2. THE MODEL AND DISCRETE-TIME SAMPLING OF A STOCK VARIABLE

The model considered by PS92 is the DTLP given by

(1) $$ \begin{align} X_t = C(L)\epsilon_t = \sum_{j=0}^\infty c_j\epsilon_{t-j}, \end{align} $$

where $C(z)=\sum _{j=0}^\infty c_jz^j$ with $0<|C(1)|<\infty $ ,

(2) $$ \begin{align} \sum_{j=0}^\infty c_j^2<\infty, \end{align} $$

and $\epsilon _t$ is an i.i.d. innovation process satisfying various moment conditions (the $\mathcal {A}$ and $\mathcal {C}$ conditions in PS92). PS92 also consider situations in which $\epsilon _t$ is a martingale difference sequence (their $\mathcal {B}$ conditions) but we will focus on the i.i.d. assumption here. PS92 demonstrate how various summability properties for the coefficients $c_j$ , as well as the different moment conditions for the innovations $\epsilon _t$ , can be utilized to derive a number of asymptotic results (such as LLNs, CLTs, and IPs) regarding normalized sums of $X_t$ and $X_t^2$ (both being our focus here) as well as other quantities of interest.

The analysis of finite variance covariance stationary linear processes in continuous time has a long history in the time series literature. Following, for example, Rozanov (Reference Rozanov1967, Thm. 3.1) and Hannan (Reference Hannan1970, p. 155), such processes, for a variable $X(t)$ , can be written

(3) $$ \begin{align} X(t)=\int_{-\infty}^t \theta(t-r)\zeta(dr), \ \ -\infty < t < \infty, \end{align} $$

where $\theta (\cdot )$ is a real-valued kernel (or weighting) function satisfying

(4) $$ \begin{align} \int_0^\infty \theta(r)^2dr < \infty, \end{align} $$

while $\zeta (dt)$ is an uncorrelated random measure satisfying the following assumption.

Further details regarding random measures can be found in Rozanov (Reference Rozanov1967) as well as Bergstrom (Reference Bergstrom, Griliches and Intriligator1984). Alternative, stronger, and conditions regarding $\zeta (dt)$ are used later in Theorems 2 and 4 when applying the PS92 results.

Ideally, it would be possible to proceed with the generality of the CTLP (3) and relate the $c_j$ coefficients in the DTLP (1) to the weighting function $\theta (\cdot )$ . Although this can, in fact, be done, the resulting innovations ( $\epsilon _t$ ) in (1) aren’t, in general, i.i.d., but instead exhibit correlations over time. Hansen and Sargent (Reference Hansen, Sargent, Hansen and Sargent1991) and Marcet (Reference Marcet, Hansen and Sargent1991) show that from (1) we can write

$$\begin{align*}\epsilon_t = C(L)^{-1}X_t = C(L)^{-1}\int_{-\infty}^t\theta(t-r)\zeta(dr) = \int_{-\infty}^t\phi(t-r)\zeta(dr), \end{align*}$$

where, writing $C(z)^{-1}=\sum _{j=0}^\infty \gamma _jz^j = \gamma (z)$ ,

$$\begin{align*}\phi(t-r) = C(L)^{-1}\theta(t-r) = \gamma(L)\theta(t-r) = \sum_{j=0}^\infty\gamma_j\theta(t-j-r). \end{align*}$$

The implication is that, in general, $\epsilon _t$ will depend on the entire past history of $\zeta (dt)$ and is, therefore, serially correlated.

Hansen and Sargent (Reference Hansen, Sargent, Hansen and Sargent1991) investigate circumstances in which $\epsilon _t$ will be uncorrelated and consider what they regard as the “ideal” form given by

$$\begin{align*}\epsilon_t = \int_{t-1}^t\theta(t-r)\zeta(dr), \end{align*}$$

so that $\epsilon _t$ only depends on $\zeta (dt)$ over the unit time interval and, moreover, the weighting function is given by $\theta (r)$ . They are able to show that it is only continuous-time processes with first-order discrete-time representations that have innovations of this form, which appears to restrict the class of continuous-time processes to which the PS92 results can be applied. However, we demonstrate below, in Theorem 1, that the class of eligible processes is somewhat broader, extending to finite-order CAR(p) and CARMA(p, q) processes.

Our focus here, then, is on a variable, $X(t)$ , that evolves in continuous time as either a CAR(p) or a CARMA(p, q) process. In the case of a CAR(p) process, we have

(5) $$ \begin{align} &dD^{p-1}X(t) \nonumber\\ &\quad =\big[a_{p-1}D^{p-1}X(t) + a_{p-2}D^{p-2}X(t) + \cdots + a_1DX(t) + a_0X(t)\big]dt + \zeta(dt), \end{align} $$

where D denotes the mean square differential operator, $a_0,a_1,\ldots ,a_{p-1}$ are real-valued coefficients, and $\zeta (dt)$ satisfies Assumption 1.

