Estimating weak periodic vector autoregressive time series

Abstract

This article develops the asymptotic distribution of the least squares estimator of the model parameters in periodic vector autoregressive time series models (hereafter PVAR) with uncorrelated but dependent innovations. When the innovations are dependent, this asymptotic distributions can be quite different from that of PVAR models with independent and identically distributed (iid for short) innovations developed (Ursu and Duchesne in J Time Ser Anal 30:70–96, 2009). Modified versions of the Wald tests are proposed for testing linear restrictions on the parameters. These asymptotic results are illustrated by Monte Carlo experiments. An application to a bivariate real financial data is also proposed.

Appendix: Proofs of the main results

The proof of Theorem 3.1 is quite technical. This is adaptation of the arguments used in Francq et al. (2011).

1.1 Proof of Theorem 3.1

The proof is quite long so we divide it in several steps.

$\diamond$ Step 1: preliminaries

In view of (3), it is easy to see that $\textbf{X}_n^{\top }(\nu )$ is a measurable function of the random vectors $\{ \varvec{\epsilon }_{ns+\nu -k}, k \ge 1 \}$. Thus, the assumption (A2) of the error term $( \varvec{\epsilon }_n^{*} )_{n\in {\mathbb {Z}}}$ allows us to show that $( \textrm{vec}\{\varvec{\epsilon }_{ns+\nu }\textbf{X}_n^{\top }(\nu ) \} )_{n\in {\mathbb {Z}}}$ is a stationary and ergodic sequence. Applying the ergodic theorem, we obtain that

$$\begin{aligned} N^{-1}\sum _{n=0}^{N-1} \textrm{vec}\{ \varvec{\epsilon }_{ns+\nu }\textbf{X}_n^{\top }(\nu ) \} {\mathop {\rightarrow }\limits ^{a.s.}}{\mathbb {E}}\left[ \textrm{vec}\{ \varvec{\epsilon }_{ns+\nu }\textbf{X}_n^{\top }(\nu ) \}\right] = \textbf{0}, \end{aligned}$$

(46)

by using the non-correlation between $\varvec{\epsilon }_{ns+\nu }$’s (see (A0)) and where $\textbf{0}$ is the $\{ d^2 p(\nu ) \} \times 1$ null vector.

$\diamond$ Step 2: convergence in distribution of $N^{-1/2}\sum _{n=0}^{N-1} \textrm{vec}\{ \varvec{\epsilon }_{ns+\nu }\textbf{X}_n^{\top }(\nu ) \}$

Using the stationarity of $( \textrm{vec}\{\varvec{\epsilon }_{ns+\nu }\textbf{X}_n^{\top }(\nu ) \} )_{n\in {\mathbb {Z}}}$, we have

$$\begin{aligned}&\hbox {var}\left\{ \frac{1}{\sqrt{N}}\sum _{n=0}^{N-1} \textrm{vec}\{ \varvec{\epsilon }_{ns+\nu }\textbf{X}_n^{\top }(\nu ) \}\right\} \\&\quad =\frac{1}{N}\sum _{n=0}^{N-1}\sum _{n'=0}^{N-1}\hbox {cov}\left\{ \textrm{vec}\{ \varvec{\epsilon }_{ns+\nu }\textbf{X}_n^{\top }(\nu ) \},\textrm{vec}\{ \varvec{\epsilon }_{n's+\nu }\textbf{X}_{n'}^{\top }(\nu ) \}\right\} \\&\quad =\frac{1}{N}\sum _{h=-N+1}^{N-1}\left( N-|h|\right) c(\nu ,h), \end{aligned}$$

where

$$\begin{aligned} c(\nu ,h)=\hbox {cov}\left( \textrm{vec}\{ \varvec{\epsilon }_{ns+\nu }\textbf{X}_n^{\top }(\nu ) \},\textrm{vec}\{ \varvec{\epsilon }_{(n-h)s+\nu }\textbf{X}_{n-h}^{\top }(\nu ) \}\right) . \end{aligned}$$

By the dominated convergence theorem, it follows that

$$\begin{aligned} \varvec{\Psi }(\nu )=\sum _{h=-\infty }^{\infty }\hbox {cov}\left( \textrm{vec}\{ \varvec{\epsilon }_{ns+\nu }\textbf{X}_n^{\top }(\nu ) \},\textrm{vec}\{ \varvec{\epsilon }_{(n-h)s+\nu }\textbf{X}_{n-h}^{\top }(\nu ) \}\right) . \end{aligned}$$

