Why Triangular Arrays Matter
Many asymptotic arguments imagine taking larger and larger samples from one fixed distribution. Econometric samples need not follow that pattern. As the sample size grows, the composition of the sample, the covariate design, or the distribution of disturbances may also change. A sequence of such samples is represented by a triangular array: row \(n\) contains the \(n\) observations used to construct the estimator at sample size \(n\), and the distribution of that row may differ from the distributions of other rows.
The target can change with the row as well. In that case, consistency means that the estimator gets close to the target for that row, not necessarily to one fixed number. Asymptotic normality likewise concerns the estimator centered at its row-specific target. These distinctions matter whenever a fixed-distribution argument silently treats changing moments or designs as if they were constant.
Consistency for a Moving Target
A common route to consistency is to minimize a sample criterion. Let \(\Theta\) be a compact parameter space, and suppose \(Q_n(\theta)=n^{-1}\sum_{i=1}^n q_{ni}(\theta)\) is the sample criterion. Its expectation is \(\bar{Q}_n(\theta)=E[Q_n(\theta)]\). Assume that \(\bar{Q}_n\) has a unique minimizer \(\theta_n\), which is allowed to depend on \(n\).
Two conditions make the minimizer of the sample criterion track \(\theta_n\). First, the sample criterion must uniformly approximate its expectation: \(\sup_{\theta\in\Theta}|Q_n(\theta)-\bar{Q}_n(\theta)|\to_p0\). Second, the expected criterion must separate the target from parameters that are a fixed distance away. Specifically, for each \(\varepsilon>0\), suppose that for all sufficiently large \(n\), \(\inf_{\|\theta-\theta_n\|\geq\varepsilon}\{\bar{Q}_n(\theta)-\bar{Q}_n(\theta_n)\}\geq c_\varepsilon\), for some \(c_\varepsilon>0\).
If \(\hat{\theta}_n\) minimizes \(Q_n\) (or has an optimization error that is \(o_p(1)\)), these conditions imply \(\hat{\theta}_n-\theta_n\to_p0\). The argument is direct. When \(\hat{\theta}_n\) is at least \(\varepsilon\) from the target, separation makes its expected criterion worse by at least \(c_\varepsilon\). A uniform approximation error smaller than, for example, \(c_\varepsilon/3\) cannot erase that gap: the sample criterion at \(\hat{\theta}_n\) would still exceed the sample criterion at \(\theta_n\). That contradicts minimization. The probability of such a large approximation error tends to zero, so the probability that the minimizer is at least \(\varepsilon\) from \(\theta_n\) also tends to zero.
Worked Example: A Mean With Changing Bernoulli Probabilities
Suppose the \(n\) observations in row \(n\) are independent Bernoulli variables with \(P(X_{ni}=1)=p_{ni}=0.2+0.4i/n\). The probabilities vary across observations and with sample size. Estimate the row-average probability \(\theta_n=n^{-1}\sum_{i=1}^n p_{ni}\) by \(\hat{\theta}_n=\bar{X}_n\).
Target and consistency. Since \(E[\bar{X}_n]=\theta_n\), independence gives \(\operatorname{Var}(\bar{X}_n)=n^{-2}\sum_{i=1}^n p_{ni}(1-p_{ni})\leq1/(4n)\). Therefore Chebyshev’s inequality gives \(P(|\bar{X}_n-\theta_n|>\varepsilon)\leq1/(4n\varepsilon^2)\to0\). The estimator is consistent for the row-specific average probability.
Calculate the target. Using \(\sum_{i=1}^n i/n=(n+1)/2\), \(\theta_n=0.2+0.4(n+1)/(2n)=0.4+0.2/n\). Thus the target itself approaches \(0.4\), but the estimator’s consistency statement is centered at \(\theta_n\).
Asymptotic distribution. The variance of the normalized sum is \(\Omega_n=n^{-1}\sum_{i=1}^n p_{ni}(1-p_{ni})\). As \(n\) grows, this Riemann sum converges to \(\int_0^1(0.2+0.4t)(0.8-0.4t)\,dt=0.2267\) (to four significant figures). Each centered observation is bounded in absolute value by 1, so for every fixed \(\varepsilon>0\), none can exceed \(\varepsilon\sqrt{n}\) for sufficiently large \(n\). The Lindeberg condition therefore holds, and the triangular-array central limit theorem yields \(\sqrt{n}(\bar{X}_n-\theta_n)\xrightarrow{d}N(0,0.2267)\).