Extending (5) in the natural way to allow for MA(q) errors results in an equation whose disturbance term contains components of the form $D^j\zeta (dt)$ for $j=1,\ldots ,q$ , but this is problematical owing to derivatives of $\zeta (dt)$ not being well-defined. However, Brockwell (Reference Brockwell, Rao and Shanbhag2001, Reference Brockwell2004) shows that a valid representation of a CARMA(p, q) process (for $p>q$ )Footnote ² is given by a state space form consisting of the transition equation

(6) $$ \begin{align} dY(t) = AY(t)dt + e_p\zeta(dt), \end{align} $$

where

$$\begin{align*}Y(t) = \left(\begin{array}{c} Y_1(t) \\ Y_2(t) \\ \vdots \\ Y_{p-1}(t) \\Y_p(t) \end{array} \right), \ \ A = \left(\begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \vdots & & & \vdots \\ 0 & 0 & 0 & \ldots & 1 \\ a_0 & a_1 & a_2 & \ldots & a_{p-1} \end{array}\right), \ \ e_p = \left(\begin{array}{c} 0 \\ 0\\ \vdots \\ 0 \\ 1 \end{array} \right), \end{align*}$$

and the measurement equation

(7) $$ \begin{align} X(t) = (1 \ b_1 \ \dots \ b_{p-2} \ b_{p-1})Y(t) = b'Y(t), \end{align} $$

where $b_j (j=1,\ldots ,p-1)$ denote the MA coefficients with $b_j=0$ for $j>q$ . The CAR(p) is obtained as a special case by setting $b_1=b_2=\cdots =b_{p-1}=0$ . In the CAR(p) case, it is the first element of $Y(t)$ that is of interest whereas in the CARMA(p, q) case interest centers on a linear combination of (potentially all of) the elements of $Y(t)$ .

We make the following assumption regarding the roots of the characteristic equation of the companion matrix A in (6).

Assumption 2 ensures that the process $Y(t)$ is covariance stationary. It would be possible to allow for repeated roots albeit at the cost of greater notational complexity but little of real substance would be gained by doing so. As the coefficients of the characteristic equation are real, the characteristic roots will also either be real or they will occur in complex conjugate pairs. It is convenient, for some of what follows, to assign the real roots to the first $p_1$ eigenvalues, followed by the $p_2$ distinct complex eigenvalues and then their $p_2$ conjugates, which gives (with $i^2=-1$ )

$$\begin{align*}\begin{array}{ll} \lambda_k = -r_k, & k=1,\ldots,p_1, \\[.5em] \lambda_k = -r_k + is_k, & k=p_1+1,\dots,p_1+p_2, \\[.5em] \lambda_k = -r_k - is_k, & k=p_1+p_2+1,\dots,p, \end{array} \end{align*}$$

so that $p=p_1+2p_2$ ; if $p_1=0$ (or $p_2=0$ ) then all roots are complex (or real). The companion matrix A then has the decomposition

(8) $$ \begin{align} A = P\Lambda P^{-1}, \end{align} $$

where $\Lambda $ is a diagonal matrix with the $\lambda _k$ on the diagonal and P is a matrix whose columns are the eigenvectors corresponding to the eigenvalues. The exponential of A, which is needed in what follows, also has the convenient representation

(9) $$ \begin{align} e^A = Pe^\Lambda P^{-1}, \end{align} $$

where $e^\Lambda $ is a diagonal matrix with $e^{\lambda _k}$ on the diagonal. It is also convenient, notationally, to define the $p\times p$ matrix function

(10) $$ \begin{align} \Theta(r) = e^{Ar} \end{align} $$

using this notation, $e^A=\Theta (1)$ .

In the remainder of this section, we shall assume that $X(t)$ is a stock variable whose skip-sampled version at regular time intervals of length $h>0$ is given by the sequence

(11) $$ \begin{align} X_{th}=X(th), \ \ t=\ldots,-1,0,1,\ldots \end{align} $$

With this in mind, the discrete-time measurement equation is

(12) $$ \begin{align} X_{th} = b'Y_{th}, \end{align} $$

where $Y_{th}=Y(th)$ . From (6), $Y(t)$ has the solution

(13)

where we note that post-multiplication of $e^A$ by $e_p$ picks out the last column (p) of that matrix. Further, by splitting the integral on the right, it is straightforward to show that $Y_{th}$ has the VAR(1) representation

(14)

these representations form the basis for the following result.

Theorem 1. Let the continuous-time process $Y(t)$ satisfy (6). Then, under Assumptions 1 and 2:

(a) $Y(t)$ has the CTLP representation

$$\begin{align*}Y(t) = \int_{-\infty}^t \Theta(t-r)\tilde\zeta(dr), \ \ -\infty < t < \infty, \end{align*}$$

where $\tilde \zeta (dt)=e_p\zeta (dt)$ and

$$\begin{align*}\int_0^\infty \|\Theta(r)\|^2dr<\infty. \end{align*}$$

(b) $X(t)$ has the CTLP representation

$$\begin{align*}X(t)=\int_{-\infty}^t \theta(t-r)\zeta(dr), \ \ -\infty < t < \infty, \end{align*}$$

where and

$$\begin{align*}\int_0^\infty \theta(r)^2dr<\infty. \end{align*}$$

(c) $Y_{th}$ has the DTLP representation

(15) $$ \begin{align} Y_{th} = \sum_{j=0}^\infty C_{h,j}\epsilon_{th-jh}, \ \ t=0,\pm1,\ldots, \end{align} $$

where $C_{h,j} = e^{Ahj}$ and $\epsilon _{th}$ , which is defined in (14), is an uncorrelated $p\times 1$ vector process with mean zero and covariance matrix

(16)

The coefficient matrices in the linear process (15) satisfy $C_{h,0}=I_p$ ,

$$\begin{align*}\|C_{h,j}\| \leq \kappa(P)\left(2\sum_{k=1}^{p_1+p_2}e^{-2hjr_k}\right)^{1/2} \ \ \mbox{and} \ \ \sum_{j=0}^\infty \|C_{h,j}\|^2 < \infty, \end{align*}$$

where $\kappa (P)=\|P\|\|P^{-1}\|\geq 1$ is the condition number of P.