The existence of the last sum is a consequence of (A3) and the Davydov (1968) inequality. Using (46) and the elementary relations $\textrm{vec}(ab^\top )=b\otimes a$ for any vectors a and b, and $(A\otimes B)(C\otimes D)= (AC)\otimes (BD)$ for matrices of appropriate sizes (see Lütkepohl 2005), it follows that

$$\begin{aligned} \varvec{\Psi }(\nu )= \sum _{h=-\infty }^{\infty }{\mathbb {E}}\left( \textbf{X}_n(\nu )\textbf{X}_{n-h}^{\top }(\nu ) \otimes \varvec{\epsilon }_{ns+\nu }\varvec{\epsilon }_{(n-h)s+\nu }^{\top }\right) . \end{aligned}$$

Let $\varvec{\epsilon }_n(\nu ) = (\varvec{\epsilon }_{ns+\nu -1}^{\top },\ldots ,\varvec{\epsilon }_{ns+\nu -p(\nu )}^{\top })^{\top }$, $n=0,1,\ldots ,N-1$, be a $\{ d p(\nu ) \} \times 1$ random vectors. In the sequel, we need the elementary identity $\textrm{vec}(ABC)=(I\otimes AB)\textrm{vec}(C)$ (see Lütkepohl 2005). In view of (3), we have for all $r\ge 0$

$$\begin{aligned} \textrm{vec}\{ \varvec{\epsilon }_{ns+\nu }\textbf{X}_n^{\top }(\nu ) \}&= \sum _{i=0}^{\infty }\left( \textbf{I}_{dp(\nu )}\otimes \varvec{\epsilon }_{ns+\nu }\varvec{\epsilon }_{n-i}^{\top }(\nu )\right) \textrm{vec}\left( \textbf{I}_{p(\nu )}\otimes \textbf{C}_i^{\top }(\nu ) \right) \nonumber \\&=\textbf{W}_{n,r}(\nu )+\textbf{U}_{n,r}(\nu ), \end{aligned}$$

(47)

where

$$\begin{aligned} \textbf{W}_{n,r}(\nu )&=\sum _{i=0}^{r}\left( \textbf{I}_{dp(\nu )}\otimes \varvec{\epsilon }_{ns+\nu } \varvec{\epsilon }_{n-i}^{\top }(\nu )\right) \textrm{vec}\left( \textbf{I}_{p(\nu )}\otimes \textbf{C}_i^{\top }(\nu ) \right) \\ \textbf{U}_{n,r}(\nu )&=\sum _{i=r+1}^{\infty }\left( \textbf{I}_{dp(\nu )}\otimes \varvec{\epsilon }_{ns+\nu }\varvec{\epsilon }_{n-i}^{\top }(\nu )\right) \textrm{vec}\left( \textbf{I}_{p(\nu )}\otimes \textbf{C}_i^{\top }(\nu ) \right) . \end{aligned}$$

The processes $(\textbf{W}_{n,r}(\nu ))_{n\in {\mathbb {Z}}}$ and $(\textbf{U}_{n,r}(\nu ))_{n\in {\mathbb {Z}}}$ are stationary and centered. Moreover, under Assumption (A3) and r fixed, the process $(\textbf{W}_{n,r}(\nu ))_{n\in {\mathbb {Z}}}$ is strongly mixing (see Theorem 14.1 in Davidson 1994), with mixing coefficients $\alpha _{\textbf{W}_r}(h)\le \alpha _{\varvec{\epsilon }}\left( \max \{0,h-1\}\right)$. Thus, (A3) implies $\sum _{h=0}^{\infty }\{\alpha _{\textbf{W}_r}(h)\}^{\kappa /(2+\kappa )}< \infty$ and using the Höder inequality, we obtain that $\Vert \textbf{W}_{n,r}(\nu )\Vert _{2+\kappa }< \infty$ for some $\kappa >0$. The central limit theorem for strongly mixing processes (see Herrndorf 1984) implies that $N^{-1/2}\sum _{n=0}^{N-1}\textbf{W}_{n,r}(\nu )$ has a limiting ${\mathcal {N}}(0,\varvec{\Psi }_r(\nu ))$ distribution with

$$\begin{aligned} \varvec{\Psi }_r(\nu )=\lim _{N\rightarrow \infty }\hbox {var}\left( \frac{1}{\sqrt{N}}\sum _{n=0}^{N-1}\textbf{W}_{n,r}(\nu )\right) =\sum _{h=-\infty }^{\infty } \hbox {cov}\left( \textbf{W}_{n,r}(\nu ),\textbf{W}_{n-h,r}(\nu )\right) . \end{aligned}$$

