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A note on the asymptotic and exact Fisher information matrices of a Markov switching VARMA process

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Abstract

We study the asymptotic and exact Fisher information (FI) matrices of Markov switching vector autoregressive moving average (MS VARMA) models. In a related paper (2017), we propose a method to derive an explicit expression in closed form for the asymptotic FI matrix of the underlying model, and use such a matrix to derive the asymptotic covariance matrix of the Gaussian maximum likelihood (ML) estimator of the parameters in the MS VARMA model. In this paper, the exact FI matrix of a Gaussian MS VARMA process is considered for a time series of length T in relation to the exact ML estimation method. Furthermore, we prove that the Gaussian exact FI matrix converges in probability to the asymptotic FI matrix when the sample size T goes to infinity.

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Acknowledgements

Work financially supported by FAR (2017) research Grant of the University of Modena and Reggio Emilia, Italy. We are grateful to the Editor-in-Chief Prof. Tommaso Proietti, and two anonymous referees for their very useful suggestions and remarks which improved the final version of the paper.

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Correspondence to Maddalena Cavicchioli.

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Appendix

Appendix

Proof of Theorem 2

Each equation is meant to be evaluated on the ML estimates of the parameters, without explicit mention. To prove the theorem we use matrix differential calculus and linear algebra of matrices. See Magnus and Neudecker (1999). From Cavicchioli (2017), Appendix (A.2), it follows that

$$\begin{aligned} {\text {plim}}_{T \rightarrow \infty } \, {{\mathcal {J}}}_T ({{\varvec{\sigma }}}_m) = {{\mathcal {F}}}_a ({{\varvec{\sigma }}}_m) \qquad \qquad {\text {plim}}_{T \rightarrow \infty } \, {{\mathcal {J}}}_T ({{\varvec{\alpha }}}_m, {{\varvec{\sigma }}}_m) = {{\mathcal {F}}}_a ({{\varvec{\alpha }}}_m, {{\varvec{\sigma }}}_m) = {{\varvec{0}}}. \end{aligned}$$

So it remains to prove the convergence in probability of \( {{\mathcal {J}}}_T ({{\varvec{\alpha }}}_m)\) to \({{\mathcal {F}}}_a ({{\varvec{\alpha }}}_m) \) when T goes to infinity. By Cavicchioli (2017), Appendix (A.2), the first derivatives of \({\text {log}}_e \eta _{m t} = {\text {log}}_e \eta _{m t} ({{\varvec{\theta }}})\) with respect to the components of \({{\varvec{\alpha }}}_m\) are given by

$$\begin{aligned} \frac{\partial \, {\text {log}}_{e} { \eta }_{m t} }{\partial \, {{\varvec{\nu }}}_m}= & {} {{\varvec{\varTheta }}}_{m}^{- 1} (1)^\top \, {{\varvec{\varOmega }}}_{m}^{- 1} \, \mathbf{u}_t \end{aligned}$$
(13)
$$\begin{aligned} \frac{\partial \, {\text {log}}_{e} { \eta }_{m t} }{\partial \, {{\varvec{\varPhi }}}_{m i}}= & {} \sum _{\ell = 0}^{ \infty } \, {{\varvec{\varPsi }}}_{m \ell }^\top \, {{\varvec{\varOmega }}}_{m}^{- 1} \, \mathbf{u}_t \, \mathbf{y}_{t - i - \ell }^\top \end{aligned}$$
(14)
$$\begin{aligned} \frac{\partial {\text {log}}_{e} {\eta }_{m t} }{\partial {{\varvec{\varTheta }}}_{m j} }= & {} \sum _{\ell , n \ge 0} \, {{\varvec{\varPsi }}}_{m n} \, \mathbf{z}_{m, t - j - \ell - n} \, \mathbf{u}_{t}^\top \, {{\varvec{\varOmega }}}_{m}^{- 1} \, {{\varvec{\varPsi }}}_{m \ell } \end{aligned}$$
(15)

for all \(i = 1, \dots , p\), \(j = 1, \dots , q\) and \(m = 1, \dots , M\). Recall Equations (A.2)–(A.4) from Cavicchioli (2017):

