Haruki Shi

26. Limit Distribution and Scale-Free Reparameterization

Sat, 22 Aug 2026 00:00:00 GMT

Settings

Background: In econometric theory proofs, eliminating a nuisance macroscopic parameter (like ) can sometimes be achieved through a mathematical trick called "Scale-Free" or "Reparameterization".

Condition: This trick works perfectly when (single variable, in [Ketz(2019)]), which allows us to utilize the ratio property of the statistic.

Objective: Explain why and how the nuisance parameter can be completely absorbed (normalized to 1) under the condition, specifically in the context of the limit distribution involving a truncated maximum function.

Derivation

The Physical Essence of the Statistic as a Ratio: The basic structure of a statistic (used for confidence intervals or hypothesis testing) is always: In the limit state (e.g., based on Equation 14):

Numerator (Estimated Error): Follows a truncated distribution . Here, is a normal random variable with mean 0 and variance .

Denominator (Standard Error): In the limit, this is the square root of the variance, .

Algebraic Transformation (Absorbing the Denominator): Dividing the limit state numerator and denominator yields the statistic in the limit: Since the denominator is a positive number, we can use the property of the maximum function () and directly pull it inside the bracket:

Perfect Absorption of (The "Scale-Free" Magic):

Right Term : A normal variable with variance divided by its standard deviation. This is standard Z-score normalization. After division, it becomes a variable that always follows the standard normal distribution , which we call . The parameter has vanished, and the variance is normalized to 1.

Left Term : This still contains . However, our ultimate goal is to find the infimum for the worst-case coverage probability (AsySz), which requires to traverse from to . Since is a variable that takes any value in , regardless of whether the constant denominator is 2 or 200, the entire term remains a new variable that takes any value in . We can reparameterize this entire term as a new relative scaling drift parameter, . The final limit statistic becomes:

Conclusion

Why is Eliminated: In the new formula, is always a standard normal distribution (variance 1, independent of ). To find the worst-case coverage, we simply let run from to . The parameter has not truly disappeared; it has been completely absorbed by . In the univariate case, changing only scales the coordinate axis, which does not affect the result of finding the infimum (the worst-case scenario).

Contrast: Why This Fails for : If (two variance parameters), the underlying is a covariance matrix. Although matrix multiplication can normalize the two diagonal elements to 1, the correlation coefficients (off-diagonal elements) will still remain and be governed by . We cannot use a simple scalar to absorb multi-dimensional correlation. This is why the perfect dimensionality reduction (normalizing the variance-covariance structure to completely rid of ) is a special property unique to the case.

25. Asymptotic T-Test and Delta Method

Fri, 21 Aug 2026 00:00:00 GMT

Settings

The classic exact t-test for an estimator is given by:

In GMM and large-sample theory, the asymptotic normality of an estimator is: where is the sample size (e.g., number of markets) and is the asymptotic variance matrix.

The model estimates the variance directly. Let the estimated variance be with its asymptotic variance given by .

Objective: Derive the asymptotic t-statistic for the standard deviation using the Delta Method.

Derivation

Asymptotic Variance of the Original Estimator: From (2), the variance of is , making the standard error . Substituting this into (1) yields the asymptotic t-statistic: This explains the presence of in the numerator of the asymptotic t-test formula.

First-Order Taylor Expansion (Delta Method): Suppose is a consistent estimator of the true parameter . For a non-linear transformation , the first-order Taylor expansion around is:

Asymptotic Variance of the Transformed Estimator: Taking the asymptotic variance on both sides of (4), since and are constants: Here is the asymptotic variance in the sense of (2), i.e., the variance of the limiting distribution of . Working in (rather than finite-sample ) is what keeps the derivation consistent with the normalization in (3) and with the definition of in Settings 3. Because and is continuous (Slutsky), we replace the unknown with the sample estimate :

Application to the Standard Deviation: We need to test the standard deviation , so our transformation function is . The derivative is . Substituting and (per Settings 3, which defines as the asymptotic variance of , not its finite-sample variance ): The asymptotic standard-deviation scale (which relates to the finite-sample SE by ; equivalently, the "SE before the factor is pulled into the numerator of the statistic") is:

Conclusion

The asymptotic t-statistic for testing the standard deviation is constructed by combining (3) and (8): Note: A condition such as if is an engineering safeguard. Since the estimated parameter appears in the denominator after applying the Delta Method, this prevents a division-by-zero error.

24. The Second Order Derivation in Ketz (2019)

Fri, 21 Aug 2026 00:00:00 GMT

1. Matrix-Calculus Chain for

Settings

Consider a BLP market with inside products. Let be the unobserved product attribute vector, the mean utility vector, the market share vector, and the th nonlinear parameter (scalar).

From the implicit function theorem applied to the share equation , the first-order derivative is:

Objective: derive the second-order cross partial by differentiating (1) once more with respect to , using only two matrix-calculus rules.

Proof

Notation shorthand. Let so that (1) collapses to (a vector). The task becomes computing .

Two matrix-calculus rules.

Product rule: for any two conformable factors and ,

Inverse-matrix derivative: for any invertible matrix , (Scalar analogue: .)