Since $N^{-1/2}\sum _{n=0}^{N-1}\textbf{W}_{n,r}(\nu )$ and $N^{-1/2}\sum _{n=0}^{N-1}\textrm{vec}\{ \varvec{\epsilon }_{ns+\nu }\textbf{X}_n^{\top }(\nu )\}$ have zero expectation, we shall have

$$\begin{aligned} \lim _{r\rightarrow \infty }\hbox {var}\left( \frac{1}{\sqrt{N}}\sum _{n=0}^{N-1}\textbf{W}_{n,r}(\nu )\right) =\hbox {var}\left\{ \frac{1}{\sqrt{N}}\sum _{n=0}^{N-1} \textrm{vec}\{ \varvec{\epsilon }_{ns+\nu }\textbf{X}_n^{\top }(\nu ) \}\right\} , \end{aligned}$$

as soon as

$$\begin{aligned} \lim _{r\rightarrow \infty }\limsup _{N\rightarrow \infty }{\mathbb {P}}\left\{ \left\| \frac{1}{\sqrt{N}} \sum _{n=0}^{N-1}\textbf{U}_{n,r}(\nu )\right\| >\varepsilon \right\} =0 \end{aligned}$$

(48)

for every $\varepsilon >0$. As a consequence we will have $\lim _{r\rightarrow \infty }\varvec{\Psi }_r(\nu )=\varvec{\Psi }(\nu )$. The result (48) follows from a straightforward adaptation of Theorem 7.7.1 and Corollary 7.7.1 of Anderson (see Anderson 1971, pp. 425–426). Indeed, by stationarity we have

$$\begin{aligned} \hbox {var}\left( \frac{1}{\sqrt{N}}\sum _{n=0}^{N-1}\textbf{U}_{n,r}(\nu )\right)= & {} \frac{1}{N}\sum _{n,n'=0}^{N-1} \hbox {cov}\left( \textbf{U}_{n,r}(\nu ),\textbf{U}_{n',r}(\nu )\right) \\= & {} \frac{1}{N}\sum _{|h|<N-1}(N-|h|)\hbox {cov}\left( \textbf{U}_{n,r}(\nu ),\textbf{U}_{n-h,r}(\nu )\right) \\\le & {} \sum _{h=-\infty }^{\infty }\left\| \hbox {cov}\left( \textbf{U}_{n,r}(\nu ),\textbf{U}_{n-h,r}(\nu )\right) \right\| . \end{aligned}$$

Because $\Vert \textbf{C}_{i}\Vert \le K\rho ^{i}$ for $\rho \in [0,1[$ and $K>0$ and in view of (47), we have

$$\begin{aligned} \left\| \textbf{U}_{n,r}(\nu )\right\| \le K\sum _{i=r+1}^{\infty }\rho ^{i}\Vert \varvec{\epsilon }_{ns+\nu }\Vert \Vert \varvec{\epsilon }_{n-i}(\nu )\Vert . \end{aligned}$$

Under (A3) we have ${\mathbb {E}}\Vert \varvec{\epsilon }_{ns+\nu }\Vert ^{4+2\kappa }<\infty$, it follows from the Hölder inequality that

$$\begin{aligned} \sup _h\left\| \hbox {cov}\left( \textbf{U}_{n,r}(\nu ),\textbf{U}_{n-h,r}(\nu )\right) \right\| =\sup _h \left\| {\mathbb {E}}\left( \textbf{U}_{n,r}(\nu )\textbf{U}_{n-h,r}^\top (\nu )\right) \right\| \le K\rho ^r. \end{aligned}$$

(49)

Let $h>0$ such that $[h/2]>r$. Write

$$\begin{aligned} \textbf{U}_{n,r}(\nu )=\textbf{U}_{n,r}^{h^-}(\nu )+\textbf{U}_{n,r}^{h^+}(\nu ), \end{aligned}$$

where

$$\begin{aligned} \textbf{U}_{n,r}^{h^-}(\nu )&=\sum _{i=r+1}^{[h/2]}\left( \textbf{I}_{dp(\nu )}\otimes \varvec{\epsilon }_{ns+\nu } \varvec{\epsilon }_{n-i}^{\top }(\nu )\right) \textrm{vec}\left( \textbf{I}_{p(\nu )}\otimes \textbf{C}_i^{\top }(\nu ) \right) , \\ \textbf{U}_{n,r}^{h^+}(\nu )&=\sum _{i=[h/2]+1}^{\infty }\left( \textbf{I}_{dp(\nu )}\otimes \varvec{\epsilon }_{ns+\nu }\varvec{\epsilon }_{n-i}^{\top }(\nu )\right) \textrm{vec}\left( \textbf{I}_{p(\nu )}\otimes \textbf{C}_i^{\top }(\nu ) \right) . \end{aligned}$$