$$\begin{aligned} \frac{\partial \, \mathbf{u}_t}{\partial \, {{\varvec{\nu }}}_{m}^\top } \,= & {} \, - \, {{\varvec{\varTheta }}}_{m}^{- 1} (1) \end{aligned}$$
(16)
$$\begin{aligned} \frac{\partial \, \mathbf{u}_{t}}{\partial \, {{\varvec{\varPhi }}}^\top _{m i}} \,= & {} \, - \, \sum _{\ell = 0}^{ \infty } \, {{\varvec{\varPsi }}}_{m \, \ell } \otimes \mathbf{y}_{t - i - \ell }^\top \end{aligned}$$
(17)
$$\begin{aligned} \frac{\partial \mathbf{u}_{t}}{\partial {{\varvec{\varTheta }}}_{m j}^\top }= & {} \sum _{\ell = 0}^{ \infty }\, \sum _{n = 0}^{ \infty }\, {{\varvec{\varPsi }}}_{m \ell }^\top \otimes (\mathbf{z}_{m, t- j - \ell - n}^\top \, {{\varvec{\varPsi }}}_{m n}^\top ) \end{aligned}$$
(18)

for all \(i = 1, \dots , p\), \(j = 1, \dots , q\) and \(m = 1, \dots , M\). From the first order conditions (FOC) of the ML estimation method, we get

$$\begin{aligned} \sum _{t = 1}^T \, \frac{\partial \, {\text {log}}_e \eta _{m t}}{\partial \, {{\varvec{\nu }}}_m} \, \xi _{m t | T} = {{\varvec{\varTheta }}}_{m}^{- 1} (1) \, {{\varvec{\varOmega }}}_{m}^{- 1} \, \sum _{t = 1}^T \, \mathbf{u}_t \, \xi _{m t | T} = {{\varvec{0}}} \end{aligned}$$

hence

$$\begin{aligned} \sum _{t =1}^T \, {\widehat{{\varvec{u}}}}_t \, \xi _{m t | T} = {{\varvec{0}}} \end{aligned}$$
(19)

where \({\widehat{{\varvec{u}}}}_t\) is the residual estimate. Now we compute the second derivatives of \({\text {log}}_e \eta _{m t}\) with respect to \({{\varvec{\nu }}}_m\) and the components of \({{\varvec{\alpha }}}_m\). By (13) and (16), we have

$$\begin{aligned} \frac{\partial ^2 \, {\text {log}}_e \eta _{m t}}{\partial \, {{\varvec{\nu }}}_m \, \partial {{\varvec{\nu }}}_{m}^\top } = {{\varvec{\varTheta }}}_{m}^{- 1} (1)^\top \, {{\varvec{\varOmega }}}_{m}^{- 1}\, \frac{\partial \, \mathbf{u}_t}{\partial \, {{\varvec{\nu }}}_{m}^\top } = \, - \, {{\varvec{\varTheta }}}_{m}^{- 1} (1)^\top \, {{\varvec{\varOmega }}}_{m}^{- 1}\, {{\varvec{\varTheta }}}_{m}^{- 1} (1). \end{aligned}$$

Thus

$$\begin{aligned} {{\mathcal {J}}}_T ({{\varvec{\nu }}}_m) \, = \, \left( \frac{1}{T} \, \sum _{t = 1}^T \, \xi _{m t | T} \right) \, [ {{\varvec{\varTheta }}}_{m}^{- 1} (1)^\top \, {{\varvec{\varOmega }}}_{m}^{- 1}\, {{\varvec{\varTheta }}}_{m}^{- 1} (1) ] \end{aligned}$$

hence \( {\text {plim}}_{T \rightarrow \infty } \, {{\mathcal {J}}}_T ({{\varvec{\nu }}}_m) \, = \, \pi _m \, {{\varvec{\varTheta }}}_{m}^{- 1} (1)^\top \, {{\varvec{\varOmega }}}_{m}^{- 1}\, {{\varvec{\varTheta }}}_{m}^{- 1} (1) = {{\mathcal {F}}}_a ({{\varvec{\nu }}}_m) \) by (7). By (13) and (17), we obtain

$$\begin{aligned} \frac{\partial ^2 \, {\text {log}}_e \eta _{m t}}{\partial {{\varvec{\nu }}}_m \, \partial {{\varvec{\varPhi }}}_{m i}^\top } = {{\varvec{\varTheta }}}_{m}^{- 1} (1)^\top \, {{\varvec{\varOmega }}}_{m}^{- 1}\, \frac{\partial \mathbf{u}_t}{\partial {{\varvec{\varPhi }}}_{m i}^\top } = \, - \, \sum _{\ell = 0}^{\infty } \, [{{\varvec{\varTheta }}}_{m}^{- 1} (1)^\top \, {{\varvec{\varOmega }}}_{m}^{- 1}\, {{\varvec{\varPsi }}}_{m \ell }] \otimes \mathbf{y}_{t - i - \ell }^\top \end{aligned}$$

