On consistency of the least squares estimators in linear errors-in-variables models with inﬁnite variance errors

: This paper deals simultaneously with linear structural and func- tional errors-in-variables models (SEIVM and FEIVM), revisiting in this context the ordinary least squares estimators (LSE) for the slope and in- tercept of the corresponding simple linear regression. It has been known that, subject to some model conditions, these estimators become weakly and strongly consistent in the linear SEIVM and FEIVM with the measurement errors having ﬁnite variances when the explanatory variables have an inﬁnite variance in the SEIVM, and a similar inﬁnite spread in the FEIVM, while otherwise, the LSE’s require an adjustment for consistency with the so-called reliability ratio. In this paper, weak and strong consistency, with and without the possible rates of convergence being determined, is proved for the LSE’s of the slope and intecept, assuming that the measurement errors are in the domain of attraction of the normal law (DAN) and thus are, for the ﬁrst time, allowed to have inﬁnite variances. Moreover, these results are obtained under the conditions that the explanatory variables are in DAN, have an inﬁnite variance, and dominate the measurement errors in terms of variation in the SEIVM, and under appropriately matching ver- sions of these conditions in the FEIVM. This duality extends a previously known interplay between SEIVM’s and FEIVM’s. signal-to-noise ratio, domain of attraction of the normal law, inﬁnite variance, slowly varying function at inﬁnity, weak and strong consistency.

We consider the linear errors-in-variables model (EIVM) where (y i , x i ) ∈ Ê 2 are vectors of observations, ξ i are unknown explanatory/latent variables, the real-valued slope β and intercept α are to be estimated, and δ i and ε i are unknown measurement error terms/variables, 1 ≤ i ≤ n, n ∈ AE.
EIVM (1.1) is also known as a measurement error model, or regression with errors in variables. It is a generalization of the simple linear regression of the form y i = βξ i + α + δ i in that in (1.1) it is assumed that, in addition to the two variables η := βξ + α and ξ being linearly related, now not only η, but also ξ, are observed with respective measurement errors δ i and ε i . This paper deals simultaneously with structural and functional versions of EIVM (1.1) (SEIVM and FEIVM). In an SEIVM the explanatory variables ξ i are assumed to be independent identically distributed (i.i.d.) random variables (r.v.'s) that are independent of the error terms, while in case of an FEIVM, one treats them as deterministic variables. The vectors of the error terms {(δ,ε),(δ i ,ε i ), i ≥ 1} are usually, and also presently, assumed to be i.i.d. mean zero random vectors.
Hereafter, the following notations will be used: where {u i , 1 ≤ i ≤ n} and {v i , 1 ≤ i ≤ n} are real-valued variables and constant c = 0, if the intercept α is known to be zero, 1, if the intercept α is unknown. (1.2)