Note that $\textbf{U}_{n,r}^{h^-}(\nu )$ belongs to the $\sigma$-field generated by $\{\varvec{\epsilon }_{ns+\nu },\varvec{\epsilon }_{ns+\nu -1},\dots ,\varvec{\epsilon }_{ns+\nu -[h/2]}\}$ and that $\textbf{U}_{n-h,r}(\nu )$ belongs to the $\sigma$-field generated by $\{\varvec{\epsilon }_{(n-h)s+\nu },\varvec{\epsilon }_{(n-h-1)s+\nu -1},,\dots \}$. By (A3), ${\mathbb {E}}\Vert \textbf{U}_{n,r}^{h^-}(\nu )\Vert ^{2+\kappa }<\infty$ and ${\mathbb {E}}\Vert \textbf{U}_{n-h,r}(\nu )\Vert ^{2+\kappa }<\infty$. Davydov’s inequality (see Davydov 1968) then entails that

$$\begin{aligned} \left\| \hbox {cov}\left( \textbf{U}_{n,r}^{h^-}(\nu ),\textbf{U}_{n-h,r}(\nu )\right) \right\| \le K\alpha _{\varvec{\epsilon }}^{\kappa /(2+\kappa )}([h/2]). \end{aligned}$$

(50)

By the argument used to show (49), we also have

$$\begin{aligned} \left\| \hbox {cov}\left( \textbf{U}_{n,r}^{h^+}(\nu ),\textbf{U}_{n-h,r}(\nu )\right) \right\| \le K\rho ^h\rho ^r. \end{aligned}$$

(51)

In view of (49), (50) and (51), we have

$$\begin{aligned} \sum _{h=0}^{\infty }\left\| \hbox {cov}\left( \textbf{U}_{n,r}(\nu ),\textbf{U}_{n-h,r}(\nu )\right) \right\| \le K\rho ^r+K\sum _{h=r}^{\infty }\alpha _{\varvec{\epsilon }}^{\kappa /(2+\kappa )}(h)\rightarrow 0 \end{aligned}$$

as $r\rightarrow \infty$ by (A3). We have the same bound for $h<0$. This implies that

$$\begin{aligned}&\sup _N\hbox {var}\left( \frac{1}{\sqrt{N}}\sum _{n=0}^{N-1}\textbf{U}_{n,r}(\nu )\right) \xrightarrow [r\rightarrow \infty ]{} 0. \end{aligned}$$

(52)

The conclusion of (48) follows from the Markov inequality.

From a standard result (see, e.g., Proposition 6.3.9 in Brockwell and Davis 1991), we deduce that

$$\begin{aligned} \frac{1}{\sqrt{N}}\sum _{n=0}^{N-1}\textrm{vec}\{ \varvec{\epsilon }_{ns+\nu }\textbf{X}_n^{\top }(\nu ) \}= \frac{1}{\sqrt{N}}\sum _{n=0}^{N-1}\textbf{W}_{n,r}(\nu )+\frac{1}{\sqrt{N}} \sum _{n=0}^{N-1}\textbf{U}_{n,r}(\nu ){\mathop {\rightarrow }\limits ^{d}}\mathcal{N}\left( 0,\varvec{\Psi }(\nu )\right) , \end{aligned}$$

which completes the proof of (14).

$\diamond$ Step 3: existence and invertibility of the matrix $\varvec{\Omega }(\nu )$

By ergodicity of the centered process $(\textbf{X}_n(\nu ))_{n\in {\mathbb {Z}}}\in {\mathbb {R}}^{ d p(\nu ) }$, we deduce that

$$\begin{aligned} \frac{1}{N} \textbf{X}_n(\nu ) \textbf{X}_n^{\top }(\nu ) {\mathop {\rightarrow }\limits ^{a.s.}}\varvec{\Omega }(\nu ){:}{=}{\mathbb {E}}\left( \textbf{X}_n(\nu )\textbf{X}_n^{\top }(\nu )\right) . \end{aligned}$$

(53)

