JEL Classification: C12, C13, C14.
Introduction
Two-way clustering is widely used in empirical economics to account for dependence among observations that share a cluster in either of two dimensions. In a firm panel, for example, outcomes may be correlated both within a firm over time and across firms in the same period. The conventional approach estimates a parameter from the full array and adjusts its standard error for dependence within rows and columns [3 ] . Even for the sample mean, however, the dependence structure can affect the form of the limiting distribution. When row or column effects dominate, the normalized full-array mean typically has a Gaussian limit. As these effects weaken, interactions between rows and columns can produce a non-Gaussian limit. Menzel [24 ] shows that the limiting law of the normalized full-array mean cannot be estimated uniformly consistently over such sequences. We approach the inference problem by constructing a different estimator of the same population mean, for which Gaussian critical values are uniformly valid.
Our estimator averages the means obtained from repeated random matchings. Taking the dimension with fewer clusters as the rows, each matching assigns a distinct column to every row, uniformly at random and independently of the data. No two selected observations share a row or a column. Under separately exchangeable, dissociated sampling, they therefore form an i.i.d. sample from the marginal distribution. A single matching permits conventional inference for the population mean but uses only one observation per row. Additional matchings recover precision: conditional on the observed array, their means are independent and centered at the full-array mean. The choice of how many matchings to use matters because averaging too many can cause the estimator to inherit the non-Gaussian limit of the full-array mean.
We make four contributions. First, we characterize the limiting distribution of the matching estimator as the number of matchings varies. An exact variance decomposition separates the sampling variance of the full-array mean from the additional variance due to matching. Increasing the number of matchings improves precision, but it can also give the full-array error enough weight to produce a non-Gaussian limit. This trade-off motivates letting the number of matchings grow more slowly than the smaller cluster count to recover precision while retaining a Gaussian limit across dependence regimes.
Second, we provide a closed-form variance estimator that makes the confidence interval straightforward to compute. It combines estimates of row, column, and intersection variation with nonnegative weights. The estimate is positive semidefinite in every sample by construction. Inference then requires only the estimated standard error and a standard normal critical value, without a bootstrap or a preliminary test of the dependence regime.
Third, we establish uniformly asymptotically exact inference for the population mean. Under our moment conditions, the studentized matching estimator admits a uniform standard normal approximation whenever the number of matchings is fixed or grows at any rate slower than the smaller cluster count. If the number of matchings grows more slowly than the square root of that count, a uniform bound on the standardized fourth moment suffices for the same conclusion. The approximation holds uniformly over sequences of data-generating processes that may vary with sample size. The strength of clustering in either dimension need not converge, and the two cluster counts may grow at different rates.
Fourth, we extend the procedure to smooth functions of generalized method of moments (GMM) parameters, including regression coefficients. Under two-way clustering, different moments can converge at different rates. Estimation error in a slowly converging coefficient can then enter the limiting distribution of a faster one through a quadratic term (Davezies et al. [13 ] , Section 3). We show that the same restriction on the number of matchings makes this term negligible relative to the standard error of the target. Applying the variance estimator to the estimated influence function therefore gives uniformly asymptotically exact inference even when the moment components follow different dependence regimes.
Our analysis complements the literature on inference based on the full array. Davezies et al. [12 ] develop empirical process theory for exchangeable arrays, and Chiang and Sasaki [11 ] give conditions under which conventional two-way standard errors yield valid Gaussian inference. MacKinnon et al. [23 ] study wild bootstrap inference for regression, while Chiang, Matsushita, and Otsu (2025) construct an empirical likelihood statistic using estimates that leave out rows or columns. Davezies et al. [13 ] develop analytic tests that control size uniformly over their class. Their tests are asymptotically exact in Gaussian regimes and can be conservative in non-Gaussian regimes. By changing the estimator, we obtain exact asymptotic coverage even for laws under which the full-array mean has a non-Gaussian limit, at a cost in precision that we quantify.
Menzel [24 ] shows that, when the observation pattern is independent of the data, observing only a vanishing fraction of intersections can yield a Gaussian limit for the sample mean. Chiang, Li, and Sasaki [9 ] establish asymptotic normality under Bernoulli subsampling for each fixed data-generating process. We select intersections by repeated matching and establish asymptotically exact coverage uniformly over laws in which row and column effects may weaken or vanish as the sample size grows. The role of the number of sampled terms in determining the limiting distribution also arises in the theory of incomplete U U U [19 , 6 ] .
Section 2 introduces the two-way clustering framework and describes how the dependence structure affects the limiting distribution of the full-array mean. Section 3 presents the matching estimator, shows how the number of matchings affects its precision and limiting distribution, and establishes uniformly asymptotically exact coverage for the resulting interval. Section 4 extends the results to smooth functions of GMM parameters. Section 5 reports Monte Carlo evidence on confidence interval coverage and length. Section 6 concludes. The appendix contains the proofs.
Two-way clustering
Sampling framework
We observe a complete array Y N T = { Y N T , i t : 1 ≤ i ≤ N , 1 ≤ t ≤ T } Y_{NT} = \{Y_{NT,it} : 1 \le i \le N, 1 \le t \le T\} Y N T = { Y N T , i t : 1 ≤ i ≤ N , 1 ≤ t ≤ T } 2 ≤ N ≤ T 2 \le N \le T 2 ≤ N ≤ T N , T → ∞ N, T \to\infty N , T → ∞
Assumption 1 (Two-way clustered sampling). At each ( N , T ) (N,T) ( N , T ) the observations admit the representation
Y N T , i t = f N T ( ξ i , η t , ζ i t ) . (1) Y_{NT,it}=f_{NT}(\xi_i,\eta_t,\zeta_{it}). \tag*{(1)} Y N T , i t = f N T ( ξ i , η t , ζ i t ) . (1) The row shocks { ξ i } \{\xi_i\} { ξ i } column shocks { η t } \{\eta_t\} { η t } and intersection shocks { ζ i t } \{\zeta_{it}\} { ζ i t } form three mutually independent families, with i.i.d. shocks within each family. The marginal mean is θ N T = E [ Y N T , i t ] \theta_{NT}=E[Y_{NT,it}] θ N T = E [ Y N T , i t ] and the marginal variance satisfies σ N T 2 = Var ( Y N T , i t ) ∈ ( 0 , ∞ ) \sigma^2_{NT}=\operatorname{Var}(Y_{NT,it})\in(0,\infty) σ N T 2 = Var ( Y N T , i t ) ∈ ( 0 , ∞ )
Equation (1) is the Aldous-Hoover-Kallenberg representation for a separately exchangeable, dissociated array [1 , 16 , 22 ] . Observations may be dependent within a row or column, while collections of observations with disjoint row and column indices are independent. Allowing f N T f_{NT} f N T ( N , T ) (N,T) ( N , T )
To describe how each shared shock affects the conditional mean, define the row projection and column projection by
a N T , i = E [ Y N T , i t ∣ ξ i ] − θ N T , b N T , t = E [ Y N T , i t ∣ η t ] − θ N T . a_{NT,i}=E[Y_{NT,it}\mid\xi_i]-\theta_{NT},\qquad b_{NT,t}=E[Y_{NT,it}\mid\eta_t]-\theta_{NT}. a N T , i = E [ Y N T , i t ∣ ξ i ] − θ N T , b N T , t = E [ Y N T , i t ∣ η t ] − θ N T . The remaining variation is the intersection residual r N T , i t r_{NT,it} r N T , i t
Y N T , i t − θ N T = a N T , i + b N T , t + r N T , i t . (2) Y_{NT,it}-\theta_{NT}=a_{NT,i}+b_{NT,t}+r_{NT,it}. \tag*{(2)} Y N T , i t − θ N T = a N T , i + b N T , t + r N T , i t . (2) All three components are centered and mutually uncorrelated. The intersection residual satisfies E [ r N T , i t ∣ ξ i ] = E [ r N T , i t ∣ η t ] = 0 E[r_{NT,it}\mid\xi_i]=E[r_{NT,it}\mid\eta_t]=0 E [ r N T , i t ∣ ξ i ] = E [ r N T , i t ∣ η t ] = 0 ξ i \xi_i ξ i η t \eta_t η t
Averaging (2) gives the corresponding decomposition of the full-array mean Y ˉ N T = \bar{Y}_{NT}= Y ˉ N T =
( N T ) − 1 ∑ i , t Y N T , i t , (NT)^{-1}\sum_{i,t}Y_{NT,it}, ( N T ) − 1 i , t ∑ Y N T , i t , Y ˉ N T − θ N T = 1 N ∑ i = 1 N a N T , i + 1 T ∑ t = 1 T b N T , t + 1 N T ∑ i = 1 N ∑ t = 1 T r N T , i t . \bar{Y}_{NT}-\theta_{NT}=\frac{1}{N}\sum_{i=1}^{N}a_{NT,i}+\frac{1}{T}\sum_{t=1}^{T}b_{NT,t}+\frac{1}{NT}\sum_{i=1}^{N}\sum_{t=1}^{T}r_{NT,it}. Y ˉ N T − θ N T = N 1 i = 1 ∑ N a N T , i + T 1 t = 1 ∑ T b N T , t + N T 1 i = 1 ∑ N t = 1 ∑ T r N T , i t . Let σ N T , R 2 = E [ a N T , i 2 ] \sigma^{2}_{NT,R}=E[a^{2}_{NT,i}] σ N T , R 2 = E [ a N T , i 2 ] σ N T , C 2 = E [ b N T , t 2 ] \sigma^{2}_{NT,C}=E[b^{2}_{NT,t}] σ N T , C 2 = E [ b N T , t 2 ] σ N T , I 2 = E [ r N T , i t 2 ] \sigma^{2}_{NT,I}=E[r^{2}_{NT,it}] σ N T , I 2 = E [ r N T , i t 2 ]
σ N T 2 = σ N T , R 2 + σ N T , C 2 + σ N T , I 2 , V N T , F = Var ( Y ˉ N T ) = σ N T , R 2 N + σ N T , C 2 T + σ N T , I 2 N T . (3) \sigma^{2}_{NT}=\sigma^{2}_{NT,R}+\sigma^{2}_{NT,C}+\sigma^{2}_{NT,I},\qquad V_{NT,F}=\operatorname{Var}(\bar{Y}_{NT})=\frac{\sigma^{2}_{NT,R}}{N}+\frac{\sigma^{2}_{NT,C}}{T}+\frac{\sigma^{2}_{NT,I}}{NT}. \tag*{(3)} σ N T 2 = σ N T , R 2 + σ N T , C 2 + σ N T , I 2 , V N T , F = Var ( Y ˉ N T ) = N σ N T , R 2 + T σ N T , C 2 + N T σ N T , I 2 . (3) Limiting distributions across dependence regimes
By (3), the row and column projections account for the following share of the variance of the full-array mean:
ν N T = σ N T , R 2 / N + σ N T , C 2 / T V N T , F , 1 − ν N T = σ N T , I 2 / ( N T ) V N T , F . (4) \nu_{NT}=\frac{\sigma^{2}_{NT,R}/N+\sigma^{2}_{NT,C}/T}{V_{NT,F}},\qquad1-\nu_{NT}=\frac{\sigma^{2}_{NT,I}/(NT)}{V_{NT,F}}. \tag*{(4)} ν N T = V N T , F σ N T , R 2 / N + σ N T , C 2 / T , 1 − ν N T = V N T , F σ N T , I 2 / ( N T ) . (4) We call a sequence nondegenerate if ν N T → 1 \nu_{NT}\to1 ν N T → 1 degenerate case, ν N T → 0 \nu_{NT}\to0 ν N T → 0 Exact degeneracy , where σ N T , R 2 = σ N T , C 2 = 0 \sigma^{2}_{NT,R}=\sigma^{2}_{NT,C}=0 σ N T , R 2 = σ N T , C 2 = 0 intermediate case, ν N T → c ∈ ( 0 , 1 ) \nu_{NT}\to c\in(0,1) ν N T → c ∈ ( 0 , 1 ) ν N T \nu_{NT} ν N T
Even under exact degeneracy, the limiting law can be Gaussian or non-Gaussian. Consider two designs with θ N T = 0 \theta_{NT}=0 θ N T = 0 σ N T , I 2 = σ N T 2 = 1 \sigma^{2}_{NT,I}=\sigma^{2}_{NT}=1 σ N T , I 2 = σ N T 2 = 1 V N T , F = 1 / ( N T ) V_{NT,F}=1/(NT) V N T , F = 1/ ( N T ) Y N T , i t = ζ i t Y_{NT,it}=\zeta_{it} Y N T , i t = ζ i t ζ i t \zeta_{it} ζ i t N T Y ˉ N T \sqrt{NT}\bar{Y}_{NT} N T Y ˉ N T Y N T , i t = ξ i η t Y_{NT,it}=\xi_i\eta_t Y N T , i t = ξ i η t Y ˉ N T \bar{Y}_{NT} Y ˉ N T
