ_2^2+ξ_2^2(β'Δ)^2+2Δ'Z'ξβ'Δ≤2u'ZΔ+2u'ξβ'Δ.

	
	We then have 
	
    [ξ_2(β'Δ)+ξ_2^-1(Δ'Z'ξ-u'ξ)]^2≤ξ_2^-2(Δ'Z'ξ-u'ξ)^2+2u'ZΔ-ZΔ_2^2.

	
	It follows that
	
    ξ_2^-2(Δ'Z'ξ-u'ξ)^2+2u'ZΔ-ZΔ_2^2≥0.

	
	Notice that Δ_1≤_1+w_1=2. Hence, 
	
    (Δ'Z'ξ-u'ξ)^2≤2(Δ'Z'ξ)^2+2(u'ξ)^2≤2Δ_1^2Z'ξ_∞^2+2c_1^2ξ_2^2≤(8λ_1^2+2c_1^2)ξ_2^2.

	
	By the above two displays, we have 
	
    8λ_1^2+2c_1^2+2u'ZΔ≥ZΔ_2^2.

	
	Moreover, we notice that |u'ZΔ|≤λ_2Δ_1
	and 
	
    ZΔ_2^2    =Δ'EZ'ZΔ+Δ'(Z'Z-EZ'Z)Δ
       ≥Δ'EZ'ZΔ-Z'Z-EZ'Z_∞Δ_1^2.

	
	By Δ_1≤2, we have that
	
    8λ_1^2+2c_1^2+4λ_2≥ TΔ'Σ_ZΔ-4λ_3.

	
	Therefore, 
	
    Δ'Σ_ZΔ≤1/T[8λ_1^2+2c_1^2+4λ_2+4λ_3].

	
	We obtain the bound on Δ_2 by λ_min(Σ_Z)≥ c_0^-1. 
	
	Step 2: show _-N≠0.
	
	We proceed by contradiction. Suppose that _-N=0. By the constraint
	of _N'=1, we have _N=Δ_N=1. 
	
	Let Δ=(Δ_-N',Δ_N)'∈^N with Δ_-N=(Δ_1,...,Δ_N-1)'∈^N-1.
	For 1≤ j≤ N, let X_(j)∈^T denote the jth column
	of X. In other words, X_(j)=θ+Z_(j) for 1≤ j≤ N-1
	and X_(N)=θ+Z_(N)+ξ. 
	
	Let Q_j(λ)=Y-X_(N)(1-λ)-X_(j)λ_2^2
	for 1≤ j≤ N-1. Notice that d/dλQ_j(λ)|_λ=0=2(Y-X_(N))'(X_(N)-X_(j)).
	It follows that (Y-X_(N))'(X_(N)-X_(j))≥0 for 1≤ j≤ N-1;
	otherwise, one could find a small λ that delivers a smaller
	value for the objective function. Notice that 
	
    Y-X_(N)=θ+Zw+u-(θ+Z_(N)+ξ)=Zw-Z_(N)+u-ξ

	and 
	
    X_(N)-X_(j)=(θ+Z_(N)+ξ)-(θ+Z_(j))=ξ+Z_(N)-Z_(j).

	
	Then 
	
    (Zw-Z_(N)+u-ξ)'(ξ+Z_(N)-Z_(j))
        =(Zw+u)'(ξ+Z_(N)-Z_(j))-(ξ+Z_(N))'(ξ+Z_(N)-Z_(j))
       ≤(Zw+u)'(ξ+Z_(N)-Z_(j))-c_2ξ_2^2
        =w'Z'ξ+u'ξ+(Zw+u)'(Z_(N)-Z_(j))-c_2ξ_2^2
       ≤w_1Z'ξ_∞+c_1ξ_2+(Zw+u)'(Z_(N)-Z_(j))-c_2ξ_2^2
       ≤(λ_1+c_1)ξ_2+(Zw_2+u_2)(Z_(N)_2+Z_(j)_2)-c_2ξ_2^2
       ≤(λ_1+c_1)ξ_2+2c_3√(T)(Zw_2+c_3√(T))-c_2ξ_2^2.

	
	Since w_j∈[0,1], we have w_j^2≤ w_j, which means
	w_2^2=∑_j=1^Nw_j^2≤∑_j=1^Nw_j=1.
	Thus, 
	
    Zw_2^2=w'Z'Zw≤ w'EZ'Zw+|w'(Z'Z-EZ'Z)w|
    ≤w_2^2Tλ_max(Σ_Z)+w_1^2Z'Z-EZ'Z_∞≤ Tc_0+λ_3.