for all \(i = 1, \dots , p\) and \(m = 1, \dots , M\). Thus

$$\begin{aligned} {{\mathcal {J}}}_T ({{\varvec{\nu }}}_m, {{\varvec{\gamma }}}_m) \, = \, \sum _{\ell = 0}^{\infty } \, [ {{\varvec{\varTheta }}}_{m}^{- 1} (1)^\top \, {{\varvec{\varOmega }}}_{m}^{- 1}\, {{\varvec{\varPsi }}}_{m \ell } ] \otimes \left( \frac{1}{T} \, \sum _{t = 1}^T \, \mathbf{x}_{t - \ell }^\top \, \xi _{m t | T}\right) . \end{aligned}$$

Since \({{\mathcal {J}}}_T ({{\varvec{\gamma }}}_m, {{\varvec{\nu }}}_m) = {{\mathcal {J}}}_T ({{\varvec{\nu }}}_m, {{\varvec{\gamma }}}_m)^\top \), we have

$$\begin{aligned} {\text {plim}}_{T \rightarrow \infty } \, {{\mathcal {J}}}_T ({{\varvec{\gamma }}}_m, {{\varvec{\nu }}}_m) \, = \, \sum _{\ell = 0}^{\infty } \, [{{\varvec{\varPsi }}}_{m \ell }^\top \, {{\varvec{\varOmega }}}_{m}^{- 1}\, {{\varvec{\varTheta }}}_{m}^{- 1} (1)] \otimes E(\mathbf{x}_{t - \ell } \, \xi _{m t | T}) = {{\mathcal {F}}}_a ({{\varvec{\gamma }}}_m, {{\varvec{\nu }}}_m) \end{aligned}$$

by (10). By (13) and (18), we get

$$\begin{aligned} \begin{aligned} \frac{\partial ^2 \, {\text {log}}_e \eta _{m t}}{\partial \, {{\varvec{\nu }}}_m \, \partial {{\varvec{\varTheta }}}_{m j}^\top }&= \, - \, [(\mathbf{u}_{t}^\top \, {{\varvec{\varOmega }}}_{m}^{- 1}) \otimes \mathbf{I}_K ]\, [{{\varvec{\varTheta }}}_{m}^{- 1} (1)^\top \otimes {{\varvec{\varTheta }}}_{m}^{- 1} (1)^\top ] - {{\varvec{\varTheta }}}_{m}^{- 1} (1)^\top \, {{\varvec{\varOmega }}}_{m}^{- 1}\, \frac{\partial \mathbf{u}_{t}}{\partial {{\varvec{\varTheta }}}_{m j}^\top } \\&= \, - \, [(\mathbf{u}_{t}^\top \, {{\varvec{\varOmega }}}_{m}^{- 1}) \otimes \mathbf{I}_K ]\, [{{\varvec{\varTheta }}}_{m}^{- 1} (1)^\top \otimes {{\varvec{\varTheta }}}_{m}^{- 1} (1)^\top ] \\&\quad \, \, -\, {{\varvec{\varTheta }}}_{m}^{- 1} (1)^\top \, {{\varvec{\varOmega }}}_{m}^{- 1}\, \sum _{\ell , n \ge 0}^{\infty } \, (\mathbf{I}_K \otimes \mathbf{z}_{m, t - j - \ell - n}^\top ) \, ({{\varvec{\varPsi }}}_{m \ell }^\top \otimes {{\varvec{\varPsi }}}_{m n}^\top ). \end{aligned} \end{aligned}$$

Using (19) gives

$$\begin{aligned} {{\mathcal {J}}}_T ({{\varvec{\nu }}}_m, {{\varvec{\delta }}}_{m j})= & {} {{\varvec{\varTheta }}}_{m}^{- 1} (1)^\top \, {{\varvec{\varOmega }}}_{m}^{- 1}\, \sum _{\ell , n \ge 0}^{\infty } \, \left[ \mathbf{I}_K \otimes \left( \frac{1}{T} \, \sum _{t = 1}^T \, \mathbf{z}_{m, t - j - \ell - n}^\top \, \xi _{m t | T}\right) \right] \\&({{\varvec{\varPsi }}}_{m \ell }^\top \otimes {{\varvec{\varPsi }}}_{m n}^\top ) \end{aligned}$$

for all \(j = 1, \dots , q\) and \(m = 1, \dots , M\). Then \( {\text {plim}}_{T \rightarrow \infty } \, {{\mathcal {J}}}_T ({{\varvec{\nu }}}_m, {{\varvec{\delta }}}_m) = {{\varvec{0}}} = {{\mathcal {F}}}_a ({{\varvec{\nu }}}_m, {{\varvec{\delta }}}_m) \) because \(E(\mathbf{z}_{m, \tau } \, \xi _{m t | T}) = {{\varvec{0}}}\).