N T Y ˉ N T = ( N ξ ˉ N ) ( T η ˉ T ) = d Z 1 Z 2 , \sqrt{NT}\bar{Y}_{NT}=(\sqrt{N}\bar{\xi}_{N})(\sqrt{T}\bar{\eta}_{T})\mathrel{\overset{d}{=}} Z_{1}Z_{2}, N T Y ˉ N T = ( N ξ ˉ N ) ( T η ˉ T ) = d Z 1 Z 2 , where Z 1 Z_1 Z 1 Z 2 Z_2 Z 2
A local shift moves the product design into the intermediate regime. Let
Y N T , i t = ξ i ( η t + δ / T ) . (5) Y_{NT,it} = \xi_i(\eta_t + \delta/\sqrt{T}). \tag*{(5)} Y N T , i t = ξ i ( η t + δ / T ) . (5) For fixed δ ≠ 0 \delta\ne0 δ = 0 a N T , i = δ ξ i / T a_{NT,i} = \delta\xi_i/\sqrt{T} a N T , i = δ ξ i / T b N T , t = 0 b_{NT,t} = 0 b N T , t = 0 r N T , i t = ξ i η t r_{NT,it} = \xi_i\eta_t r N T , i t = ξ i η t σ N T , R 2 = δ 2 / T → 0 \sigma^2_{NT,R} = \delta^2/T \to0 σ N T , R 2 = δ 2 / T → 0 δ 2 / ( N T ) \delta^2/(NT) δ 2 / ( N T ) V N T , F V_{NT,F} V N T , F 1 / ( N T ) 1/(NT) 1/ ( N T ) ν N T = δ 2 / ( 1 + δ 2 ) \nu_{NT} = \delta^2/(1+\delta^2) ν N T = δ 2 / ( 1 + δ 2 )
Y ˉ N T − θ N T V N T , F = d Z 1 ( Z 2 + δ ) 1 + δ 2 . \frac{\bar{Y}_{NT}-\theta_{NT}}{\sqrt{V_{NT,F}}} \overset{d}{=} \frac{Z_1(Z_2+\delta)}{\sqrt{1+\delta^2}}. V N T , F Y ˉ N T − θ N T = d 1 + δ 2 Z 1 ( Z 2 + δ ) . Thus a column shift of order T − 1 / 2 T^{-1/2} T − 1/2 δ = 0 \delta= 0 δ = 0 δ = 1 \delta= 1 δ = 1 [24 Proposition 4.1] . The next section uses random matching to estimate the same θ N T \theta_{NT} θ N T
Matching and uniform inference
One matching
A matching selects one observation from each row by assigning a different column to every row. To draw the first matching, start with all T T T T T T i = 2 , … , N i = 2,\ldots,N i = 2 , … , N T − i + 1 T-i+1 T − i + 1 π 1 ( i ) \pi_1(i) π 1 ( i ) i i i ( N , T ) (N,T) ( N , T )
i ≠ j ⟹ π 1 ( i ) ≠ π 1 ( j ) . (6) i \ne j \;\Longrightarrow\; \pi_1(i) \ne\pi_1(j). \tag*{(6)} i = j ⟹ π 1 ( i ) = π 1 ( j ) . (6) Conditional on the draw, the N N N (1) . They are therefore conditionally i.i.d. with the same marginal law as Y N T , 11 Y_{NT,11} Y N T , 11
The matching mean and its sample variance are
M N T , 1 = 1 N ∑ i = 1 N Y N T , i π 1 ( i ) , S N T , 1 2 = 1 N − 1 ∑ i = 1 N ( Y N T , i π 1 ( i ) − M N T , 1 ) 2 . (7) M_{NT,1} = \frac{1}{N}\sum_{i=1}^{N}Y_{NT,i\pi_1(i)}, \qquad S^2_{NT,1} = \frac{1}{N-1}\sum_{i=1}^{N}\left(Y_{NT,i\pi_1(i)}-M_{NT,1}\right)^2. \tag*{(7)} M N T , 1 = N 1 i = 1 ∑ N Y N T , i π 1 ( i ) , S N T , 1 2 = N − 1 1 i = 1 ∑ N ( Y N T , i π 1 ( i ) − M N T , 1 ) 2 . (7) For q ≥ 2 q \ge2 q ≥ 2 K < ∞ K < \infty K < ∞ Q N T , q ( K ) Q_{NT,q}(K) Q N T , q ( K ) E ∣ Y N T , 11 − θ N T ∣ q ≤ K σ N T q E|Y_{NT,11}-\theta_{NT}|^q \le K\sigma_{NT}^q E ∣ Y N T , 11 − θ N T ∣ q ≤ K σ N T q Φ \Phi Φ z p = Φ − 1 ( p ) z_p = \Phi^{-1}(p) z p = Φ − 1 ( p )
Proposition 1 (One matching). As N , T → ∞ N,T \to\infty N , T → ∞ N ≤ T N \le T N ≤ T
sup P ∈ Q N T , 3 ( K ) sup x ∈ R ∣ Pr P ( N ( M N T , 1 − θ N T ) S N T , 1 ≤ x ) − Φ ( x ) ∣ → 0. \sup_{P\in Q_{NT,3}(K)}\sup_{x\in\mathbb{R}}\left|\Pr_P\left(\frac{\sqrt{N}(M_{NT,1}-\theta_{NT})}{S_{NT,1}}\le x\right)-\Phi(x)\right|\to0. P ∈ Q N T , 3 ( K ) sup x ∈ R sup P Pr ( S N T , 1 N ( M N T , 1 − θ N T ) ≤ x ) − Φ ( x ) → 0. By Proposition 1, the coverage probability of the one-matching interval M N T , 1 ± z 1 − α / 2 S N T , 1 / N M_{NT,1}\pm z_{1-\alpha/2}S_{NT,1}/\sqrt{N} M N T , 1 ± z 1 − α /2 S N T , 1 / N 1 − α 1-\alpha 1 − α ν N T \nu_{NT} ν N T V N T , 1 = Var ( M N T , 1 ) = σ N T 2 / N V_{NT,1}=\operatorname{Var}(M_{NT,1})=\sigma_{NT}^2/N V N T , 1 = Var ( M N T , 1 ) = σ N T 2 / N (3) gives
V N T , 1 − V N T , F = T − N N T σ N T , C 2 + T − 1 N T σ N T , I 2 ≥ 0. (8) V_{NT,1}-V_{NT,F}=\frac{T-N}{NT}\sigma_{NT,C}^2+\frac{T-1}{NT}\sigma_{NT,I}^2\ge0. \tag*{(8)} V N T , 1 − V N T , F = N T T − N σ N T , C 2 + N T T − 1 σ N T , I 2 ≥ 0. (8) The row variance cancels because one matching uses every row. On a square array (N = T N=T N = T V N T , 1 = V N T , F V_{NT,1}=V_{NT,F} V N T , 1 = V N T , F σ N T , I 2 = 0 \sigma_{NT,I}^2=0 σ N T , I 2 = 0 N < T N<T N < T σ N T , I 2 = 0 \sigma^{2}_{NT,I}=0 σ N T , I 2 = 0 σ N T , C 2 = 0 \sigma^{2}_{NT,C}=0 σ N T , C 2 = 0
The precision loss is particularly clear when the intersections are independent. One matching then averages N N N N T NT N T V N T , 1 = T V N T , F V_{NT,1}=T V_{NT,F} V N T , 1 = T V N T , F
Repeated matching and trade-off
Fix the number of matchings B = B N T ≥ 1 B=B_{NT}\ge1 B = B N T ≥ 1 B B B π b ( i ) \pi_b(i) π b ( i ) i i i b b b
M N T , b = 1 N ∑ i = 1 N Y N T , i π b ( i ) , θ ^ N T , B = 1 B ∑ b = 1 B M N T , b . M_{NT,b}=\frac{1}{N}\sum_{i=1}^{N}Y_{NT,i\pi_b(i)},\qquad\hat{\theta}_{NT,B}=\frac{1}{B}\sum_{b=1}^{B}M_{NT,b}. M N T , b = N 1 i = 1 ∑ N Y N T , i π b ( i ) , θ ^ N T , B = B 1 b = 1 ∑ B M N T , b . Conditional on the observed array Y N T Y_{NT} Y N T 1 / T 1/T 1/ T E [ M N T , b ∣ Y N T ] = Y ˉ N T E[M_{NT,b}\mid Y_{NT}]=\bar{Y}_{NT} E [ M N T , b ∣ Y N T ] = Y ˉ N T
θ ^ N T , B − θ N T = ( Y ˉ N T − θ N T ) + 1 B ∑ b = 1 B ( M N T , b − Y ˉ N T ) . \hat{\theta}_{NT,B}-\theta_{NT}=(\bar{Y}_{NT}-\theta_{NT})+\frac{1}{B}\sum_{b=1}^{B}(M_{NT,b}-\bar{Y}_{NT}). θ ^ N T , B − θ N T = ( Y ˉ N T − θ N T ) + B 1 b = 1 ∑ B ( M N T , b − Y ˉ N T ) . The first term is the sampling error of the full-array mean. It has variance V N T , F V_{NT,F} V N T , F (3) and can have a non-Gaussian limit. The second term is the error from drawing the matchings. Conditional on Y N T Y_{NT} Y N T B B B B B B
Proposition 2 (Variance of repeated matching). For every N ≤ T N\le T N ≤ T B ≥ 1 B\ge1 B ≥ 1 E [ θ ^ N T , B ] = θ N T E[\hat{\theta}_{NT,B}]=\theta_{NT} E [ θ ^ N T , B ] = θ N T
V N T , B : = Var ( θ ^ N T , B ) = V N T , F + V N T , 1 − V N T , F B . (9) V_{NT,B}:=\operatorname{Var}(\hat{\theta}_{NT,B})=V_{NT,F}+\frac{V_{NT,1}-V_{NT,F}}{B}. \tag*{(9)} V N T , B := Var ( θ ^ N T , B ) = V N T , F + B V N T , 1 − V N T , F . (9) Proposition 2 holds for any N ≤ T N \le T N ≤ T N = T N = T N = T (3) , (8) , and (9) ,
V N T , B = σ N T , R 2 + σ N T , C 2 N + σ N T , I 2 N 2 ⏟ full-array variance + σ N T , I 2 N B ( 1 − 1 N ) ⏟ matching variance . (10) V_{NT,B} =
\underbrace{\frac{\sigma^{2}_{NT,R}+\sigma^{2}_{NT,C}}{N}+\frac{\sigma^{2}_{NT,I}}{N^{2}}}_{\text{full-array variance}}
+
\underbrace{\frac{\sigma^{2}_{NT,I}}{NB}\left(1-\frac{1}{N}\right)}_{\text{matching variance}}. \tag*{(10)} V N T , B = full-array variance N σ N T , R 2 + σ N T , C 2 + N 2 σ N T , I 2 + matching variance N B σ N T , I 2 ( 1 − N 1 ) . (10) The matching variance is proportional to 1 / B 1/B 1/ B B B B (10) and (4) , the full-array share of the total variance is
w N T , B = V N T , F V N T , B = B B + ( N − 1 ) ( 1 − ν N T ) . (11) w_{NT,B}=\frac{V_{NT,F}}{V_{NT,B}}=\frac{B}{B+(N-1)(1-\nu_{NT})}. \tag*{(11)} w N T , B = V N T , B V N T , F = B + ( N − 1 ) ( 1 − ν N T ) B . (11) Suppose the marginal fourth-moment bound of Proposition 3 holds. In the degenerate and intermediate cases (ν N T → c < 1 \nu_{NT} \to c < 1 ν N T → c < 1 B = o ( N ) B = o(N) B = o ( N ) w N T , B w_{NT,B} w N T , B B / N → λ ∈ ( 0 , ∞ ] B/N \to\lambda\in(0,\infty] B / N → λ ∈ ( 0 , ∞ ] ν N T → 1 \nu_{NT} \to1 ν N T → 1 B B B
The choice of B B B (10) gives
V N T , B ≍ σ N T 2 ( 1 N B + 1 N 2 ) . V_{NT,B}\asymp\sigma^{2}_{NT}\left(\frac{1}{NB}+\frac{1}{N^{2}}\right). V N T , B ≍ σ N T 2 ( N B 1 + N 2 1 ) . When σ N T → σ ∈ ( 0 , ∞ ) \sigma_{NT} \to\sigma\in(0,\infty) σ N T → σ ∈ ( 0 , ∞ ) N B \sqrt{NB} N B B = o ( N ) B = o(N) B = o ( N ) B → ∞ B \to\infty B → ∞ B = N B = N B = N N N N V N T , B / V N T , F → 2 − c V_{NT,B}/V_{NT,F} \to2-c V N T , B / V N T , F → 2 − c B B B B / N → ∞ B/N \to\infty B / N → ∞ V N T , B / V N T , F → 1 V_{NT,B}/V_{NT,F} \to1 V N T , B / V N T , F → 1
Example 1: Pure product model. Consider the pure product model Y N T , i t = ξ i η t Y_{NT,it} = \xi_i\eta_t Y N T , i t = ξ i η t ν N T = 0 \nu_{NT} = 0 ν N T = 0 w N T , B = B / ( B + N − 1 ) w_{NT,B} = B/(B + N - 1) w N T , B = B / ( B + N − 1 ) Z 1 Z 2 Z_1Z_2 Z 1 Z 2 B B B
θ ^ N T , B − θ N T V N T , B ⇒ { Z 3 , B / N → 0 , ( Z 1 Z 2 + Z 3 ) / 2 , B = N , Z 1 Z 2 , B / N → ∞ . \frac{\widehat{\theta}_{NT,B} - \theta_{NT}}{\sqrt{V_{NT,B}}}
\Rightarrow
\begin{cases}
Z_3, & B/N \to0, \\
(Z_1Z_2 + Z_3)/\sqrt{2}, & B = N, \\
Z_1Z_2, & B/N \to\infty.
\end{cases} V N T , B θ N T , B − θ N T ⇒ ⎩ ⎨ ⎧ Z 3 , ( Z 1 Z 2 + Z 3 ) / 2 , Z 1 Z 2 , B / N → 0 , B = N , B / N → ∞. where Z 1 , Z 2 , Z 3 Z_1, Z_2, Z_3 Z 1 , Z 2 , Z 3 Y ‾ N T − θ N T \overline{Y}_{NT} - \theta_{NT} Y N T − θ N T
The limiting distribution as B B B
In the square-array example, the share w N T , B w_{NT,B} w N T , B N ≤ T N \le T N ≤ T N < T N < T N < T N N N T T T (8) shows. For any array dimensions, (9) gives the full-array variance share
w N T , B = V N T , F V N T , B = B B + V N T , 1 / V N T , F − 1 . (12) w_{NT,B} = \frac{V_{NT,F}}{V_{NT,B}} = \frac{B}{B + V_{NT,1}/V_{NT,F} - 1}. \tag*{(12)} w N T , B = V N T , B V N T , F = B + V N T , 1 / V N T , F − 1 B . (12) Proposition 3 shows how this share weights the two errors in the limiting distribution.
Proposition 3 (Limiting law for any number of matchings). Let N , T → ∞ N, T \to\infty N , T → ∞ N ≤ T N \le T N ≤ T Q N T , 4 ( K ) Q_{NT,4}(K) Q N T , 4 ( K ) K K K B = B N T ≥ 1 B = B_{NT} \ge1 B = B N T ≥ 1 w N T , B → w w_{NT,B} \to w w N T , B → w
If w = 0 w = 0 w = 0 ( θ ^ N T , B − θ N T ) / V N T , B ⇒ N ( 0 , 1 ) (\widehat{\theta}_{NT,B} - \theta_{NT})/\sqrt{V_{NT,B}} \Rightarrow\mathcal{N}(0,1) ( θ N T , B − θ N T ) / V N T , B ⇒ N ( 0 , 1 )
If w > 0 w > 0 w > 0 ( Y ‾ N T − θ N T ) / V N T , F ⇒ L F (\overline{Y}_{NT} - \theta_{NT})/\sqrt{V_{NT,F}} \Rightarrow L_F ( Y N T − θ N T ) / V N T , F ⇒ L F
θ ^ N T , B − θ N T V N T , B ⇒ w L F + 1 − w Z , \frac{\widehat{\theta}_{NT,B} - \theta_{NT}}{\sqrt{V_{NT,B}}} \Rightarrow\sqrt{w}L_F + \sqrt{1-w}Z, V N T , B θ N T , B − θ N T ⇒ w L F + 1 − w Z , where Z Z Z L F L_F L F
When w = 0 w = 0 w = 0 0 < w < 1 0 < w < 1 0 < w < 1 w = 1 w = 1 w = 1 B B B ν N T → 1 \nu_{NT} \to1 ν N T → 1
The role of the longer dimension. In the degenerate and intermediate cases, it is T T T
V N T , 1 T V N T , F = σ N T 2 N T V N T , F = 1 − ν N T + σ N T , R 2 + σ N T , C 2 N T V N T , F , 0 ≤ σ N T , R 2 + σ N T , C 2 N T V N T , F ≤ ν N T N . \frac{V_{NT,1}}{T V_{NT,F}} = \frac{\sigma^2_{NT}}{NTV_{NT,F}} = 1 - \nu_{NT} + \frac{\sigma^2_{NT,R} + \sigma^2_{NT,C}}{NTV_{NT,F}}, \qquad0 \le\frac{\sigma^2_{NT,R} + \sigma^2_{NT,C}}{NTV_{NT,F}} \le\frac{\nu_{NT}}{N}. T V N T , F V N T , 1 = N T V N T , F σ N T 2 = 1 − ν N T + N T V N T , F σ N T , R 2 + σ N T , C 2 , 0 ≤ N T V N T , F σ N T , R 2 + σ N T , C 2 ≤ N ν N T . If ν N T → c < 1 \nu_{NT} \to c < 1 ν N T → c < 1 ( V N T , 1 / V N T , F − 1 ) / T → 1 − c (V_{NT,1}/V_{NT,F} - 1)/T \to1 - c ( V N T , 1 / V N T , F − 1 ) / T → 1 − c B = o ( T ) B = o(T) B = o ( T ) w N T , B → 0 w_{NT,B} \to0 w N T , B → 0 B / T → ∞ B/T \to\infty B / T → ∞ w N T , B → 1 w_{NT,B} \to1 w N T , B → 1 B / T → λ ∈ ( 0 , ∞ ) B/T \to\lambda\in(0,\infty) B / T → λ ∈ ( 0 , ∞ ) w N T , B → λ / ( λ + 1 − c ) w_{NT,B} \to\lambda/(\lambda+ 1 - c) w N T , B → λ / ( λ + 1 − c ) Table 1 gives the corresponding rates and limits. For example, with B = N B = N B = N T / N → ∞ T/N \to\infty T / N → ∞ N N N N T NT N T
Table 1. Rates and limiting laws of the matching estimator for N ≤ T N \le T N ≤ T
Table 1 gives two routes to a Gaussian limit. When ν N T → c < 1 \nu_{NT} \to c < 1 ν N T → c < 1 B = o ( T ) B = o(T) B = o ( T ) ν N T → 1 \nu_{NT} \to1 ν N T → 1 B B B
Notes. The scale multiplies θ ^ N T , B − θ N T \hat{\theta}_{NT,B}-\theta_{NT} θ ^ N T , B − θ N T L F L_F L F Z Z Z L F L_F L F
example in Appendix C illustrates the role of the Lindeberg condition in this second case.