	
	Therefore, we have 
	
    (Zw-Z_(N)+u-ξ)'(ξ+Z_(N)-Z_(j))
       ≤(λ_1+c_1)ξ_2+2c_3√(T)(√(Tc_0+λ_3)+c_3√(T))-c_2ξ_2^2
       ≤(λ_1+c_1)ξ_2+2c_3√(T)(√(Tc_0)+√(λ_3)+c_3√(T))-c_2ξ_2^2
       (i)≤1/3c_2ξ_2^2+1/3c_2ξ_2^2-c_2ξ_2^2<0,

	where (i) follows by the definition of _2. This contradicts
	(Y-X_(N))'(X_(N)-X_(j))≥0 for 1≤ j≤ N-1. Hence,
	_-N≠0. 
	
	Step 3: bound β'Δ.

	
	Let J=(_-N). By Step 2, J≠∅. Since (w)⊆{1,...,N-1},
	we have Δ_N=_N. Notice that β'Δ=Δ_N.
	
	If Δ_N=0, the result clearly holds. Thus, now we assume
	Δ_N=_N>0. Let =(_1,....,_N)'∈^N.
	For any j∈ J, define L_j(λ)=Y-X^(j)(λ)_2^2,
	where ^(j)(λ)=(_1^(j)(λ),....,_N^(j)(λ))'∈^N
	with 
	
    _k^(j)(λ)=_j+λ   if k=j
    _N-λ   if k=N
    _j   otherwise.

	
	By the first-order condition, we have that d/dλL_j(λ)|_λ=0=0;
	otherwise, for some small λ, ^(j)(λ) satisfies
	all the required constraints and delivers a smaller value of the objective
	function. Notice that d/dλL_j(λ)|_λ=0=2(X_(j)-X_(N))'(u-XΔ).
	Hence, we have 
	
    (X_(j)-X_(N))'(u-XΔ)=0.

	
	We notice that 
	
    X_(j)-X_(N)=θ+Z_(j)-(θ+Z_(N)+ξ)=Z_(j)-Z_(N)-ξ

	and 
	
    XΔ=ZΔ+ξβ'Δ=Z_(-N)Δ_-N+(Z_(N)+ξ)Δ_N.

	
	Therefore, (Z_(j)-Z_(N)-ξ)'[Z_(-N)Δ_-N+(Z_(N)+ξ)Δ_N]=0,
	which means 
	
    Δ_N=-(Z_(j)-Z_(N)-ξ)'Z_(-N)Δ_-N/(Z_(j)-Z_(N)-ξ)'(Z_(N)+ξ).

	
	Let a=(0,⋯,0,1,0,⋯,0,-1)'∈^N with the jth entry
	taking value 1. Notice that
	
    |ξ'Z_(-N)Δ_-N|≤Z'ξ_∞Δ_1≤2λ_1ξ_2

	and 
	
    |(Z_(j)-Z_(N))'Z_(-N)Δ_-N|     =|a'Z'Z[ Δ_-N;    0 ]|
       ≤|a'(Z'Z-EZ'Z)[ Δ_-N;    0 ]|+|a'EZ'Z[ Δ_-N;    0 ]|
       ≤a_1Δ_1Z'Z-EZ'Z_∞+Tλ_max(Σ_Z)a_2Δ_2
       ≤4λ_3+√(2)c_0TΔ_2.

	
	It follows that 
	
    |Δ_N|    ≤2λ_1ξ_2+4λ_3+√(2)c_0TΔ_2/|(Z_(j)-Z_(N)-ξ)'(Z_(N)+ξ)|
       ≤2λ_1ξ_2+4λ_3+√(2)c_0TΔ_2/c_2ξ_2^2
       ≤2λ_1ξ_2+4λ_3+√(2)c_0T×√(c_0/T[8λ_1^2+2c_1^2+4λ_2+4λ_3])/c_2ξ_2^2
       ≤2λ_1ξ_2+4λ_3+√(2)c_0√(c_0T)×[√(8λ_1^2)+√(2c_1^2)+√(4λ_2)+√(4λ_3)]/c_2ξ_2^2.

	
	The proof is complete.



	Let Δ_(k)=_(k)-w. Notice that
	
    _k-τ    =T_1^-1∑_t=T_0+1^T(Y_t-X_t'_(k))-1/|H_k|∑_t∈ H_k(Y_t-X_t'_(k))-τ
        =T_1^-1∑_t=T_0+1^T(α_t+u_t-X_t'Δ_(k))-1/|H_k|∑_t∈ H_k(u_t-X_t'Δ_(k))-τ
        =T_1^-1∑_t=T_0+1^T(u_t-X_t'Δ_(k))-1/|H_k|∑_t∈ H_k(u_t-X_t'Δ_(k))
        =T_1^-1∑_t=T_0+1^Tu_t-1/|H_k|∑_t∈ H_ku_t+1/|H_k|∑_t∈ H_kX_t'Δ_(k)-T_1^-1∑_t=T_0+1^TX_t'Δ_(k).