Theorem 1 has three components. Part (a) shows that the continuous-time process $Y(t)$ , satisfying (6), has a valid CTLP representation with square-integrable norms for the matrix filter function. Part (b) shows that the process of interest, $X(t)$ , which is a linear combination of $Y(t)$ , also has a valid CTLP representation with square-integrable weighting function. The final part, (c), presents the DTLP representation for $Y_{th}$ and demonstrates the square-summability of the matrix norms in this representation as well as providing an upper bound for the individual matrix norms themselves.

A conspicuous absence from Theorem 1 is a result containing a DTLP for the observed process $X_{th}=b'Y_{th}$ . In view of $X_{th}$ being covariance stationary, a Wold representation certainly exists but the challenge here is relating its coefficients and innovations back to the underlying continuous-time processes, $X(t)$ or $Y(t)$ . Essentially what would be required is a result for a linear combination, $X_{th}=b'Y_{th}$ , of the elements of a multivariate MA( $\infty $ ) process, $Y_{th}$ . Results that exist tend to focus on the resulting order of the contemporaneously aggregated process rather than the relationship between the coefficients and also concentrate on finite-order VARMA(p, q) processes (see, e.g., Lütkepohl, Reference Lütkepohl1987, Reference Lütkepohl2005).

In order to utilize the PS92 results, it is necessary to verify that the relevant innovation and summability conditions are satisfied. In the case of the former, Table 1 contains the relevant PS92 innovation conditions and the corresponding continuous-time conditions relating to $\zeta (dt)$ which represent a strengthening of Assumption 1. The continuous-time conditions contain a superscript “c” so that, for example, $\mathcal {A}_1^c$ is the continuous-time counterpart to the PS92 condition $\mathcal {A}_1$ for DTLPs. In the absence of a DTLP, for observations on $X(t),$ we have to rely instead on properties of the vector DTLP for $Y_{th}$ and then relate them to $X(t)$ . The required summability conditions on the coefficient matrices $C_{h,j}$ are contained in the following proposition.

Proposition 1. Let $Y(t)$ satisfy (6) so that $Y_{th}$ has the DTLP representation (15) with coefficient matrices $C_{h,j}=e^{Ahj}$ . Then, under Assumption 2:

$$\begin{align*}\begin{array}{llll} \mathcal{S}_1^c: &\hspace{-6pt} \sum_{j=1}^\infty j^2\|C_{h,j}\|^2<\infty; & \mathcal{S}_3^c: &\hspace{-6pt} \sum_{j=1}^\infty j\|C_{h,j}\|<\infty; \\[1.5em] \mathcal{S}_5^c: &\hspace{-6pt} \sum_{j=1}^\infty j\|C_{h,j}\|^2<\infty; & \mathcal{S}_6^c: &\hspace{-6pt} \sum_{j=1}^\infty \log j\|C_{h,j}\|<\infty. \end{array} \end{align*}$$

Table 1

Innovation conditions

Table 1 Long description

The table is divided into two main sections.

Section 1: P S 9 2 conditions.

- Label A sub 1: epsilon sub t is i.i.d. with zero mean and E absolute value of epsilon sub 0 is less than infinity.

- Label A sub 2: epsilon sub t is i.i.d. with zero mean and sigma sub epsilon squared equals E of epsilon sub 0 squared which is less than infinity.

- Label A sub 3: epsilon sub t is i.i.d. with zero mean and finite fourth cumulant kappa sub 4.

Section 2: Continuous-time conditions.

- Label A sub 1 super c: zeta (d t) is i.i.d. with zero mean and E of absolute value of zeta (d t) is less than or equal to K sub 0 d t.

- Label A sub 2 super c: zeta (d t) is i.i.d. with zero mean and E of zeta (d t) squared equals sigma squared d t.

- Label A sub 3 super c: zeta (d t) is i.i.d. with zero mean and E of zeta (d t) to the fourth power equals K sub 4 d t.

A footer note specifies that in conditions A sub 1 super c through A sub 3 super c, the constants K sub 0, sigma squared, and K sub 4 are all finite.

Note: In $\mathcal {A}_1^c$ – $\mathcal {A}_3^c$ , the constants $K_0$ , $\sigma ^2$ , and $K_4$ are all finite.

The numbering of these summability conditions conforms to the numbering in PS92; they are used in the proof of Theorem 2 below. Of particular note is the fact that no further restrictions are required on the parameters of (6) and (7) in order for the required summability conditions to hold.

We shall now consider a finite sample of skip-sampled observations of the form

(17) $$ \begin{align} X_h, X_{2h},\ldots,X_{nh}, \end{align} $$

where n denotes the sample size. It is convenient to write the linear process (15) in the form

$$\begin{align*}Y_{th} = C_h(L^h)\epsilon_{th} \ \ \mbox{where} \ \ C_h(z)=\sum_{j=0}^\infty C_{h,j}z^j. \end{align*}$$

This representation is particularly useful for the BN decomposition of $Y_{th}$ given by

(18) $$ \begin{align} Y_{th} = C_h(1)\epsilon_{th} + \tilde\epsilon_{th-h} - \tilde\epsilon_{th}, \end{align} $$

where

$$\begin{align*}\tilde\epsilon_{th} = \sum_{j=0}^{\infty}\tilde C_{h,j}\epsilon_{th-jh}, \ \ \tilde C_{h,j} = \sum_{k=j+1}^\infty C_{h,j} \end{align*}$$

(see PS92 (p.976)). As shown by PS92, the BN decomposition is particularly useful in deriving asymptotic results for sums of the variable of interest, such as

$$\begin{align*}\sum_{t=1}^nY_{th} = C_h(1)\sum_{t=1}^n\epsilon_{th} + \tilde\epsilon_0 - \tilde\epsilon_{nh}, \end{align*}$$

because they can typically be based on results for sums of the uncorrelated process $\epsilon _{th}$ provided $\tilde \epsilon _0$ and $\tilde \epsilon _{nh}$ are asymptotically negligible, while from Theorem 1(c),

$$\begin{align*}C_h(1) = \sum_{j=0}^\infty C_{h,j} = \sum_{j=0}^\infty e^{Ahj} = \big(I_p-e^{Ah}\big)^{-1}. \end{align*}$$

The results in PS92, suitably modified for the multivariate nature of $Y_{th}$ , enable the following results to be determined.

