# Asymptotic Variance of Test Statistics in the ML and QML Frameworks

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## Abstract

In this study, we consider the test statistics that can be written as the sample average of data and derive their limiting distributions under the maximum likelihood (ML) and the quasi-maximum likelihood (QML) frameworks. We first generalize the asymptotic variance formula suggested in Pierce (Ann Stat 10(2):475–478, 1982) in the ML framework and illustrate its applications through some well-known test statistics: (1) the skewness statistic, (2) the kurtosis statistic, (3) the Cox statistic, (4) the information matrix test statistic, and (5) the Durbin’s h-statistic. We next provide a similar result in the QML setting and illustrate its applications by providing two examples. Illustrations show the simplicity and the effectiveness of our results for the asymptotic variance of test statistics, and therefore, they are recommended for practical applications.

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## Notes

1. Note that in these assumptions, the mixing coefficients and all expectations are defined with respect to $$\text {P}_0$$.

2. Note that the setting in Gallant and White [12] does not require that $$\mathbb {E}\left( L_n(\theta )\right)$$ converges to a limit. Thus, they require that $$\theta _\star \equiv \{\theta ^\star _n\}$$ are identifiably unique with respect to $$\mathbb {E}\left( L_n(\theta )\right)$$. For notational simplicity, we suppress n in stating the maximizers of $$\mathbb {E}\left( L_n(\theta )\right)$$.

## References

1. Andreou E, Werker BJM (2012) An alternative asymptotic analysis of residual-based statistics. Rev Econ Stat 94(1):88–99

2. Andrews DWK (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59(3):817–858

3. Bates C, White H (1985) A unified theory of consistent estimation for parametric models. Econom Theory 1(2):151–178

4. Bera AK, Zuo X-L (1996) Specification test for a linear regression model with ARCH process. J Stat Plan Inference 50(2):283–308

5. Bontemps C (2019) Moment-based tests under parameter uncertainty. Rev Econ Stat 101(1):146–159

6. Borjas GJ, Sueyoshi GT (1994) A two-stage estimator for probit models with structural group effects. J Econom 64(1):165–182

7. Cox DR (1961) Tests of separate families of hypotheses. In: Proceedings of the fourth Berkeley symposium on mathematical statistics and probability, volume 1: contributions to the theory of statistics. University of California Press, Berkeley, California, pp 105–123

8. Cox DR (1962) Further results on tests of separate families of hypotheses. J R Stat Soc Ser B Methodol 24(2):406–424

9. Durbin J (1970) Testing for serial correlation in least-squares regression when some of the regressors are lagged dependent variables. Econometrica 38(3):410–421

10. Eicker F (1963) Asymptotic normality and consistency of the least squares estimators for families of linear regressions. Ann Math Stat 34(2):447–456

11. Eicker F (1967) Limit theorems for regressions with unequal and dependent errors. In: LeCam LM, Neyman J (eds) Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, volume 1: statistics. University of California Press, pp 59–82

12. Gallant AR, White H (1988) A unified theory of estimation and inference for nonlinear dynamic models. Blackwell, Oxford

13. Gonçalves S, White H (2004) Maximum likelihood and the bootstrap for nonlinear dynamic models. J Econom 119(1):199–219

14. Gorodnichenko Y, Mikusheva A, Ng S (2012) Estimators for persistent and possibly nonstationary data with classical properties. Econom Theory 28(5):1003–1036

15. Gourieroux C, Monfort A, Trognon A (1984) Pseudo maximum likelihood methods: theory. Econometrica 52(3):681–700

16. Harvey A, Stephen T (2016) Testing against changing correlation. Recent developments in financial econometrics and empirical finance. J Empir Finance 38:575–589

17. Huber PJ (1967) The behavior of maximum likelihood estimates under nonstandard conditions. In: Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, volume 1: statistics. University of California Press, Berkeley, California, pp 221–233

18. Jarque CM, Bera AK (1987) A test for normality of observations and regression residuals. Int Stat Rev 55(2):163–172

19. Koopmans TC, Rubin H, Leipnik RB (1950) Measuring the equation systems of dynamic economics. In: Koopmans TC (ed) Statistical inference in dynamic economic models by Cowles commission monograph no 10. Wiley, New York, pp 53–237

