Once the model is estimated, only some information at low frequency on the residual (the combination $\mathrm{\Theta}$) is known. Equation (10) allows to *smooth* this residual at the higher frequency. As $\epsilon $ is an exogenous and unexplained component, in practice one may want to limit its influence on the profile of the interpolated stock within the years. Otherwise, interpretation or econometric results based on interpolated stocks could be discredited by potential statistical artifacts. The typical issue for quarterly national accounts is a residual with a jump every first quarter. As I show in Section 3.1 and the example in 3.2, the statistically optimal distribution of the high-frequency residual in dynamic models can induce undesirable features such as this jump every first quarter.

*Theoretical variance-covariance matrix of the estimated residual* Given formula (10), the estimated residual is equal to $\stackrel{\u02c6}{E}=\mathrm{\Omega}{{M}_{2}}^{\prime}{\left({M}_{2}\mathrm{\Omega}{{M}_{2}}^{\prime}\right)}^{-1}\mathrm{\Theta}$ while by definition of ${M}_{2}$, $\mathrm{\Theta}={M}_{2}E$. Thus the theoretical variance-covariance matrix of the estimated process $\stackrel{\u02c6}{E}$ is:

$\mathbb{E}\left(\stackrel{\u02c6}{E}{\stackrel{\u02c6}{E}}^{\prime}\right)={\mathrm{\Omega}}^{E}=\mathrm{\Omega}{{M}_{2}}^{\prime}{\left({M}_{2}\mathrm{\Omega}{{M}_{2}}^{\prime}\right)}^{-1}{M}_{2}\mathrm{\Omega}$(19)

It has the following property:

${M}_{2}{\mathrm{\Omega}}^{E}={M}_{2}\mathrm{\Omega}$(20)

${\mathrm{\Omega}}^{E}{{M}_{2}}^{\prime}=\mathrm{\Omega}{{M}_{2}}^{\prime}$(21)

If $\mathrm{\Omega}={\mathrm{\Omega}}^{E}$ implies eqs. (20) and (21), the reciprocal is false since ${M}_{2}^{\prime}{M}_{2}$, contrary to ${M}_{2}{{M}_{2}}^{\prime}$, is not invertible.

Hence, $\stackrel{\u02c6}{E}$ does not have the same stochastic properties as $E$, but combines the assumption made for $E$ (through the matrix $\mathrm{\Omega}$) with the dynamics of the model (${M}_{2}$).

*An application of the Wiener-Kolmogorov optimal signal extraction theory* The previous result can be linked to the signal extraction framework developed by Wiener and Kolmogorov.

Conditional on the values of the parameters to be estimated, the problem can be written as follows: ${\theta}_{fn}=\sum _{i=1}^{f}{\omega}_{n,i}\phantom{\rule{1em}{0ex}}\text{with}\phantom{\rule{1em}{0ex}}{\omega}_{n,i}={\rho}^{f-i}{\epsilon}_{f(n-1)+i}$(22)

${\theta}_{fn}$ observed every $f$ period is the sum of $f$ signals ${\left({\omega}_{n,i}\right)}_{1\le n\le N}$ (Figure 1).

Figure 1: Timeline example with quarterly data ($f=4$).

On a two-sided infinite sample, the optimal filter to extract each of these signals takes the form:

${\stackrel{\u02c6}{\omega}}_{n,i}=\sum _{-\mathrm{\infty}}^{\mathrm{\infty}}{\gamma}_{s,i}{\theta}_{f(y-s)}={\gamma}_{i}(L){\theta}_{fy}$(23)

with ${\gamma}_{i}(L)=\sum _{-\mathrm{\infty}}^{\mathrm{\infty}}{\gamma}_{s,i}{L}^{s}$, $L$ the lag operator for **annual** data.

We know from signal extraction theory (e.g. Whittle 1963) that ${\gamma}_{i}$ depends on the covariance generating functions of the $f+1$ processes ${\left({\theta}_{fn},{\omega}_{n,1},\cdots ,{\omega}_{n,f}\right)}_{n\in \mathbb{Z}}$.