Theorem 2. Let $X_{th}$ satisfy (12) and let $Y(t)$ satisfy (6). Then, under Assumptions 1 and 2 and the additional conditions stated below, as $n\rightarrow \infty $ , the discrete-time sample (17) satisfies the following properties:

(a) If either $\mathcal {A}_1^c$ or $\mathcal {A}_2^c$ holds, then

$$\begin{align*}\frac{1}{n}\sum_{t=1}^nX_{th} \stackrel{a.s.}{\rightarrow} 0. \end{align*}$$

(b) If $\mathcal {A}_2^c$ holds, then

$$\begin{align*}\frac{1}{n^{1/2}}\sum_{t=1}^nX_{th} \stackrel{d}{\rightarrow} N\left(0,\sigma_h^2\right) \ \ \mbox{and} \ \ \frac{1}{n^{1/2}}\sum_{t=1}^{[nr]}X_{th} \stackrel{d}{\rightarrow}\sigma_hW(r), \end{align*}$$

where $\sigma _h^2 = b'C_h(1)\Omega _h C_h(1)'b$ .

(c) If $\mathcal {A}_2^c$ holds, then

$$\begin{align*}\frac{1}{n}\sum_{t=1}^nX_{th}^2 \stackrel{a.s.}{\rightarrow} \gamma_{h,0} = E(X_{th}^2) = b'\sum_{k=0}^\infty C_{h,k}\Omega_h C_{h,k}'b. \end{align*}$$

(d) If $\mathcal {A}_3^c$ holds, then

$$\begin{align*}\frac{1}{n^{1/2}}\sum_{t=1}^n\left(X_{th}^2-\gamma_{h,0}\right) \stackrel{d}{\rightarrow} N\left(0,v_h(0)\right) \ \ \mbox{and} \ \ \frac{1}{n^{1/2}}\sum_{t=1}^{[nr]}\left(X_{th}^2-\gamma_{h,0}\right) \stackrel{d}{\rightarrow} v_h(0)^{1/2}W(r), \end{align*}$$

where

$$ \begin{align*} v_h(0) &=\sum_{i=1}^p\sum_{j=1}^p\sum_{k=1}^p\sum_{l=1}^p \left(K_4\phi_{h,ijkl} + \sigma^4(\phi_{h,ij}\phi_{h,kl} + \phi_{h,ik}\phi_{h,jl} + \phi_{h,il}\phi_{h,jk})\right)\\[.5em] & \quad \times F_{0,ij}(1)F_{0,kl}(1) + 4\mbox{tr}\{\Omega_h^\dagger\Omega_h\} - \gamma_{h,0}^2, \\[.5em] \phi_{h,ijkl} &= \int_0^h \Theta_{ip}(r)\Theta_{jp}(r)\Theta_{kp}(r)\Theta_{lp}(r)dr, \ \ \phi_{h,ij} = \int_0^h \Theta_{ip}(r)\Theta_{jp}(r)dr, \\[.5em] F_r(1) &= \sum_{k=0}^\infty C_{h,k+r}'bb'C_{h,k} \ \ (r=0,1,2,\ldots), \ \ \Omega_h^\dagger = \sum_{r=1}^\infty F_r(1)'\Omega_h F_r(1), \end{align*} $$

and $K_4$ is defined in Table 1.

Theorem 2 provides a useful set of asymptotic results for discrete-time samples of stock variables that are justified by conditions on the innovation in the underlying continuous-time process in (6); no further assumptions are required on the process beyond Assumption 2 as it has been shown in Proposition 1 that the required summability conditions on the DTLP are satisfied under this assumption. A potential application of the results in Theorem 2 is provided in Section 4.

3. DISCRETE-TIME SAMPLING OF A FLOW VARIABLE

Not all variables can be observed in the same way as stock variables so the sample that a researcher is faced with may be different from that represented in (17). In this section, we will investigate how, if at all, the conditions underpinning the asymptotic results might change when the discrete-time sample represents observations on a flow variable.

The temporal aggregation process that yields a flow variable is of the form

$$\begin{align*}X^f(th) = \int_{th-h}^{th}X(r)dr = b'\int_{th-h}^{th}Y(r)dr = b'Y^f(th), \end{align*}$$

where

$$\begin{align*}Y^f(th) = \int_{th-h}^{th}Y(r)dr. \end{align*}$$

The sequence of discrete-time values of interest for a flow variable are the skip-sampled values of $X^f(th)$ , that is,

(19) $$ \begin{align} X_{th}^f = X^f(th) = b'Y_{th}^f, \ \ t=\ldots,-1,0,1,\ldots, \end{align} $$

where $Y_{th}^f$ denotes the skip-sampled values of $Y^f(th)$ . The analog of Theorem 1 for flow variables is presented below.