There, ν N T → 1 \nu_{NT} \to1 ν N T → 1
With B = N B=N B = N B = o ( T ) B=o(T) B = o ( T )
Variance estimation and uniform Gaussian inference
We take B = o ( N ) B=o(N) B = o ( N ) ν N T \nu_{NT} ν N T
To accommodate sequences along which the full-array error remains important, we estimate the total variance V N T , B = ( 1 − 1 / B ) V N T , F + σ N T 2 / ( N B ) V_{NT,B}=(1-1/B)V_{NT,F}+\sigma^2_{NT}/(NB) V N T , B = ( 1 − 1/ B ) V N T , F + σ N T 2 / ( N B ) Y ˉ N T , i ⋅ \bar{Y}_{NT,i\cdot} Y ˉ N T , i ⋅ Y ˉ N T , ⋅ t \bar{Y}_{NT,\cdot t} Y ˉ N T , ⋅ t (3) are
σ ^ N T , R 2 = 1 N ∑ i ( Y ˉ N T , i ⋅ − Y ˉ N T ) 2 , σ ^ N T , C 2 = 1 T ∑ t ( Y ˉ N T , ⋅ t − Y ˉ N T ) 2 , r ^ N T , i t = Y N T , i t − Y ˉ N T , i ⋅ − Y ˉ N T , ⋅ t + Y ˉ N T , σ ^ N T , I 2 = 1 N T ∑ i , t r ^ N T , i t 2 . (13) \begin{aligned}
\hat{\sigma}^{2}_{NT,R} &= \frac{1}{N}\sum_i\left(\bar{Y}_{NT,i\cdot}-\bar{Y}_{NT}\right)^2, &
\hat{\sigma}^{2}_{NT,C} &= \frac{1}{T}\sum_t\left(\bar{Y}_{NT,\cdot t}-\bar{Y}_{NT}\right)^2,\\
\hat{r}_{NT,it} &= Y_{NT,it}-\bar{Y}_{NT,i\cdot}-\bar{Y}_{NT,\cdot t}+\bar{Y}_{NT}, &
\hat{\sigma}^{2}_{NT,I} &= \frac{1}{NT}\sum_{i,t}\hat{r}^{2}_{NT,it}. \tag*{(13)}
\end{aligned} σ ^ N T , R 2 r ^ N T , i t = N 1 i ∑ ( Y ˉ N T , i ⋅ − Y ˉ N T ) 2 , = Y N T , i t − Y ˉ N T , i ⋅ − Y ˉ N T , ⋅ t + Y ˉ N T , σ ^ N T , C 2 σ ^ N T , I 2 = T 1 t ∑ ( Y ˉ N T , ⋅ t − Y ˉ N T ) 2 , = N T 1 i , t ∑ r ^ N T , i t 2 . (13) The marginal variance estimate σ ^ N T 2 = ( N T ) − 1 ∑ i , t ( Y N T , i t − Y ˉ N T ) 2 \hat{\sigma}^{2}_{NT}=(NT)^{-1}\sum_{i,t}(Y_{NT,it}-\bar{Y}_{NT})^2 σ ^ N T 2 = ( N T ) − 1 ∑ i , t ( Y N T , i t − Y ˉ N T ) 2 σ ^ N T , R 2 + σ ^ N T , C 2 + σ ^ N T , I 2 \hat{\sigma}^{2}_{NT,R}+\hat{\sigma}^{2}_{NT,C}+\hat{\sigma}^{2}_{NT,I} σ ^ N T , R 2 + σ ^ N T , C 2 + σ ^ N T , I 2
Substitution in (9) gives variance estimator
V ^ N T , B = ( 1 − 1 B ) ( σ ^ N T , R 2 N + σ ^ N T , C 2 T + σ ^ N T , I 2 N T ) + σ ^ N T 2 N B . (14) \widehat{V}_{NT,B}=\left(1-\frac{1}{B}\right)\left(\frac{\widehat{\sigma}^{2}_{NT,R}}{N}+\frac{\widehat{\sigma}^{2}_{NT,C}}{T}+\frac{\widehat{\sigma}^{2}_{NT,I}}{NT}\right)+\frac{\widehat{\sigma}^{2}_{NT}}{NB}. \tag*{(14)} V N T , B = ( 1 − B 1 ) ( N σ N T , R 2 + T σ N T , C 2 + N T σ N T , I 2 ) + N B σ N T 2 . (14) The estimator V ^ N T , B \widehat{V}_{NT,B} V N T , B [3 , 7 ] .
The matching estimator uses the selected observations, while its standard error V ^ N T , B \sqrt{\widehat{V}_{NT,B}} V N T , B
I N T , B ( 1 − α ) = [ θ ^ N T , B − z 1 − α / 2 V ^ N T , B , θ ^ N T , B + z 1 − α / 2 V ^ N T , B ] . (15) I_{NT,B}(1-\alpha)=\left[\widehat{\theta}_{NT,B}-z_{1-\alpha/2}\sqrt{\widehat{V}_{NT,B}},\ \widehat{\theta}_{NT,B}+z_{1-\alpha/2}\sqrt{\widehat{V}_{NT,B}}\right]. \tag*{(15)} I N T , B ( 1 − α ) = [ θ N T , B − z 1 − α /2 V N T , B , θ N T , B + z 1 − α /2 V N T , B ] . (15) For any deterministic B B B w N T , B → 0 w_{NT,B}\to0 w N T , B → 0 V ^ N T , B / V N T , B → 1 \widehat{V}_{NT,B}/V_{NT,B}\to1 V N T , B / V N T , B → 1
Assumption 2 (Moment bounds). For a constant K K K , E [ a N T , 1 4 ] ≤ K σ N T 2 σ N T , R 2 E[a_{NT,1}^{4}]\le K\sigma_{NT}^{2}\sigma_{NT,R}^{2} E [ a N T , 1 4 ] ≤ K σ N T 2 σ N T , R 2 and E [ b N T , 1 4 ] ≤ K σ N T 2 σ N T , C 2 E[b_{NT,1}^{4}]\le K\sigma_{NT}^{2}\sigma_{NT,C}^{2} E [ b N T , 1 4 ] ≤ K σ N T 2 σ N T , C 2
Let P N T ( K ) P_{NT}(K) P N T ( K ) Q N T , 4 ( K ) Q_{NT,4}(K) Q N T , 4 ( K )
Theorem 1 (Uniform Gaussian inference). As N , T → ∞ N,T\to\infty N , T → ∞ N ≤ T N\le T N ≤ T B = B N T ≥ 1 B=B_{NT}\ge1 B = B N T ≥ 1
deterministic with B = o ( N ) B=o(N) B = o ( N ) ϵ > 0 \epsilon>0 ϵ > 0
sup P ∈ P N T ( K ) Pr P ( ∣ V ^ N T , B V N T , B − 1 ∣ > ϵ ) → 0 , (16) \sup_{P\in\mathcal{P}_{NT}(K)}\Pr_{P}\left(\left|\frac{\widehat{V}_{NT,B}}{V_{NT,B}}-1\right|>\epsilon\right)\to0, \tag*{(16)} P ∈ P N T ( K ) sup P Pr ( V N T , B V N T , B − 1 > ϵ ) → 0 , (16) sup P ∈ P N T ( K ) sup x ∈ R ∣ Pr P ( θ ^ N T , B − θ N T V ^ N T , B ≤ x ) − Φ ( x ) ∣ → 0. (17) \sup_{P\in\mathcal{P}_{NT}(K)}\sup_{x\in\mathbb{R}}\left|\Pr_{P}\left(\frac{\widehat{\theta}_{NT,B}-\theta_{NT}}{\sqrt{\widehat{V}_{NT,B}}}\leq x\right)-\Phi(x)\right|\to0. \tag*{(17)} P ∈ P N T ( K ) sup x ∈ R sup P Pr V N T , B θ N T , B − θ N T ≤ x − Φ ( x ) → 0. (17) If B = o ( N ) B=o(\sqrt{N}) B = o ( N ) Q N T , 4 ( K ) \mathcal{Q}_{NT,4}(K) Q N T , 4 ( K )
For B = o ( N ) B=o(N) B = o ( N ) P N T ( K ) \mathcal{P}_{NT}(K) P N T ( K ) ν N T \nu_{NT} ν N T Table 1 to obtain a Gaussian limit for unrestricted B B B ν N T → 1 \nu_{NT}\to1 ν N T → 1 1
Remark 1. Theorem 1 permits any deterministic sequence B = o ( N ) B=o(N) B = o ( N ) B B B B = ⌊ N / log N ⌋ B=\lfloor N/\log N\rfloor B = ⌊ N / log N ⌋ σ N T log N / N \sigma_{NT}\sqrt{\log N}/N σ N T log N / N σ N T / N \sigma_{NT}/N σ N T / N ( log N ) − 1 / 2 (\log N)^{-1/2} ( log N ) − 1/2
The weaker moment condition in Theorem 1 places a separate restriction on B B B B = o ( N ) B=o(N) B = o ( N ) B / N B/\sqrt{N} B / N B = o ( N ) B=o(\sqrt{N}) B = o ( N )
Extension to generalized method of moments
The population mean solves the moment condition E ( Y N T , i t − θ N T ) = 0 E(Y_{NT,it}-\theta_{NT})=0 E ( Y N T , i t − θ N T ) = 0 § 3 extends to smooth functions of the GMM parameter even when the sample moments converge at different rates.
Let X N T , i t = f N T ( ξ i , η t , ζ i t ) \mathbf{X}_{NT,it}=f_{NT}(\xi_i,\eta_t,\zeta_{it}) X N T , i t = f N T ( ξ i , η t , ζ i t ) m N T ( X N T , i t , β ) \mathbf{m}_{NT}(\mathbf{X}_{NT,it},\beta) m N T ( X N T , i t , β ) r r r d d d β 0 , N T \beta_{0,NT} β 0 , N T E [ m N T ( X N T , i t , β 0 , N T ) ] = 0 E[\mathbf{m}_{NT}(\mathbf{X}_{NT,it},\beta_{0,NT})]=0 E [ m N T ( X N T , i t , β 0 , N T )] = 0 r ≥ d r\ge d r ≥ d ℓ ( β 0 , N T ) \ell(\beta_{0,NT}) ℓ ( β 0 , N T ) [15 ] , the population moments identify β 0 , N T \beta_{0,NT} β 0 , N T B B B β \beta β
m ^ N T , B ( β ) = 1 N B ∑ b = 1 B ∑ i = 1 N m N T ( X N T , i π b ( i ) , β ) , \widehat{\mathbf{m}}_{NT,B}(\beta)=\frac{1}{NB}\sum_{b=1}^{B}\sum_{i=1}^{N}\mathbf{m}_{NT}(\mathbf{X}_{NT,i\pi_b(i)},\beta), m N T , B ( β ) = N B 1 b = 1 ∑ B i = 1 ∑ N m N T ( X N T , i π b ( i ) , β ) , β ^ N T , B ∈ arg min β ∈ Θ m ^ N T , B ( β ) T W ^ N T m ^ N T , B ( β ) . (18) \widehat{\beta}_{NT,B}\in\mathop{\arg\min}_{\beta\in\Theta}\widehat{\mathbf{m}}_{NT,B}(\beta)^{\mathsf{T}}\widehat{W}_{NT}\widehat{\mathbf{m}}_{NT,B}(\beta). \tag*{(18)} β N T , B ∈ arg min β ∈ Θ m N T , B ( β ) T W N T m N T , B ( β ) . (18) The weight W ^ N T \widehat{W}_{NT} W N T
Let m ˙ N T \dot{\mathbf{m}}_{NT} m ˙ N T r × d r\times d r × d
D N T = E m ˙ N T ( X N T , 11 , β 0 , N T ) , A N T = ( D N T T W N T D N T ) − 1 D N T T W N T , D_{NT}=E\dot{\mathbf{m}}_{NT}(\mathbf{X}_{NT,11},\beta_{0,NT}),\qquad A_{NT}=(D_{NT}^{\mathsf{T}}W_{NT}D_{NT})^{-1}D_{NT}^{\mathsf{T}}W_{NT}, D N T = E m ˙ N T ( X N T , 11 , β 0 , N T ) , A N T = ( D N T T W N T D N T ) − 1 D N T T W N T , where W N T W_{NT} W N T ( i , t ) (i,t) ( i , t ) ℓ ( β ^ N T , B ) \ell(\widehat{\beta}_{NT,B}) ℓ ( β N T , B )
u N T , i t = − ∇ ℓ ( β 0 , N T ) T A N T m N T ( X N T , i t , β 0 , N T ) . (19) u_{NT,it}=-\nabla\ell(\beta_{0,NT})^{\mathsf{T}}A_{NT}\mathbf{m}_{NT}(\mathbf{X}_{NT,it},\beta_{0,NT}). \tag*{(19)} u N T , i t = − ∇ ℓ ( β 0 , N T ) T A N T m N T ( X N T , i t , β 0 , N T ) . (19) For the population mean, this contribution equals Y N T , i t − θ N T Y_{NT,it}-\theta_{NT} Y N T , i t − θ N T (14) to the scalar contributions in (19) . To obtain their fitted values u ^ N T , i t \widehat{u}_{NT,it} u N T , i t β ^ N T , B \widehat{\beta}_{NT,B} β N T , B m ˙ N T \dot{m}_{NT} m ˙ N T β ^ N T , B \widehat{\beta}_{NT,B} β N T , B W ^ N T \widehat{W}_{NT} W N T β 0 , N T \beta_{0,NT} β 0 , N T D N T D_{NT} D N T W N T W_{NT} W N T D ^ N T , B \widehat{D}_{NT,B} D N T , B V ^ N T , B ( x ) \widehat{V}_{NT,B}(x) V N T , B ( x ) x x x V ^ N T , B ( u ^ N T ) \sqrt{\widehat{V}_{NT,B}(\widehat{u}_{NT})} V N T , B ( u N T )
Assumption 3 (GMM regularity). The following conditions hold uniformly over array sizes and laws.