Finally, we compute the second derivatives of \({\text {log}}_e \eta _{m t}\) with respect to the pairs \(({{\varvec{\gamma }}}_m, {{\varvec{\gamma }}}_m)\), \(({{\varvec{\gamma }}}_m, {{\varvec{\delta }}}_m)\), and \(({{\varvec{\delta }}}_m, {{\varvec{\delta }}}_m)\). By (14) and (17), we have

$$\begin{aligned} \frac{\partial ^2 \, {\text {log}}_e \eta _{m t}}{\partial \, {{\varvec{\varPhi }}}_{m i} \, \partial {{\varvec{\varPhi }}}_{m j}^\top }= & {} \, \sum _{\ell = 0}^{\infty } \, [( {{\varvec{\varPsi }}}_{m \ell }^\top \, {{\varvec{\varOmega }}}_{m}^{- 1}) \otimes \mathbf{y}_{t - i - \ell } ]\, \frac{\partial \mathbf{u}_{t}}{\partial {{\varvec{\varPhi }}}_{m j}^\top } \, \\= & {} \, - \, \sum _{\ell , n \ge 0}^{\infty } \, ({{\varvec{\varPsi }}}_{m \ell }^\top \, {{\varvec{\varOmega }}}_{m}^{- 1} {{\varvec{\varPsi }}}_{m n}) \otimes (\mathbf{y}_{t - i - \ell }\, \mathbf{y}_{t - j - n}^\top ) \end{aligned}$$

for all \(i, j = 1, \dots , p\) and \(m = 1, \dots , M\). Thus

$$\begin{aligned} {{\mathcal {J}}}_T ({{\varvec{\gamma }}}_m) = \, \sum _{\ell , n \ge 0}^{\infty } \, ({{\varvec{\varPsi }}}_{m \ell }^\top \, {{\varvec{\varOmega }}}_{m}^{- 1} {{\varvec{\varPsi }}}_{m n}) \otimes \left( \frac{1}{T} \, \sum _{t = 1}^T \, \mathbf{x}_{t - \ell }\, \mathbf{x}_{t - n}^\top \, \xi _{m t | T}\right) \end{aligned}$$

which implies \( {\text {plim}}_{T \rightarrow \infty } {{\mathcal {J}}}_T ({{\varvec{\gamma }}}_m) = {{\mathcal {F}}}_a ({{\varvec{\gamma }}}_m) \) by (8). By (14) and (18), we get

$$\begin{aligned} \begin{aligned} \frac{\partial ^2 \, {\text {log}}_e \eta _{m t}}{\partial \, {{\varvec{\varPhi }}}_{m i} \, \partial {{\varvec{\varTheta }}}_{m j}^\top }&= \sum _{\ell = 0}^{\infty } \, [( {{\varvec{\varPsi }}}_{m \ell }^\top \, {{\varvec{\varOmega }}}_{m}^{- 1}) \otimes \mathbf{y}_{t - i - \ell }]\, \frac{\partial \mathbf{u}_{t}}{\partial {{\varvec{\varTheta }}}_{m j}^\top } \\&= \, \sum _{\ell , n, r \ge 0}^{\infty } \, ({{\varvec{\varPsi }}}_{m \ell }^\top \, {{\varvec{\varOmega }}}_{m}^{- 1} {{\varvec{\varPsi }}}_{m r}^\top ) \otimes (\mathbf{y}_{t - i - \ell } \, \mathbf{z}_{m, t - j - n - r}^\top \, {{\varvec{\varPsi }}}_{m n}^\top ) \end{aligned} \end{aligned}$$

for every \( i = 1, \dots , p\), \( j = 1, \dots , q\) and \(m = 1, \dots , M\). Thus

$$\begin{aligned} {{\mathcal {J}}}_T ({{\varvec{\gamma }}}_m, {{\varvec{\delta }}}_m)= & {} \, - \, \sum _{\ell , n, r \ge 0}^{\infty } \, ({{\varvec{\varPsi }}}_{m \ell }^\top \, {{\varvec{\varOmega }}}_{m}^{- 1} {{\varvec{\varPsi }}}_{m r}^\top ) \otimes \left[ \left( \frac{1}{T} \, \sum _{t = 1}^T \, \mathbf{x}_{t - \ell } \, \mathbf{w}_{m, t - r - n}^\top \, {\xi }_{m t | T} \right) \right. \\&\left. (\mathbf{I}_q \otimes {{\varvec{\varPsi }}}_{m n}^\top ) \right] . \end{aligned}$$