20. Lange N, Ryan L (1989) Assessing normality in random effects models. Ann Stat 17(2):624–642

21. Newey WK (1985a) Generalized method of moments specification testing. J Econom 29(3):229–256

22. Newey WK (1985b) Maximum likelihood specification testing and conditional moment tests. Econometrica 53(5):1047–1070

23. Newey WK, West KD (1987) A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55(3):703–708

24. Neyman J (1935) Sur la vérification des hypothèses statistiques composées, French. Bulletin de la Sociéeté Mathéematique de France 63:246–266

25. Neyman J (1957) Current problems of mathematical statistics. Statistical Laboratory, University of California, Berkeley

26. Neyman J (1959) Optimal asymptotic tests of composite statistical hypotheses. In: Grenander U (ed) Probability and statistics, the Harald Cramer volume. Wiley, New York, pp 416–444

27. Pierce DA (1982) The asymptotic effect of substituting estimators for parameters in certain types of statistics. Ann Stat 10(2):475–478

28. Rao CR (1973) Linear statistical inference and its applications, 2nd edn. Wiley series in probability and mathematical statistics: probability and mathematical statistics. Wiley, New York

29. Student (1908) The probable error of a mean. Biometrika 6(1):1–25

30. Tauchen G (1985) Diagnostic testing and evaluation of maximum likelihood models. J Econom 30(1):415–443

31. Tse YK (2002) Residual-based diagnostics for conditional heteroscedasticity models. Econom J 5(2):358–373

32. Vella F, Verbeek M (1999) Two-step estimation of panel data models with censored endogenous variables and selection bias. J Econom 90(2):239–263

33. White H (1982a) Maximum likelihood estimation of misspecified models. Econometrica 50(1):1–25

34. White H (1982) Regularity conditions for Cox’s test of non-nested hypotheses. J Econom 19(2):301–318

35. White H (1987) Specification testing in dynamic models. In: Bewley T (ed) Advances in econometrics—fifth world congress, vol 1: econometric society monographs. Cambridge University Press, New York, pp 1–58

36. White H (1994) Estimation, inference and specification analysis: econometric society monographs no 22. Cambridge University Press, Cambridge

37. Yang Z, Tse YK, Bai Z (2007) Statistics with estimated parameters. Stati Sin 17(2):817–837

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Correspondence to Osman Doğan.

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We would like to thank the Editor and two anonymous referees for many helpful comments that assisted us to revise and improve the paper. An earlier version of this paper was presented at the 4th Workshop on Goodness-of-Fit, Change-Point and Related Problems, Trento, Italy, 6–8 September 2019. We are grateful to the Workshop participants for constructive comments and suggestions. Any remaining shortcomings and errors are, of course, ours.

## Appendices

### A Lemmas

In this section, we provide two useful lemmas that are required in the proofs of Propositions 1 and 2. The first lemma gives the conditions for the uniform law of large numbers for functions of mixing sequences. The second lemma shows how a consistent estimator of a term that obeys the uniform law of large numbers can be formulated.

### Lemma 1

Assume that Assumption 1holds, and let $$\{V_t\}$$ be a mixing process with $$\phi _m$$ of size $$-r/(2r-2)$$, $$r\ge 2$$, or $$\alpha _m$$ of size $$-r/(r-2)$$, $$r>2$$. Suppose that

1. (1)

$$\log f_{nt}(Y^t_n,\theta )$$ is Lipschitz continuous on $$\Theta$$, where $$\Theta$$ is compact set.

2. (2)

$$\log f_{nt}(Y^t_n,\theta )$$ is r-dominated on $$\Theta$$ uniformly in t and n.

3. (3)

$$\{\log f_{nt}(Y^t_n,\theta )\}$$ is near epoch dependent on $$\{V_t\}$$ of size $$-1/2$$ uniformly on $$(\Theta ,\varrho )$$, where $$\varrho$$ is a metric on $$\Theta$$.

Then, (a) $$\mathbb {E}(\log f_{nt}(Y^t_n,\theta ))$$ is continuous on $$\Theta$$ uniformly in n and t, and (b) $$\log f_{nt}(Y^t_n,\theta ) -\mathbb {E}\left( \log f_{nt}(Y^t_n,\theta )\right) \xrightarrow {}0$$ a.s. -$$\text {P}_0$$ uniformly on $$\Theta$$.