The parameter space Θ \Theta Θ β 0 , N T \beta_{0,NT} β 0 , N T Θ \Theta Θ D N T D_{NT} D N T ϵ > 0 \epsilon> 0 ϵ > 0 c ϵ > 0 c_{\epsilon} > 0 c ϵ > 0 ∥ E m N T ( X N T , 11 , β ) ∥ ≥ c ϵ \lVert E m_{NT}(X_{NT,11},\beta)\rVert\ge c_{\epsilon} ∥ E m N T ( X N T , 11 , β )∥ ≥ c ϵ β ∈ Θ \beta\in\Theta β ∈ Θ ∥ β − β 0 , N T ∥ ≥ ϵ \lVert\beta-\beta_{0,NT}\rVert\ge\epsilon ∥ β − β 0 , N T ∥ ≥ ϵ
For every x x x m N T , j ( x , ⋅ ) m_{NT,j}(x,\cdot) m N T , j ( x , ⋅ ) j = 1 , … , r j=1,\ldots,r j = 1 , … , r Θ \Theta Θ Θ \Theta Θ F N T ( x ) F_{NT}(x) F N T ( x ) E [ F N T ( X N T , 11 ) 2 ] ≤ K E[F_{NT}(X_{NT,11})^2] \le K E [ F N T ( X N T , 11 ) 2 ] ≤ K ℓ \ell ℓ Θ \Theta Θ
The weight W ^ N T \widehat{W}_{NT} W N T ∥ W ^ N T − W N T ∥ = O P ( N − 1 / 2 ) \lVert\widehat{W}_{NT}-W_{NT}\rVert=O_P(N^{-1/2}) ∥ W N T − W N T ∥ = O P ( N − 1/2 ) W N T W_{NT} W N T
Assumption 3 ensures that the population moments uniformly identify β 0 , N T \beta_{0,NT} β 0 , N T β 0 , N T \beta_{0,NT} β 0 , N T N \sqrt{N} N
For q ≥ 2 q \ge2 q ≥ 2 K K K v > 0 v > 0 v > 0 Q N T , G M M , q ( K ) Q_{NT,\mathrm{GMM},q}(K) Q N T , GMM , q ( K ) P P P Var P ( u N T , 11 ) ≥ v \operatorname{Var}_P(u_{NT,11}) \ge v Var P ( u N T , 11 ) ≥ v u N T u_{NT} u N T Q N T , q ( K ) Q_{NT,q}(K) Q N T , q ( K ) P N T , G M M ( K ) \mathcal{P}_{NT,\mathrm{GMM}}(K) P N T , GMM ( K ) Q N T , G M M , 4 ( K ) Q_{NT,\mathrm{GMM},4}(K) Q N T , GMM , 4 ( K ) u N T u_{NT} u N T
Theorem 2 (Matching GMM). Let N , T → ∞ N,T \to\infty N , T → ∞ N ≤ T N \le T N ≤ T B = o ( N ) B=o(N) B = o ( N )
sup P ∈ P N T , G M M ( K ) sup x ∈ R ∣ Pr P ( ℓ ( β ^ N T , B ) − ℓ ( β 0 , N T ) V ^ N T , B ( u ^ N T ) ≤ x ) − Φ ( x ) ∣ → 0. (20) \sup_{P\in\mathcal{P}_{NT,\mathrm{GMM}}(K)}\sup_{x\in\mathbb{R}}\left|\Pr_{P}\left(\frac{\ell(\widehat{\beta}_{NT,B})-\ell(\beta_{0,NT})}{\sqrt{\widehat{V}_{NT,B}(\widehat{u}_{NT})}}\le x\right)-\Phi(x)\right|\to0. \tag*{(20)} P ∈ P N T , GMM ( K ) sup x ∈ R sup P Pr V N T , B ( u N T ) ℓ ( β N T , B ) − ℓ ( β 0 , N T ) ≤ x − Φ ( x ) → 0. (20) If B = o ( N ) B=o(\sqrt{N}) B = o ( N ) (20) holds uniformly over the larger class Q N T , G M M , 4 ( K ) \mathcal{Q}_{NT,\mathrm{GMM},4}(K) Q N T , GMM , 4 ( K )
The lower variance bound v v v u N T , 11 u_{NT,11} u N T , 11 β 0 , N T \beta_{0,NT} β 0 , N T χ d 2 \chi^2_d χ d 2
Remark 2. For nonlinear GMM, a quadratic term in the estimation error can enter the limiting distribution under two-way clustering [13 ] . In the present construction the estimation remainder is O P ( N − 1 ) O_P(N^{-1}) O P ( N − 1 ) u N T u_{NT} u N T Var ( u N T , 11 ) / ( N B ) \sqrt{\operatorname{Var}(u_{NT,11})/(NB)} Var ( u N T , 11 ) / ( N B ) B = o ( N ) B=o(N) B = o ( N ) β ^ N T , B \widehat{\beta}_{NT,B} β N T , B
Example 2: GMM moments with different dependence regimes. On a square array, observe X N T , i t = ( x 1 , i , x 2 , i t ) ⊤ X_{NT,it}=(x_{1,i},x_{2,it})^\top X N T , i t = ( x 1 , i , x 2 , i t ) ⊤ { x 1 , i } i = 1 N \{x_{1,i}\}_{i=1}^{N} { x 1 , i } i = 1 N { x 2 , i t } i , t = 1 N \{x_{2,it}\}_{i,t=1}^{N} { x 2 , i t } i , t = 1 N m N T ( X N T , i t ; β ) = ( x 1 , i − β 1 x 2 , i t + β 1 2 − β 2 ) \boldsymbol{m}_{NT}(X_{NT,it};\beta)=\begin{pmatrix} x_{1,i}-\beta_1 \\ x_{2,it}+\beta_1^2-\beta_2 \end{pmatrix} m N T ( X N T , i t ; β ) = ( x 1 , i − β 1 x 2 , i t + β 1 2 − β 2 )
The first population moment gives β 1 = E [ x 1 , i ] = 0 \beta_1=E[x_{1,i}]=0 β 1 = E [ x 1 , i ] = 0 β 2 = E [ x 2 , i t ] + β 1 2 = 0 \beta_2=E[x_{2,it}]+\beta_1^2=0 β 2 = E [ x 2 , i t ] + β 1 2 = 0 β 0 , N T = 0 \beta_{0,NT}=0 β 0 , N T = 0 ℓ ( β ) = β 2 \ell(\beta)=\beta_2 ℓ ( β ) = β 2
Let x ˉ 1 = N − 1 ∑ i x 1 , i \bar{x}_1=N^{-1}\sum_i x_{1,i} x ˉ 1 = N − 1 ∑ i x 1 , i x ˉ 2 = N − 2 ∑ i , t x 2 , i t \bar{x}_2=N^{-2}\sum_{i,t}x_{2,it} x ˉ 2 = N − 2 ∑ i , t x 2 , i t x ˉ 1 \bar{x}_1 x ˉ 1 x ˉ 2 + x ˉ 1 2 \bar{x}_2+\bar{x}_1^2 x ˉ 2 + x ˉ 1 2 N x ˉ 2 N\bar{x}_2 N x ˉ 2 N x ˉ 1 \sqrt{N}\bar{x}_1 N x ˉ 1
N β ^ N T , F , 2 = N x ˉ 2 + ( N x ˉ 1 ) 2 + o P ( 1 ) ⇒ Z 0 + Z 1 2 , N\widehat{\beta}_{NT,F,2}=N\bar{x}_2+(\sqrt{N}\bar{x}_1)^2+o_P(1)\Rightarrow Z_0+Z_1^2, N β N T , F , 2 = N x ˉ 2 + ( N x ˉ 1 ) 2 + o P ( 1 ) ⇒ Z 0 + Z 1 2 , where Z 0 Z_0 Z 0 Z 1 Z_1 Z 1 [13 ] Section 3): the estimation error in the slower first coefficient enters the limiting law of the faster second coefficient.
Every matching uses all N N N x ˉ 1 \bar{x}_1 x ˉ 1 x ˉ 2 \bar{x}_2 x ˉ 2 x ˉ 2 , B = 1 N B ∑ b = 1 B ∑ i = 1 N x 2 , i π b ( i ) \bar{x}_{2,B} = \frac{1}{NB} \sum_{b=1}^{B} \sum_{i=1}^{N} x_{2,i}\pi_b(i) x ˉ 2 , B = N B 1 ∑ b = 1 B ∑ i = 1 N x 2 , i π b ( i ) B = o ( N ) B = o(N) B = o ( N ) Table 1 gives N B x ˉ 2 , B ⇒ N ( 0 , 1 ) \sqrt{NB}\,\bar{x}_{2,B} \Rightarrow\mathcal{N}(0,1) N B x ˉ 2 , B ⇒ N ( 0 , 1 ) B / N → 0 \sqrt{B/N} \to0 B / N → 0
N B β ^ N T , B , 2 = N B x ˉ 2 , B + B N ( N x ˉ 1 ) 2 + o P ( 1 ) ⇒ N ( 0 , 1 ) . \sqrt{NB}\,\hat{\beta}_{NT,B,2} = \sqrt{NB}\,\bar{x}_{2,B} + \sqrt{\frac{B}{N}}\left(\sqrt{N}\,\bar{x}_1\right)^2 + o_P(1) \Rightarrow\mathcal{N}(0,1). N B β ^ N T , B , 2 = N B x ˉ 2 , B + N B ( N x ˉ 1 ) 2 + o P ( 1 ) ⇒ N ( 0 , 1 ) . The Gaussian approximation remains valid even when different components of the GMM moment vector exhibit different dependence regimes.
Monte Carlo evidence for the mean
We return to the sample mean to measure how much repeated matching shortens an interval and how closely each method attains nominal coverage. Each design has N = T N = T N = T θ N T = 0 \theta_{NT} = 0 θ N T = 0 ξ i \xi_i ξ i η t \eta_t η t ζ i t \zeta_{it} ζ i t
Independent intersections \quad Y N T , i t = ζ i t Y_{NT,it} = \zeta_{it} Y N T , i t = ζ i t Y N T , i t = ξ i + η t Y_{NT,it} = \xi_i + \eta_t Y N T , i t = ξ i + η t
Pure product \quad Y N T , i t = ξ i η t Y_{NT,it} = \xi_i\eta_t Y N T , i t = ξ i η t Y N T , i t = ξ i ( η t + N − 1 / 2 ) Y_{NT,it} = \xi_i(\eta_t + N^{-1/2}) Y N T , i t = ξ i ( η t + N − 1/2 )
Independent intersections and additive effects give two Gaussian benchmarks with quite different costs of matching. The two product designs have the same intersection variance, σ N T , I 2 = 1 \sigma^2_{NT,I} = 1 σ N T , I 2 = 1 σ N T , R 2 = 0 \sigma^2_{NT,R} = 0 σ N T , R 2 = 0 σ N T , R 2 = 1 / N \sigma^2_{NT,R} = 1/N σ N T , R 2 = 1/ N ν N T = 1 / 2 \nu_{NT} = 1/2 ν N T = 1/2 δ = 0 \delta= 0 δ = 0 δ = 1 \delta= 1 δ = 1 (5) , the local comparison in Menzel [24 ] .
For each array we construct five nominal 95% confidence intervals. For one matching we use the ordinary sample standard error of the N N N (15) with B = ⌊ N 1 / 3 ⌋ B = \lfloor N^{1/3} \rfloor B = ⌊ N 1/3 ⌋ B = ⌊ N / log N ⌋ B = \lfloor N/\log N \rfloor B = ⌊ N / log N ⌋ B = o ( N ) B = o(N) B = o ( N ) B = o ( N ) B = o(\sqrt{N}) B = o ( N ) B B B
The other two intervals use the full-array mean. For the two-way cluster-robust interval of Cameron et al. [3 ] (CGM), the degrees-of-freedom-adjusted variance estimate on the square array is
V ^ C G M = σ ^ N T , R 2 + σ ^ N T , C 2 N − 1 − σ ^ N T 2 N 2 − 1 . \widehat{V}_{\mathrm{CGM}} = \frac{\widehat{\sigma}^{2}_{NT,R} + \widehat{\sigma}^{2}_{NT,C}}{N-1} - \frac{\widehat{\sigma}^{2}_{NT}}{N^{2}-1}. V CGM = N − 1 σ N T , R 2 + σ N T , C 2 − N 2 − 1 σ N T 2 . The interval is Y ‾ N T ± z 0.975 max { 0 , V ^ C G M } \overline{Y}_{NT} \pm z_{0.975}\sqrt{\max\{0,\widehat{V}_{\mathrm{CGM}}\}} Y N T ± z 0.975 max { 0 , V CGM } Y ‾ N T \overline{Y}_{NT} Y N T
The bootstrap interval uses the model-selection procedure (BS-S) of Menzel [24 ] and its equal-tailed, nonstudentized interval (REG). Table 2 uses the selection threshold log ( N ) / N \log(N)/N log ( N ) / N 2
Table 2. Coverage and length of nominal 95% confidence intervals. Notes. One uses a single matching. The repeated matching uses the N 1 / 3 N^{1/3} N 1/3 N / log N N/\log N N / log N
Coverage with one matching approaches the nominal 95% level across the sample sizes in Table 2 , in line with Proposition 1. Repeated matching recovers much of the precision lost by using only one matching. Under independence and in both product designs, the cube-root rule gives intervals about half as long as those from one matching at N = 100 N = 100 N = 100 B B B ⌊ N / log N ⌋ \lfloor N/\log N \rfloor ⌊ N / log N ⌋
Adding the local row effect changes coverage little for either repeated-matching rule, as the full-array error is asymptotically negligible in both product designs. BS-S gives shorter intervals in both designs and achieves close to nominal coverage under the pure product. Under the local product, its coverage still appears reasonably close to nominal in these samples, although it falls to about 92% at N = 100 N = 100 N = 100 1 / N 1/N 1/ N log ( N ) / N \log(N)/N log ( N ) / N
Based on these results, we recommend B = ⌊ N 1 / 3 ⌋ B = \lfloor N^{1/3} \rfloor B = ⌊ N 1/3 ⌋ B = ⌊ N / log N ⌋ B = \lfloor N/\log N \rfloor B = ⌊ N / log N ⌋
Conclusion
A single matching produces an i.i.d. sample because every selected observation has a different row index and a different column index. We average matching means and use a closed-form standard error with the usual Gaussian critical value. Coverage is uniformly asymptotically exact even when the limiting law of the normalized full-array mean cannot be estimated uniformly. Repeated matching recovers much of the precision of the full array. The same variance formula yields inference for smooth functions of GMM parameters. With B = o ( N ) B = o(N) B = o ( N ) [13 ] .