From (47) we obtain that

$$\begin{aligned} {\mathbb {E}}\left( \textbf{X}_n(\nu )\textbf{X}_n^{\top }(\nu )\right)&={\mathbb {E}}\left[ \left( \sum _{i=0}^{\infty }\left( \textbf{I}_{p(\nu )}\otimes \textbf{C}_i(\nu ) \right) \varvec{\epsilon }_{n-i}(\nu )\right) \left( \sum _{j=0}^{\infty }\left( \textbf{I}_{p(\nu )}\otimes \textbf{C}_j(\nu ) \right) \varvec{\epsilon }_{n-j}(\nu )\right) ^\top \right] \\&\quad =\sum _{i=0}^{\infty }\sum _{j=0}^{\infty } \left( \textbf{I}_{p(\nu )}\otimes \textbf{C}_i(\nu ) \right) {\mathbb {E}}\left[ \varvec{\epsilon }_{n-i}(\nu )\varvec{\epsilon }_{n-j}^\top (\nu ) \right] \left( \textbf{I}_{p(\nu )}\otimes \textbf{C}_j^\top (\nu ) \right) \\&\quad =\sum _{i=0}^{\infty } \left( \textbf{I}_{p(\nu )}\otimes \textbf{C}_i(\nu ) \right) \left( \textbf{I}_{p(\nu )}\otimes \varvec{\Sigma }_{\varvec{\epsilon }}(\nu ) \right) \left( \textbf{I}_{p(\nu )}\otimes \textbf{C}_i^\top (\nu ) \right) \\&\quad \le K\, \sum _{i\ge 0}\rho ^i <\infty . \end{aligned}$$

Therefore, the matrix $\varvec{\Omega }(\nu )$ exists almost surely.

If the matrix $\varvec{\Omega }(\nu )$ is not invertible, there exists some real constants $c_1,\dots ,c_{dp(\nu )}$ not all equal to zero such that $\textbf{c}^\top \varvec{\Omega }(\nu )\textbf{c}=0$, where $\textbf{c}=(c_1,\dots ,c_{dp(\nu )})^\top$. For $i=1,\dots ,dp(\nu )$, let $\textbf{X}_{i,n}(\nu )$ be the i-th component of $\textbf{X}_{n}(\nu )$ and denotes by $\varvec{\Omega }_{ji}(\nu )$ the (i, j)-th component of $\varvec{\Omega }(\nu )$. We obtain that

$$\begin{aligned} \sum _{i=1}^{dp(\nu )}\sum _{j=1}^{dp(\nu )}c_j\varvec{\Omega }_{ji}(\nu )c_i&= \sum _{i=1}^{dp(\nu )}\sum _{j=1}^{dp(\nu )}{\mathbb {E}}\left[ \left( c_j\textbf{X}_{j,n}(\nu )\right) \left( c_i\textbf{X}_{i,n}(\nu )\right) \right] \\&= {\mathbb {E}}\left[ \left( \sum _{k=1}^{dp(\nu )}c_k\textbf{X}_{k,n}(\nu )\right) ^2 \right] =0, \end{aligned}$$

which implies that

$$\begin{aligned}&\sum _{k=1}^{dp(\nu )}c_k\textbf{X}_{k,n}(\nu )=0 \ \ \mathrm {a.s.} \text { or equivalent }\ \ \textbf{c}^\top \textbf{X}_n(\nu )\\&\quad =\sum _{i=0}^{\infty }\textbf{c}^\top \left( \textbf{I}_{p(\nu )}\otimes \textbf{C}_i(\nu ) \right) \varvec{\epsilon }_{n-i}(\nu )=0 \ \ \mathrm {a.s.} \end{aligned}$$

This is in contradiction with the assumption that $\varvec{\Sigma }_{\varvec{\epsilon }}(\nu )$ is not equal to zero. Therefore, $\textbf{c}^\top \textbf{X}_n(\nu )$ is not almost surely equal to zero and $\varvec{\Omega }(\nu )$ is almost surely invertible.