Since \( {{\mathcal {J}}}_T ({{\varvec{\delta }}}_m, {{\varvec{\gamma }}}_m) = {{\mathcal {J}}}_T ({{\varvec{\gamma }}}_m, {{\varvec{\delta }}}_m)^\top \), we have

$$\begin{aligned} {\text {plim}}_{T \rightarrow \infty } \, {{\mathcal {J}}}_T ({{\varvec{\delta }}}_m, {{\varvec{\gamma }}}_m)= & {} - \sum _{\ell , n, r \ge 0}^{\infty } \, ({{\varvec{\varPsi }}}_{m r} \, {{\varvec{\varOmega }}}_{m}^{- 1} {{\varvec{\varPsi }}}_{m \ell }) \otimes \left[ (\mathbf{I}_q \otimes {{\varvec{\varPsi }}}_{m n}) \right. \\&\left. E(\mathbf{w}_{m, t - r - n} \, \mathbf{x}_{t - \ell }^\top \, {\xi }_{m t | T}) \right] \end{aligned}$$

which is equal to \( {{\mathcal {F}}}_a ({{\varvec{\delta }}}_m, {{\varvec{\gamma }}}_m)\) by (11). By (15) and (18), we get

$$\begin{aligned} \frac{\partial ^2 \, {\text {log}}_e \eta _{m t}}{\partial \, {{\varvec{\varTheta }}}_{m i} \, \partial {{\varvec{\varTheta }}}_{m j}^\top } = \, - \, \sum _{\ell , n, h, k \ge 0}^{\infty } \, ({{\varvec{\varPsi }}}_{m \ell } \, {{\varvec{\varOmega }}}_{m}^{- 1} {{\varvec{\varPsi }}}_{m h}^\top ) \otimes ({{\varvec{\varPsi }}}_{m n} \, \mathbf{z}_{m, t - i- \ell - n}\, \mathbf{z}_{m, t - j- h - k}^\top \, {{\varvec{\varPsi }}}_{m k}^\top ) \end{aligned}$$

for every \(i, j = 1,\dots , q\) and \(m = 1, \dots , M\). Hence

$$\begin{aligned} {{\mathcal {J}}}_T ({{\varvec{\delta }}}_m) = \, \sum _{\ell , n, h, k \ge 0}^{\infty } \, ({{\varvec{\varPsi }}}_{m \ell } \, {{\varvec{\varOmega }}}_{m}^{- 1} {{\varvec{\varPsi }}}_{m h}^\top ) \otimes [ (\mathbf{I}_q \otimes {{\varvec{\varPsi }}}_{m n}) \, \mathbf{R}_{m, T } (\ell + n, h + k)\, (\mathbf{I}_q \otimes {{\varvec{\varPsi }}}_{m k}^\top )] \end{aligned}$$

where

$$\begin{aligned} \mathbf{R}_{m, T } (\ell + n, h + k) = \frac{1}{T} \, \sum _{t = 1}^T \, \mathbf{w}_{m, t - \ell - n}\, \mathbf{w}_{m, t - h - k}^\top \, \xi _{m t | T}. \end{aligned}$$

Since \({\text {plim}}_{T \rightarrow \infty } \mathbf{R}_{m, T} (\ell + n, h + k) = \mathbf{R}_{m} (\ell + n, h + k)\), it follows that \( {\text {plim}}_{T \rightarrow \infty } \, {{\mathcal {J}}}_T ({{\varvec{\delta }}}_m) = {{\mathcal {F}}}_a ({{\varvec{\delta }}}_m) \) by (9). This completes the proof. \(\square \)

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Cavicchioli, M. A note on the asymptotic and exact Fisher information matrices of a Markov switching VARMA process. Stat Methods Appl 29, 129–139 (2020). https://doi.org/10.1007/s10260-019-00472-y

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