One matching and the exact variance
Proof of Proposition 1. Condition on the random matching π 1 \pi_1 π 1 (6) , the selected observations use disjoint triples of shocks. Since the draw is independent of the array, for any z 1 , … , z N z_1,\ldots,z_N z 1 , … , z N
Pr ( ⋂ i = 1 N { Y N T , i , π 1 ( i ) ≤ z i } | π 1 ) = ∏ i = 1 N Pr ( Y N T , 11 ≤ z i ) . \Pr\left(\bigcap_{i=1}^{N}\{Y_{NT,i,\pi_1(i)} \le z_i\}\middle|\pi_1\right)=\prod_{i=1}^{N}\Pr(Y_{NT,11}\le z_i). Pr ( i = 1 ⋂ N { Y N T , i , π 1 ( i ) ≤ z i } π 1 ) = i = 1 ∏ N Pr ( Y N T , 11 ≤ z i ) . The right-hand side is the same for every draw, so the product law also holds unconditionally. Write X N T , i = ( Y N T , i , π 1 ( i ) − θ N T ) / σ N T X_{NT,i}=(Y_{NT,i,\pi_1(i)}-\theta_{NT})/\sigma_{NT} X N T , i = ( Y N T , i , π 1 ( i ) − θ N T ) / σ N T Q N T , 3 ( K ) \mathcal{Q}_{NT,3}(K) Q N T , 3 ( K ) X N T , i X_{NT,i} X N T , i E ∣ X N T , i ∣ 3 ≤ K E|X_{NT,i}|^3\le K E ∣ X N T , i ∣ 3 ≤ K N − 1 / 2 ∑ i X N T , i N^{-1/2}\sum_i X_{NT,i} N − 1/2 ∑ i X N T , i C K / N CK/\sqrt{N} C K / N X N T , i 2 X_{NT,i}^2 X N T , i 2 L 2 L^2 L 2 E [ X N T , i 2 1 { ∣ X N T , i ∣ > L } ] ≤ K / L E[X_{NT,i}^2\mathbf{1}\{|X_{NT,i}|>L\}]\le K/L E [ X N T , i 2 1 { ∣ X N T , i ∣ > L }] ≤ K / L L → ∞ L\to\infty L → ∞ N − 1 ∑ i X N T , i 2 → 1 N^{-1}\sum_i X_{NT,i}^2\to1 N − 1 ∑ i X N T , i 2 → 1
Since E [ X ˉ N T 2 ] = 1 / N E[\bar{X}_{NT}^{2}]=1/N E [ X ˉ N T 2 ] = 1/ N
S N T , 1 2 σ N T 2 = N N − 1 ( 1 N ∑ i X N T , i 2 − X ˉ N T 2 ) → P 1. \frac{S_{NT,1}^{2}}{\sigma_{NT}^{2}}=\frac{N}{N-1}\left(\frac{1}{N}\sum_i X_{NT,i}^{2}-\bar{X}_{NT}^{2}\right)\xrightarrow{\mathrm{P}}1. σ N T 2 S N T , 1 2 = N − 1 N ( N 1 i ∑ X N T , i 2 − X ˉ N T 2 ) P 1. In particular, the probability of a zero denominator vanishes. Slutsky’s theorem and continuity of Φ \Phi Φ [2 ] .
Proof of Proposition 2. Each column is equally likely to be assigned to row i i i E [ M N T , b ∣ Y N T ] = N − 1 ∑ i T − 1 ∑ t Y N T , i t = Y ˉ N T E[M_{NT,b}\mid Y_{NT}]=N^{-1}\sum_i T^{-1}\sum_t Y_{NT,it}=\bar{Y}_{NT} E [ M N T , b ∣ Y N T ] = N − 1 ∑ i T − 1 ∑ t Y N T , i t = Y ˉ N T E θ ^ N T , B = E Y ˉ N T = θ N T E\hat{\theta}_{NT,B}=E\bar{Y}_{NT}=\theta_{NT} E θ ^ N T , B = E Y ˉ N T = θ N T B B B
Var ( θ ^ N T , B ) = Var ( Y ˉ N T ) + 1 B E Var ( M N T , 1 ∣ Y N T ) , \operatorname{Var}(\hat{\theta}_{NT,B})=\operatorname{Var}(\bar{Y}_{NT})+\frac{1}{B}E\operatorname{Var}(M_{NT,1}\mid Y_{NT}), Var ( θ ^ N T , B ) = Var ( Y ˉ N T ) + B 1 E Var ( M N T , 1 ∣ Y N T ) , E Var ( M N T , 1 ∣ Y N T ) = Var ( M N T , 1 ) − Var ( Y ˉ N T ) . E\operatorname{Var}(M_{NT,1}\mid Y_{NT})=\operatorname{Var}(M_{NT,1})-\operatorname{Var}(\bar{Y}_{NT}). E Var ( M N T , 1 ∣ Y N T ) = Var ( M N T , 1 ) − Var ( Y ˉ N T ) . Substitution gives (9).
Proof of Theorem 1
Fix any sequence of laws in P N T ( K ) \mathcal{P}_{NT}(K) P N T ( K ) N , T → ∞ N,T\to\infty N , T → ∞ N ≤ T N\leq T N ≤ T θ N T \theta_{NT} θ N T σ N T \sigma_{NT} σ N T θ N T = 0 \theta_{NT}=0 θ N T = 0 σ N T 2 = 1 \sigma_{NT}^{2}=1 σ N T 2 = 1 ( N , T ) (N,T) ( N , T ) K K K
V N T , B ≥ V N T , F , V N T , B ≥ 1 N B , V N T , F = σ N T , R 2 N + σ N T , C 2 T + σ N T , I 2 N T . (21) V_{NT,B}\geq V_{NT,F},\qquad V_{NT,B}\geq\frac{1}{NB},\qquad V_{NT,F}=\frac{\sigma_{NT,R}^{2}}{N}+\frac{\sigma_{NT,C}^{2}}{T}+\frac{\sigma_{NT,I}^{2}}{NT}. \tag*{(21)} V N T , B ≥ V N T , F , V N T , B ≥ N B 1 , V N T , F = N σ N T , R 2 + T σ N T , C 2 + N T σ N T , I 2 . (21) Write a ˉ N T \bar{a}_{NT} a ˉ N T b ˉ N T \bar{b}_{NT} b ˉ N T r ˉ N T \bar{r}_{NT} r ˉ N T
Variance estimation. Distinct residuals are uncorrelated, so their row, column and full-array averages satisfy
E r ˉ N T , i ⋅ 2 = σ N T , I 2 T , E r ˉ N T , ⋅ t 2 = σ N T , I 2 N , E r ˉ N T 2 = σ N T , I 2 N T . E\bar{r}_{NT,i\cdot}^{2} = \frac{\sigma_{NT,I}^{2}}{T}, \qquad E\bar{r}_{NT,\cdot t}^{2} = \frac{\sigma_{NT,I}^{2}}{N}, \qquad E\bar{r}_{NT}^{2} = \frac{\sigma_{NT,I}^{2}}{NT}. E r ˉ N T , i ⋅ 2 = T σ N T , I 2 , E r ˉ N T , ⋅ t 2 = N σ N T , I 2 , E r ˉ N T 2 = N T σ N T , I 2 . To bound the row variance estimate, write its centered row means in terms of the true projections and residuals:
σ ^ N T , R 2 = 1 N ∑ i [ ( a N T , i − a ˉ N T ) + ( r ˉ N T , i ⋅ − r ˉ N T ) ] 2 . \widehat{\sigma}_{NT,R}^{2} = \frac{1}{N}\sum_i \left[(a_{NT,i}-\bar{a}_{NT})+(\bar{r}_{NT,i\cdot}-\bar{r}_{NT})\right]^2. σ N T , R 2 = N 1 i ∑ [ ( a N T , i − a ˉ N T ) + ( r ˉ N T , i ⋅ − r ˉ N T ) ] 2 . The mean square of the residual contrasts is bounded by N − 1 ∑ i r ˉ N T , i ⋅ 2 = O P ( σ N T , I 2 / T ) N^{-1}\sum_i\bar{r}_{NT,i\cdot}^{2}=O_{\mathrm{P}}(\sigma_{NT,I}^{2}/T) N − 1 ∑ i r ˉ N T , i ⋅ 2 = O P ( σ N T , I 2 / T ) O P ( σ N T , R 2 ) O_{\mathrm{P}}(\sigma_{NT,R}^{2}) O P ( σ N T , R 2 ) O P ( σ N T , R 2 σ N T , I 2 / T ) O_{\mathrm{P}}(\sqrt{\sigma_{NT,R}^{2}\sigma_{NT,I}^{2}/T}) O P ( σ N T , R 2 σ N T , I 2 / T )
Var ( N − 1 ∑ i a N T , i 2 ) ≤ K σ N T , R 2 N , E a ˉ N T 2 = σ N T , R 2 N . \operatorname{Var}\left(N^{-1}\sum_i a_{NT,i}^{2}\right)\leq\frac{K\sigma_{NT,R}^{2}}{N}, \qquad E\bar{a}_{NT}^{2}=\frac{\sigma_{NT,R}^{2}}{N}. Var ( N − 1 i ∑ a N T , i 2 ) ≤ N K σ N T , R 2 , E a ˉ N T 2 = N σ N T , R 2 . Applying the same argument to the columns gives the bounds
∣ σ ^ N T , R 2 − σ N T , R 2 ∣ = O P ( σ N T , R 2 N + σ N T , R 2 σ N T , I 2 T + σ N T , I 2 T ) , \left|\widehat{\sigma}_{NT,R}^{2}-\sigma_{NT,R}^{2}\right|=O_{\mathrm{P}}\left(\sqrt{\frac{\sigma_{NT,R}^{2}}{N}}+\sqrt{\frac{\sigma_{NT,R}^{2}\sigma_{NT,I}^{2}}{T}}+\frac{\sigma_{NT,I}^{2}}{T}\right), σ N T , R 2 − σ N T , R 2 = O P N σ N T , R 2 + T σ N T , R 2 σ N T , I 2 + T σ N T , I 2 , ∣ σ ^ N T , C 2 − σ N T , C 2 ∣ = O P ( σ N T , C 2 T + σ N T , C 2 σ N T , I 2 N + σ N T , I 2 N ) . (22) \left|\widehat{\sigma}_{NT,C}^{2}-\sigma_{NT,C}^{2}\right|=O_{\mathrm{P}}\left(\sqrt{\frac{\sigma_{NT,C}^{2}}{T}}+\sqrt{\frac{\sigma_{NT,C}^{2}\sigma_{NT,I}^{2}}{N}}+\frac{\sigma_{NT,I}^{2}}{N}\right). \tag*{(22)} σ N T , C 2 − σ N T , C 2 = O P T σ N T , C 2 + N σ N T , C 2 σ N T , I 2 + N σ N T , I 2 . (22) By Jensen’s inequality and the marginal fourth-moment bound, E r N T , 11 4 Er_{NT,11}^{4} E r N T , 11 4 r N T , i t 2 r_{NT,it}^{2} r N T , i t 2 r N T , i ′ t ′ 2 r_{NT,i't'}^{2} r N T , i ′ t ′ 2 i ≠ i ′ i\ne i' i = i ′ t ≠ t ′ t\ne t' t = t ′ Var ( ( N T ) − 1 ∑ i , t r N T , i t 2 ) = O ( 1 / N + 1 / T ) \operatorname{Var}((NT)^{-1}\sum_{i,t}r_{NT,it}^{2})=O(1/N+1/T) Var (( N T ) − 1 ∑ i , t r N T , i t 2 ) = O ( 1/ N + 1/ T )
0 ≤ 1 N T ∑ i , t r N T , i t 2 − σ ^ N T , I 2 = 1 N ∑ i r ˉ N T , i ⋅ 2 + 1 T ∑ t r ˉ N T , ⋅ t 2 − r ˉ N T 2 = O P ( 1 / N ) . 0\leq\frac{1}{NT}\sum_{i,t}r_{NT,it}^{2}-\widehat{\sigma}_{NT,I}^{2}=\frac{1}{N}\sum_i\bar{r}_{NT,i\cdot}^{2}+\frac{1}{T}\sum_t\bar{r}_{NT,\cdot t}^{2}-\bar{r}_{NT}^{2}=O_{\mathrm{P}}(1/N). 0 ≤ N T 1 i , t ∑ r N T , i t 2 − σ N T , I 2 = N 1 i ∑ r ˉ N T , i ⋅ 2 + T 1 t ∑ r ˉ N T , ⋅ t 2 − r ˉ N T 2 = O P ( 1/ N ) . Consequently, σ ^ N T , I 2 − σ N T , I 2 = O P ( N − 1 / 2 ) \widehat{\sigma}^{2}_{NT,I}-\sigma^{2}_{NT,I}=O_P(N^{-1/2}) σ N T , I 2 − σ N T , I 2 = O P ( N − 1/2 ) Y N T , i t 2 Y_{NT,it}^{2} Y N T , i t 2 E [ Y ‾ N T 2 ] = V N T , F ≤ 1 / N E[\overline{Y}_{NT}^{2}]=V_{NT,F}\leq1/N E [ Y N T 2 ] = V N T , F ≤ 1/ N σ ^ N T 2 − 1 = O P ( N − 1 / 2 ) \widehat{\sigma}^{2}_{NT}-1=O_P(N^{-1/2}) σ N T 2 − 1 = O P ( N − 1/2 )
By (21) , N V N T , B ≥ max { σ N T , R 2 , 1 / B } NV_{NT,B}\geq\max\{\sigma^{2}_{NT,R},1/B\} N V N T , B ≥ max { σ N T , R 2 , 1/ B } T V N T , B ≥ max { σ N T , C 2 , 1 / B } TV_{NT,B}\geq\max\{\sigma^{2}_{NT,C},1/B\} T V N T , B ≥ max { σ N T , C 2 , 1/ B } x / max { x , 1 / B } ≤ B \sqrt{x}/\max\{x,1/B\}\leq\sqrt{B} x / max { x , 1/ B } ≤ B x ≥ 0 x\geq0 x ≥ 0 (22) by N V N T , B NV_{NT,B} N V N T , B T V N T , B TV_{NT,B} T V N T , B