$\diamond$ Step 4: convergence in probability of ${\hat{\varvec{\beta }}}(\nu )$

Using the relation (13), we can write:

$$\begin{aligned} {\hat{\textbf{B}}}(\nu ) - \textbf{B}(\nu ) = N^{-1} \textbf{E}(\nu ) \textbf{X}^{\top }(\nu ) \{ N^{-1} \textbf{X}(\nu ) \textbf{X}^{\top }(\nu ) \}^{-1}. \end{aligned}$$

Noting that $\sum _{n=0}^{N-1} \textrm{vec}\{ \varvec{\epsilon }_{ns+\nu } \textbf{X}_n^{\top }(\nu ) \} = \textrm{vec}\{ \textbf{E}(\nu )\textbf{X}^{\top }(\nu ) \}$, from (14), it follows that $N^{-1/2}\textrm{vec}\{ \textbf{E}(\nu ) \textbf{X}^{\top }(\nu ) \} {\mathop {\rightarrow }\limits ^{d}}N_{d^2p(\nu )}(\textbf{0}, \varvec{\Psi }(\nu ) )$. Applying the ergodic theorem and from (46), we have $N^{-1}\textrm{vec}\{ \textbf{E}(\nu ) \textbf{X}^{\top }(\nu ) \} {\mathop {\rightarrow }\limits ^{a.s.}}\textbf{0}$, where the dimension of $\textbf{0}$ is $\{ d^2 p(\nu ) \} \times 1$, and also $\{ N^{-1}\textbf{X}(\nu )\textbf{X}^{\top }(\nu ) \}^{-1} {\mathop {\rightarrow }\limits ^{a.s.}}\varvec{\Omega }^{-1}(\nu )$; these results show (15).

$\diamond$ Step 5: convergence in distribution of $N^{1/2}\{ {\hat{\varvec{\beta }}}(\nu ) - \varvec{\beta }(\nu ) \}$

Since

$$\begin{aligned} N^{1/2}\{ {\hat{\varvec{\beta }}}(\nu ) - \varvec{\beta }(\nu ) \}&= \left[ \{ N^{-1}\textbf{X}(\nu ) \textbf{X}^{\top }(\nu ) \}^{-1} \otimes \textbf{I}_d \right] N^{-1/2} \{ \textbf{X}(\nu ) \otimes \textbf{I}_d \} \textbf{e}(\nu ),\nonumber \\&= \left[ \{ N^{-1}\textbf{X}(\nu ) \textbf{X}^{\top }(\nu ) \}^{-1} \otimes \textbf{I}_d \right] N^{-1/2} \textrm{vec}\{ \textbf{E}(\nu ) \textbf{X}^\top (\nu ) \} \end{aligned}$$

(54)

Slutsky’s theorem and relation (14) give (16), using the following argument:

$$\begin{aligned} \varvec{\Theta }(\nu )&=\left( \varvec{\Omega }^{-1}(\nu ) \otimes \textbf{I}_d\right) \sum _{h=-\infty }^{\infty }{\mathbb {E}}\left( \textbf{X}_n(\nu )\textbf{X}_{n-h}^{\top }(\nu ) \otimes \varvec{\epsilon }_{ns+\nu }\varvec{\epsilon }_{(n-h)s+\nu }^{\top }\right) \left( \varvec{\Omega }^{-1}(\nu ) \otimes \textbf{I}_d\right) \\&\quad =\sum _{h=-\infty }^{\infty }{\mathbb {E}}\left[ \varvec{\Omega }^{-1}(\nu )\textbf{X}_n(\nu )\textbf{X}_{n-h}^{\top }(\nu )\varvec{\Omega }^{-1}(\nu ) \otimes \varvec{\epsilon }_{ns+\nu }\varvec{\epsilon }_{(n-h)s+\nu }^{\top }\right] . \end{aligned}$$

The joint asymptotic normality of $N^{1/2} \{ {\hat{\varvec{\beta }}}^\top (1) - \varvec{\beta }^\top (1), \ldots , {\hat{\varvec{\beta }}}^\top (s) - \varvec{\beta }^\top (s) \}$ follows using the same kind of manipulations as those for a single season $\nu$. We also have

$$\begin{aligned} N^{1/2}\{ {\hat{\varvec{\beta }}} - \varvec{\beta }\} {\mathop {\rightarrow }\limits ^{d}}N_{sd^2p(\nu )}\left( \textbf{0}, \varvec{\Theta }\right) , \end{aligned}$$

where the asymptotic covariance matrix $\varvec{\Theta }$ is a block matrix, with the asymptotic variances given by $\varvec{\Theta }(\nu )$, $\nu =1,\dots ,s$, and the asymptotic covariances given by:

$$\begin{aligned}{} & {} \lim _{N\rightarrow \infty }\text {cov}\left( N^{1/2}\{ {\hat{\varvec{\beta }}}(\nu ) - \varvec{\beta }(\nu ) \},N^{1/2}\{ {\hat{\varvec{\beta }}}(\nu ') - \varvec{\beta }(\nu ') \}\right) \\{} & {} \quad =\left( \varvec{\Omega }^{-1}(\nu ) \otimes \textbf{I}_d\right) \sum _{h=-\infty }^{\infty }{\mathbb {E}}\left( \textbf{X}_n(\nu )\textbf{X}_{n-h}^{\top }(\nu ') \otimes \varvec{\epsilon }_{ns+\nu }\varvec{\epsilon }_{(n-h)s+\nu '}^{\top }\right) \left( \varvec{\Omega }^{-1}(\nu ') \otimes \textbf{I}_d\right) , \end{aligned}$$

for $\nu \ne \nu '$ and $\nu , \nu ' = 1,\ldots ,s$.

1.2 Proof of Theorem 4.2

Observe that

$$\begin{aligned} {\hat{\varvec{\Psi }}}^\textrm{HAC}(\nu )-\varvec{\Psi }(\nu )&=\sum _{h=-T_N}^{T_N}f(hb_N)\left( {\hat{\Lambda }}_{h}(\nu )-{\Lambda }_{h}(\nu )\right) +\sum _{h=T_N}^{T_N}\left\{ f(hb_N)-1\right\} {\Lambda }_{h}(\nu )\\&\quad -\sum _{|h|> T_N}{\Lambda }_{h}(\nu ). \end{aligned}$$

By the triangular inequality, for any multiplicative norm, we have

$$\begin{aligned} \left\| {\hat{\varvec{\Psi }}}^\textrm{HAC}(\nu )-\varvec{\Psi }(\nu )\right\|\le & {} g_1+g_2+g_3, \end{aligned}$$

where

$$\begin{aligned} g_1&=\sup _{|h|<N}\left\| {\hat{\Lambda }}_{h}(\nu )-{\Lambda }_{h}(\nu )\right\| \sum _{|h|\le T_N}\left| f(hb_N)\right| ,&\\g_2&=\sum _{|h|\le T_N}\left| f(hb_N)-1\right| \left\| {\Lambda }_{h}(\nu )\right\| \quad \hbox {and}\quad g_3=\sum _{|h|>T_N}\left\| {\Lambda }_{h}(\nu )\right\| . \end{aligned}$$

In view of this last inequality, to prove the convergence in probability of ${\hat{\varvec{\Psi }}}^\textrm{HAC}(\nu )$ to $\varvec{\Psi }(\nu )$, it suffices to show that the probability limit of $g_1$, $g_2$ and $g_3$ is 0.

$\diamond$ Step 1: convergence in probability of $\sup _{|h|<N}\left\| {\hat{\Lambda }}_{h}(\nu )-{\Lambda }_{h}(\nu )\right\|$ to 0

Let $\Lambda ^{*}_h(\nu )$ be the matrix defined, for $0\le h<N$, by

$$\begin{aligned} \Lambda ^{*}_h(\nu )=\frac{1}{N}\sum _{n=0}^{N-h-1}\textbf{W}_{n}(\nu )\textbf{W}_{n-h}^{\top }(\nu )\quad \text {and}\quad \Lambda ^{*}_{-h}(\nu )=\Lambda ^{*\top }_h(\nu ). \end{aligned}$$

Observe that

$$\begin{aligned} \sup _{|h|<N}\left\| {\hat{\Lambda }}_{h}(\nu )-{\Lambda }_{h}(\nu )\right\| \le \sup _{|h|<N}\left\| {\hat{\Lambda }}_{h}(\nu )-{\Lambda }^{*}_{h}(\nu )\right\| +\sup _{|h|<N}\left\| {\Lambda }^{*}_{h}(\nu )-{\Lambda }_{h}(\nu )\right\| . \end{aligned}$$

By the ergodic theorem, we have

$$\begin{aligned} \Lambda ^{*}_h(\nu )&{\mathop {\rightarrow }\limits ^{a.s.}}&\Lambda _h(\nu ). \end{aligned}$$

(55)

A Taylor expansion of $\textrm{vec}\{{\hat{\Lambda }}_h(\nu )\}$ around $\varvec{\beta }$ and (14) give

$$\begin{aligned} \textrm{vec}\{{\hat{\Lambda }}_h(\nu )\}&=\textrm{vec}\{\Lambda ^{*}_h(\nu )\}+\frac{\partial \textrm{vec}\{\Lambda ^{*}_h(\nu )\}}{\partial \varvec{\beta }^\top (\nu )} ({{\hat{\varvec{\beta }}}}(\nu )-\varvec{\beta }(\nu ))+\textrm{O}_{\mathbb {P}}\left( \frac{1}{N}\right) . \end{aligned}$$