### Proof

See Gallant and White [12, Theorem 3.18]. $$\square$$

### Lemma 2

Let $$\{Q_n:\Omega \times \Theta \xrightarrow {}\mathbb {R}\}$$ be a sequence of continuous function on $$\Theta$$ a.s.-$$P_0$$, and let $$\{\hat{\theta }_n:\Omega \xrightarrow {}\Theta \}$$ be a sequence satisfying $$\hat{\theta }_n-\theta ^*_n\xrightarrow {}0$$ a.s.-$$P_0$$. Suppose that $$\sup _{\theta \in \Theta }\left| Q_n(\cdot ,\theta )-\bar{Q}_n(\theta )\right| \xrightarrow {}0$$ a.s.-$$P_0$$, where $$\{\bar{Q}_n:\Theta \xrightarrow {}\mathbb {R}\}$$ is continuous on $$\Theta$$ uniformly in n. Then,

\begin{aligned} Q_n(\cdot ,\hat{\theta }_n)-\bar{Q}_n(\theta ^*_n)\xrightarrow {}0\quad {\textit{ a.s.-}}P_0. \end{aligned}

### Proof

See White [36, Corollary 3.8]. $$\square$$

### 1.1 B.1 Proof of Proposition 1

Define the following vector

\begin{aligned} \xi _{nt}(Y^t_n, \theta ,\rho )= \begin{pmatrix} \frac{1}{n}\frac{\partial \log f_{nt}(Y^t_n,\theta )}{\partial \theta }\\ \frac{1}{n}(\rho _{nt}(Y^t_n,\theta )-\rho ) \end{pmatrix}. \end{aligned}
(B.1)

Under assumption that $$\hat{\theta }_n-\theta _0\xrightarrow {}0$$ a.s. $$\text {P}_0$$, we have $$\sum _{t=1}^n\xi _{nt}(Y^t_n, \hat{\theta }_n,\hat{\rho }_n)=0$$ a.s. $$\text {P}_0$$ by Lemmas 1 and 2. Also note that

\begin{aligned} \mathrm {Var}\left( \sqrt{n}\sum _{t=1}^n\xi _{nt}(Y^t_n, \theta _0,\rho _0)\right) = \begin{pmatrix} \mathcal {A}_n(\theta _0)&{}\mathcal {P}_n(\theta _0,\rho _0)\\ \mathcal {P}^{'}_n(\theta _0,\rho _0)&{}\mathcal {C}_n(\theta _0,\rho _0) \end{pmatrix}, \end{aligned}
(B.2)

where $$\mathcal {P}_n(\theta _0,\rho _0)=\mathbb {E}\left( \left( \frac{1}{\sqrt{n}}\sum _{t=1}^n\frac{\partial \log f_{nt}(Y^t_n,\theta _0)}{\partial \theta }\right) \times \left( \frac{1}{\sqrt{n}}\sum _{t=1}^n(\rho _{nt}(Y^t_n,\theta _0)-\rho _0)\right) ^{'}\right)$$ and $$\mathcal {C}(\theta _0,\rho _0)=\mathrm {Var}\left( \frac{1}{\sqrt{n}}\sum _{t=1}^n(\rho _{nt}(Y^t_n,\,\theta _0)-\rho _0)\right)$$. Let $$\nabla _\delta$$ be the gradient with respect to $$\delta$$. Then, taking a mean value expansion of $$\xi _{nt}(Y^t_n, \hat{\theta }_n,\hat{\rho }_n)$$ around $$\theta _0$$ and $$\rho _0$$ gives

\begin{aligned} \sum _{t=1}^n\sqrt{n}\xi _{nt}(Y^t_n, \theta _0,\rho _0)=\left( -\sum _{t=1}^n\nabla _{\theta \rho }\xi _{nt}(Y^t_n, \tilde{\theta }_n,\tilde{\rho }_n)\right) \begin{pmatrix} \sqrt{n}(\hat{\theta }_n-\theta _0)\\ \sqrt{n}\left( T_n(Y^n_n,\hat{\theta }_n)-\rho _0\right) \end{pmatrix} \quad a.s. \text {-P}_0, \end{aligned}
(B.3)

where $$\tilde{\theta }_n$$ and $$\tilde{\rho }_n$$ are the mean values. Note that