∣ σ ^ N T , R 2 − σ N T , R 2 ∣ N V N T , B + ∣ σ ^ N T , C 2 − σ N T , C 2 ∣ T V N T , B = O P ( B N + B N ) . \frac{\left|\widehat{\sigma}^{2}_{NT,R}-\sigma^{2}_{NT,R}\right|}{NV_{NT,B}}+\frac{\left|\widehat{\sigma}^{2}_{NT,C}-\sigma^{2}_{NT,C}\right|}{TV_{NT,B}}=O_P\left(\sqrt{\frac{B}{N}}+\frac{B}{N}\right). N V N T , B σ N T , R 2 − σ N T , R 2 + T V N T , B σ N T , C 2 − σ N T , C 2 = O P ( N B + N B ) . The estimation errors for σ N T , I 2 \sigma^{2}_{NT,I} σ N T , I 2 σ ^ N T 2 \widehat{\sigma}^{2}_{NT} σ N T 2 V N T , B V_{NT,B} V N T , B o P ( 1 ) o_P(1) o P ( 1 ) N T V N T , B ≥ T / B NTV_{NT,B}\geq T/B N T V N T , B ≥ T / B N B V N T , B ≥ 1 NBV_{NT,B}\geq1 N B V N T , B ≥ 1
Variation in the full-array mean. The identity Y ‾ N T = a ‾ N T + b ‾ N T + r ‾ N T \overline{Y}_{NT}=\overline{a}_{NT}+\overline{b}_{NT}+\overline{r}_{NT} Y N T = a N T + b N T + r N T E r ‾ N T 2 / V N T , B ≤ B / T → 0 E\overline{r}_{NT}^{2}/V_{NT,B}\leq B/T\to0 E r N T 2 / V N T , B ≤ B / T → 0 V N T , P = σ N T , R 2 / N + σ N T , C 2 / T = ν N T V N T , F V_{NT,P}=\sigma^{2}_{NT,R}/N+\sigma^{2}_{NT,C}/T=\nu_{NT}V_{NT,F} V N T , P = σ N T , R 2 / N + σ N T , C 2 / T = ν N T V N T , F ( a ‾ N T + b ‾ N T ) / V N T , B (\overline{a}_{NT}+\overline{b}_{NT})/\sqrt{V_{NT,B}} ( a N T + b N T ) / V N T , B
1 V N T , B 2 ( E a ‾ N T , 1 4 N 3 + E b ‾ N T , 1 4 T 3 ) ≤ K V N T , P N 2 V N T , B 2 ≤ K B N → 0. \frac{1}{V_{NT,B}^{2}}\left(\frac{E\overline{a}_{NT,1}^{4}}{N^{3}}+\frac{E\overline{b}_{NT,1}^{4}}{T^{3}}\right)\leq\frac{KV_{NT,P}}{N^{2}V_{NT,B}^{2}}\leq\frac{KB}{N}\to0. V N T , B 2 1 ( N 3 E a N T , 1 4 + T 3 E b N T , 1 4 ) ≤ N 2 V N T , B 2 K V N T , P ≤ N K B → 0. From any subsequence take a further subsequence with V N T , P / V N T , B → ℓ ∈ [ 0 , 1 ] V_{NT,P}/V_{NT,B}\to\ell\in[0,1] V N T , P / V N T , B → ℓ ∈ [ 0 , 1 ] ℓ > 0 \ell>0 ℓ > 0 Y ‾ N T / V N T , B ⇒ N ( 0 , ℓ ) \overline{Y}_{NT}/\sqrt{V_{NT,B}}\Rightarrow N(0,\ell) Y N T / V N T , B ⇒ N ( 0 , ℓ ) ℓ = 0 \ell=0 ℓ = 0
Variation from drawing the matchings. Put D N T , b = M N T , b − Y ‾ N T D_{NT,b}=M_{NT,b}-\overline{Y}_{NT} D N T , b = M N T , b − Y N T Y N T Y_{NT} Y N T D N T , b D_{NT,b} D N T , b
ω N T 2 ( Y N T ) : = N Var ( D N T , 1 ∣ Y N T ) = T T − 1 ( σ ^ N T , I 2 + ( 1 − N / T ) σ ^ N T , C 2 ) . (23) \omega^{2}_{NT}(Y_{NT}):=N\operatorname{Var}(D_{NT,1}\mid Y_{NT})=\frac{T}{T-1}\left(\widehat{\sigma}^{2}_{NT,I}+(1-N/T)\widehat{\sigma}^{2}_{NT,C}\right). \tag*{(23)} ω N T 2 ( Y N T ) := N Var ( D N T , 1 ∣ Y N T ) = T − 1 T ( σ N T , I 2 + ( 1 − N / T ) σ N T , C 2 ) . (23) To verify (23) , pad the N N N T − N T-N T − N T × T T \times T T × T A A A g t = Y ˉ N T , t − Y ˉ N T g_t=\bar{Y}_{NT,t}-\bar{Y}_{NT} g t = Y ˉ N T , t − Y ˉ N T t t t A A A ( N / T ) g t (N/T)g_t ( N / T ) g t
Let C C C A A A
( N T ) − 1 ∑ i , t ≤ T A i t 2 = σ ^ N T , I 2 + σ ^ N T , C 2 (NT)^{-1}\sum_{i,t\leq T} A_{it}^{2}=\hat{\sigma}_{NT,I}^{2}+\hat{\sigma}_{NT,C}^{2} ( N T ) − 1 i , t ≤ T ∑ A i t 2 = σ ^ N T , I 2 + σ ^ N T , C 2 and
T − 1 ∑ t g t 2 = σ ^ N T , C 2 , T^{-1}\sum_{t}g_t^{2}=\hat{\sigma}_{NT,C}^{2}, T − 1 t ∑ g t 2 = σ ^ N T , C 2 , 1 N T ∑ i , t ≤ T C i t 2 = σ ^ N T , I 2 + σ ^ N T , C 2 − 1 N ∑ t = 1 T ( N T g t ) 2 = σ ^ N T , I 2 + ( 1 − N / T ) σ ^ N T , C 2 . \frac{1}{NT}\sum_{i,t\leq T}C_{it}^{2}=\hat{\sigma}_{NT,I}^{2}+\hat{\sigma}_{NT,C}^{2}-\frac{1}{N}\sum_{t=1}^{T}\left(\frac{N}{T}g_t\right)^{2}=\hat{\sigma}_{NT,I}^{2}+\left(1-N/T\right)\hat{\sigma}_{NT,C}^{2}. N T 1 i , t ≤ T ∑ C i t 2 = σ ^ N T , I 2 + σ ^ N T , C 2 − N 1 t = 1 ∑ T ( T N g t ) 2 = σ ^ N T , I 2 + ( 1 − N / T ) σ ^ N T , C 2 . Complete the matching to a uniform permutation of the T T T C C C N D N T , 1 ND_{NT,1} N D N T , 1 S C = ∑ i , t ≤ T C i t 2 S_C=\sum_{i,t\leq T}C_{it}^{2} S C = ∑ i , t ≤ T C i t 2 ∑ j ≠ i , s ≠ t C j s = C i t \sum_{j\ne i,\ s\ne t}C_{js}=C_{it} ∑ j = i , s = t C j s = C i t 1 / T 1/T 1/ T 1 / [ T ( T − 1 ) ] 1/[T(T-1)] 1/ [ T ( T − 1 )]
Var ( N D N T , 1 ∣ Y N T ) = S C T + ∑ i , t ≤ T C i t ∑ j ≠ i , s ≠ t C j s T ( T − 1 ) = S C T − 1 . \operatorname{Var}(ND_{NT,1}\mid Y_{NT})=\frac{S_C}{T}+\frac{\sum_{i,t\leq T}C_{it}\sum_{j\ne i,\ s\ne t}C_{js}}{T(T-1)}=\frac{S_C}{T-1}. Var ( N D N T , 1 ∣ Y N T ) = T S C + T ( T − 1 ) ∑ i , t ≤ T C i t ∑ j = i , s = t C j s = T − 1 S C . which proves (23) . Total variance gives E ω N T 2 ( Y N T ) = 1 − N V N T , F = ( 1 − N / T ) σ N T , C 2 + ( 1 − 1 / T ) σ N T , I 2 E\omega_{NT}^{2}(Y_{NT})=1-NV_{NT,F}=(1-N/T)\sigma_{NT,C}^{2}+(1-1/T)\sigma_{NT,I}^{2} E ω N T 2 ( Y N T ) = 1 − N V N T , F = ( 1 − N / T ) σ N T , C 2 + ( 1 − 1/ T ) σ N T , I 2 (23) and variance component consistency, ω N T 2 ( Y N T ) − E ω N T 2 ( Y N T ) = o P ( 1 ) \omega_{NT}^{2}(Y_{NT})-E\omega_{NT}^{2}(Y_{NT})=o_P(1) ω N T 2 ( Y N T ) − E ω N T 2 ( Y N T ) = o P ( 1 )
ω N T 2 ( Y N T ) N B V N T , B = 1 − V N T , F V N T , B + o P ( 1 ) ⟶ P ρ : = 1 − ℓ . \frac{\omega_{NT}^{2}(Y_{NT})}{NBV_{NT,B}}=1-\frac{V_{NT,F}}{V_{NT,B}}+o_P(1)\mathrel{\mathop{\longrightarrow}^{\mathrm{P}}}\rho:=1-\ell. N B V N T , B ω N T 2 ( Y N T ) = 1 − V N T , B V N T , F + o P ( 1 ) ⟶ P ρ := 1 − ℓ . If ρ = 0 \rho=0 ρ = 0 ( θ ^ N T , B − Y ˉ N T ) / V N T , B (\hat{\theta}_{NT,B}-\bar{Y}_{NT})/\sqrt{V_{NT,B}} ( θ ^ N T , B − Y ˉ N T ) / V N T , B ρ > 0 \rho>0 ρ > 0 (21) gives Pr ( ω N T 2 ( Y N T ) > ρ / 2 ) → 1 \Pr(\omega_{NT}^{2}(Y_{NT})>\rho/2)\to1 Pr ( ω N T 2 ( Y N T ) > ρ /2 ) → 1
Consider first B → ∞ B\to\infty B → ∞ ρ > 0 \rho>0 ρ > 0 N N N
E [ ( N M N T , 1 ) 4 ] ≤ K / N + 3. E[(\sqrt{N}M_{NT,1})^{4}]\leq K/N+3. E [( N M N T , 1 ) 4 ] ≤ K / N + 3. As Y ˉ N T = E [ M N T , 1 ∣ Y N T ] \bar{Y}_{NT}=E[M_{NT,1}\mid Y_{NT}] Y ˉ N T = E [ M N T , 1 ∣ Y N T ] E [ N 2 D N T , 1 4 ∣ Y N T ] = O P ( 1 ) E[N^{2}D_{NT,1}^{4}\mid Y_{NT}]=O_P(1) E [ N 2 D N T , 1 4 ∣ Y N T ] = O P ( 1 )
∑ b = 1 B E [ D N T , b 4 ∣ Y N T ] { ∑ b = 1 B Var ( D N T , b ∣ Y N T ) } 2 = E [ N 2 D N T , 1 4 ∣ Y N T ] B ( ω N T 2 ( Y N T ) ) 2 = o P ( 1 ) . \frac{\sum_{b=1}^{B}E[D_{NT,b}^{4}\mid Y_{NT}]}{\left\{\sum_{b=1}^{B}\operatorname{Var}(D_{NT,b}\mid Y_{NT})\right\}^{2}}
=\frac{E[N^{2}D_{NT,1}^{4}\mid Y_{NT}]}{B(\omega_{NT}^{2}(Y_{NT}))^{2}}=o_P(1). { ∑ b = 1 B Var ( D N T , b ∣ Y N T ) } 2 ∑ b = 1 B E [ D N T , b 4 ∣ Y N T ] = B ( ω N T 2 ( Y N T ) ) 2 E [ N 2 D N T , 1 4 ∣ Y N T ] = o P ( 1 ) . For a fixed number of matchings, apply Theorem 1.1 of Chen and Fang [5 ] to a single permutation of C \mathcal{C} C s 2 = N ω N T 2 ( Y N T ) s^2=N\omega_{NT}^2(Y_{NT}) s 2 = N ω N T 2 ( Y N T )
sup x ∣ Pr ( N D N T , 1 / s ≤ x ∣ Y N T ) − Φ ( x ) ∣ ≤ C 0 T s 3 ∑ i , t ≤ T ∣ C i t ∣ 3 = o P ( 1 ) . \sup_x \left|\Pr(ND_{NT,1}/s\le x\mid Y_{NT})-\Phi(x)\right|\le\frac{C_0}{Ts^3}\sum_{i,t\le T}|C_{it}|^3=o_P(1). x sup ∣ Pr ( N D N T , 1 / s ≤ x ∣ Y N T ) − Φ ( x ) ∣ ≤ T s 3 C 0 i , t ≤ T ∑ ∣ C i t ∣ 3 = o P ( 1 ) . Here C 0 C_0 C 0 N N N T − 1 ∑ ∣ C i t ∣ 3 = O P ( N ) T^{-1}\sum|C_{it}|^3=O_P(N) T − 1 ∑ ∣ C i t ∣ 3 = O P ( N ) s 3 / N 3 / 2 s^3/N^{3/2} s 3 / N 3/2
Every integer sequence B B B u u u
E [ exp { i u ( θ ^ N T , B − Y ‾ N T ) / V N T , B } ∣ Y N T ] ⟶ P exp ( − ρ u 2 / 2 ) . E\left[\exp\left\{iu(\widehat{\theta}_{NT,B}-\overline{Y}_{NT})/\sqrt{V_{NT,B}}\right\}\mid Y_{NT}\right]\overset{\mathrm{P}}{\longrightarrow}\exp(-\rho u^2/2). E [ exp { i u ( θ N T , B − Y N T ) / V N T , B } ∣ Y N T ] ⟶ P exp ( − ρ u 2 /2 ) . The conditional characteristic functions are bounded, so convergence also holds in L 1 L^1 L 1 exp ( i u Y ‾ N T / V N T , B ) \exp(iu\overline{Y}_{NT}/\sqrt{V_{NT,B}}) exp ( i u Y N T / V N T , B ) exp [ − ( ℓ + ρ ) u 2 / 2 ] = exp ( − u 2 / 2 ) \exp[-(\ell+\rho)u^2/2]=\exp(-u^2/2) exp [ − ( ℓ + ρ ) u 2 /2 ] = exp ( − u 2 /2 ) (16) and Slutsky’s theorem give (17) , since a continuous limiting distribution gives convergence in Kolmogorov distance. The argument applies to every sequence of laws and thus establishes uniformity.
