The MLE of Aigner, Amemiya, and Poirier is not the expectile MLE

Abstract

This article compares two asymmetric Gaussian likelihood models and their corresponding estimators. Recently, there has been confusion in the literature regarding these models and (1) whether they are the same, or (2) whether both of them can be used to estimate expectiles. After the comparison, it becomes clear that they are not the same and only one of these models is appropriate for that purpose. The similarity between these models is purely superficial. The historical origin of expectiles has also been disputed: some degree of credit can be shared between two papers.

Keywords:

• Expectile regression
• generalized quantile regression
• maximum likelihood estimation
• asymmetric normal distribution

JEL CLASSIFICATION:

• C0
• C21
• C46

Acknowledgment

We would like to thank Minchul Shin, Sebastian Laumer, and two anonymous referees for their helpful comments.

Notes

1 To enrich the physics analogy, the first moment is the point of balance of a lever where the torque from the right and left sides of the fulcrum are equal. The τth expectile is the point of imbalance where the torque from the left of the fulcrum is $\tau /(1-\tau )$ times the torque on the right.

2 For non-degenerate F, Newey and Powell (1987, Theorem 1) prove that the expectile relation $\mu \left(\tau \right):(0,1)\to \left\{y\right|0<F\left(y\right)<1\}$ is strictly monotone increasing in τ and injective. Add the convention that $\mu \left(0\right):=\inf \left(Y\right)$ and $\mu \left(1\right):=\sup \left(Y\right),$ and $\mu \left(\tau \right)$ is a surjective and injective mapping from the closed unit interval to ${\mathcal{S}}_Y\in \mathbb{R},$ the convex hull of the support of Y. Thus, every value in the interval $\left[\inf \right(Y),\sup (Y\left)\right]$ is an expectile of Y.

3 The exact origin of this asymmetric normal distribution is not obvious. The same distribution was studied previously by Kato et al. (2002) under a different parametrization. It had also appeared elsewhere prior to that date (Liess et al., 1997).

4 The estimator in (2.10(2.10) $${\widehat{\beta }}_{\tau }={\left(\sum |\tau -I(y_i<{\mathbf{x}}_i^{\prime }{\beta }_{\tau }\left)\right|{\mathbf{x}}_i{\mathbf{x}}_i^{\prime }\right)}^{-1}\left(\sum |\tau -I(y_i<{\mathbf{x}}_i^{\prime }{\beta }_{\tau }\left)\right|{\mathbf{x}}_i{y}_i\right).$$(2.10) ) has the standard linear form typical of a generalized least squares estimator, $$\widehat{{\beta }_{\tau }}={(\mathbf{X}\prime W\mathbf{X})}^{-1}\mathbf{X}\prime W\mathbf{y}.$$ The matrix W is a diagonal matrix of weights ${\left[W\right]}_{ii}=w_i,$ with $w_i=\tau$ if $y_i\ge {\mathbf{x}}_i^{\prime }{\beta }_{\tau }$ and $w_i=1-\tau$ otherwise.

5 Alternately, we might make the comparison between equation 3.4(3.4) $${ϵ}_i=\{\begin{array}{cc}\frac{1}{\sqrt{1-\theta }}{ϵ}^{*}\hfill & \text{if}\mathrm{ }{ϵ}_i^{*}>0\hfill \\ \frac{1}{\sqrt{\theta }}{ϵ}^{*}\hfill & \text{if}\mathrm{ }{ϵ}_i^{*}\le 0\hfill \end{array}$$(3.4) and equation 2.2(2.2) $${ϵ}_i\sim \{\begin{array}{cc}HN(0,\frac{{\sigma }_{\tau }^2}{2\tau })\hfill & \mathrm{w}.\mathrm{p}.\frac{\sqrt{\tau }}{(\sqrt{\tau }+\sqrt{1-\tau })}\hfill \\ -HN(0,\frac{{\sigma }_{\tau }^2}{2(1-\tau )})\hfill & \mathrm{w}.\mathrm{p}.\frac{\sqrt{1-\tau }}{(\sqrt{\tau }+\sqrt{1-\tau })}\hfill \end{array}$$(2.2) even clearer by writing $${ϵ}_i=\{\begin{array}{cc}HN(0,\frac{{\sigma }^2}{1-\theta })\hfill & \mathrm{w}.\mathrm{p}.\frac{1}{2}\hfill \\ -HN(0,\frac{{\sigma }^2}{\theta })\hfill & \mathrm{w}.\mathrm{p}.\frac{1}{2}.\hfill \end{array}$$

6 For $\theta \ne .5$ and arbitrary ${\sigma }^2,$ a simple calculation will show that $$n\int {\left(f_{\theta }{\left(y\right|\mu ={\mu }_o)}^{1/2}-f_{\theta }{\left(y\right|\mu ={\mu }_o+h/\sqrt{n})}^{1/2}\right)}^{2}dy\to \infty$$ which violates the result in Lemma 12.2.1 in Lehmann and Romano (2006). See also van der Vaart, (2000, p. 64). This demonstrates that the AAP distribution is not differentiable in quadratic mean.

7 Hirano and Porter (2012) also require differentiability in quadratic mean, which is not applicable to our discontinuous density problem.

8 As an aside, it is also worth noting that the difference between the two distributions persists in the “external” case of τ = 0 or τ = 1, where we have $$\underset{\theta \to 0}{\lim }{F}_{\theta }\left(y\right|\mu ,\theta ,{\sigma }^2)=\frac{1}{2}\mathcal{U}(-\infty ,\mu )+\frac{1}{2}\delta (\mu \left)\underset{\tau \to 0}{\lim }{F}_{\tau }\right(y|{\mu }_{\tau },\tau ,{\sigma }_{\tau }^2)=\mathcal{U}(-\infty ,{\mu }_{\tau }).$$ Here, δ is the Dirac delta function.

9 Write $\widehat{\tau }=\frac{\sqrt{1-{\widehat{\theta }}_{\mathit{MLE}}}}{\sqrt{1-{\widehat{\theta }}_{\mathit{MLE}}}+\sqrt{{\theta }_{\mathit{MLE}}}},$ and it is trivial to show that ${\widehat{\mu }}_{\widehat{\tau }}\stackrel{p}{\to }{\widehat{\mu }}_{\tau }.$

10 We would quickly point out that this tradeoff will exist for other parametric models with discontinuous density at the location parameter, so long as that parameter is not the minimum or maximum of the distribution. If the model has two moments, (1) the location parameter will be an expectile ${\mu }_{\tau }$ for some $\tau \in (0,1)$ and (2) the sample expectile ${\widehat{\mu }}_{\tau }$ will be consistent and asymptotically normal (Holzmann et al., 2016).