Theorem 3. Let the continuous-time process $Y(t)$ satisfy (6). Then, under Assumptions 1 and 2:

(a) $Y^f(th)$ has the CTLP representation

$$\begin{align*}Y^f(th) = \int_{-\infty}^{th} \Psi(th-r)\tilde\zeta(dr), \ \ -\infty < t < \infty, \end{align*}$$

where

(20) $$ \begin{align} \Psi(r)=\left\{ \begin{array}{ll} \int_0^r\Theta(s)ds, & 0\leq r<h, \\[1em] \int_{r-h}^r\Theta(s)ds, & h\leq r<\infty, \end{array}\right. \ \ \int_0^\infty \|\Psi(r)\|^2dr<\infty, \end{align} $$

and $\tilde \zeta (dt)$ is defined in Theorem 1(a).

(b) $X^f(th)$ has the CTLP representation

$$\begin{align*}X^f(th)=\int_{-\infty}^{th} \psi(th-r)\zeta(dr), \ \ -\infty < t < \infty, \end{align*}$$

where and

$$\begin{align*}\int_0^\infty \psi(r)^2dr<\infty. \end{align*}$$

(c) $Y_{th}^f$ has the DTLP representation

(21) $$ \begin{align} Y_{th}^f = \sum_{j=0}^\infty B_{h,j}\xi_{th-jh} + \eta_{th}, \ \ t=0,\pm1,\ldots, \end{align} $$

where

$$\begin{align*}\xi_{th} = A^{-1}\epsilon_{th}, \ \ \eta_{th} = -A^{-1}e_p\int_{th-h}^{th}\zeta(dr), \end{align*}$$

are uncorrelated random disturbances with zero means and covariance matrices

$$\begin{align*}\Omega_h^f = A^{-1}\Omega_h (A^{-1})' \ \ \mbox{and} \ \ \Omega_h^\eta = \sigma^2hA^{-1}e_pe_p'(A^{-1})', \end{align*}$$

respectively, $B_{h,0}=I_p$ , $B_{h,j} = C_{h,j}-C_{h,j-1} (j=1,2,\ldots )$ ,

$$\begin{align*}\|B_{h,j}\| \leq \lambda_h\|C_{h,j}\| \ \ \mbox{and} \ \ \sum_{j=0}^\infty \|B_{h,j}\|^2 < \infty, \end{align*}$$

where $\lambda _h = p^{1/2}+\kappa (P)\left (2\sum _{k=1}^{p_1+p_2}e^{2hr_k}\right )^{1/2}$ and $\kappa (P)$ is defined in Theorem 1(c).

Theorem 3 shows that, although the temporal aggregation inherent in flow variables affects the precise form of the LP representations, the substantive properties of the representations remain broadly unchanged, with one exception. That is, the process $Y^f(th)$ has a CTLP representation with weighting matrix function having square-integrable norm, the scalar process of interest, $X^f(th)$ , also has a CTLP representation with square-integrable weighting function, while the vector process $Y_{th}^f$ , from which the observed flows are obtained, has a DTLP representation whose weighting matrices are bounded and square-summable. The exception manifests itself in the additional uncorrelated disturbance term, $\eta _{th}$ , in the DTLP representation for $Y_{th}^f$ , and its presence is important in the derivation of the asymptotic distributions in Theorem 4 below.

The matrix weighting function in the CTLP representation of $Y^f(th)$ is more complicated than in the case of a stock variable as it involves integrals of $\Theta (r)$ —this arises solely due to the integration inherent in a flow variable. It is also worth noting in passing that the DTLP representation for $Y_{th}^f$ in Theorem 3(c) can be obtained in at least four different ways. The proof provided in Appendix A is based on integrating (6) over $(th-h,th]$ , yielding the integral on the right-hand side and the difference $Y(th)-Y(th-h)$ on the left, and then using the DTLP in Theorem 1(c) for $Y(th)$ and its lag. Alternatively, it is possible to obtain the same result by integrating (14); by integrating, the DTLP in Theorem 1(c) directly; or by integrating, the equation

$$\begin{align*}Y(th-h+s) = e^{As}Y(th-h) + \int_{th-h}^{th-h+s}e^{A(th-h+s-r)}e_p\zeta(dr) \end{align*}$$

over $s\in (th-h,th]$ , as in the state space approaches of Harvey and Stock (Reference Harvey and Stock1988) and Zadrozny (Reference Zadrozny1988). All four methods yield the same DTLP in Theorem 3(c).

In Theorem 3(c), the disturbance terms, $\xi _{th}$ and $\eta _{th}$ , both involve the inverse of the companion matrix A. The determinant of A is equal to $(-1)^{p+1}a_0$ which is clearly nonzero.Footnote ³ If we write A in the form

$$\begin{align*}A = \left(\begin{array}{cc} 0_{p-1} & I_{p-1} \\[.5em] a_0 & a' \end{array}\right), \end{align*}$$

where $a'=(a_1,a_2,\ldots ,a_{p-1})$ , it is straightforward to show that

$$\begin{align*}A^{-1} = \left(\begin{array}{cc} -\frac{a'}{a_0} & \frac{1}{a_0} \\[1em] I_{p-1} & 0_{p-1} \end{array}\right). \end{align*}$$

In terms of the definition of $\eta _{th}$ , which involves $A^{-1}e_p$ (the last column of $A^{-1}$ ), this implies that

$$\begin{align*}\eta_{th}' = \left(-\frac{1}{a_0},0,\ldots,0\right)\zeta_{th} \ \ \mbox{where} \ \ \zeta_{th} = \int_{th-h}^{th}\zeta(dr). \end{align*}$$

Furthermore, as $X_{th}^f=b'Y_{th}^f$ , we will need to consider $b'\eta _{th}$ which is simply

$$\begin{align*}b'\eta_{th} = -\frac{1}{a_0}\zeta_{th}. \end{align*}$$

Note that, under Assumption 1, $\zeta _{th}$ is an uncorrelated zero mean process with variance $\sigma ^2h$ .