For example, at \(n=100\), the target is \(0.402\); the asymptotic standard error is approximately \(\sqrt{0.2267/100}=0.0476\). The limiting distribution describes error around the row-specific target, not an assertion that all observations share one common Bernoulli probability.
Asymptotic Normality from Estimating Equations
A general way to obtain an asymptotic distribution is to express an estimator as a solution to an estimating equation. Suppose \(\hat{\theta}_n\) satisfies \(n^{-1}\sum_{i=1}^n\psi_{ni}(\hat{\theta}_n)=0\), where \(\psi_{ni}(\theta)\) is a \(d\)-dimensional estimating function. The row-specific target \(\theta_n\) is defined by \(n^{-1}\sum_{i=1}^n E[\psi_{ni}(\theta_n)]=0\). The observations and their estimating functions may be independent but non-identically distributed within each row.
The usual square-root-\(n\) result requires more than consistency. The average Jacobian (the derivative matrix of the expected estimating function) must approach a nonsingular matrix, and the centered estimating functions must have a nondegenerate limiting covariance and satisfy a central limit condition. For a differentiable estimating function, define \(A_n=n^{-1}\sum_{i=1}^n E[\partial\psi_{ni}(\theta_n)/\partial\theta']\) and \(\Omega_n=n^{-1}\sum_{i=1}^n\operatorname{Var}(\psi_{ni}(\theta_n))\).
Here \(A^{-T}\) means the transpose of \(A^{-1}\). The covariance \(\Omega\) describes variability in the estimating functions; \(A^{-1}\) translates that variability into parameter units. It is the same “bread–meat–bread” structure that appears in sandwich covariance calculations, but all its ingredients are limits along the triangular array.
Why the Normal Limit Has Sandwich Form
The key step is a Taylor expansion of the estimating equation around the row-specific target. Write \(\bar{\psi}_n(\theta)=n^{-1}\sum_i\psi_{ni}(\theta)\). Since the target solves the expected equation, and the observations are centered there, the expansion gives \(0=\bar{\psi}_n(\theta_n)+\hat{A}_n(\hat{\theta}_n-\theta_n)+o_p(n^{-1/2})\), where \(\hat{A}_n\) is the average sample Jacobian along the segment joining \(\theta_n\) and \(\hat{\theta}_n\). Multiplying by \(\sqrt n\) and rearranging, \[ \sqrt n(\hat{\theta}_n-\theta_n) =-\hat{A}_n^{-1}\frac{1}{\sqrt n}\sum_{i=1}^n \{\psi_{ni}(\theta_n)-E[\psi_{ni}(\theta_n)]\}+o_p(1). \]
The multivariate central limit theorem for the triangular array supplies the normal limit for the sum. Since \(\hat{A}_n\to_p A\) and \(A\) is nonsingular, Slutsky’s theorem permits replacing \(\hat{A}_n^{-1}\) by \(A^{-1}\) in the limit. Premultiplying a mean-zero normal vector by \(-A^{-1}\) gives covariance \(A^{-1}\Omega A^{-T}\); the minus sign does not change covariance. This proves the stated limit. The approximation error must be \(o_p(n^{-1/2})\) on the original equation’s scale for it to disappear after multiplication by \(\sqrt n\).
Worked Example: Through-Origin Regression With Changing Design
For even \(n\), consider independent observations satisfying \(Y_{ni}=x_{ni}\beta+\varepsilon_{ni}\), where the nonrandom regressor is \(x_{ni}=1\) for half the observations and \(x_{ni}=2\) for the other half. The errors are independent normal variables with mean zero and variance \(1\) when \(x_{ni}=1\), and variance \(4\) when \(x_{ni}=2\). The design and error distribution are allowed to depend on the row. Estimate \(\beta\) by minimizing \(\sum_i(Y_{ni}-x_{ni}b)^2\).
Estimator and consistency. Differentiating the criterion gives the estimating equation \(\sum_i x_{ni}(Y_{ni}-x_{ni}\hat{\beta}_n)=0\), hence \(\hat{\beta}_n-\beta=(\sum_i x_{ni}^2)^{-1}\sum_i x_{ni}\varepsilon_{ni}\). Here \(\sum_i x_{ni}^2=(n/2)(1)+(n/2)(4)=2.5n\). The estimator is unbiased, and its variance is \[ \operatorname{Var}(\hat{\beta}_n-\beta) =\frac{(n/2)(1^2\cdot1)+(n/2)(2^2\cdot4)}{(2.5n)^2} =\frac{8.5}{6.25n} =\frac{1.36}{n}. \] The variance tends to zero, so Chebyshev’s inequality establishes consistency.