${\gamma}_{i}(L)=\frac{{f}_{{\omega}_{i},{\omega}_{i}}(L)+\sum _{j\ne i}{f}_{{\omega}_{i},{\omega}_{j}}(L)}{{f}_{\theta ,\theta}(L)}$(24)

In the most simple case where $\epsilon $ is an i.i.d white noise of variance ${\sigma}^{2}$ the formula simplifies to:

${\gamma}_{i}(L)=\frac{{\rho}^{2(f-i)}{\sigma}^{2}}{\sum _{i=1}^{f}{\rho}^{2(f-i)}{\sigma}^{2}}$(25)

$=\frac{{\rho}^{2(f-i)}}{\frac{1-{\rho}^{2f}}{1-{\rho}^{2}}}$(26)

which yields

${\stackrel{\u02c6}{\omega}}_{n,i}=\frac{1-{\rho}^{2}}{1-{\rho}^{2f}}{\rho}^{2(f-i)}{\theta}_{fn}$(27)

${\stackrel{\u02c6}{\epsilon}}_{f(n-1)+i}=\frac{1-{\rho}^{2}}{1-{\rho}^{2f}}{\rho}^{f-i}{\theta}_{fn}$(28)

Given the i.i.d hypothesis, this result is identical to the solution in finite sample detailed in Section 3.2.1.

In the static case, ${M}_{2}(\rho )$ should be replaced by ${M}_{2}(1)$ in the definition of $\mathrm{\Theta}$, in other words $\rho =1$ in the definition of ${\omega}_{n,i}$, thus eq. (25) simplifies into ${\gamma}_{i}(L)=\frac{1}{f}$ which yields the standard result ${\stackrel{\u02c6}{\omega}}_{n,i}={\stackrel{\u02c6}{\epsilon}}_{f(n-1)+i}=\frac{{\theta}_{fn}}{f}$ exemplified by Figure 3b.

Since if ${y}_{t}=A(L){x}_{t}$ then ${f}_{y,y}(L)=A(L)A({L}^{-1}){f}_{x,x}(L)$, the autocovariance generating function of ${\stackrel{\u02c6}{\omega}}_{n,i}$ is:

${f}_{{\stackrel{\u02c6}{\omega}}_{i},{\stackrel{\u02c6}{\omega}}_{i}}(L)={\gamma}_{i}(L){\gamma}_{i}({L}^{-1}){f}_{\theta ,\theta}(L)$(29)
$=\frac{\sum _{j=1}^{f}{f}_{{\omega}_{i},{\omega}_{j}}(L)\sum _{j=1}^{f}{f}_{{\omega}_{i},{\omega}_{j}}({L}^{-1})}{\sum _{i=1}^{f}\sum _{j=1}^{f}{f}_{{\omega}_{i},{\omega}_{j}}({L}^{-1})}$(30)from which the property ${f}_{{\stackrel{\u02c6}{\omega}}_{i},{\stackrel{\u02c6}{\omega}}_{i}}={f}_{{\omega}_{i},{\omega}_{i}}$ is not true in general.

The covariance generating function of ${\stackrel{\u02c6}{\omega}}_{i}$ with ${\stackrel{\u02c6}{\omega}}_{k}$ is:
${f}_{{\stackrel{\u02c6}{\omega}}_{i},{\stackrel{\u02c6}{\omega}}_{k}}(L)={\gamma}_{i}(L){\gamma}_{k}({L}^{-1}){f}_{\theta ,\theta}(L)$(31)
$=\frac{\sum _{j=1}^{f}{f}_{{\omega}_{i},{\omega}_{j}}(L)\sum _{j=1}^{f}{f}_{{\omega}_{k},{\omega}_{j}}({L}^{-1})}{\sum _{i=1}^{f}\sum _{j=1}^{f}{f}_{{\omega}_{i},{\omega}_{j}}({L}^{-1})}$(32)
$=\frac{\sum _{j=1}^{f}{f}_{{\omega}_{i},{\omega}_{j}}(L)\sum _{j=1}^{f}{f}_{{\omega}_{j},{\omega}_{k}}(L)}{\sum _{i=1}^{f}\sum _{j=1}^{f}{f}_{{\omega}_{i},{\omega}_{j}}({L}^{-1})}$(33)

and here again, the property ${f}_{{\stackrel{\u02c6}{\omega}}_{i},{\stackrel{\u02c6}{\omega}}_{k}}={f}_{{\omega}_{i},{\omega}_{k}}$ is not true in general.

In particular in the white noise case, even though we assumed that there is no correlation between sub-periods within and across the years (${f}_{{\omega}_{i},{\omega}_{k}}=0$), the estimated process shows correlation within the years (${f}_{{\stackrel{\u02c6}{\omega}}_{i},{\stackrel{\u02c6}{\omega}}_{k}}=\frac{1-{\rho}^{2}}{1-{\rho}^{2f}}{\rho}^{2(2f-i-k)}$).

This property does not invalidate the estimation procedure but is a general result of optimal signal extraction in unobserved component models. This property is also not due to the sample size.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.