\begin{aligned} \mathbb {E}\left( \sum _{t=1}^n\nabla _{\theta \rho }\xi _{nt}(Y^t_n, \theta _0,\rho _0)\right) =\mathbb {E}\begin{pmatrix} n^{-1}\sum _{t=1}^n\frac{\partial ^2\log f_{nt}(Y^t_n,\theta _0)}{\partial \theta \partial \theta ^{'}}&{}0\\ n^{-1}\sum _{t=1}^n\frac{\partial \rho _{nt}(Y^t_n,\theta _0)}{\partial \theta ^{'}}&{}-I_q \end{pmatrix} = \begin{pmatrix} -\mathcal {A}_n(\theta _0)&{}0\\ \mathcal {D}_n(\theta _0)&{}-I_q \end{pmatrix}, \end{aligned}
(B.4)

where $$\mathcal {D}_n(\theta _0)=\mathbb {E}\left( n^{-1}\sum _{t=1}^n\frac{\partial \rho _{nt}(Y^t_n,\,\theta _0)}{\partial \theta ^{'}}\right)$$ and $$I_q$$ is the $$q\times q$$ identity matrix. Our Lemmas 1 and 2 ensure that

\begin{aligned} -\left( \sum _{t=1}^n\nabla _{\theta \rho }\xi _{nt}(Y^t_n, \tilde{\theta }_n,\tilde{\rho }_n)\right) -\begin{pmatrix} -\mathcal {A}_n(\theta _0)&{}0\\ \mathcal {D}_n(\theta _0)&{}-I_q \end{pmatrix} \xrightarrow {}0\quad a.s.\text {P}_0. \end{aligned}
(B.5)

Since $$\mathcal {A}_n(\theta _0)$$ is uniformly non-singular, it follows from (B.5) that $$\left( \sum _{t=1}^n\nabla _{\theta \rho }\xi _{nt}(Y^t_n, \tilde{\theta }_n,\tilde{\rho }_n)\right)$$ is non-singular $$a.s.\text {P}_0$$. Thus,

\begin{aligned} \begin{pmatrix} \sqrt{n}(\hat{\theta }_n-\theta _0)\\ \sqrt{n}\left( T_n(Y^n_n,\hat{\theta }_n)-\rho _0\right) \end{pmatrix} =-\left( \sum _{t=1}^n\nabla _{\theta \rho }\xi _{nt}(Y^t_n, \tilde{\theta }_n,\tilde{\rho }_n)\right) ^{-1}\sum _{t=1}^n\sqrt{n}\xi _{nt}(Y^t_n, \theta _0,\rho _0)\quad a.s.\text {-P}_0. \end{aligned}
(B.6)

Our stated assumptions ensure the conditions of the central limit theorem for near epoch-dependent functions of a mixing process (White [36], Theorem A.3.7, p. 358). Thus, we have

\begin{aligned} \sum _{t=1}^n\sqrt{n}\xi _{nt}(Y^t_n, \theta _0,\rho _0){\mathop {\sim }\limits ^{A}} N\left[ 0,\, \begin{pmatrix} \mathcal {A}_n(\theta _0)&{}\mathcal {P}_n(\theta _0,\rho _0)\\ \mathcal {P}^{'}_n(\theta _0,\rho _0)&{}\mathcal {C}_n(\theta _0,\rho _0) \end{pmatrix} \right] . \end{aligned}
(B.7)

Then, using (B.7) in (B.6), we obtain

\begin{aligned} \begin{pmatrix} \sqrt{n}(\hat{\theta }_n-\theta _0)\\ \sqrt{n}\left( T_n(Y^n_n,\hat{\theta }_n)-\rho _0\right) \end{pmatrix} {\mathop {\sim }\limits ^{A}} N\left[ 0,\, \begin{pmatrix} \mathcal {A}^{-1}_n(\theta _0)&{}\mathcal {V}^{'}_n(\theta _0,\rho _0)\\ \mathcal {V}_n(\theta _0,\rho _0)&{}\mathcal {S}_n(\theta _0,\rho _0) \end{pmatrix} \right] , \end{aligned}
(B.8)