Least squares estimators for the slope and intercept in SEIVM's
It is well-known that the ordinary least squares estimators (LSE's) of the slope and intercept of the simple linear regression y i = βx i + α + δ i , 1 ≤ i ≤ n, that isβ n = S xy S xx andα n = y n −β n x n , (1.3) are inconsistent in SEIVM (1.1) when 0 < Var ξ, Var δ, Var ε < ∞. However, if E(δε) = 0, using the so-called reliability ratio k ξ that is defined via what is known as the signal-to-noise ratio k as k := Eξ 2 − c(Eξ) 2 Var ε and k ξ := k k + 1 = Eξ 2 − c(Eξ) 2 Eξ 2 − c(Eξ) 2 + Var ε , (1.4) one can adjustβ n andα n and obtain consistent estimators for β and α as follows:β n = k −1 ξβ n andα n = y n −β n x n . (1.5) The reliability ratio k ξ is a measure of relative spread of the explanatory variables ξ i to that of the observables x i , and, clearly, 0 < k ξ < 1. Larger values of k lead to larger values of k ξ and to that ξ i are more dominant over the measurement errors ε i , and to that x i , and hence the statistical inference in SEIVM (1.1), are more precise.
To ensure identifiability and the possibility of consistent estimation of unknown parameters in the model (1.1), such as β and α for example, it is common in the literature to make use of some side conditions in this regard. Assuming prior knowledge of k ξ of (1.4) in SEIVM (1.1) is one of the several standard conditions of this kind (cf. Cheng and Van Ness (1999) for further details on identifiability in (1.1)). In practice, this assumption is usually unrealistic. Hence, (consistent) estimation of the reliability ratio k ξ has become a standard practice in SEIVM (1.1). The estimatorsβ n andα n in (1.5), with known or estimated k ξ , are also known as the correction-for-attenuation estimators for β and α.
A new type of SEIVM (1.1), with new asymptotic methodologies and results, was introduced in Martsynyuk (2004), and then studied also in Martsynyuk (2005Martsynyuk ( , 2007aMartsynyuk ( , 2007bMartsynyuk ( , 2009, where the explanatory variables ξ i are, for the first time, assumed to belong to the domain of attraction of the normal law (DAN) with a possibly infinite variance. In particular, this enriched the traditional twomoment space of the explanatory variables that had been used for consistency and central limit theorems studies in SEIVM (1.1) in the literature.
Example 1.1. From Remark 1.1, all the distributions with finite positive variances are in DAN. As to some examples of the distributions in DAN that have infinite variances, a Pareto distribution and its modification that has a somewhat heavier tail, with the respective probability density functions (pdf's) were shown to belong to DAN and to have the following slowly varying functions at infinity in the respective norming constants b n as in Remark 1.1: ℓ 1 (n) = log n and ℓ 2 (n) = log n √ 2 (cf. Martsynyuk (2013, Example 1)).
Among other things, it was observed in Martsynyuk (2005, Remark 1.1.6) that the LSE'sβ n andα n of (1.3), as well as the estimators β n = S yy S xy andα n = y n −β n x n , (1.6) are strongly consistent in SEIVM (1.1) with 0 < Var δ < ∞ and 0 < Var ε < ∞ when Var ξ = ∞ (independently of whether E(δε) = 0 or not). Thus, unlike in the traditional model with 0 < Var ξ < ∞, the LSE's do not require any adjustments for consistency if Var ξ = ∞, when one can formally put k ξ := 1. This can be interpreted as follows: the impact of the finite variance measurement errors ε i in the observables x i is negligible as compared to that of the infinite variance explanatory variables ξ i , so much so that the model becomes close in spirit to, and behaves as if it were, the simple linear regression The LSE's of (1.3) and estimators in (1.6) in SEIVM (1.1) with Var ξ = ∞ add to a handful of examples of consistent estimators in special SEIVM's that do not require any additional information, such as prior knowledge of k ξ for example (cf. Van Montfort (1988) and texts Kendall and Stuart (1979) and Cheng and Van Ness (1999) for details on these examples). It is interesting to note that the existence of these consistent estimators for β implies that β is identifiable, and the latter fact, when (δ, ε) has a normal distribution, can also be concluded from Reiersøl (1950), as it accordingly holds if and only if ξ is not normally distributed.

Least squares estimators in FEIVM's
We now turn our attention to FEIVM (1.1), a companion to SEIVM (1.1), and describe a parallel picture on the LSE's of (1.3) in it. When 0 < Var δ < ∞ and 0 < Var ε < ∞, just like in SEIVM (1.1) with 0 < Var ξ < ∞, the LSE's of (1.3) are inconsistent in FEIVM (1.1) with the deterministic explanatory variables {ξ i } i≥1 satisfying the assumptions lim n→∞ ξ n < ∞ and 0 < lim in place of k ξ of (1.4), are adjustments of the LSE's for strong consistency when E(δε) = 0, where, similarly to k ξ , the ratio in (1.8) usually requires estimation and may be viewed as a measure of relative spread of the explanatory variables to that of the error terms.

Model assumptions and introduction to main results
The results of Martsynyuk (2004Martsynyuk ( , 2005, Liu and Chen (2005) and Miao et al. (2011) in connection with consistency of the LSE's of (1.3) in EIVM (1.1), which were discussed in sections 1.2 and 1.3, are all for the model with 0 < Var δ < ∞ and 0 < Var ε < ∞. In contrast, in this paper we deal with SEIVM and FEIVM (1.1) where both measurement errors δ and ε are, for the first time, allowed to have infinite variances via assuming that mean zero random vectors with δ, ε ∈ DAN and the respective slowly varying functions at infinity ℓ δ (n) and ℓ ε (n) that are such that Concerning our conditions on the explanatory variables, throughout the paper, and the slowly varying function at infinity ℓ ξ (n) as in Remark 1.1, and ξ is independent of (δ, ε), in SEIVM (1.1), Sometimes, we will also assume that The main Theorems 1.1 and 1.2 of the present paper prove respectively weak and strong consistency of the LSE'sβ n andα n under (A1)-(A3), and some additional assumptions that ensure that the explanatory variables dominate the measurement errors in terms of variation, a natural requirement for obtaining meaningful inference in the model (1.1). In our main Theorems 1.3 and 1.4, we refine the results of Theorems 1.1 and 1.2 and establish possible rates of weak and strong consistency ofβ n andα n .
To the best of our knowledge, EIVM (1.1) with the explanatory variables having an infinite variance or spread (as in (A2)), and with the error terms possibly having infinite variances (as in (A1)), as well as estimation problems in this model, have not been previously studied in the literature. On the other hand, various authors have studied practical and theoretical aspects of linear regression when both errors and regressors may have infinite variances, and established asymptotics for the LSE's for the slope and intercept in it. Initial work in this regard was offered in Blattberg and Sargent (1971) and Smith (1973), under the condition that the errors followed stable laws. Andrews (1987aAndrews ( , 1987b provides, among other things, a complete list of references for applications of infinite variance regression, particularly in economics. Assuming that the regressors are in a stable domain of attraction (in particular, in DAN), Cline (1989) considers the LSE's for the slope and intercept in linear regression and determines necessary and sufficient conditions for their weak consistency in terms of a relationship between the regressors' and errors' distributions. The latter relationship roughly amounts to a certain asymptotic dominance of the tail probabilities of the regressors over those of the errors. For some further related works on infinite variance linear regression models and asymptotics for the LSE's in them, we refer to a useful survey of the literature in Cline (1989).
Remark 1.5. We note that in the special case of (A1) when Var δ, Var ε < ∞, Theorems 1.1 and 1.2 hold true simply under (A2) and, in case of consistency of α n , also under (A3). Hence, Theorems 1.1 and 1.2 extend weak and strong consistency results forβ n andα n in SEIVM and FEIVM (1.1) that were previously obtained in Martsynyuk (2004Martsynyuk ( , 2005 and Liu and Chen (2005) (cf. sections 1.2 and 1.3).
The following theorem provides refinements of (1.14) and (1.15) of Theorem 1.1, under some stronger model assumptions than those used in Theorem 1.1.