Apply the product rule to :

Substitute (4) into the first term of (5):

Distribute the outer negative sign (double negation on the first term):

Restore the original symbols using (2). The four building blocks are:

(a matrix)

(a matrix)

(a vector)

(a vector)

Substituting into (7):

Dimension check. The two terms on the right-hand side of (8) both evaluate to :

Term 1:

Term 2:

matching the dimension of itself.

Conclusion

The second-order derivative of with respect to two nonlinear parameters is a purely algebraic consequence of the product rule (3) and the inverse-matrix derivative (4) applied to the first-order expression (1). No additional economic assumption enters: the same identity holds for any implicit function defined by with an invertible share Jacobian. Q.E.D.

2. Off-Diagonal Entry of the Share Hessian ()

Settings

The matrix result (8) reduces to four building blocks. One of them, , is the share Hessian whose entry is .

Starting point. From the individual-choice representation , the first-order derivative w.r.t. is

Objective: differentiate the integrand of (10) once more w.r.t. for the off-diagonal case , using the product rule together with the Logit own/cross derivative rules. The diagonal case is analogous and produces the corresponding "own" formula.

Proof

Compute . For , the Logit cross-derivative gives

Compute . The term is constant in . Differentiating the sum requires splitting from :

Plugging in the own-rule and the cross-rule for , then re-absorbing the term back into the full sum: Restoring the leading minus from :

Assemble . Both pieces carry the common factor :

Adding them, the two identical sums merge into :

Reinstate the integral tail to recover the full entry.

Conclusion

The off-diagonal entry of is Symmetry in inside the parenthesis reflects the interchangeability of the two shares in the cross term . This entry populates the off-diagonal of the share-Hessian block that feeds into (8).

3. Diagonal Entry of the Share Hessian ()

Settings

Companion to Section 2. The same block has diagonal entries , obtained by setting in the derivative w.r.t. .

Starting point. Identical to (10):

Objective: differentiate the integrand of (10) once more w.r.t. for the diagonal case , using the product rule together with the Logit own derivative rule (which replaces the cross-rule used in Section 2).

Proof

Compute . For , the Logit own-derivative gives

Compute . The term is constant in . Differentiating the sum requires splitting from :

Plugging in the own-rule and the cross-rule for , then re-absorbing the term back into the full sum: Restoring the leading minus from :

Assemble . Both pieces share the deviation factor :

Adding them, the contributions accumulate:

Reinstate the integral tail to recover the full entry.

Conclusion

The diagonal entry of is Consistency check with (17): naively substituting into the raw expansion (16) gives ; the true diagonal formula (23) adds the extra that arises because switches from the cross form to the own form (their difference being exactly ). contributes no such correction because the diagonal and naive-substitution values of coincide. This entry populates the diagonal of the share-Hessian block that feeds into (8).

4. Clairaut–Schwarz Theorem (Symmetry of Mixed Partials) 克莱罗定理

Theorem. Let be open and . Then for all and all , Application. is real-analytic in , and by the implicit function theorem wherever is invertible. Hence so (8) and its share-Hessian block are symmetric; the derivation order used in Sections 2–3 is interchangeable.

23. Derivation of the Bivariate Projected Asymptotic Distribution

Fri, 21 Aug 2026 00:00:00 GMT

Settings

Consider the case where there are unconstrained parameters (corresponding to the block ) and parameters constrained by a lower bound (corresponding to the block ).

Let and the unconstrained normal random vector be .

Let the symmetric information matrix be , where is , is , and is .

Objective: Find the constrained minimizer by projecting onto the constrained parameter space . This requires minimizing the quadratic distance function .

Proof

Step 1: Block Expansion of the Quadratic Form

Define the difference vector with . The quadratic form can be expanded using block matrix multiplication: Multiply the block matrix by the column block vector: Multiply the row block vector by the resulting column vector: Since (a scalar equals its transpose), combine the symmetric cross-terms and substitute back:

Step 2: First-Order Condition for the Unconstrained Block

Since is unconstrained, take the gradient of with respect to and set it to zero: Since is positive definite (hence invertible), solve for to obtain its optimal value conditional on :

Step 3: Boundary Truncation for the Constrained Block

Substitute equation (6) into the objective to concentrate out . From (6), . Let and , so . Substitute each term of equation (4):

Term 1: (using by symmetry of ).

Term 2: .

Term 3: . Summing the three terms cancels one copy of the cross-quadratic and leaves: Restoring yields the concentrated quadratic in : The bracketed matrix is the Schur complement (舒尔补) of in , and inherits positive definiteness from . Because of the componentwise hard boundary , the optimal solution is the unconstrained minimizer truncated at the lower bound : where the max is applied component-wise.

Step 4: Substitution and Simplification

Substitute the truncated solution back into equation (6) to find the final estimator for : Apply the componentwise identity (letting ): Substitute this identity back into equation (10):

Conclusion

By minimizing the generalized quadratic distance subject to the parameter space constraints, the asymptotic distributions of the estimators (parameterized by the true values ) are precisely given by:

Q.E.D.