where

\begin{aligned}&\begin{pmatrix} \mathcal {A}^{-1}_n(\theta _0)&{}\mathcal {V}^{'}_n(\theta _0,\rho _0)\\ \mathcal {V}_n(\theta _0,\rho _0)&{}\mathcal {S}_n(\theta _0,\rho _0) \end{pmatrix} \\&\quad = \begin{pmatrix} -\mathcal {A}_n(\theta _0)&{}0\\ \mathcal {D}_n(\theta _0)&{}-I_q \end{pmatrix}^{-1} \begin{pmatrix} \mathcal {A}_n(\theta _0)&{}\mathcal {P}_n(\theta _0,\rho _0)\\ \mathcal {P}^{'}_n(\theta _0,\rho _0)&{}\mathcal {C}_n(\theta _0,\rho _0) \end{pmatrix} \begin{pmatrix} -\mathcal {A}_n(\theta _0)&{}\mathcal {D}^{'}_n(\theta _0)\\ 0&{}-I_q \end{pmatrix}^{-1}. \end{aligned}

An application of the inverse partitioned matrix formula gives

\begin{aligned}&\begin{pmatrix} -\mathcal {A}_n(\theta _0)&{}0\\ \mathcal {D}_n(\theta _0)&{}-I_q \end{pmatrix}^{-1} =\begin{pmatrix} -\mathcal {A}^{-1}_n(\theta _0)&{}0\\ -\mathcal {D}_n(\theta _0)\mathcal {A}^{-1}_n(\theta _0)&{}-I_q \end{pmatrix}. \end{aligned}
(B.9)

Then, using (B.9), we can express $$\mathcal {V}_n(\theta _0,\rho _0)$$ and $$\mathcal {S}_n(\theta _0,\rho _0)$$ in (B.8) as

\begin{aligned}&\mathcal {V}_n(\theta _0,\rho _0)=\mathcal {D}_n(\theta _0)\mathcal {A}^{-1}_n(\theta _0)+\mathcal {P}^{'}_n(\theta _0,\rho _0)\mathcal {A}^{-1}_n(\theta _0), \end{aligned}
(B.10)
\begin{aligned}\mathcal {S}_n(\theta _0,\rho _0)=&\,\mathcal {C}_n(\theta _0,\rho _0)+\mathcal {D}_n(\theta _0)\mathcal {A}^{-1}_n(\theta _0)\mathcal {D}^{'}_n(\theta _0) +\mathcal {P}^{'}_n(\theta _0,\rho _0) \mathcal {A}^{-1}_n(\theta _0)\mathcal {D}^{'}_n(\theta _0)\nonumber \\& +\,\mathcal {D}_n(\theta _0)\mathcal {A}^{-1}_n(\theta _0)\mathcal {P}_n(\theta _0,\rho _0). \end{aligned}
(B.11)

Next, we show how $$\mathcal {S}_n(\theta _0,\rho _0)$$ is simplified under Assumption 11(2). The assumption that $$\mathbb {E}(\rho _{nt}(Y^t_n,\theta _0))$$ is independent of $$\theta _0$$ implies that

\begin{aligned}\frac{\partial \mathbb {E}\left( T(Y^n_n,\theta )\right) }{\partial \theta ^{'}}\big |_{\theta _0}&=n^{-1}\sum _{t=1}^n\frac{\partial \mathbb {E}(\rho _{nt}(Y^t_n,\theta ))}{\partial \theta ^{'}}\big |_{\theta _0}\nonumber \\&\quad =\int \left( n^{-1}\sum _{t=1}^n\frac{\partial \rho _{nt}(Y^t_n,\theta )}{\partial \theta ^{'}}\bigg |_{\theta _0}\right) \times f_n(Y^n_n,\theta _0)\text {d}\mu ^n\nonumber \\&\quad +\,\int \left( \frac{1}{\sqrt{n}} \sum _{t=1}^n \rho _{nt}(Y^t_n,\theta )\right) \left( \frac{1}{\sqrt{n}}\frac{\partial \log f_n(Y^n_n,\theta )}{\partial \theta }\bigg |_{\theta _0}\right) ^{'} f_n(Y^n_n,\theta _0)\text {d}\mu ^n=0. \end{aligned}
(B.12)