The sample of observations of the flow variable is of the form

(22) $$ \begin{align} X_h^f,X_{2h}^f,\ldots,X_{nh}^f, \end{align} $$

where, as before, n denotes the sample size. In what follows, it is convenient to express (21) in the form

$$\begin{align*}Y_{th}^f = B_h(L^h)\xi_{th} + \eta_{th}, \ \ \mbox{where} \ \ B_h(z)=\sum_{j=0}^\infty B_{h,j}z^j. \end{align*}$$

An interesting property of this representation manifests itself when we consider the BN decomposition given by

$$\begin{align*}Y_{th}^f = B_h(1)\xi_{th} + \tilde\xi_{th-h} - \tilde\xi_{th} + \eta_{th}, \end{align*}$$

where, analogously to the stock case,

$$\begin{align*}\tilde\xi_{th} = \sum_{j=0}^{\infty}\tilde B_{h,j}\xi_{th-jh}, \ \ \tilde B_{h,j} = \sum_{k=j+1}^\infty B_{h,j}. \end{align*}$$

There are two features of this representation that warrant highlighting. The first is the presence of the uncorrelated term $\eta _{th}$ in addition to the usual components. The second is not immediately obvious but concerns the term $B_h(1)$ . Given that $B_{h,j} = C_{h,j}-C_{h,j-1}$ and $B_{h,0}=C_{h,0}=I_p$ , we find that

$$\begin{align*}B_h(1) = \sum_{j=0}^\infty B_{h,j} = I_p + \sum_{j=1}^\infty\big(C_{h,j}-C_{h,j-1}\big) = 0, \end{align*}$$

and hence, the BN decomposition becomes

(23) $$ \begin{align} Y_{th}^f = \tilde\xi_{th-h} - \tilde\xi_{th} + \eta_{th}. \end{align} $$

It follows that sums of $Y_{th}^f$ , for example, are of the form

$$\begin{align*}\sum_{t=1}^nY_{th}^f = \tilde\xi_0 - \tilde\xi_{nh} + \sum_{t=1}^n\eta_{th} \end{align*}$$

so that it is suitably-normalized sums of $\eta _{th}$ , rather than of $\xi _{th}$ , that determine the asymptotics (assuming that $\tilde \xi _0$ and $\tilde \xi _{nh}$ are asymptotically negligible).

In order to use the PS92 results with a flow variable, it is necessary to verify that the required summability conditions apply for the DTLP in Theorem 3(c). This, in fact, follows straightforwardly from Proposition 1 and the bound relating $\|B_{h,j}\|$ to $\|C_{h,j}\|$ , resulting in the following proposition.

Proposition 2. Let $Y(t)$ satisfy (6) so that $Y_{th}^f$ has the DTLP representation (21) with coefficient matrices $B_{h,j}$ defined in Theorem 3(c). Then, under Assumption 2,

$$\begin{align*}\begin{array}{llll} \mathcal{S}_1^f: &\hspace{-6pt} \sum_{j=1}^\infty j^2\|B_{h,j}\|^2<\infty; & \mathcal{S}_3^f: &\hspace{-6pt} \sum_{j=1}^\infty j\|B_{h,j}\|<\infty; \\[1.5em] \mathcal{S}_5^f: &\hspace{-6pt} \sum_{j=1}^\infty j\|B_{h,j}\|^2<\infty; & \mathcal{S}_6^f: &\hspace{-6pt} \sum_{j=1}^\infty \log j\|B_{h,j}\|<\infty. \end{array} \end{align*}$$

The numbering of these summability conditions once again conforms to the numbering in PS92, and an “f” superscript is used to distinguish them from those in Proposition 1. The asymptotic results in the case of a flow variable are presented in Theorem 4.

Theorem 4. Let $X_{th}$ satisfy (19) and let $Y(t)$ satisfy (6). Then, under Assumptions 1 and 2 and the additional conditions stated below, as $n\rightarrow \infty $ , the discrete-time sample (22) satisfies the following properties:

(a) If either $\mathcal {A}_1^c$ or $\mathcal {A}_2^c$ holds, then

$$\begin{align*}\frac{1}{n}\sum_{t=1}^nX_{th}^f \stackrel{a.s.}{\rightarrow} 0. \end{align*}$$

(b) If $\mathcal {A}_2^c$ holds, then

$$\begin{align*}\frac{1}{n^{1/2}}\sum_{t=1}^nX_{th}^f \stackrel{d}{\rightarrow} N\left(0,\sigma_f^2\right) \ \ \mbox{and} \ \ \frac{1}{n^{1/2}}\sum_{t=1}^{[nr]}X_{th}^f \stackrel{d}{\rightarrow}\sigma_fW(r), \end{align*}$$

where $\sigma _f = -\sigma /a_0$ .