In view of (55) and by (A3), we then deduce that

$$\begin{aligned} \lim _{N\rightarrow \infty }\sup _{|h|<N}\left\| \frac{\partial \textrm{vec}\{\Lambda ^{*}_h(\nu )\}}{\partial \varvec{\beta }^\top (\nu )}\right\| <\infty ,\quad a.s. \end{aligned}$$

(56)

By the ergodic theorem, (14) and (56), for any multiplicative norm, we have

$$\begin{aligned} \sup _{|h|<N}\left\| \textrm{vec}\left( {\hat{\Lambda }}_h(\nu )-\Lambda ^{*}_h(\nu )\right) \right\|{} & {} \le \lim _{N\rightarrow \infty }\sup _{|h|<N}\left\| \frac{\partial \textrm{vec}\{\Lambda ^{*}_h(\nu )\}}{\partial \varvec{\beta }^\top (\nu )}\right\| \left\| {{\hat{\varvec{\beta }}}}(\nu )-\varvec{\beta }(\nu )\right\| \nonumber \\{} & {} \quad +\textrm{O}_{\mathbb {P}}\left( \frac{1}{N}\right) =\textrm{O}_{\mathbb {P}}\left( \frac{1}{\sqrt{N}}\right) . \end{aligned}$$

(57)

From (55) and (57), we deduce that

$$\begin{aligned} \sup _{|h|<N}\left\| {\hat{\Lambda }}_{h}(\nu )-{\Lambda }_{h}(\nu )\right\| =\textrm{O}_{\mathbb {P}} \left( \frac{1}{\sqrt{N}}\right) =\textrm{o}_{{\mathbb {P}}}(1), \end{aligned}$$

(58)

the conclusion is complete.

$\diamond$ Step 2: convergence in probability of $g_1$, $g_2$ and $g_3$ to 0

By (A3), ${\mathbb {E}}\Vert \textbf{W}_{n}\Vert ^{2+\kappa }<\infty$. Davydov’s inequality (see Davydov 1968) then entails that

$$\begin{aligned} \left\| {\Lambda }_{h}(\nu )\right\| =\left\| \hbox {cov}\left( \textbf{W}_{n}(\nu ),\textbf{W}_{n-h}(\nu )\right) \right\| \le K\alpha _{\varvec{\epsilon }}^{\kappa /(2+\kappa )}([h/2]). \end{aligned}$$

(59)

In view of (A3), we thus have $g_3\rightarrow 0$ as $N\rightarrow \infty$. Let m be a fixed integer and we write $g_2\le s_1+s_2$, where

$$\begin{aligned} s_1=\sum _{|h|\le m}\left| f(hb_N)-1\right| \left\| {\Lambda }_{h}(\nu )\right\| \quad \text {and}\quad s_2=\sum _{m<|h|\le T_N}\left| f(hb_N)-1\right| \left\| {\Lambda }_{h}(\nu )\right\| . \end{aligned}$$

For $|h|\le m$, we have $hb_N\rightarrow 0$ as $N\rightarrow \infty$ and $f(hb_N)\rightarrow 1$, it follows that $s_1\rightarrow 0$. If we choose m sufficiently large, $s_2$ becomes small. Using (59) and the fact that $f(\cdot )$ is bounded, it follows that $g_2\rightarrow 0$.

In view of (35) and (58), we have

$$\begin{aligned} g_1&=\sup _{|h|<N}\left\| {\hat{\Lambda }}_{h}(\nu )-{\Lambda }_{h}(\nu )\right\| \sum _{|h|\le T_N}\left| f(hb_N)\right| ,\\&= \frac{1}{b_N}\sup _{|h|<N}\left\| {\hat{\Lambda }}_{h}(\nu )-{\Lambda }_{h}(\nu )\right\| b_N\sum _{|h|\le T_N}\left| f(hb_N)\right| ,\\&\le \frac{1}{b_N}\sup _{|h|<N}\left\| {\hat{\Lambda }}_{h}(\nu )-{\Lambda }_{h}(\nu )\right\| \textrm{O}(1)= \textrm{O}_{\mathbb {P}}\left( \frac{1}{b_N\sqrt{N}}\right) =\textrm{o}_{\mathbb {P}}(1), \end{aligned}$$

since $Nb_N^2\rightarrow \infty$, in view of (34). The proof is complete.