	
	We notice that 
	
    ∑_t∈ H_kX_t'Δ_(k)=∑_t∈ H_kZ_t'Δ_(k)+(∑_t∈ H_kξ_t)(β'Δ_(k)).

	
	Since Δ_(k) depends on {(θ_t,Z_t,ξ_t)}_t∈ H_(-k)
	and {Z_t}_t∈ H_(-k) is independent of {(θ_t,ξ_t)}_t=1^T,
	it follows that conditional on {(θ_t,ξ_t)}_t=1^T,
	we can still use the coupling argument in the proof of Theorem <ref>
	and obtain 
	
    ∑_t∈ H_kZ_t'Δ_(k)=O_P(ρ_Tγ_TΔ_(k)_1+√(|H_k|)·Δ_(k)_2).

	
	By Assumption <ref>, we apply Lemma <ref>
	and obtain 
	
    Δ_(k)_2=O_P(√(T_0^-1[log N+√(T_0log N)]))=O_P((T_0^-1log N)^1/4)=o_P(1)

	and 
	
    |β'Δ_(k)|=O_P(√(log N/∑_t∈ H_(-k)ξ_t^2)+T_0^3/4(log N)^1/4/∑_t∈ H_(-k)ξ_t^2).

	
	Therefore, 
	
    √(min{T_0,T_1})/|H_k|∑_t∈ H_kX_t'Δ_(k)
        =√(min{T_0,T_1})/|H_k|∑_t∈ H_kZ_t'Δ_(k)+√(min{T_0,T_1})/|H_k|(∑_t∈ H_kξ_t)(β'Δ_(k))
        =√(min{T_0,T_1})/|H_k|O_P(ρ_Tγ_TΔ_(k)_1+√(|H_k|)·Δ_(k)_2)
           +√(min{T_0,T_1})/|H_k|O_P((∑_t∈ H_kξ_t)√(log N/∑_t∈ H_(-k)ξ_t^2)+(∑_t∈ H_kξ_t)T_0^3/4(log N)^1/4/∑_t∈ H_(-k)ξ_t^2)
       (i)=o_P(1)+T_1^-1/2O_P(√(log N)∑_t∈ H_kξ_t/√(∑_t∈ H_(-k)ξ_t^2)+∑_t∈ H_kξ_t/∑_t∈ H_(-k)ξ_t^2× T^3/4(log N)^1/4)
       (ii)=o_P(1)+T_1^-1/2O_P(√(log N)· T_1T^-1/2+T_1T^-3/2× T^3/4(log N)^1/4)(ii)=o_P(1),

	where (i) follows by (<ref>), min{T_1,T_0}≍ T_1,
	|H_k|≍ T_1, T_0≍ T, T_1^-1/2ρ_Tγ_T=o(1)
	(Assumption <ref>), (ii) follows by Assumption <ref>,
	and (iii) follows by T_1≪ T_0/log N.
	
	Similarly, we can show that
	
	
    ∑_t=T_0+1^TZ_t'Δ_(k)=O_P(ρ_Tγ_TΔ_(k)_1+√(T_1)Δ_(k)_2)

	and thus 
	
    √(min{T_0,T_1})/T_1∑_t=T_0+1^TX_t'Δ_(k)
        =√(min{T_0,T_1})/T_1∑_t=T_0+1^TZ_t'Δ_(k)+√(min{T_0,T_1})/T_1(∑_t=T_0+1^Tξ_t)(β'Δ_(k))
        =√(min{T_0,T_1})/T_1O_P(ρ_Tγ_TΔ_(k)_1+√(T_1)Δ_(k)_2)
           +√(min{T_0,T_1})/T_1O_P((∑_t=T_0+1^Tξ_t)√(log N/∑_t∈ H_(-k)ξ_t^2)+(∑_t=T_0+1^Tξ_t)T_0^3/4(log N)^1/4/∑_t∈ H_(-k)ξ_t^2)
       (i)=o_P(1)+T_1^-1/2O_P(√(log N)∑_t=T_0+1^Tξ_t/√(∑_t∈ H_(-k)ξ_t^2)+∑_t=T_0+1^Tξ_t/∑_t∈ H_(-k)ξ_t^2× T^3/4(log N)^1/4)
       (ii)=o_P(1)+T_1^-1/2O_P(√(log N)· T_1T^-1/2+T_1T^-3/2× T^3/4(log N)^1/4)(iii)=o_P(1),

	where (i) follows by T_1≪ T_0/log N and Assumption <ref>,
	(ii) follows by Assumption <ref> and
	(iii) follows by T_1≪ T_0/log N.
	