Asymptotic distribution. With score \(\psi_{ni}(b)=x_{ni}(Y_{ni}-x_{ni}b)\), the Jacobian is \(-n^{-1}\sum_i x_{ni}^2=-2.5\), while the variance of the normalized score sum is \(n^{-1}\sum_i x_{ni}^2\operatorname{Var}(\varepsilon_{ni})=8.5\). Normal errors give the score sum a normal distribution in every row; its variance after division by \(\sqrt n\) is \(8.5\). Consequently, \(\sqrt n(\hat{\beta}_n-\beta)\xrightarrow{d}N(0,8.5/2.5^2)=N(0,1.36)\).
The result is in units of the regression coefficient: for large even \(n\), the estimator’s standard deviation is approximately \(\sqrt{1.36/n}\) coefficient units. The calculation uses the actual row design and error variances; treating all observations as if they had the same regressor or variance would give a different, incorrect covariance.
When the Usual Rate Fails
Triangular-array assumptions are substantive. In particular, a nondegenerate square-root-\(n\) normal limit generally requires information that remains sufficiently strong as the sample grows. A changing distribution can make each observation increasingly uninformative about the target. Consistency in absolute error may still hold in that situation, while the standard rate and normal approximation do not.
Worked Example: Rare Events and a Degenerate Limit
In row \(n\), let \(X_{n1},\ldots,X_{nn}\) be independent Bernoulli variables with probability \(p_n=1/n\). Estimate \(p_n\) by \(\hat{p}_n=\bar{X}_n\). This model has a moving distribution and a moving target.
Absolute consistency. The estimator is unbiased and \(\operatorname{Var}(\hat{p}_n)=p_n(1-p_n)/n=(1/n)(1-1/n)/n\leq1/n^2\). Chebyshev’s inequality therefore implies \(\hat{p}_n-p_n\to_p0\). But this absolute consistency does not show that the estimator has relative accuracy when \(p_n\) itself is shrinking.
Failure of the standard normal limit. The variance of \(\sqrt n(\hat{p}_n-p_n)\) is \(p_n(1-p_n)\to0\), so the square-root-\(n\) error tends to zero in probability rather than to a nondegenerate normal distribution. In fact, the count \(n\hat{p}_n\) has a binomial distribution with parameters \(n\) and \(1/n\), which converges to a Poisson distribution with mean 1. Thus \(n\hat{p}_n\), not \(\sqrt n(\hat{p}_n-p_n)\), has a nondegenerate limiting scale in this example.
The usual theorem cannot be applied with a positive limiting score variance: here the variance of one centered score, \(p_n(1-p_n)\), vanishes. This is not a minor technicality. It signals that the square-root-\(n\) approximation and its normal-based uncertainty calculations do not describe this rare-event regime.
Common Mistakes and Practical Checks
- Centering at a fixed target without justification. If the target is \(\theta_n\), the basic limit is centered at \(\theta_n\). Replacing it with a fixed \(\theta\) requires checking that \(\sqrt n(\theta_n-\theta)\to0\); mere convergence \(\theta_n\to\theta\) is not enough.
- Using one row’s variance as the limiting variance. In a triangular array, the relevant covariance is the limit of the row-average covariance \(\Omega_n\). Individual distributions can differ within and across rows.
- Assuming consistency implies a normal limit. Consistency alone gives no rate or limiting shape. A nonsingular Jacobian, a nondegenerate score covariance, and a suitable central limit condition are additional requirements.
- Ignoring the design in a regression covariance. The Jacobian and score covariance both depend on the row’s design. They must be calculated from that design rather than borrowed from a fixed-design calculation without checking its conditions.
Check Your Understanding
Use these questions to check both the logic and the conditions behind the results.
- What does consistency mean when the target parameter depends on sample size?
- Why do uniform approximation and criterion separation together imply consistency of a sample-criterion minimizer?
- In the estimating-equation result, what roles do the limiting Jacobian \(A\) and score covariance \(\Omega\) play?
- For a changing design, why must the score covariance be computed using the row-specific regressors and error variances?
- In the rare-event example, which condition needed for a nondegenerate square-root-\(n\) normal limit fails?