29)
and, provided also that for some d, θ > 0,  1) with δ, ε ∈ DAN to (1.10) that is due to Miao et al. (2011, Theorems 2.1 and 2.2) and proved under E(|δ| p + |ε| p ) < ∞ for some p ≥ 2. Thus, if p = 2, then the speed of strong consistency for the LSEβ n in (1.10) is S ξξ , which is the maximum possible rate b n in (1.29) of Theorem 1.4 when Var ε, Var δ < ∞, in view of having lim sup n→∞ b n / S ξξ < ∞ in (1.28). Both consistency results hold true in this case simply if lim n→∞ S ξξ = ∞. If at least one of the error variances is assumed to be infinite in Theorem 1.4, then b n in (1.29) for strong consistency ofβ n is slower than S ξξ (cf. (1.28)). As to the respective rates of strong consistency of the LSEα n in (1.10) and (1.32), we note that while they are both slower than √ n, the rate b n in (1.32) can sometimes be a bit faster than the rate of n 1−θ in (1.10), where θ ∈ (1/2, 1]: for example, when Var ε, Var δ < ∞, we can have b n = n 1/2 /(ln n) q/2 in (1.32), with q > 1, under (A2), (A3) and (1.28). Moreover, when (A3) is satisfied and, in particular, Var ε, Var δ < ∞ and b n = n 1−a for some a ∈ (1/2, 1] in Theorem 1.4, then (1.32) holds true under the conditions of lim n→∞ S ξξ = ∞ and n 1−a / S ξξ = O(1), and this amounts to (1.10) forα n .
Remark 1.11. By adapting accordingly the conditions of Theorems 1.1-1.4, we can also prove weak and strong consistency, with and without determining the respective possible rates of convergence, for the estimatorsβ n andα n of (1.6). The statements and proofs of these results are omitted here.

Auxiliary results
Lemma 2.1. In FEIVM (1.1), let {δ, δ i } i≥1 be i.i.d. mean zero r.v.'s and δ ∈ DAN with Var δ = ∞, and let S ξξ ≥ const > 0 for all n ≥ 1. (2.1) Assume that one of the following conditions (2.2)-(2.5) is satisfied: (2.5) If the intercept α of (1.1) is not known to be zero, suppose also that (A3) holds true. Then, Proof of Lemma 2.1. The proof of (2.6) reduces to showing that on account of having and, if α = 0 (and c of (1.2) is 1), also applying the strong law of large numbers (SLLN) for δ n and conditions (2.1) and (A3). If (2.2) holds true, then (2.7) follows directly from Lemma A.2 of Appendix (with r = 1), using only that E|δ| < ∞, while the rest of the proof is dedicated to establishing (2.7) when one of the conditions (2.3)-(2.5) is satisfied.
In the rest of the proof, we will establish (2.16), and thus (2.7) as well, assuming that one of the conditions (2.3)-(2.5) is satisfied.
If (2.3) holds true with some b > 0, then (2.16) is on account of the Borel-Cantelli lemma and having The Hájek-Rényi inequality (cf. Lemma A.1 of Appendix) and steps similar to those in Kounias and Weng (1969) that were used for concluding Lemma A.2 of Appendix, give (2.16) when (2.4) is valid.
Consenquently, (2.5) holds true in this case as well, both when Var δ < ∞ and Var δ = ∞. We also note that the series in (