	Now we combine (<ref>) with (<ref>)
	and (<ref>), obtaining 
	
    √(min{T_0,T_1})(_k-τ)
        =√(min{T_0,T_1}/T_1)(T_1^-1/2∑_t=T_0+1^Tu_t)-√(min{T_0,T_1}/|H_k|)(|H_k|^-1/2∑_t∈ H_ku_t)
           +√(min{T_0,T_1})/|H_k|∑_t∈ H_kX_t'Δ_(k)-√(min{T_0,T_1})/T_1∑_t=T_0+1^TX_t'Δ_(k)
        =√(min{T_0,T_1}/T_1)(T_1^-1/2∑_t=T_0+1^Tu_t)-√(min{T_0,T_1}/|H_k|)(|H_k|^-1/2∑_t∈ H_ku_t)+o_P(1).

	
	Since T_1≪ T_0 and |H_k|=min{T_0/K,T_1}, we have
	min{T_0,T_1}/T_1→1 and min{T_0,T_1}/|H_k|→1.
	The first claim of the theorem follows.
	
	For the second claim, we notice that (T_1^-1/2∑_t=T_0+1^Tu_t,|H_1|^-1/2∑_t∈ H_1u_t,...,|H_K|^-1/2∑_t∈ H_ku_t)'
	converges in distribution to the normal distribution with mean zero
	and variance matrix I_K+1σ^2, where σ^2 is the
	LRV of u_t. Notice that 
	
    𝕋_K=√(KT_1/|H_1|K+T_1)·(-τ)/√(1/K-1∑_k=1^K(_(k)-)^2).

	
	By the same argument as in the proof of Theorem  <ref>, asymptotically
	the numerator has N(0,1) distribution and is independent of the
	denominator, which asymptotically has a χ_K-1^2 distribution
	divided by K-1. Therefore, 𝕋_K converges in distribution
	to t_K-1. The
	proof is complete.






  §.§.§ Proof of Theorem <ref>

We define the mapping ψ:^K→ by 

    ψ(v_1,...,v_K)=√(K)v̅/√((K-1)^-1∑_k=1^K(v_k-v̅)^2)    with    v̅=K^-1∑_j=1^Kv_j.


We define Σ_T,K to be the covariance matrix of √(G)(_1,...,_K).
We also define the matrix Σ_K to be the diagonal matrix
whose diagonal entries are all equal to TE(_1^2). Finally,
we define ξ∼ N(0,I_K). Hence, 𝐓_K has the same distribution
as ψ(Σ_T,K^1/2ξ). Moreover, ψ(Σ_K^1/2ξ)
follows student's t-distribution with K-1 degrees of freedom.
It suffices to show that 

    |P(ψ(Σ_T,K^1/2ξ)≤ a)-P(ψ(Σ_K^1/2ξ)≤ a)|=O(T^-1)    ∀ a∈.


Clearly, ψ(·) is Lipschitz. Since the student's t-distribution
with K-1 degrees of freedom has bounded density, we only need to
show that Σ_T,K^1/2ξ-Σ_K^1/2ξ_2=O_P(T^-1). 

Now we do so by showing that Σ_T,K-Σ_K=O(T^-1).
The diagonal entries in both matrices are the same. We consider the
off-diagonal entries. Fix k,l∈{1,...,K} with l>k≥1. Then

    |(Σ_T,K)_k,l|
        =G|E(_k_l)|
        =G^-1|∑_t_1∈ H_k∑_t_2∈ H_lEu_t_1u_t_2|
        =G^-1|∑_t_1=(k-1)G+1^kG∑_t_2=(l-1)G+1^lGEu_t_1u_t_2|
        =G^-1|∑_t_1=(k-1)G+1^kG∑_t_2=(l-1)G+1^lGγ(t_2-t_1)|
        =G^-1|∑_t_1=(k-1)G+1^kG∑_h=(l-1)G+1-t_1^lG-t_1γ(h)|
        =G^-1|∑_h=(l-2)G+1^(l-1)Gγ(h)[min{ (l-k+1)G-h,G} -max{ (l-k)G+1-h,1}]|
       ≤ G^-1∑_h=(l-2)G+1^(l-1)G|γ(h)|·|[min{ (l-k+1)G-h,G} -max{ (l-k)G+1-h,1}]|
       ≤ G^-1∑_h=(l-2)G+1^(l-1)G|γ(h)|h≤ KT^-1∑_h=1^∞|γ(h)|h.


Since ∑_h=1^∞|γ(h)|h is bounded, we have (Σ_T,K)_k,l=O(T^-1).
Therefore, Σ_T,K-Σ_K=O(T^-1). The proof is complete.  