Since $$\mathbb {E}\left( \frac{\partial \log f_n(Y^n_n,\theta )}{\partial \theta }\big |_{\theta _0}\right) =0$$ by Assumption 7, (B.12) can be expressed as

\begin{aligned}&\int \left( n^{-1}\sum _{t=1}^n\frac{\partial \rho _{nt}(Y^t_n,\theta )}{\partial \theta ^{'}}\bigg |_{\theta _0}\right) \times f_n(Y^t_n,\theta _0)\text {d}\mu ^n\nonumber \\&\quad +\,\int \left( \frac{1}{\sqrt{n}} \sum _{t=1}^n \rho _{nt}(Y^t_n,\theta )-\rho _0\right) \left( \frac{1}{\sqrt{n}}\sum _{t=1}^n\frac{\partial \log f_{nt}(Y^t_n,\theta )}{\partial \theta }\bigg |_{\theta _0}\right) ^{'} f_n(Y^n_n,\theta _0)\text {d}\mu ^n=0, \end{aligned}
(B.13)

which implies that

\begin{aligned} \mathcal {P}^{'}_n(\theta _0,\rho _0)=-\mathcal {D}_n(\theta _0). \end{aligned}
(B.14)

Using (B.14) in (B.10) and (B.11) for $$\mathcal {V}_n(\theta _0,\rho _0)$$ and $$\mathcal {S}_n(\theta _0,\rho _0)$$, respectively, yields the desired results.

### 1.2 B.2 Proof of Proposition 2

The proof is similar to that of Proposition 1. Consider the following vector

\begin{aligned} \xi _{nt}(Y^t_n, \theta ,\rho )= \begin{pmatrix} \frac{1}{n}\frac{\partial \log f_{nt}(Y^t_n,\theta )}{\partial \theta }\\ \frac{1}{n}(\rho _{nt}(Y^t_n,\theta )-\rho ) \end{pmatrix}. \end{aligned}
(B.15)

Because $$\hat{\theta }_n-\theta _\star \xrightarrow {}0$$ a.s. $$\text {P}_0$$, our assumptions ensure that $$\sum _{t=1}^n\xi _{nt}(Y^t_n, \hat{\theta }_n,\hat{\rho }_n)=0$$ a.s. -$$\text {P}_0$$ by Lemmas 1 and 2. Here, it is important to note that

\begin{aligned} \mathrm {Var}\left( \sqrt{n}\sum _{t=1}^n\xi _{nt}(Y^t_n, \theta _\star ,\rho _\star )\right) = \begin{pmatrix} \mathcal {B}_n(\theta _\star )&{}\mathcal {P}_n(\theta _\star ,\rho _\star )\\ \mathcal {P}^{'}_n(\theta _\star ,\rho _\star )&{}\mathcal {C}_n(\theta _\star ,\rho _\star ) \end{pmatrix}, \end{aligned}
(B.16)

where $$\mathcal {P}_n(\theta _\star ,\rho _\star )=\mathbb {E}\left( \left( \frac{1}{\sqrt{n}}\sum _{t=1}^n\frac{\partial \log f_{nt}(Y^t_n,\theta _\star )}{\partial \theta }\right) \times \left( \frac{1}{\sqrt{n}}\sum _{t=1}^n(\rho _{nt}(Y^t_n,\theta _\star )-\rho _\star )\right) ^{'}\right)$$ and $$\mathcal {C}(\theta _\star )=\mathrm {Var}\left( \frac{1}{\sqrt{n}}\sum _{t=1}^n(\rho _{nt}(Y^t_n,\,\theta _\star )-\rho _\star )\right)$$. Similarly, it can be shown that

\begin{aligned} \mathbb {E}\left( \sum _{t=1}^n\nabla _{\theta \rho }\xi _{nt}(Y^t_n, \theta _\star ,\rho _\star )\right) =\mathbb {E}\begin{pmatrix} n^{-1}\sum _{t=1}^n\frac{\partial ^2\log f_{nt}(Y^t_n,\theta _\star )}{\partial \theta \partial \theta ^{'}}&{}0\\ n^{-1}\sum _{t=1}^n\frac{\partial \rho _{nt}(Y^t_n,\theta _\star )}{\partial \theta ^{'}}&{}-I_q \end{pmatrix} = \begin{pmatrix} -\mathcal {A}_n(\theta _\star )&{}0\\ \mathcal {D}_n(\theta _\star )&{}-I_q \end{pmatrix}, \end{aligned}
(B.17)

where $$\mathcal {D}_n(\theta _\star )=\mathbb {E}\left( n^{-1}\sum _{t=1}^n\frac{\partial \rho _{nt}(Y^t_n,\,\theta _\star )}{\partial \theta ^{'}}\right)$$ and $$I_q$$ is the $$q\times q$$ identity matrix. Also, Lemmas 1 and 2 ensure that

\begin{aligned} -\left( \sum _{t=1}^n\nabla _{\theta \rho }\xi _{nt}(Y^t_n, \hat{\theta }_n,\hat{\rho }_n)\right) - \mathbb {E}\left( \sum _{t=1}^n\nabla _{\theta \rho }\xi _{nt}(Y^t_n, \theta _\star ,\rho _\star )\right) \xrightarrow {}0\quad a.s.\text {P}_0. \end{aligned}
(B.18)