(c) If $\mathcal {A}_2^c$ holds, then

$$\begin{align*}\frac{1}{n}\sum_{t=1}^n\big(X_{th}^f\big)^2 \stackrel{a.s.}{\rightarrow} \gamma_{h,0}^f, \end{align*}$$

where

(d) If $\mathcal {A}_3^c$ holds, then

$$\begin{align*} &\frac{1}{n^{1/2}}\sum_{t=1}^n\left(\big(X_{th}^f\big)^2-\gamma_{h,0}^f\right) \stackrel{d}{\rightarrow} N\big(0,v_h^f(0)\big) \ \ \mbox{and} \ \ \\ &\quad \frac{1}{n^{1/2}}\sum_{t=1}^{[nr]}\left(\big(X_{th}^f\big)^2-\gamma_{h,0}^f\right) \stackrel{d}{\rightarrow} v_h^f(0)^{1/2}W(r), \end{align*}$$

where

$$\begin{align*}v_h^f(0) = \mu_1 + \mu_2 + \mu_3 + \mu_4 + \mu_6 + \mu_7 + \mu_9 + \mu_{10}- (\gamma_{h,0}^f)^2 \end{align*}$$

and the $\mu _i$ quantities are defined in Table 2.

Table 2

Definitions of $\mu _i$ quantities in Theorem 4

Table 2 Long description

The table is organized into two columns: the left column lists the variable name and the right column provides its mathematical definition.

* mu sub 1: Summation from i equals 1 to p, summation from j equals 1 to p, summation from k equals 1 to p, summation from l equals 1 to p of the quantity { K sub 4 phi sub h,ijkl + sigma super 4 (phi sub h,ij phi sub h,kl + phi sub h,ik phi sub h,jl + phi sub h,il phi sub h,jk) } F sub A,ij F sub A,kl.

* mu sub 2: 4 trace { Omega sub h super f dagger Omega sub h super f }.

* mu sub 3: K sub 4 h all over a sub 0 super 4.

* mu sub 4: 4 all over a sub 0 squared b prime A inverse ((K sub 4 + sigma super 4 h) Phi sub h + 2 sigma super 4 phi sub h, dot p phi sub h, dot p prime ) (A inverse) prime b + 4 sigma super 4 h all over a sub 0 squared b prime summation from r equals 1 to infinity B sub h,r A inverse Phi sub h (A inverse) prime B sub h,r prime b.

* mu sub 6: 2 all over a sub 0 squared trace { F sub 0 (1) A inverse ((K sub 4 + sigma super 4 h) Phi sub h + 2 sigma super 4 phi sub h, dot p phi sub h, dot p prime ) (A inverse) prime }.

* mu sub 7: minus 4 all over a sub 0 summation from i equals 1 to p summation from j equals 1 to p summation from k equals 1 to p F sub A,ij b sub k (K sub 4 phi sub h,ijk + sigma super 4 (phi sub h,ij phi sub h,k + phi sub h,ik phi sub h,j + phi sub h,i phi sub h,jk) ) + 4 sigma squared gamma sub h,0 super f all over a sub 0 b prime A inverse phi sub h, dot p.

* mu sub 9: minus 8 sigma squared all over a sub 0 b prime summation from r equals 1 to infinity B sub h,r Omega sub h super f F sub s (1) A inverse phi sub h, dot p.

* mu sub 10: minus 4 all over a sub 0 cubed (K sub 4 + 3 h sigma super 4) b prime A inverse phi sub h, dot p.

Additional definitions at the bottom include:

* Omega sub h super f dagger: Summation from s equals 1 to infinity F sub s (1) prime Omega sub h super f F sub s (1).

* F sub A: (A inverse) prime F sub 0 (1) A inverse.

* Phi sub h: Integral from 0 to h of Theta sub dot p (s) Theta sub dot p (s) prime d s.

* phi sub h, dot p: Integral from 0 to h of Theta sub dot p (s) d s.

* phi sub h,ijk: Integral from 0 to h of Theta sub ip (s) Theta sub jp (s) Theta sub kp (s) d s.

Note: $K_4$ is defined in Table 1 and $F_s(1)$ is defined in Theorem 2.

Theorem 4 provides analogous results to Theorem 2 but this time for the case of a flow variable. The LLN, CLT, and IP for the level $X_{th}^f$ hold under the same conditions as for a stock variable although the long run variance is of a simpler form ( $\sigma _f^2=\sigma ^2/a_0^2$ ). The corresponding results for the square $\big (X_{th}^f\big )^2$ also hold under the same conditions as for a stock although the expressions are of a more complicated nature, some of which are contained in Table 2 for ease of presentation.

4. AN ILLUSTRATIVE APPLICATION

As an illustrative example of how the results developed earlier can be utilized, we consider a simple model that lies at the heart of many tests for unit roots in time series. Abstracting from any deterministic time trends so as to focus on the stochastic component,Footnote ⁴ we consider the discrete-time model

(24) $$ \begin{align} y_t = \alpha y_{t-1} + v_t, \ \ v_t = \sum_{j=0}^\infty \delta_je_{t-j}, \end{align} $$

where $e_t$ is an innovation term which is assumed to be an i.i.d. process in Ng and Perron (Reference Ng and Perron2001) and is, in addition, assumed to be normally distributed in Elliott, Rothenberg, and Stock (Reference Elliott, Rothenberg and Stock1996, Condition A). Both sets of authors impose the summability condition

(25) $$ \begin{align} \sum_{j=0}^\infty j|\delta_j|<\infty, \end{align} $$

which is condition $\mathcal {S}_3$ of PS92. The objective is to test the null hypothesis that $\alpha =1$ against the alternative that $\alpha <1$ .