Uniformity under a marginal fourth-moment bound. For P ∈ Q N T , 4 ( K ) P\in Q_{NT,4}(K) P ∈ Q N T , 4 ( K ) E a N T , 1 4 , E b N T , 1 4 ≤ K Ea_{NT,1}^4,Eb_{NT,1}^4\le K E a N T , 1 4 , E b N T , 1 4 ≤ K O P ( N − 1 / 2 ) O_P(N^{-1/2}) O P ( N − 1/2 ) σ ^ N T , R 2 \widehat{\sigma}^2_{NT,R} σ N T , R 2 σ ^ N T , C 2 \widehat{\sigma}^2_{NT,C} σ N T , C 2 O P ( B / N ) O_P(B/\sqrt{N}) O P ( B / N )
1 V N T , B 2 ( E a N T , 1 4 N 3 + E b N T , 1 4 T 3 ) ≤ 2 K N 3 V N T , B 2 ≤ 2 K B 2 N . \frac{1}{V_{NT,B}^2}\left(\frac{Ea_{NT,1}^4}{N^3}+\frac{Eb_{NT,1}^4}{T^3}\right)\le\frac{2K}{N^3V_{NT,B}^2}\le\frac{2KB^2}{N}. V N T , B 2 1 ( N 3 E a N T , 1 4 + T 3 E b N T , 1 4 ) ≤ N 3 V N T , B 2 2 K ≤ N 2 K B 2 . Both bounds vanish when B = o ( N ) B=o(\sqrt{N}) B = o ( N ) □ \square □
Limiting distributions and a boundary example
Proof of Proposition 3. Center the observations and set σ N T 2 = 1 \sigma_{NT}^{2}=1 σ N T 2 = 1 D N T , b = M N T , b − Y ˉ N T D_{NT,b}=M_{NT,b}-\bar{Y}_{NT} D N T , b = M N T , b − Y ˉ N T ω N T 2 = N Var ( D N T , 1 ∣ Y N T ) \omega_{NT}^{2}=N\operatorname{Var}(D_{NT,1}\mid Y_{NT}) ω N T 2 = N Var ( D N T , 1 ∣ Y N T ) (23) . Total variance and (9) give
E ω N T 2 = N ( V N T , 1 − V N T , F ) = N B V N T , B ( 1 − w N T , B ) , N B V N T , B ≥ 1. E\omega_{NT}^{2}=N(V_{NT,1}-V_{NT,F})=NBV_{NT,B}(1-w_{NT,B}), \qquad NBV_{NT,B}\geq1. E ω N T 2 = N ( V N T , 1 − V N T , F ) = N B V N T , B ( 1 − w N T , B ) , N B V N T , B ≥ 1. The marginal fourth-moment bound implies σ ^ N T , C 2 − σ N T , C 2 = o P ( 1 ) \hat{\sigma}_{NT,C}^{2}-\sigma_{NT,C}^{2}=o_{P}(1) σ ^ N T , C 2 − σ N T , C 2 = o P ( 1 ) σ ^ N T , I 2 − σ N T , I 2 = o P ( 1 ) \hat{\sigma}_{NT,I}^{2}-\sigma_{NT,I}^{2}=o_{P}(1) σ ^ N T , I 2 − σ N T , I 2 = o P ( 1 ) B B B (23) gives ω N T 2 − E ω N T 2 = o P ( 1 ) \omega_{NT}^{2}-E\omega_{NT}^{2}=o_{P}(1) ω N T 2 − E ω N T 2 = o P ( 1 )
If w < 1 w<1 w < 1 lim inf E ω N T 2 ≥ 1 − w > 0 \liminf E\omega_{NT}^{2}\geq1-w>0 lim inf E ω N T 2 ≥ 1 − w > 0 B → ∞ B\to\infty B → ∞ E [ N 2 D N T , 1 4 ∣ Y N T ] = O P ( 1 ) E[N^{2}D_{NT,1}^{4}\mid Y_{NT}]=O_{P}(1) E [ N 2 D N T , 1 4 ∣ Y N T ] = O P ( 1 ) O P ( 1 / B ) O_{P}(1/B) O P ( 1/ B ) B B B (23) gives a conditional normal approximation to each draw, since ω N T 2 \omega_{NT}^{2} ω N T 2 B B B ω N T 2 / E ω N T 2 → 1 \omega_{NT}^{2}/E\omega_{NT}^{2}\to1 ω N T 2 / E ω N T 2 → 1 t t t
E [ exp { i t θ ^ N T , B − Y ˉ N T V N T , B } ∣ Y N T ] ⟶ exp { − t 2 ( 1 − w ) / 2 } in L 1 . E\left[\left.\exp\left\{it\frac{\hat{\theta}_{NT,B}-\bar{Y}_{NT}}{\sqrt{V_{NT,B}}}\right\}\right|Y_{NT}\right]\longrightarrow\exp\{-t^{2}(1-w)/2\}\quad\text{in }L^{1}. E [ exp { i t V N T , B θ ^ N T , B − Y ˉ N T } Y N T ] ⟶ exp { − t 2 ( 1 − w ) /2 } in L 1 . Here convergence in probability implies L 1 L^{1} L 1 w = 1 w=1 w = 1 exp { i t ( Y ˉ N T − θ N T ) / V N T , B } \exp\{it(\bar{Y}_{NT}-\theta_{NT})/\sqrt{V_{NT,B}}\} exp { i t ( Y ˉ N T − θ N T ) / V N T , B }
E e i t ( θ ^ N T , B − θ N T ) / V N T , B = e − t 2 ( 1 − w ) / 2 E e i t w N T , B ( Y ˉ N T − θ N T ) / V N T , F + o ( 1 ) . E e^{it(\widehat{\theta}_{NT,B}-\theta_{NT})/\sqrt{V_{NT,B}}}=e^{-t^{2}(1-w)/2}E e^{it\sqrt{w_{NT,B}}(\bar{Y}_{NT}-\theta_{NT})/\sqrt{V_{NT,F}}}+o(1). E e i t ( θ N T , B − θ N T ) / V N T , B = e − t 2 ( 1 − w ) /2 E e i t w N T , B ( Y ˉ N T − θ N T ) / V N T , F + o ( 1 ) . For w > 0 w>0 w > 0 w = 0 w=0 w = 0
To establish studentization when w = 0 w=0 w = 0 V ^ N T , F = σ ^ N T , R 2 / N + σ ^ N T , C 2 / T + σ ^ N T , I 2 / ( N T ) \widehat{V}_{NT,F}=\widehat{\sigma}^{2}_{NT,R}/N+\widehat{\sigma}^{2}_{NT,C}/T+\widehat{\sigma}^{2}_{NT,I}/(NT) V N T , F = σ N T , R 2 / N + σ N T , C 2 / T + σ N T , I 2 / ( N T ) E σ ^ N T , R 2 = ( 1 − 1 / N ) ( σ N T , R 2 + σ N T , I 2 / T ) E\widehat{\sigma}^{2}_{NT,R}=(1-1/N)(\sigma^{2}_{NT,R}+\sigma^{2}_{NT,I}/T) E σ N T , R 2 = ( 1 − 1/ N ) ( σ N T , R 2 + σ N T , I 2 / T ) E σ ^ N T , C 2 = ( 1 − 1 / T ) ( σ N T , C 2 + σ N T , I 2 / N ) E\widehat{\sigma}^{2}_{NT,C}=(1-1/T)(\sigma^{2}_{NT,C}+\sigma^{2}_{NT,I}/N) E σ N T , C 2 = ( 1 − 1/ T ) ( σ N T , C 2 + σ N T , I 2 / N ) E σ ^ N T , I 2 ≤ σ N T , I 2 E\widehat{\sigma}^{2}_{NT,I}\leq\sigma^{2}_{NT,I} E σ N T , I 2 ≤ σ N T , I 2 E V ^ N T , F ≤ σ N T , R 2 / N + σ N T , C 2 / T + 3 σ N T , I 2 / ( N T ) ≤ 3 V N T , F E\widehat{V}_{NT,F}\leq\sigma^{2}_{NT,R}/N+\sigma^{2}_{NT,C}/T+3\sigma^{2}_{NT,I}/(NT)\leq3V_{NT,F} E V N T , F ≤ σ N T , R 2 / N + σ N T , C 2 / T + 3 σ N T , I 2 / ( N T ) ≤ 3 V N T , F ϵ > 0 \epsilon>0 ϵ > 0
Pr ( V ^ N T , F / V N T , B > ϵ ) ≤ 3 V N T , F ϵ V N T , B = 3 w N T , B ϵ ⟶ 0. \Pr\left(\widehat{V}_{NT,F}/V_{NT,B}>\epsilon\right)\leq\frac{3V_{NT,F}}{\epsilon V_{NT,B}}=\frac{3w_{NT,B}}{\epsilon}\longrightarrow0. Pr ( V N T , F / V N T , B > ϵ ) ≤ ϵ V N T , B 3 V N T , F = ϵ 3 w N T , B ⟶ 0. By Appendix § 8 , σ ^ N T 2 / σ N T 2 → 1 \widehat{\sigma}^{2}_{NT}/\sigma^{2}_{NT}\to1 σ N T 2 / σ N T 2 → 1
σ N T 2 N B V N T , B = 1 − ( 1 − 1 / B ) w N T , B → 1 , \frac{\sigma^{2}_{NT}}{NBV_{NT,B}}=1-(1-1/B)w_{NT,B}\to1, N B V N T , B σ N T 2 = 1 − ( 1 − 1/ B ) w N T , B → 1 , (14) gives V ^ N T , B / V N T , B → 1 \widehat{V}_{NT,B}/V_{NT,B}\to1 V N T , B / V N T , B → 1
Scales in Table 1 . If ν N T → c < 1 \nu_{NT}\to c<1 ν N T → c < 1 § 3.3 gives V N T , 1 / ( T V N T , F ) → 1 − c V_{NT,1}/(TV_{NT,F})\to1-c V N T , 1 / ( T V N T , F ) → 1 − c V N T , 1 = σ N T 2 / N V_{NT,1}=\sigma^{2}_{NT}/N V N T , 1 = σ N T 2 / N (9) , we obtain
N T V N T , F σ N T 2 ⟶ 1 1 − c , N T ( V N T , B − V N T , F ) σ N T 2 = T B ( 1 − N V N T , F σ N T 2 ) . \frac{NTV_{NT,F}}{\sigma^{2}_{NT}}\longrightarrow\frac{1}{1-c},\qquad\frac{NT(V_{NT,B}-V_{NT,F})}{\sigma^{2}_{NT}}=\frac{T}{B}\left(1-\frac{NV_{NT,F}}{\sigma^{2}_{NT}}\right). σ N T 2 N T V N T , F ⟶ 1 − c 1 , σ N T 2 N T ( V N T , B − V N T , F ) = B T ( 1 − σ N T 2 N V N T , F ) . Thus for B / T → λ ∈ ( 0 , ∞ ) B/T\to\lambda\in(0,\infty) B / T → λ ∈ ( 0 , ∞ ) N T / σ N T \sqrt{NT}/\sigma_{NT} N T / σ N T 1 / 1 − c 1/\sqrt{1-c} 1/ 1 − c 1 / λ 1/\sqrt{\lambda} 1/ λ B / T → ∞ B/T\to\infty B / T → ∞ B = o ( T ) B=o(T) B = o ( T ) N B / σ N T \sqrt{NB}/\sigma_{NT} N B / σ N T N B V N T , F / σ N T 2 → 0 NBV_{NT,F}/\sigma_{NT}^{2} \to0 N B V N T , F / σ N T 2 → 0 N B V N T , B / σ N T 2 = 1 + ( B − 1 ) N V N T , F / σ N T 2 → 1 NBV_{NT,B}/\sigma_{NT}^{2} = 1 + (B - 1)NV_{NT,F}/\sigma_{NT}^{2} \to1 N B V N T , B / σ N T 2 = 1 + ( B − 1 ) N V N T , F / σ N T 2 → 1
When the projections dominate. Let V N T , P = σ N T , R 2 / N + σ N T , C 2 / T = ν N T V N T , F V_{NT,P} = \sigma_{NT,R}^{2}/N + \sigma_{NT,C}^{2}/T = \nu_{NT}V_{NT,F} V N T , P = σ N T , R 2 / N + σ N T , C 2 / T = ν N T V N T , F ν N T → 1 \nu_{NT} \to1 ν N T → 1 V N T , F \sqrt{V_{NT,F}} V N T , F 1 − ν N T → 0 1-\nu_{NT} \to0 1 − ν N T → 0 1 / N 1/N 1/ N 1 / T 1/T 1/ T ϵ > 0 \epsilon> 0 ϵ > 0
E [ a N T , 1 2 1 { ∣ a N T , 1 ∣ > ϵ N V N T , P } ] N V N T , P + E [ b N T , 1 2 1 { ∣ b N T , 1 ∣ > ϵ T V N T , P } ] T V N T , P → 0. \frac{E[a_{NT,1}^{2}1\{|a_{NT,1}| > \epsilon N\sqrt{V_{NT,P}}\}]}{NV_{NT,P}} + \frac{E[b_{NT,1}^{2}1\{|b_{NT,1}| > \epsilon T\sqrt{V_{NT,P}}\}]}{TV_{NT,P}} \to0. N V N T , P E [ a N T , 1 2 1 { ∣ a N T , 1 ∣ > ϵ N V N T , P }] + T V N T , P E [ b N T , 1 2 1 { ∣ b N T , 1 ∣ > ϵ T V N T , P }] → 0. Under this condition the normalized projection sum is asymptotically standard normal. Since V N T , P / V N T , F → 1 V_{NT,P}/V_{NT,F} \to1 V N T , P / V N T , F → 1 P N T ( K ) \mathcal{P}_{NT}(K) P N T ( K )
E [ a N T , 1 4 ] / N 3 + E [ b N T , 1 4 ] / T 3 V N T , P 2 ≤ K σ N T 2 N 2 V N T , P ≤ K T N ( 1 N + 1 − ν N T ν N T ) . \frac{E[a_{NT,1}^{4}]/N^{3} + E[b_{NT,1}^{4}]/T^{3}}{V_{NT,P}^{2}} \le\frac{K\sigma_{NT}^{2}}{N^{2}V_{NT,P}} \le K\frac{T}{N}\left(\frac{1}{N} + \frac{1-\nu_{NT}}{\nu_{NT}}\right). V N T , P 2 E [ a N T , 1 4 ] / N 3 + E [ b N T , 1 4 ] / T 3 ≤ N 2 V N T , P K σ N T 2 ≤ K N T ( N 1 + ν N T 1 − ν N T ) . This tends to zero if T / N T/N T / N T = N 2 T=N^{2} T = N 2 Y N T , i t = I N T , i − 1 / N + ζ i t Y_{NT,it}=I_{NT,i}-1/N+\zeta_{it} Y N T , i t = I N T , i − 1/ N + ζ i t ( 1 / N ) (1/N) ( 1/ N ) σ N T , R 2 = ( 1 − 1 / N ) / N \sigma_{NT,R}^{2}=(1-1/N)/N σ N T , R 2 = ( 1 − 1/ N ) / N σ N T , C 2 = 0 \sigma_{NT,C}^{2}=0 σ N T , C 2 = 0 σ N T , I 2 = 1 \sigma_{NT,I}^{2}=1 σ N T , I 2 = 1 P N T ( K ) \mathcal{P}_{NT}(K) P N T ( K ) K K K σ N T 2 ≥ 1 \sigma_{NT}^{2}\ge1 σ N T 2 ≥ 1 N 2 V N T , F = 1 N^{2}V_{NT,F}=1 N 2 V N T , F = 1 ν N T = 1 − 1 / N → 1 \nu_{NT}=1-1/N\to1 ν N T = 1 − 1/ N → 1 N Y ‾ N T ⇒ Poisson ( 1 ) − 1 N\overline{Y}_{NT}\Rightarrow\operatorname{Poisson}(1)-1 N Y N T ⇒ Poisson ( 1 ) − 1 1 / N 1/N 1/ N