Then, following the argument in the proof of Proposition 1, it follows that

\begin{aligned} \begin{pmatrix} \sqrt{n}(\hat{\theta }_n-\theta _\star )\\ \sqrt{n}(T_n(Y^n_n,\hat{\theta }_n)-\rho _\star ) \end{pmatrix} {\mathop {\sim }\limits ^{A}} N\left[ 0,\, \begin{pmatrix} \mathcal {A}^{-1}_n(\theta _\star )\mathcal {B}_n(\theta _\star )\mathcal {A}^{-1}_n(\theta _\star )&{}\mathcal {V}^{'}_n(\theta _\star ,\rho _\star )\\ \mathcal {V}_n(\theta _\star ,\rho _\star )&{}\mathcal {S}_n(\theta _\star ,\rho _\star ) \end{pmatrix} \right] , \end{aligned}
(B.19)

where

\begin{aligned}&\begin{pmatrix} \mathcal {A}^{-1}_n(\theta _\star )\mathcal {B}_n(\theta _\star )\mathcal {A}^{-1}_n(\theta _\star )&{}\mathcal {V}^{'}_n(\theta _\star ,\rho _\star )\\ \mathcal {V}_n(\theta _\star ,\rho _\star )&{}\mathcal {S}_n(\theta _\star ,\rho _\star ) \end{pmatrix} \nonumber \\&\quad = \begin{pmatrix} -\mathcal {A}_n(\theta _\star )&{}0\\ \mathcal {D}_n(\theta _\star )&{}-I_q \end{pmatrix}^{-1} \begin{pmatrix} \mathcal {B}_n(\theta _\star )&{}\mathcal {P}_n(\theta _\star ,\rho _\star )\\ \mathcal {P}^{'}_n(\theta _\star ,\rho _\star )&{}\mathcal {C}_n(\theta _\star ,\rho _\star ) \end{pmatrix} \begin{pmatrix} -\mathcal {A}_n(\theta _\star )&{}\mathcal {D}^{'}_n(\theta _\star )\\ 0&{}-I_q \end{pmatrix}^{-1}. \end{aligned}
(B.20)

Using the inverse partitioned matrix formula (see (B.9)) in (B.20), it can be shown that

\begin{aligned} \mathcal {V}_n(\theta _\star ,\rho _\star )&=\mathcal {D}_n(\theta _\star )\mathcal {A}^{-1}_n(\theta _\star )\mathcal {B}_n(\theta _\star )\mathcal {A}^{-1}_n(\theta _\star ) +\mathcal {P}^{'}_n(\theta _\star ,\rho _\star )\mathcal {A}^{-1}_n(\theta _\star ), \end{aligned}
(B.21)
\begin{aligned} \mathcal {S}_n(\theta _\star ,\rho _\star )&=\mathcal {C}_n(\theta _\star ,\rho _\star )+\mathcal {D}_n(\theta _\star )\mathcal {A}^{-1}_n(\theta _\star )\mathcal {B}_n(\theta _\star ) \mathcal {A}^{-1}_n(\theta _\star )\mathcal {D}^{'}_n(\theta _\star )+\mathcal {P}^{'}_n(\theta _\star ,\rho _\star ) \mathcal {A}^{-1}_n(\theta _\star )\mathcal {D}^{'}_n(\theta _\star )\nonumber \\&\quad +\,\mathcal {D}_n(\theta _\star )\mathcal {A}^{-1}_n(\theta _\star )\mathcal {P}_n(\theta _\star ,\rho _\star ). \end{aligned}
(B.22)

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Bera, A.K., Doğan, O. & Taşpınar, S. Asymptotic Variance of Test Statistics in the ML and QML Frameworks. J Stat Theory Pract 15, 2 (2021). https://doi.org/10.1007/s42519-020-00137-0