A continuous time version of the model outlined above can be written

(26) $$ \begin{align} dy(t) = \big(\alpha_c y(t) + v(t)\big)dt, \ \ v(t) = \int_{-\infty}^t\delta(t-r)\zeta(dr), \end{align} $$

where $\zeta (dt)$ is an uncorrelated random measure satisfying Assumption 1 which is assumed to drive the CARMA(p, q) process $v(t)$ so that $\delta (r)=b'e^{Ar}e_p$ , that is, $v(t)$ satisfies the continuous-time measurement equation $v(t)=b'Y(t)$ , as in (7), and the dynamics of $Y(t)$ are governed by the transition equation (6). In this model, the null hypothesis of interest is that $\alpha _c=0$ , the stationary alternative being $\alpha _c<0$ . Assuming that $y(t)$ is skip-sampled, as in (17), the observations satisfy the discrete-time modelFootnote ⁵

(27) $$ \begin{align} y_{th} = \beta_h y_{th-h} + u_{th}, \ \ u_{th} = \int_{th-h}^{th}e^{\alpha(th-r)}v(r)dr, \ \ t=1,\ldots,n, \end{align} $$

where $\beta _h=e^{\alpha _c h}$ . In this example, testing for a zero root can be based on $\hat \alpha _c = h^{-1}\log \hat \beta _h$ , where $\hat \beta _h$ denotes the ordinary least squares estimator of $\beta _h$ using (27), and some results in this vein can be found in Chambers (Reference Chambers2015). Noting that $\log x = (x-1) + O\big ((x-1)^2\big ) (0<x\leq 2)$ ,Footnote ⁶ we thus consider the approximation $\log \hat \beta _h \approx \hat \beta _h-1$ . We have

$$\begin{align*}\hat\beta_h = \frac{\sum_{t=1}^n y_{th-h}y_{th}}{\sum_{t=1}^n y_{th-h}^2} = \beta_h + \frac{\sum_{t=1}^n y_{th-h}u_{th}}{\sum_{t=1}^n y_{th-h}^2}. \end{align*}$$

Under the null hypothesis ( $\alpha _c=0$ ), it follows that $\beta _h=1$ and so standard arguments lead us to consider

$$\begin{align*}n\hat\alpha_c \approx \frac{n}{h}(\hat\beta_h-1) = \frac{\frac{1}{nh}\sum_{t=1}^n y_{th-h}u_{th}}{\frac{1}{n^2}\sum_{t=1}^n y_{th-h}^2}. \end{align*}$$

Also under the null hypothesis, the form of $u_{th}$ simplifies to

$$\begin{align*}u_{th} = \int_{th-h}^{th}v(r)dr = b'\int_{th-h}^{th}Y(r)dr = b'Y_{th}^f \end{align*}$$

using the notation of Section 3 whose results are relevant here. For example, Theorem 3(c) shows that $Y_{th}^f$ can be represented as the sum of a DTLP in $\xi _{th}$ plus an additional random disturbance $\eta _{th}$ , while Proposition 2 shows that a summability condition of the form (25) applies to the matrix norms in the DTLP component. Furthermore, we find from Theorem 4(b) that

$$\begin{align*}\frac{1}{n^{1/2}}\sum_{t=1}^{[nr]}u_{th} \stackrel{d}{\rightarrow} \sigma_fW(r), \end{align*}$$

where $W(r)$ is a Wiener process and $\sigma _f=-\sigma /a_0$ . This IP for $u_{th}$ leads, via standard manipulations,Footnote ⁷ to the following asymptotics:

$$ \begin{align*} \frac{1}{n}\sum_{t=1}^n y_{th-h}u_{th} &\stackrel{d}{\rightarrow} \sigma_f^2\int_0^1W(r)dW(r) + \frac{1}{2}(\sigma_f^2 - \gamma_{h,0}^f), \\[.5em] \frac{1}{n^2}\sum_{t=1}^n y_{th-h}^2 &\stackrel{d}{\rightarrow} \sigma_f^2\int_0^1W(r)^2dr, \end{align*} $$

where $\gamma _{h,0}^f$ is defined in Theorem 4(c). It then follows that

$$\begin{align*}n\hat\alpha_c \stackrel{d}{\rightarrow} \frac{\int_0^1W(r)dW(r) + \frac{1}{2}\left(1 - \frac{\gamma_{h,0}^f}{\sigma_f^2}\right)}{h\int_0^1W(r)^2dr}. \end{align*}$$

In order to use this statistic for inference it is, of course, necessary to deal with the nuisance parameters $\gamma _{h,0}^f$ and $\sigma _f^2$ , and a variety of methods can be used to achieve this, many of which are described in Choi (Reference Choi2015). A similar approach can also be taken in the case of a flow variable.

5. CONCLUDING COMMENTS

In this article, we have considered covariance stationary ARMA processes in continuous time that are sampled at discrete-time intervals and have examined conditions on the continuous-time process such that some of the relevant results of Phillips and Solo (Reference Phillips and Solo1992) can be utilized for normalized sums of the discrete-time observations. The coefficient summability conditions of PS92 are satisfied for the stationary ARMA process although some strengthening of the assumptions regarding the random measure that drives the process is required in order that the discrete-time innovations satisfy the PS92 conditions. The results cover both discretely-sampled stock and flow variables and an illustrative example of their use has been provided. The results could also be extended to the case where the underlying CARMA process is a vector rather than a scalar, a process which is already facilitated by the need to consider a state vector in the proofs here.

Although we have focused on continuous-time counterparts to the PS92 i.i.d. $\mathcal {A}$ innovation conditions, it nevertheless would appear possible to extend the results to continuous time versions of their martingale difference sequence conditions ( $\mathcal {B}$ ) and stable law i.i.d. conditions ( $\mathcal {C}$ ) when the innovations may have undefined means. Such extensions would be particularly fruitful for CARMA models for financial data where these alternative assumptions are commonly employed.