Sharpness under a marginal fourth-moment bound. Suppose B = o ( N ) B=o(N) B = o ( N ) B / N B/\sqrt{N} B / N B ≥ c N B\ge c\sqrt{N} B ≥ c N c > 0 c>0 c > 0 N ≤ T N\le T N ≤ T
Y N T , i t = a N T , i + e i t , a N T , i = N / B ( I N T , i − 1 / N ) , I N T , i ∼ i . i . d . Bernoulli ( 1 / N ) , Y_{NT,it}=a_{NT,i}+e_{it}, \qquad a_{NT,i}=\sqrt{N/B}\left(I_{NT,i}-1/N\right), \qquad I_{NT,i}\mathrel{\overset{\mathrm{i.i.d.}}{\sim}}\operatorname{Bernoulli}(1/N), Y N T , i t = a N T , i + e i t , a N T , i = N / B ( I N T , i − 1/ N ) , I N T , i ∼ i.i.d. Bernoulli ( 1/ N ) , where the e i t e_{it} e i t I N T , i I_{NT,i} I N T , i θ N T = 0 \theta_{NT}=0 θ N T = 0 σ N T , R 2 = ( 1 − 1 / N ) / B \sigma^2_{NT,R}=(1-1/N)/B σ N T , R 2 = ( 1 − 1/ N ) / B σ N T , C 2 = 0 \sigma^2_{NT,C}=0 σ N T , C 2 = 0 σ N T , I 2 = 1 \sigma^2_{NT,I}=1 σ N T , I 2 = 1 E [ a N T , i 4 ] ∼ N / B 2 = O ( 1 ) E[a^4_{NT,i}]\sim N/B^2=O(1) E [ a N T , i 4 ] ∼ N / B 2 = O ( 1 ) E [ a N T , i 4 ] / ( σ N T 2 σ N T , R 2 ) ∼ N / B → ∞ E[a^4_{NT,i}]/(\sigma^2_{NT}\sigma^2_{NT,R})\sim N/B\to\infty E [ a N T , i 4 ] / ( σ N T 2 σ N T , R 2 ) ∼ N / B → ∞
Each matching uses every row once, so the row contribution is
N B N ∑ i a N T , i = ∑ i = 1 N I N T , i − 1 ⇒ Q − 1 , Q ∼ Poisson ( 1 ) . \frac{\sqrt{NB}}{N}\sum_i a_{NT,i}=\sum_{i=1}^{N}I_{NT,i}-1\Rightarrow Q-1,\qquad Q\sim\operatorname{Poisson}(1). N N B i ∑ a N T , i = i = 1 ∑ N I N T , i − 1 ⇒ Q − 1 , Q ∼ Poisson ( 1 ) . Let K N T , i t K_{NT,it} K N T , i t ( i , t ) (i,t) ( i , t ) i i i b , b ′ b,b' b , b ′ 1 / T 1/T 1/ T N N N B ( B − 1 ) B(B-1) B ( B − 1 )
E ∑ i , t K N T , i t ( K N T , i t − 1 ) = N B ( B − 1 ) T . E\sum_{i,t}K_{NT,it}(K_{NT,it}-1)=\frac{NB(B-1)}{T}. E i , t ∑ K N T , i t ( K N T , i t − 1 ) = T N B ( B − 1 ) . Since ∑ i , t K N T , i t = N B \sum_{i,t}K_{NT,it}=NB ∑ i , t K N T , i t = N B B = o ( N ) B=o(N) B = o ( N ) N ≤ T N\le T N ≤ T
E [ ∑ i , t K N T , i t 2 N B − 1 ] = B − 1 T → 0. E\left[\frac{\sum_{i,t}K^2_{NT,it}}{NB}-1\right]=\frac{B-1}{T}\to0. E [ N B ∑ i , t K N T , i t 2 − 1 ] = T B − 1 → 0. The quantity in brackets is nonnegative, so ∑ i , t K N T , i t 2 / ( N B ) → 1 \sum_{i,t}K^2_{NT,it}/(NB)\to1 ∑ i , t K N T , i t 2 / ( N B ) → 1 N B θ ^ N T , B \sqrt{NB}\hat{\theta}_{NT,B} N B θ ^ N T , B N B θ ^ N T , B ⇒ Q − 1 + Z \sqrt{NB}\hat{\theta}_{NT,B}\Rightarrow Q-1+Z N B θ ^ N T , B ⇒ Q − 1 + Z Z Z Z Q Q Q
The variance estimate V ^ N T , B \hat{V}_{NT,B} V ^ N T , B θ ^ N T , B \hat{\theta}_{NT,B} θ ^ N T , B
E Φ ( 1 − Q ) ≥ e − 1 { Φ ( 1 ) + 1 2 + 1 2 Φ ( − 1 ) } > 0.52. E\Phi(1-Q)\ge e^{-1}\left\{\Phi(1)+\frac{1}{2}+\frac{1}{2}\Phi(-1)\right\}>0.52. E Φ ( 1 − Q ) ≥ e − 1 { Φ ( 1 ) + 2 1 + 2 1 Φ ( − 1 ) } > 0.52. Since Φ ( 0 ) = 1 / 2 \Phi(0)=1/2 Φ ( 0 ) = 1/2 B B B B = ⌊ N ⌋ B=\lfloor\sqrt{N}\rfloor B = ⌊ N ⌋
Proof of Theorem 2
Work along any admissible sequence of laws. Applying (9) to a scalar function x x x Var ( x N T , 11 ) / N \operatorname{Var}(x_{NT,11})/N Var ( x N T , 11 ) / N β 0 , N T \beta_{0,NT} β 0 , N T
m ^ N T , B ( β 0 , N T ) = O P ( N − 1 / 2 ) , ∂ m ^ N T , B ( β 0 , N T ) ∂ β ⊤ − D N T = O P ( N − 1 / 2 ) . \widehat{m}_{NT,B}(\beta_{0,NT}) = O_{\mathrm{P}}(N^{-1/2}), \qquad\frac{\partial\widehat{m}_{NT,B}(\beta_{0,NT})}{\partial\beta^{\top}} - D_{NT} = O_{\mathrm{P}}(N^{-1/2}). m N T , B ( β 0 , N T ) = O P ( N − 1/2 ) , ∂ β ⊤ ∂ m N T , B ( β 0 , N T ) − D N T = O P ( N − 1/2 ) . Applied on a finite net of Θ \Theta Θ O P ( 1 ) O_{\mathrm{P}}(1) O P ( 1 ) O ( 1 ) O(1) O ( 1 ) Θ \Theta Θ β ^ N T , B − β 0 , N T = o P ( 1 ) \widehat{\beta}_{NT,B} - \beta_{0,NT} = o_{\mathrm{P}}(1) β N T , B − β 0 , N T = o P ( 1 )
Write Δ N T , B = β ^ N T , B − β 0 , N T \Delta_{NT,B} = \widehat{\beta}_{NT,B} - \beta_{0,NT} Δ N T , B = β N T , B − β 0 , N T D ‾ N T , B \overline{D}_{NT,B} D N T , B D ^ N T , B − D N T = o P ( 1 ) \widehat{D}_{NT,B} - D_{NT} = o_{\mathrm{P}}(1) D N T , B − D N T = o P ( 1 ) D ‾ N T , B − D N T = o P ( 1 ) \overline{D}_{NT,B} - D_{NT} = o_{\mathrm{P}}(1) D N T , B − D N T = o P ( 1 )
0 = D ^ N T , B ⊤ W ^ N T { m ^ N T , B ( β 0 , N T ) + D ‾ N T , B Δ N T , B } , 0 = \widehat{D}_{NT,B}^{\top}\widehat{W}_{NT}\left\{\widehat{m}_{NT,B}(\beta_{0,NT}) + \overline{D}_{NT,B}\Delta_{NT,B}\right\}, 0 = D N T , B ⊤ W N T { m N T , B ( β 0 , N T ) + D N T , B Δ N T , B } , Δ N T , B = − A ~ N T , B m ^ N T , B ( β 0 , N T ) = O P ( N − 1 / 2 ) . \Delta_{NT,B} = -\widetilde{A}_{NT,B}\widehat{m}_{NT,B}(\beta_{0,NT}) = O_{\mathrm{P}}(N^{-1/2}). Δ N T , B = − A N T , B m N T , B ( β 0 , N T ) = O P ( N − 1/2 ) . where A ~ N T , B = ( D ^ N T , B ⊤ W ^ N T D ‾ N T , B ) − 1 D ^ N T , B ⊤ W ^ N T \widetilde{A}_{NT,B} = (\widehat{D}_{NT,B}^{\top}\widehat{W}_{NT}\overline{D}_{NT,B})^{-1}\widehat{D}_{NT,B}^{\top}\widehat{W}_{NT} A N T , B = ( D N T , B ⊤ W N T D N T , B ) − 1 D N T , B ⊤ W N T O P ( N − 1 / 2 ) O_{\mathrm{P}}(N^{-1/2}) O P ( N − 1/2 ) A ~ N T , B − A N T = O P ( N − 1 / 2 ) \widetilde{A}_{NT,B} - A_{NT} = O_{\mathrm{P}}(N^{-1/2}) A N T , B − A N T = O P ( N − 1/2 ) ℓ \ell ℓ u N T , i u_{NT,i} u N T , i (19) give
ℓ ( β ^ N T , B ) − ℓ ( β 0 , N T ) = 1 N B ∑ b , i u N T , i π b ( i ) + R N T , B , \ell(\widehat{\beta}_{NT,B}) - \ell(\beta_{0,NT}) = \frac{1}{NB}\sum_{b,i}u_{NT,i}\pi_{b(i)} + R_{NT,B}, ℓ ( β N T , B ) − ℓ ( β 0 , N T ) = N B 1 b , i ∑ u N T , i π b ( i ) + R N T , B , ∣ R N T , B ∣ ≤ C ( ∥ Δ N T , B ∥ 2 + ∥ A ~ N T , B − A N T ∥ ∥ m ^ N T , B ( β 0 , N T ) ∥ ) = O P ( N − 1 ) . \lvert R_{NT,B}\rvert\leq C\left(\lVert\Delta_{NT,B}\rVert^{2} + \lVert\widetilde{A}_{NT,B} - A_{NT}\rVert\lVert\widehat{m}_{NT,B}(\beta_{0,NT})\rVert\right) = O_{\mathrm{P}}(N^{-1}). ∣ R N T , B ∣ ≤ C ( ∥ Δ N T , B ∥ 2 + ∥ A N T , B − A N T ∥ ∥ m N T , B ( β 0 , N T )∥ ) = O P ( N − 1 ) . By (9) , V N T , B ( u N T ) ≥ Var ( u N T , 11 ) / ( N B ) ≥ v / ( N B ) V_{NT,B}(u_{NT}) \ge\operatorname{Var}(u_{NT,11})/(NB) \ge v/(NB) V N T , B ( u N T ) ≥ Var ( u N T , 11 ) / ( N B ) ≥ v / ( N B ) R N T , B / V N T , B ( u N T ) = O P ( B / N ) = o P ( 1 ) R_{NT,B}/\sqrt{V_{NT,B}(u_{NT})} = O_P(\sqrt{B/N}) = o_P(1) R N T , B / V N T , B ( u N T ) = O P ( B / N ) = o P ( 1 )
For feasible studentization, the full-array average of ( u ^ N T , i t − u N T , i t ) 2 (\widehat{u}_{NT,it}-u_{NT,it})^2 ( u N T , i t − u N T , i t ) 2 O P ( N − 1 ) O_P(N^{-1}) O P ( N − 1 ) (19) has estimation error O P ( N − 1 / 2 ) O_P(N^{-1/2}) O P ( N − 1/2 ) O P ( 1 ) O_P(1) O P ( 1 ) (14) is a sum of squared linear functions with nonnegative coefficients, so its square root is a seminorm. The row, column, and residual sums of squares add to the centered marginal sum of squares. With N ≤ T N \le T N ≤ T V ^ N T , B ( x ) ≤ N − 1 ( N T ) − 1 ∑ i , t x N T , i t 2 \widehat{V}_{NT,B}(x) \le N^{-1}(NT)^{-1}\sum_{i,t}x_{NT,it}^2 V N T , B ( x ) ≤ N − 1 ( N T ) − 1 ∑ i , t x N T , i t 2
∣ V ^ N T , B ( u ^ N T ) − V ^ N T , B ( u N T ) ∣ ≤ V ^ N T , B ( u N T − u N T ) ≤ ( 1 N 2 T ∑ i , t ( u ^ N T , i t − u N T , i t ) 2 ) 1 / 2 = O P ( N − 1 ) . \begin{aligned}
\left\lvert\sqrt{\widehat{V}_{NT,B}(\widehat{u}_{NT})}-\sqrt{\widehat{V}_{NT,B}(u_{NT})}\right\rvert\le\sqrt{\widehat{V}_{NT,B}(u_{NT}-u_{NT})} \\
&\le\left(\frac{1}{N^2T}\sum_{i,t}(\widehat{u}_{NT,it}-u_{NT,it})^2\right)^{1/2}=O_P(N^{-1}).
\end{aligned} V N T , B ( u N T ) − V N T , B ( u N T ) ≤ V N T , B ( u N T − u N T ) ≤ ( N 2 T 1 i , t ∑ ( u N T , i t − u N T , i t ) 2 ) 1/2 = O P ( N − 1 ) . Applying Theorem 1 to u N T u_{NT} u N T (20) , including the assertion under B = o ( N ) B=o(\sqrt{N}) B = o ( N )
Joint inference. For a unit vector c c c ℓ ( β ) = c ⊤ β \ell(\beta)=c^\top\beta ℓ ( β ) = c ⊤ β u N T ( c ) u_{NT}(c) u N T ( c ) c c c S N T , B S_{NT,B} S N T , B S ^ N T , B \widehat{S}_{NT,B} S N T , B c ⊤ S N T , B c = V N T , B ( u N T ( c ) ) c^\top S_{NT,B}c=V_{NT,B}(u_{NT}(c)) c ⊤ S N T , B c = V N T , B ( u N T ( c )) c ⊤ S ^ N T , B c = V ^ N T , B ( u ^ N T ( c ) ) c^\top\widehat{S}_{NT,B}c=\widehat{V}_{NT,B}(\widehat{u}_{NT}(c)) c ⊤ S N T , B c = V N T , B ( u N T ( c )) S N T , B − 1 / 2 c S_{NT,B}^{-1/2}c S N T , B − 1/2 c S N T , B − 1 / 2 ( β ^ N T , B − β 0 , N T ) ⇒ N ( 0 , I d ) S_{NT,B}^{-1/2}(\widehat{\beta}_{NT,B}-\beta_{0,NT})\Rightarrow\mathcal{N}(0,I_d) S N T , B − 1/2 ( β N T , B − β 0 , N T ) ⇒ N ( 0 , I d ) S N T , B − 1 / 2 S ^ N T , B S N T , B − 1 / 2 → I d S_{NT,B}^{-1/2}\widehat{S}_{NT,B}S_{NT,B}^{-1/2}\to I_d S N T , B − 1/2 S N T , B S N T , B − 1/2 → I d ( β ^ N T , B − β 0 , N T ) ⊤ S ^ N T , B − 1 ( β ^ N T , B − β 0 , N T ) ⇒ χ d 2 (\widehat{\beta}_{NT,B}-\beta_{0,NT})^\top\widehat{S}_{NT,B}^{-1}(\widehat{\beta}_{NT,B}-\beta_{0,NT})\Rightarrow\chi_d^2 ( β N T , B − β 0 , N T ) ⊤ S N T , B − 1 ( β N T , B − β 0 , N T ) ⇒ χ d 2
